This chapter contains sections titled: Works cited.

continent. 1.2 (2011): 78-91. This article consists of three parts. First, I will review the major themes of Quentin Meillassoux’s After Finitude . Since some of my readers will have read this book and others not, I will try to strike a balance between clear summary and fresh critique. Second, I discuss an unpublished book by Meillassoux unfamiliar to all readers of this article, except those scant few that may have gone digging in the microfilm archives of the École (...) normale supérieure. The book in question is Meillassoux’s revised doctoral dissertation L’Inexistence divine (or The Divine Inexistence ), with its seemingly bizarre vision of a God who does not yet exist but might exist in the future. Without literally accepting this view, I will claim that it is philosophically interesting in ways that even a hardened sceptic might be able to appreciate. Third and finally, I will speculate on the possible future of Meillassoux’s speculative materialism itself. And here I mean its future development not by Meillassoux, but by those readers who might be inspired by his book. Plato could never have predicted the emergence of Aristotle’s philosophy, despite the obvious debt of the latter to the former. Nor could Descartes have predicted Spinoza and Leibniz, nor Kant the German Idealists, and neither could Husserl in 1901 have foreseen the later emergence of Heidegger. How are the works of interesting philosophers transformed by later thinkers of comparable importance? While it may seem that there are countless ways to do this, I think there are only two basic ways in which this happens: you can radicalize your predecessors, or you can reverse them. I will close this article with a few words about these two methods, and try to imagine how Meillassoux might be radicalized or reversed by some future admirer. My view is that the more important thinkers are, the easier they are to radicalize or reverse. This helps explain why the great philosophers of the West have so often appeared in clusters, succeeding one another at relatively brief intervals during periods of especial ferment. 1. After Finitude After Finitude is unusually short for such an influential book of philosophy: running to just 178 pages in the original French, and an even more compact 128 pages in the English version, despite the introduction of roughly eight pages of new material for the English edition. Rather than summarizing Meillassoux’s book in the order he intended, I will focus on six points that strike me as the pillars of his debut book. Along the way, I will offer a few criticisms as well. The first pillar of the book is Meillassoux’s own term “correlationism.”1 Although he introduces this term as the name for an enemy, it is striking that Meillassoux remains impressed by correlationism much more than his fellow speculative realists are. This continued appreciation for his great enemy influences the shape of his own ontology. Is there a world outside our thinking of it, or does the world consist entirely in being thought? Traditionally, this dispute between realism and idealism has been dismissed in continental philosophy as a “pseudo-problem,” in a strategy pioneered by Husserl and extended by Heidegger. We cannot be realists, since following Kant we have no direct access to things-in-themselves. But neither are we idealists, since the human being is always already outside itself, aiming at objects in intentional experience, deeply engaged with practical implements, or stationed in some particular world-disclosing mood. The centuries-old dispute between realism and idealism is dissolved by saying that we cannot think either real or ideal in isolation from the other. There is neither human without world nor world without human, but only a primordial correlation or rapport between the two. This is what “correlationism” means: philosophy trapped in a permanent meditation on the human-world correlate, trying to find the best model of the correlate: is it language, intentionality, embodiment, or some other form of correlation between human and world? Among other problems, this generates some friction between philosophy and the literal meaning of science. When cosmologists say that the universe originated 13.5 billion years ago, they do not mean “13.5 billion years ago for us ,” but literally 13.5 billion years ago, well before conscious life existed, and thus at a time when there was no such thing as a correlate. Meillassoux also coins the term “ancestrality” (10) for the reality that predated the correlate, and later expands this term to “dia-chronicity,” (112) to refer to events occurring after the extinction of human beings no less than to those occurring before we existed. Up to this point, Meillassoux’s focus on ancestral entities existing prior to consciousness might seem like a straightforward realist who wants to unmask correlationism as just another form of idealism. Yet Meillassoux also admires the correlationist maneuver, which can obviously be traced back to Kant. Unlike a thinker such as Whitehead, Meillassoux feels no nostalgia for the pre-critical realism that came before Kant: “we cannot but be heirs of Kantianism,” he says (29). What impresses Meillassoux about correlationism is something both simple and familiar. If we attempt to think a tree outside thought, this is itself a thought . Any form of realism which thinks it can simply and directly address the world the way it is fails to escape the correlational circle, since the attempt to think something outside of thought is itself nothing other than a thought, and thereby collapses back into the very human-world correlate that it pretends to escape. For Meillassoux this step, suggested by Kant but first refined by the ensuing figures of German Idealism, marks decisive forward progress in the history of philosophy that must not be abolished. Any attempt to break free from the correlate must first acknowledge its mighty intellectual power. Realist though he may seem, Meillassoux’s works are filled with praise of such figures as Fichte and Hegel, not of so-called “naïve realists.” It is also the case that for Meillassoux, not all correlationisms are the same. The second pillar of his book is a distinction between various positions that I have termed “Meillassoux’s Spectrum,” though of course he is never so immodest as to name it after himself. He distinguishes between at least six different possible positions, and perhaps we could add even subtler variations if we wished. But in its simplest form, Meillassoux’s Spectrum allows for just four basic outlooks on the question of realism vs. anti-realism. Three of these are easy to understand, since we have already been discussing them. At one extreme is so-called “naïve realism,” which holds that a world exists outside the mind, and that we can know this world. Meillassoux rejects this naïve realism as having been overthrown by Kant’s critical philosophy. At the other extreme is subjective idealism, in which nothing exists outside the mind. For to think a dog outside thought immediately turns it into a thought, and therefore there cannot be anything outside; the very notion is meaningless. In between these two is what we have called correlationism. And here comes a crucial moment for Meillassoux, since he distinguishes between the two forms of “weak” and “strong” correlationism, and chooses the strong form as the launching pad for his own philosophy. Weak correlationism is easy to explain, since we all know it from the philosophy of Kant. The things-in-themselves can be thought but not known. They certainly must exist, since there cannot be appearances without something that appears. And we can think about them, which idealism holds to be impossible. They are simply unknowable due to the finitude of human thought. Strong correlationism is the new position introduced by Meillassoux (though he sees it at work in numerous twentieth century thinkers), midway between weak correlationism and subjective idealism. The major difference between the three positions is as follows. Weak correlationism says: “The things-in-themselves exist, but we cannot know them.” The subjective idealist says: “This is a contradiction in terms, since when we think the things-in-themselves, we already turn them into thoughts.” But the strong correlationist says: “Just because ‘things-in-themselves’ is a meaningless notion does not mean that they cannot exist. No one has ever traveled to the world-in-itself and come back to make a report on it. Thus, the fact that we cannot think things-in-themselves without contradiction does not prove that they do not exist anyway. There may be things-in-themselves, we simply are not capable of thinking them without contradiction form within the correlational circle.” This step is crucial for Meillassoux, since strong correlationism is the position he attempts to radicalize into his own new standpoint: speculative materialism. As I see it, this step of the argument fails. Strong correlationism cannot avoid collapsing into subjective idealism, since the statements of the strong correlationist are rendered meaningless from within. All three of the other positions in the Spectrum make perfectly good sense even for those who disagree with them. The naïve realist says that things-in-themselves exist and we can know them; the meaning of this statement is clear. The weak correlationist can say that things-in-themselves exist but lie forever beyond our grasp; this too makes perfect sense, even though the German Idealists try to show a contradiction at work here. We can also understand the claim of the subjective idealist that to think anything outside thought turns it into a thought, and that for this reason we cannot think the unthought. The strong correlationist, alone among the four, speaks nonsense . This person says “I cannot think the unthought without turning it into a thought, and yet the unthought might exist anyway.” But notice that the final phrase “the unthought might exist anyway” is fruitless for this purpose. For we have already heard that to think any unthought turns it into a thought. But now the strong correlationist wants to do two incompatible things simultaneously with this unthought. On the one hand, he neutralizes the unthought by showing that it instantly changes into just another thought. But on the other hand, he wants to appeal to the unthought as a haunting residue that might exist outside thought, thereby undercutting the absolute status of the human-world correlate found in idealism. But this is impossible. If you accept the argument that thinking the unthought turns it into a thought, you cannot also add “but maybe there is something outside that prevents this conversion from being absolutely true,” because this “something outside” is immediately converted into nothing but a thought for us. In short, Meillassoux here seems to be offering a kind of Zen koan: his “strong correlationism” is reminiscent of the gateless gate, or the sound of one hand clapping, or the command to punch Hegel in the jaw when meeting him on the road. We cannot at the same time both destroy the realist challenge of the things-in-themselves in order to undercut realism and reintroduce that very realist sense in order to undercut idealism. In a world where everything is instantly converted into thought, we cannot claim that there might be something extra-mental anyway, because this “might be something” is itself converted into a thought by the same rules that condemned dogs, trees, and houses to the idealist prison. This brings us to the third pillar of Meillassoux’s argument, which is the key to all the rest: the necessity of contingency. His strategy is to transform our supposed ignorance of things-in-themselves into an absolute knowledge that they exist without reason, and that the laws of nature can change at any time for no reason at all. In this way the cautious agnosticism of Kantian philosophies is avoided, but so is the collapse of reality into thought as found in German Idealism. Meillassoux does try to prove the existence of things-in-themselves existing outside thought; he simply holds that they must be proven after passing through the rigors of the correlationist challenge, not just arbitrarily decreed to exist in the manner of naïve realism. As he puts it, “Everything could actually collapse: from trees to stars, from stars to laws, from physical laws to logical laws; and this not by virtue of some superior law whereby everything is destined to perish, but by virtue of the absence of any superior law capable of preserving anything, no matter what, from perishing” (53). If idealism thinks that the human-world correlate is absolute, for Meillassoux it is the facticity of the correlate that is absolute. He tries to show this with a nice brief dialogue between five separate characters (55-59) which is covered in detail in my forthcoming book,2 but which I will simplify here for reasons of time. In this simplified version, we first imagine a dogmatic realist arguing with a dogmatic idealist. The realist says that we can know the truth about the things-in-themselves; the idealist counters that we can only the truth about thought, since all statements about reality must be turned into statements concerning our thoughts about reality. Here the correlationist enters and proclaims that both of these positions are equally dogmatic. For although we have access to nothing but thoughts, we cannot be sure that these thoughts are all that exist; there could be a reality outside thought, there is simply no way to know for sure. And this latter position is the one that Meillassoux attempts to transform from an agnostic, skeptical point into an ontological claim about the contingency of everything. Consider it this way. How does the correlationist defeat the idealist? The idealist holds that the existence of anything outside thought is impossible. The correlationist, by contrast, holds that something might exist outside the human-world correlate. But this “something might” has to be an absolute possibility. It cannot mean that “something outside thought might exist for thought ,” because that is what the idealist already says. No, the correlationist must mean that something might exist outside thought quite independently of thought. In other words, the correlationist says that idealism might be wrong, and this means it is absolutely true that idealism might be wrong. Thus, correlationism is no longer just a skeptical position. It holds that all the possibilities of the world are absolute possibilities. We have absolute knowledge that any of the possibilities about the existence or non-existence of things-in-themselves might be true, and this means that correlationism flips into Meillassoux’s own position: speculative materialism. As Meillassoux sees it, there are only two options here. Option A is to absolutize the human-world correlate, which is what the idealist does: there absolutely cannot be anything outside thought. Option B, by contrast, is to absolutize the facticity of the correlate: its character of simply being given to us, without any inherent necessity. The correlationist cannot have it both ways by saying: “there absolutely might be something outside thought, yet maybe this is absolutely impossible.” In other words, once we escape dogmatism we can only be idealists or speculative materialists, not correlationists. The human-world correlate is merely a fact, not an absolute necessity. But this facticity itself cannot be merely factical: it must be absolute. Here Meillassoux coins the French neologism factualité , which has been suitably translated into the English neologism “factiality.” (7, 122-3) Factialty means that for everything that exists, it is absolutely possible that it might be otherwise, not just that we cannot know whether or not it might be otherwise. Just as Kant transformed philosophy into a meditation on the categories governing human finitude, Meillassoux wishes to turn philosophy into a meditation on the necessary conditions of factiality, which he calls “figures”—a new technical term for him. (80) One such figure is that the law of non-contradiction must be true, and for an unusual reason. Since everything is proven to be contingent, nothing that exists can be contradictory, for whatever is contradictory has no opposite into which it might be transformed, and thus contingency would be impossible.3 Another such figure is that there must be something rather than nothing: for since contingency exists, something must exist in order to be contingent. It is a daring high-wire act, one that sacrifices realism to the correlational circle in order to rebuild it from out of its own ashes. Some might conclude that the lack of reason in things is a byproduct of the ignorance of finite humans, Meillassoux is making precisely the opposite point. For in fact, the doctrine of finitude usually leads directly to belief in a hidden reason. The fact that it lies beyond human comprehension merely increases our belief in this arbitrarily chosen concealed ground. By defending anew the concept of absolute knowledge Meillassoux evacuates the world of everything hidden. The reason for things having no reason is not that the reason is hidden, but that no reason exists. Thus, even while insisting on the necessity of non-contradiction, he rejects the other Leibnizian principle: sufficient reason. Everything simply is what it is, in purely immanent form, without deeply hidden causes. Or as Meillassoux puts it: “There is nothing beneath or beyond the manifest gratuitousness of the given—nothing but the limitless and lawless power of its destruction, emergence, or persistence” (63).The world is a “hyper-chaos”(64). But this is not the same thing as flux. For the chaos of the world is such that stability might occur just as easily as constant, turbulent change. Let’s now digress a bit, and return to the question of ancestrality, which Meillassoux transforms later in the book into “dia-chronicity.” Correlationism holds that all talk of a world outside the correlate is immediately recuperated by the correlate. The phrase “13.5 billion years ago” becomes “13.5 billion years ago for us ,” and the phrase “the universe following the extinction of humans” becomes “the universe following the extinction of humans for humans .” But notice that whether we talk about the world before or after humans, in both cases it is time that is used to challenge the correlate. Meillassoux has no interest in challenges that might be posed by space. For example, what about a vase in a lonely country house that topples to the floor and smashes when no one is there to watch it? Isn’t this also a challenge to correlationism, no less than the Big Bang or the heat death of the universe long after humans have vanished? In an eight-page supplement to the English translation of After Finitude ,4 possibly in response to my own 2007 review of the French original,5 Meillassoux bluntly denies that space is of any relevance to the question. Spatial distance is a merely harmless challenge to the human-world correlate. After all, even though no one is there in the lonely country house to witness the shattering of the vase, we can say that had there been an observer , that observer would have witnessed the toppling and destruction of the vase. For this event still occurs in a world in which the human-world correlate already exists, whereas the diachronicity of events both before and after the existence of humans makes it impossible to say that had there been an observer they would have witnessed the Big Bang occurring in such and such a fashion. However, it seems to me that Meillassoux merely asserts that the temporal simultaneity of our existence with that of the vase in the lonely country house is enough to render it harmless. It is true that the house does not exist prior to the correlate, but nonetheless it exists outside the correlate, and that is enough to make the same challenge. It is difficult to see why the “had there been an observer” maneuver succeeds in the case of a vase in the countryside in April 2011 but fails in the case of the Big Bang. This is not just a matter of nitpicking Meillassoux’s argumentative style: the fact that he bases his argument on time has at least two important consequences for his position. For in the first place, even though Meillassoux insists that the laws of nature are absolutely contingent, this turns out to be true only in a temporal sense. That is to say, it is a paradoxical feature of Meillassoux’s philosophy that he does allow for the existence of laws of nature, and simply believes that they can change at any moment without reason. Within any given moment, laws of nature do exist. He never suggests that different parts of the universe can have different laws at the same time, nor does he have any interest in the laws of part/whole composition that take place within any given instant. Could it be the case that rather than being made of gold atoms, a small chunk of gold could be made of silver atoms, cotton, horses, or that this same small piece of gold could be made of gigantic vaults filled with even more gold? These are not topics that draw Meillassoux’s attention, since he is focused solely on how the laws of nature might change or endure from one moment to the next . Another implication for Meillassoux’s system is that his concept of things-in-themselves turns out to be to be inadequate. For when he proves that things-in-themselves can exist without humans, this turns out to be true only in a temporal sense as well. Namely, things-in-themselves existed ten billion years ago, and they will continue to exist after all humans have succeeded in exterminating themselves. However, being able to exist before our births and after our deaths is just one small part of what it means to be a thing-in-itself. The more important part is that even if a thing is sitting on a table right now, in front of me, even if I stroke it lovingly or press my face up against it directly, I am still dealing only with a phenomenal version of the thing; the thing-in-itself continues to withdraw from all access. Yet no such thing is acknowledged by Meillassoux. For him finitude is a disaster, and absolute knowledge is in fact possible. Meillassoux’s thing-in-itself exists in independence only of the human lifespan , not of human knowledge. The fifth pillar of Meillassoux’s argument is his use of Cantor’s transfinite mathematics to show that even if the laws of nature are contingent, they need not be unstable, and thus we cannot use the apparent stability of nature to disprove his metaphysics of absolute contingency. What Cantor showed is that there are different sizes of infinity, and that all these infinities cannot be totalized in a single infinite number of infinities. Meillassoux sees this as crucial, since it allows him to discredit any “probabilistic” argument against his theory. The probabilistic argument (as defended quite clearly by Jean-René Vernes)6 would say this: given that the laws of nature seem so stable, it it is extremely improbable that there is no hidden reason for their remaining so stable. As Meillassoux sees it, probability is of value only when we can index an accessible total of cases. These can even be infinite: for example, there are an infinite number of points where a rope can break when stretched tight, but this does not stop us from calculating probabilities for various sections of the rope to break. By contrast, there is no way to sum up the number of possible laws of nature. For here there is no way to totalize; we cannot stand outside of nature and calculate the possible number of laws so as to determine the probabilities that any one of them might change. Therefore, although we can speak of probability when dealing with intraworldly events such as elections, horse races, and coin-flips, we cannot use the words “probable” or “improbable” when describing alterations at the level of nature as a whole. Rather than commenting on the validity of this argument and its use of Cantor, let me simply note that it once again creates a dualistic ontology. We already saw that Meillassoux treats time differently from space. In analogous fashion, he now treats the level of world differently from that of intraworldly events. The emergence of worlds is purely contingent and virtual and governed by no probability at all, while events within the world necessarily follow laws (even if these laws can change at any moment without reason), and thus their probabilities can be calculated. It is a strategy deeply reminiscent of Badiou’s (2005) own dualism between the normal “state of the situation” and the rare and intermittent “event.” The sixth and final pillar of Meillassoux’s book can be dealt with briefly, since we have already touched on it elsewhere. It comes at the very beginning of the book, when Meillassoux says that we must revive the distinction between primary and secondary qualities, and that the primary qualities are the ones that can be mathematized. He admits that he has not yet published a proof of this idea, though in fact it is already known as one of his primary doctrines. And here we encounter the familiar problem with Meillassoux’s inadequate conception of things-in-themselves. “Primary qualities” refers to those qualities that a thing has independently of its relations with us or anything else. But if the primary qualities can be mathematized, this means that they are not entirely independent of us, since our knowledge can get right to the bottom of them. The mathematized qualities of things are independent of us only in Meillassoux’s sense that they will still have those qualities even when all humans are dead. But to repeat, autonomy from the human lifespan is not the same as autonomy from human access. Here once more Meillassoux is concerned only with independence from the human-world correlate across time, not in any given instant. 2. L’Inexistence divine In 1997, the same year in which he turned thirty years old, Meillassoux earned his doctorate at the École normale supériuere with a brazen dissertation entitled L’Inexistence divine ( The Divine Inexistence ). The work was substantially revised in 2003. But even then, with typical fastidiousness, Meillassoux decided that the work was not yet ready for press. It has now been scrapped in favor of some future, multi-volume work bearing the same title. While writing my book on Meillassoux for Edinburgh University Press, I was permitted to translate excerpts from this unpublished work for use as an appendix in my own book; in total, the appendix contains approximately twenty percent of Meillassoux’s 2003 manuscript, the first time any of it will be published in any language. Nonetheless, a portion of the argument was already tested in the article “Spectral Dilemma,” published in English in the journal Collapse (2008: 261-75). There the philosophical motives for the virtual God are already made clear. What troubles us most are early deaths, brutal deaths, deaths of especial injustice– the sorts of deaths in which the brutal twentieth century was so abundant. And here, neither the atheist nor the believer can help us. The atheist can offer nothing but a sad and cynical resignation when reflecting on the victims of these terrible crimes. The believer does little better, being unable to explain how God could have allowed such things to happen, due to the famous intractability of the problem of evil. The solution offered to this dilemma by Meillassoux is bold, and all the more so given that he emerges from such a deeply Leftist, materialist, and unreligious background. His solution is that God does not yet exist, and therefore is not blameworthy for these catastrophes. Given that everything is contingent in Meillassoux’s philosophy, this God and divine justice might never exist, but they can at least exist as an object of hope. Let’s begin by jumping to the end of L’Inexistence divine , where the alternatives are laid out so nicely. There are four basic attitudes that humans can have towards God, Meillassoux says. First, we can believe in God because he exists. This is the classical theist attitude, rejected for the simple reason that it would be amoral and blasphemous to believe in a God who allows children to be eaten by dogs, to use Dostoevsky’s example. Second, we can disbelieve in God because he does not exist: the classical atheist attitude. But this leads to sadness, cynicism, and a sneering contempt for the greatness of human capacity. The third option, rather more complex, is to disbelieve in God because he does exist: in other words, to exist in rebellion against God as the one who must be blamed for the evils of the earth. The examples here might range from Lucifer himself, to the more human figure of Captain Ahab in Melville’s Moby-Dick , to Werner Herzog’s even more recent catchphrase, “Every man for himself, and God against all.” That leaves only the fourth option: believing in God because he does not exist. Meillassoux closes his book by saying that the fourth option has now been tried (namely, in the course of his own book), and that now that all four have been specified, we must choose. The first reaction to this theory of the inexistent God will be laughter. Few readers will ever be literally convinced by it, and probably none will immediately be convinced by it. But if we ask ourselves why we laugh, the answer is because it sounds so improbable that an inexistent God might suddenly emerge and resurrect the dead. It obviously sounds more like a gullible theology than a rigorous piece of philosophical work. Yet two things need to be kept in mind. First, Meillassoux’s theories are hardly more unlikely than those of great philosophers of the past such as Plato, Plotinus, Avicenna, Malebranche, Spinoza, Leibniz, Nietzsche, or Whitehead. We read the great philosophers not because their systems are plausible in commonsense terms that can be measured by the laws of probability. Instead, we read them precisely because they shatter the existing framework of common sense and open up new window on the universe. Second, and even more importantly, Meillassoux has already rejected probability as a valid measuring stick in philosophy. Or rather, he accepts probability in the intra-worldly realm (where it is linked with potentiality), and rejects it at the level of the world itself (where potentiality is replaced with what he calls virtuality). The virtual God can appear at any moment for no reason at all, just as any other new configuration of laws of nature can appear: in a manner that the laws of probability cannot calculate. Responding to those who might ridicule the idea of a sudden emergence of God and a resurrection of the dead, Meillassoux cites Pascal, who asserts that the resurrection of the dead would be far less incredible than the fact that we were born in the first place. This shifts philosophy onto new ground. Rather than concerning ourselves with what is likely to happen in the world as we know it, we focus instead on the most important things that could happen. For this reason, the expected objection that a virtual God is no more likely to appear than a virtual unicorn or a virtual flying spaghetti monster misses the point. Unicorns and spaghetti monsters could also appear, just like any other non-contradictory thing. But these would just be novel bizarre entities among others, not the heralds of completely new worlds. For Meillassoux, the emergence of matter, life, and thought have been the three truly amazing advents of the world so far, each of them dependent on the advent(s) preceding them. As he sees it, there can be no greater intraworldly entity than the human beings who already exist, since nothing in the world is better than the absolute knowledge of which humans alone are capable. This means that the next great advent must be something that perfects human beings rather than superseding them. And this can only be the world of justice, in which the dead are resurrected and their horrible deaths partially cancelled (Meillassoux never considers the possibility of a God who would literally erase the pre-divine past so that it never happened at all). The only immortality worth having is an immortality of this life, not an existence in some ill-defined afterworld. Human existence, he holds, must always be governed by a “symbol” that gives us the “immanent and comprehensible inscription of values in a world.” And just as cosmic history made the three great contingent leaps of matter, life, and thought, with a leap to justice as the only one still to come, a similar structure occurs within human culture and its symbols, which consist so far of the cosmological, naturalistic, and historical symbols, with a “factial” symbol still to come. We can review each of these symbols briefly. The cosmological symbol refers to the ancient dualism between the terrestrial and celestial spheres. Here below everything is conflict, corruption, and decay; but in the heavens nothing is perishable, all movement is circular, and everything is arranged in mutual harmony. This symbol is ended by modern physics when Galileo discovers such blemishes as sunspots and craters on the moon, and when Newton integrates both celestial and terrestrial movement into a single gravitational law. Next comes the naturalistic or romantic symbol, in which perfection comes not from the sky but from nature itself. The world is filled with pretty flowers (Meillassoux claims that the ancients never discussed the beauty of flowers until Plotinus in the third century) and with living creatures naturally moved by pity, at least until society corrupts them. This symbol collapses in the face of reality as we know it, since pity is no more common than war, corruption, and violence. This brings us to the historical symbol, which only now is passing away. Bad things may happen, but history has an inner logic of its own, such that everything works out in the end. The ultimate form of the historical symbol is the economic symbol, whether in a Marxist or neo-liberal form. Just as the Marxist holds that the inner economic logic of the capitalists will inexorably lead them to self-destruction, the neo-liberal assumes that the sum total of individual selfish actions will lead, in the long run, to the greatest possible good. We worship the economy and let it guide history for us, just as the ancients worshipped celestial bodies and held them to be free from blemish. The final remaining symbol is the factial symbol, which Meillassoux hopes will now emerge. Factiality, we recall, is his term for the absolute contingency of everything that exists. Once we have grasped this absolute contingency, we are free to expect the dramatic advent of the coming fourth World: the world of justice, inaugurated by a virtual God and even mediated by a messianic human figure. There is the added feature, however, that this messiah must abandon all claims to special status once the messianic realm of justice is achieved. The messianic figure will then be no more special than any person on the street, since a reign of human equality will have arisen. Although this focus on human being might seem like a return to standard humanism, Meillassoux holds that human pre-eminence has never truly been maintained. Previously, humans have been treated as special only because they contemplate the Good, because they resemble their omnipotent creator, or because they happen to be the temporary victors in a cruel Darwinian death-match between millions of living species. For Meillassoux, by contrast, humans have value because they know the eternal. But it is not the eternal that is important, since this merely represents the blind, anonymous contingency of each thing. What is important is not knowledge of the eternal , but knowledge of the eternal. We should not admire Prometheus for stealing fire from the gods; Prometheus is simply as bad as all the gods, no matter how much he increased our power. Feuerbach and Marx were wrong to say that God is a projection of the human essence, since for Meillassoux the usual concept of God represents the degradation of the human essence. If the traditional God was allowed to inflict plagues and tsunamis on the human race, the Promethean human of the twentieth century simply assumes the right to inflict death camps and atomic fireballs instead. In this respect, we have simply begun to imitate the degradation of humanity that was formerly invested in an omnipotent and arbitrary God. In response to charges that absolute contingency might lead to political quietism, Meillassoux counters that the World of justice would mean nothing unless we had already hoped for it beforehand. A World of justice that came along at random would merely be an improved third World of thought: indeed, a perfect one. But it would have satisfied no craving, and would therefore have no redemptive power. For this reason, we must actively hope for the fourth World of justice for such a fourth World ever to arise. Not only justice, but beauty is dependent on such hope: for Meillassoux, who is here somewhat dependent on Kant, beauty means an accord between our human symbolization and the actual world, which could never be present in a World of the blessed any more than justice could. And just as a messianic figure is needed to incarnate our hope and then abandon power once the World of justice is realized, it is the figure of the child whose fragile contingency shows us a dignity and a demand for justice beyond all power. 3. Meillassoux Radicalized or Reversed Given the promising reception of Meillassoux’s first book, it would not be groundless to engage in early speculation about what it might take to earn him a place in the history of philosophy. Maybe this will never happen—who knows?—but quite possibly it will: his lucid argumentative methods and sheer philosophical imagination at least make him a good candidate to be read well into the future, especially following further elaboration in print of his mature system. Philosophy is often practiced as thought it were nothing more than the amassing of “knockdown arguments.” But this is no more insightful than saying that good architecture is the amassing of steal beams. It is true that poorly constructed building cannot stand for long, but sound construction is merely the first, indispensable step in building. In fact, I am inclined to say that what really makes a philosopher important is not being right, but being wrong . I mean this in a very specific sense. I once heard the interesting remark about twentieth century culture that “you have to remember that the sixties really happened in the seventies.” That is to say, it was in the 1970’s rather than the more honored 1960’s that civil rights, free love, long hair, and the rock and roll drug culture really took root. With respect to the history of philosophy, we might just as easily say: “you have to remember that Plato really happened in Aristotle,” that “Kant really happened in Hegel” or “Hume really happened in Kant,” or that “Husserl’s phenomenology first achieved its truth in Heidegger.” One becomes an important philosopher not by being right, but by attracting rebellious admirers who tell you that you are wrong , even as their own careers silently orbit around your own. To recruit faithful disciples may be comforting and flattering, but the greatest thinkers have generally had to experience refutation at the hands of their most talented heirs. For this reason, I would propose that we size up the magnitude of living thinkers not by deciding how many times they are right and wrong, but by asking instead: who would take the trouble to refute this author? For this reason I do not ask: “Is Meillassoux right?”, since I do not believe in the virtual God myself, nor am I convinced by any important aspect of Meillassoux’s philosophy. Instead, I ask if there are interesting ways to overturn him. Only by being overturned, by no longer remaining a contemporary, does one become a classic. Let’s begin with a simple model of refutation, which can be refined further at a later date once the basic point is established. One kind of refutation simply consists in saying: “This author is a complete idiot.” The refuter now walks away in celebration, and no link between the present and the future is built; all is reduced to rubble. But this sort of mediocre triumphalism is generally practiced by those who achieve little of their own, and is not especially interesting. Much more interesting is the sort of refutation that does not take its target to be a complete idiot. I would like to suggest that there are just two basic ways in which this can be done: radicalization and reversal . It has not escaped my notice that this is a fairly good match for the Deleuzian distinction between irony and humor. Whereas irony critiques and adopts the opposite principle of what it attacks, humor accepts what it confronts but pushes it into highly exaggerated form. The ironist is like the worker who sows chaos by rebeling and contradicting the boss, while the humorist is like the worker who follows orders to an absurdly literal degree, with equally chaotic results. Let’s start with a few examples. In Aristotle’s treatment of Plato, and Heidegger’s of Husserl, we find reversal. Plato’s eidei are transformed by Aristotle into mere secondary substances, and the individual worldly things despised by Plato become what is primary. For Husserl what is primary is whatever is present to consciousness, while for Heidegger this is precisely what is secondary, since the primary stuff of the world withdraws from any form of presence at all. As for radicalization, it is most easily found in the transformation of Kant by German Idealism: “Kant was right to wall off the things-in-themselves from human access, and simply should have realized that the thought of the Ding an sich is also a thought, and thereby the noumena are just special cases of the phenomena,” with much following from this discovery. It would also be easy to read Spinoza as a radicalizer of Descartes, and Berkeley and Hume as radicalized versions of Locke. Perhaps the distinction is now sufficiently clear. Admiring refutations are not those that say “Professor X is an idiot,” which is merely the flip side of the eager disciple’s fruitless “Professor X got everything right.” Instead, it will be some variant of one of the following two options: “Professor X is important, but got it backwards,” or “Professor X is important, but didn’t push things far enough.” In the history of philosophy these two latter cases have often been painful in purely human terms: Aristotle expresses sadness at refuting Plato, Kant is openly annoyed at Fichte, and Husserl feels betrayed and used by Heidegger. Rude handling from later figures almost seems to be the sine qua non of being a great philosopher. Now, it has already been claimed that Meillassoux is an emerging philosopher of the first importance, and by no less a figure than Alain Badiou: “It would be no exaggeration to say that Quentin Meillassoux has opened up a new path in the history of philosophy…” (Preface, vii). But rather than taking Badiou’s word for it, or rejecting his word, we might experiment by asking how Meillassoux could be radicalized or reversed. Are there interesting ways of doing this that might launch whole new schools of philosophy, unexpected or even condemned by Meillassoux himself? While no one can see the future, the present is poor when it is not riddled with virtual futures. The relation between philosophers and their predecessors and successors is always somewhat complicated, of course. But generally there is one central divergence at stake, which might be taken as the key to all the others. On this basis we could say that new thinkers primarily radicalize or primarily reverse the main ideas of their chief philosophical forerunner. There may be specific historical conditions and perhaps even personality traits connected with these two types, but this question can be left aside for now. More important for us is that radicalizers will generally be followed by reversers, and vice versa. Consider the textbook example of a reversal in the history of philosophy: Kant’s Copernican Revolution, which inverts the so-called dogmatic tradition that addresses the world itself, and makes the world revolve instead around the conditions by which it is known. While it is not completely impossible that Kant’s successors might have re-reversed this principle back into a new and stronger dogmatic realism, conditions were premature for such a move. Anyone doing this too early would likely have been an angry anti-Kantian reactionary rather than an original thinker in command of a genuinely new realist principle. The far more likely outcome is the one that actually happened: Kant’s reversal of his predecessors was viewed as incomplete, or as retaining lamentable bits of the traditional view, which despite his admirable breakthrough he was unable to shake off. This was the view of German Idealism, anyway. In similar fashion, Spinoza could also be viewed as a radicalizer of Descartes, who is equally accused of preserving various Scholastic dogmas in an otherwise radical project of philosophical reversal. The point is this: reversals in the history of thought tend to be followed soon thereafter by radicalizations of those reversals. The same may hold true in reverse: radicalizations might generally be followed by reversals, given that it is not always possible to be more radical than the radicals have already been. Consider the case of Husserl, who radicalizes Brentano’s early vagueness about what lies beyond immanent objectivity, and Twardowski’s assertion that there must be an external object lying outside the intentional content, by collapsing everything into the intentional sphere: there is no difference between the Berlin in my consciousness and the actual Berlin that is home to millions of people. It is difficult to see how one could be even more radical than Husserl’s idealist turn here. And thus the road is paved to Heidegger’s reversal of classical phenomenology, in which the key point is what lies deeper than any presence to consciousness: the Sein whose power and obscurity cannot be made exhaustively present, but only sends itself in historical epochs. In similar fashion we might also read Leibniz as a reverser of Spinoza’s radicalization, retrieving a strong sense of individual substance and a certain validity of what the Scholastics had said. Returning to Meillassoux, we might ask which kind of philosopher he is: a radicalizer or a reverser? At present, Meillassoux looks to me like a radicalizer (though for now his future remains shrouded in mist). He takes the correlationist tradition, which allows us to speak only of the relation between human and world, and tries to raise it into an even more extreme claim about the absolute contingency of everything. But whereas German Idealism did this by trying to collapse the distinction between thought and world entirely into the “thought” side, Meillassoux does it by trying to shift the non-absolute contingency of the thought-world correlate from epistemology to ontology. It is no longer a question of the inability of human knowledge to know what lies outside the correlate, but the inability of reality itself to be rooted in any definite laws. Furthermore, if we look at the various features of Meillassoux’s philosophy identified earlier tonight, all but one are already so radical that there is no obvious way to push them further. The one exception would be his claim that the world as a whole can change for no reason at any moment, coupled with the inconsistent claim that within a given world there are laws of nature that everything must follow. If gravitational attraction between all masses is a current law in our world, then for Meillassoux there can be no exceptions to this law for as long as it remains in force. A toppled vase will fall to the floor every time for sure,unless there is a cosmic change by which the laws of nature as a whole have altered. (This is reminiscent of the late medieval distinction between the absolute and ordained power of God, according to which God has the power to set or change the laws of nature, but not to contravene those laws locally once they are set.) On this point, to radicalize Meillassoux would simply be to say: there are no laws of nature even in the local sense. Everything that happens, even in the world here and now, is purely contingent and not governed by even a trace of law. And while this would be a more consistent development of Meillassoux’s thoughts on contingency, it is difficult to see how it could lead to a new philosophy. Instead, the admiring successors of Meillassoux are more likely to reverse one of his already sufficiently radical points. At least four candidates come to mind: *First, we have seen that Meillassoux thinks correlationism is challenged by a time before or after consciousness, but not by a space lying outside it. Perhaps this could be reversed into saying that spatial exteriority is the really crucial point. The arguments on this point are perhaps the least convincing in After Finitude (and do not even occur in the original French edition), and therefore it might be a candidate for the “blind spot” of which no philosopher is ever free. *Second, Meillassoux uses Cantor to claim that the contingency of laws of nature would not entail that they are unstable. A successor of Meillassoux might claim that it does make them unstable, and celebrate this fact. This person would then have to explain why common sense seems to encounter a relatively stable world despite its truly rampant instability. Whereas Meillassoux’s problem is to show how stability might exist despite contingency, this successor’s problem would be slightly different: to show why actual, full-blown instability might have the appearance of stability. *Third, Meillassoux claims that the primary qualities of things are those that can be mathematized. He might be reversed by a successor who says the opposite: the mathematizable qualities are the secondary ones, and the primary ones are those that elude symbolic formulation. While this is a perfectly valid possible objection to Meillassoux, it is one that is made in advance by some of his predecessors and is still made by some of his peers, making it less interesting for futurology than some of his other points. *Fourth and finally, whereas Meillassoux claims that God does not exist but might exist in the future, a successor might argue even more bizarrely that God has always existed but might vanish in the future. Let’s arbitrarily select the first of these possibilities, and imagine briefly where it might lead, if pursued in the future by admiring detractors. Meillassoux comes from the circle of Badiou, and some of Badiou’s most ardent admirers are found in Latin America. So, let’s imagine that towards mid-century some ingenious reversers of Meillassoux emerge in that portion of the Spanish-speaking world. Just for fun, let’s call them Castro and Chávez. And in order to avoid any confusion with the present-day politicians of those names, we will stipulate that Meillassoux’s great successors are both women. The philosopher Castro (we will suppose she comes from Peru) reverses Meillassoux’s argument that the ancestral or diachronic are what most threaten the human-world correlate. Instead, she claims that the diachronic does not threaten the correlate at all, and that we must instead look at space as what ruins the correlate and demands a strange new realism. What would such a philosophy look like? In order to determine this, we might ask what price Meillassoux pays for doing it the opposite way. As I see it, he pays in two separate ways. One is that laws of nature for him are contingent over time . The laws of nature apply to the universe as a whole at any given moment, and would be changed globally if they are ever changed at all. The second price he pays is that Meillassoux has no mereology , or theory of parts and wholes. Everything for him is on the level of the given, or immanent in experience, with the sole proviso that the laws governing this immanence might change without notice at any given moment. In reversing Meillassoux, Castro makes the following claims in the preface to her stunning debut book of 2045, The Cosmos and its Neighborhoods , rapidly translated from Spanish into all the languages of the world: Despite his brilliant analysis of the contingency of laws of nature over time, Meillassoux gets two important assumptions wrong. First, he allows for only one set of contingent laws to govern nature as a whole. Second, he allows laws to govern only the world that is immanent in experience, and thereby fails to explore the contingency among part-whole relations. In this book I will argue, first, that the laws of nature vary in any given instant between one region of the universe and the next; and second, that the world is made up of layers of parts and wholes that are also contingent with respect to one another. Those are the words of Castro. This may sound like a hopeless free-for-all of chaos, yet the book somehow succeeds in drawing some compelling deductions about how laws must vary from one place or level in the world to the next. Trapped in the limited horizon of 2011, and not yet inspired by the heavily balkanized political and technological situation of 2050 that somehow lends additional credence to Castro’s vision, we can only vaguely grasp what such a philosophy might look like. After this reversal of Meillassoux by Castro, the usual pattern leads us to expect a radicalization by Chávez, a young Argentine student of Castro. How could the already strange theories of Castro be radicalized? Perhaps as follows, in a disturbing new book entitled The Implosion of the Neighborhoods , which argues as follows: Castro was right to shift the Meillassouxian framework of contingency from time to space. However, in this respect she retained a surprisingly traditional opposition between the two. In this book I will show that time and space collapse into one another. This may sound too much like the discredited four-dimensional block universe of twentieth century physics and philosophy. However, the four-dimensional universe is a model biased in favor of space, merely adding an extra dimension to the commonsense spatial continuum while stipulating that the serial passage of time is an illusion. In this book I will argue instead for a one-dimensional space-time modeled after our experience of time, in which there is no simultaneous co-existence at all between different parts of the universe, or ‘neighborhoods’ as my esteemed teacher Castro has called them. Instead, the various portions of the universe link to one another by succession rather than by coexistence. Buenos Aires, New York, and Amsterdam do not exist simultaneously in the same landscape, but one after the other in the mind of some observer, and this observer can only be an observer much larger than any human. Against Meillassoux’s notion of a virtual God that does not exist now but might exist in the future, I will argue for an actual God that surveys the universe in sequence, thereby generating the illusion of spatial diversity and even the illusion of individual minds located within that diversity. Once this divine observer dies, the universe as a whole must perish. Again, these ideas are so bizarre that we of 2011 can barely comprehend them, just as Aristotle would have had a difficult time grasping the theories of Descartes. We could then perhaps imagine a further reversal of this theory, emanating from the intellectually resurgent Philippines of the twenty-second century. The Filipino School might argue that the universe is already dead, given the collapse of its spatial richness into the serial observations of a flimsy and mortal God. The virtual universe does not yet exist, but might exist in fully spatial form in the future, and this would require the death of God and the resulting liberation of God’s succession of images as independent, spatially situated realities. With a bit of sharpening, we might be able to make all of these imaginary thinkers more intuitively clear. Along with the history of philosophy, there might arise a new discipline generating imaginary futures for philosophy. The richness of Meillassoux’s system comes not from the fact that he is plausibly right about so many things, but because his philosophy offers such a treasury of bold statements ripe for being radicalized or reversed. He is a rich target for many still-unborn intellectual heirs, and this is what gives him the chance to be an important figure. NOTES 1. Quentin Meillassoux, After Finitude . Trans. R Brassier. (London: Continuum, 2008.) Page 5. The word “correlationism” does not appear in his doctoral thesis. As Meillassoux informed me in an email of February 8, 2011, he first coined this term in 2003 or 2004, while editing for publication a lecture he had given at the École normale supérieure on a day devoted to the theme of “Philosophy and Mathematics,” an event including Alain Badiou as one of the participants. 2. Graham Harman, Quentin Meillassoux: Philosophy in the Making . (Edinburgh: Edinburgh University Press, forthcoming 2011). 3. In an email of December 6, 2010, Meillassoux clarifies that in After Finitude he only deduces the impossibility of a “universal contradiction,” not of a determinate contradiction. In the same email he suggests that he can also prove the latter, though the proof is somewhat lengthier than the one found in After Finitude . 4. After Finitude (18-26), in the passage falling between the two sets of triple asterisks. These pages were sent by Meillassoux to translator Ray Brassier (in French) during the translation process, and do not appear in the original French version of the book. 5. Graham Harman, “Quentin Meillassoux: A New French Philosopher.” Philosophy Today 51.1 (2007): Pages 104-117. The passage where I raise the question of space can be found in the first column of page 107. 6. Vernes is first cited on p. 95 of After Finitude . See Jean-René Vernes, Critique de la raison aléatoire, ou Descartes contra Kant . (Paris: Aubier, 1982).  . (shrink)

William Lane Craig has argued that there cannot be actual infinities because inverse operations are not well-defined for infinities. I point out that, in fact, there are mathematical systems in which inverse operations for infinities are well-defined. In particular, the theory introduced in John Conway's *On Numbers and Games* yields a well-defined field that includes all of Cantor's transfinite numbers.

There are two main objections against epistemological foundation of logical principles: 1. Every argument for them must necessarily make use of them. 2. Logical principles cannot be abstracted from experience because they imply elements of meaning that exceed in principle our finite experience (like universality & necessity). In opposition to these objections I argue for Husserl's thesis that logic needs a theory of experience as a foundation. To show the practicability of his attempt I argue that he is able to (...) avoid the two circles mentioned. Motivated by this investigation the Appendix presents some mathematical doubts concerning the proofs by Cantor of the transfiniteness of the set of real numbers. (shrink)

Georg Cantor argued that pure mathematics would be better-designated “free mathematics” since mathematical inquiry need not justify its discoveries through some extra-mental standard. Even so, he spent much of his later life addressing ancient and scholastic objections to his own transfinite number theory. Some philosophers have argued that Cantor need not have bothered. Thomas Aquinas at least, and perhaps Aristotle, would have consistently embraced developments in number theory, including the transfinite numbers. The author (...) of this paper asks whether the restriction of arithmetic to the natural numbers that is apparently assumed by Aristotle and Aquinas is necessary in the light of their stated principles. The author concludes that, while some texts from Aristotle and Thomas suggest that such discoveries as zero, rational, and real numbers, and even Cantor’s own transfinite numbers, are legitimate objects of scientific knowledge, a careful analysis shows that they are incompatible with the ultimate arithmetical principle, the unit. (shrink)

Ortega y Gasset is known for his philosophy of life and his effort to propose an alternative to both realism and idealism. The goal of this article is to focus on an unfamiliar aspect of his thought. The focus will be given to Ortega’s interpretation of the advancements in modern mathematics in general and Cantor’s theory of transfinite numbers in particular. The main argument is that Ortega acknowledged the historical importance of the Cantor’s Set Theory, analyzed (...) it and articulated a response to it. In his writings he referred many times to the advancements in modern mathematics and argued that mathematics should be based on the intuition of counting. In response to Cantor’s mathematics Ortega presented what he defined as an ‘absolute positivism’. In this theory he did not mean to naturalize cognition or to follow the guidelines of the Comte’s positivism, on the contrary. His aim was to present an alternative to Cantor’s mathematics by claiming that mathematicians are allowed to deal only with objects that are immediately present and observable to intuition. Ortega argued that the infinite set cannot be present to the intuition and therefore there is no use to differentiate between cardinals of different infinite sets. (shrink)

§1. Iterated Gödelian extensions of theories. The idea of iterating ad infinitum the operation of extending a theory T by adding as a new axiom a Gödel sentence for T, or equivalently a formalization of “T is consistent”, thus obtaining an infinite sequence of theories, arose naturally when Godel's incompleteness theorem first appeared, and occurs today to many non-specialists when they ponder the theorem. In the logical literature this idea has been thoroughly explored through two main approaches. One is that (...) initiated by Turing in his “ordinal logics” and taken very much further in Feferman's work on transfinite progressions, which also introduced the more general study of extensions by reflection principles, of which consistency statements are a special case. This approach starts from an assignment of theories to ordinal notations, and extracts sequences of theories through a suitable choice of a path in the set of ordinal notations. The second approach, illustrated in particular by the work of Schmerl and Beklemishev, starts instead from a suitably well-behaved primitive recursive well-ordering, which is used to define a sequence of theories. This second approach has led to precise results about the relative proof-theoretical strength of sequences of theories obtained by iterating different reflection principles. The Turing-Feferman approach, on the other hand, lends itself well to an investigation in qualitative and philosophical terms of the relevance of such iterated reflection extensions to mathematical knowledge, in particular because of two developments associated with this approach. (shrink)

This textbook is a second edition of the successful, Mathematical Logic: On Numbers, Sets, Structures, and Symmetry. It retains the original two parts found in the first edition, while presenting new material in the form of an added third part to the textbook. The textbook offers a slow introduction to mathematical logic, and several basic concepts of model theory, such as first-order definability, types, symmetries, and elementary extensions. Part I, Logic Sets, and Numbers, shows how (...) mathematical logic is used to develop the number structures of classical mathematics. All necessary concepts are introduced exactly as they would be in a course in mathematical logic; but are accompanied by more extensive introductory remarks and examples to motivate formal developments. The second part, Relations, Structures, Geometry, introduces several basic concepts of model theory, such as first-order definability, types, symmetries, and elementary extensions, and shows how they are used to study and classify mathematical structures. The added Part III to the book is closer to what one finds in standard introductory mathematical textbooks. Definitions, theorems, and proofs that are introduced are still preceded by remarks that motivate the material, but the exposition is more formal, and includes more advanced topics. The focus is on the notion of countable categoricity, which analyzed in detail using examples from the first two parts of the book. This textbook is suitable for graduate students in mathematical logic and set theory and will also be of interest to mathematicians who know the technical aspects of the subject, but are not familiar with its history and philosophical background. (shrink)

This paper continues investigations of the monotone fixed point principle in the context of Feferman's explicit mathematics begun in [14]. Explicit mathematics is a versatile formal framework for representing Bishop-style constructive mathematics and generalized recursion theory. The object of investigation here is the theory of explicit mathematics augmented by the monotone fixed point principle, which asserts that any monotone operation on classifications (Feferman's notion of set) possesses a least fixed point. To be more precise, the new axiom not merely (...) postulates the existence of a least solution, but, by adjoining a new constant to the language, it is ensured that a fixed point is uniformly presentable as a function of the monotone operation. Let T 0 + UMID denote this extension of explicit mathematics. [14] gave lower bounds for the strength of two subtheories of T 0 + UMID in relating them to fragments of second order arithmetic based on Π 1 2 comprehension. [14] showed that T $_0 \upharpoonright$ + UMID and T $_0 \upharpoonright$ + IND N + UMID have at least the strength of (Π 1 2 - CA) $\upharpoonright$ and (Π 1 2 - CA), respectively. Here we are concerned with the exact reversals. Let UMID N be the monotone fixed-point principle for subclassifications of the natural numbers. Among other results, it is shown that T $_0 \upharpoonright$ + UMID N and T $_0 \upharpoonright$ + IND N + UMID N have the same strength as (Π 1 2 - CA) $\upharpoonright$ and (Π 1 2 - CA), respectively. The results are achieved by constructing set-theoretic models for the aforementioned systems of explicit mathematics in certain extensions of Kripke-Platek set theory and subsequently relating these set theories to subsystems of second arithmetic. (shrink)

The main purpose of the paper concerns the question of the existence of hard mathematical facts as truth-makers of mathematical sentences. The paper defends the standpoint according to which hard mathematical facts do not exist in semantic models of mathematical theories. The argumentative line in favour of the defended thesis proceeds as follows: slingshot arguments supply us with some reasons to reject various ontological theories of mathematical facts; there are two ways of blocking (...) these arguments: through the rejection of the principle of extensionality for individual terms or through the rejection of the principle of Wittgenstein; the first way cannot be accepted because it leads to the practice of softening mathematical facts; the second way, called fine-graining facts, cannot be accepted because it also results in the practice of softening facts. Hence, only soft mathematical facts can be introduced into semantic models of mathematical theories. Because they should be rather interpreted as mental representations of mathematical objects, they no longer satisfy the semantic role of truth-makers in relation to mathematical sentences. The argument formulated in the paper is based on the analysis of mathematical extensions of the basic non-Fregean logic. Sixty-two definitions of potential fact-identity connectives are also presented. The paper's conclusion may be interpreted as undermining the situational paradigm of building semantics for mathematical languages. (shrink)

A number of authors have noted that the key steps in Fitch’s argument are not intuitionistically valid, and some have proposed this as a reason for an anti-realist to accept intuitionistic logic (e.g. Williamson 1982, 1988). This line of reasoning rests upon two assumptions. The first is that the premises of Fitch’s argument make sense from an anti-realist point of view – and in particular, that an anti-realist can and should maintain the principle that all truths are knowable. The (...) second is that we have some independent reason for thinking that classical logic is not appropriate in this area. This paper explores these two assumptions in the context of Michael Dummett’s version of anti-realism, with particular reference to the argument from indefinite extensibility developed at various points in Dummett’s writings (e.g. Dummett 1991 Ch. 24). -/- Dummett argues that certain concepts, the indefinitely extensible concepts, are such that we cannot form a clear and determinate conception of all the objects that fall under them. The most familiar examples of indefinitely extensible concepts are mathematical. Dummett discusses the concepts ordinal number, real number, and natural number, which are indefinitely extensible because any conception that one might form of their complete extension can be extended to a more inclusive conception (as, for example, in Cantor’s proof of the non-denumerability of the set of real numbers). This paper argues that the concept of a truth is indefinitely extensible. This gives a Dummettian anti-realist an independent motivation for rejecting the classical understanding of the quantifiers in this area. At the same time, however, it places in doubt the admissibility of the knowability principle, which seems to involve quantification over the “totality” of truths. As Dummett is at pains to point out (1991: 316), some sentences that purport to quantify over the extension of an indefinitely extensible concept plainly have a truth-value (we can truly say, for example, that every ordinal number has a successor, even though when we say that we are not quantifying over the set of all ordinals). But is the knowability principle one of these sentences? (shrink)

One of the earliest applications of transfinite numbers is in the construction of derived sequences by Cantor [2]. In [6], the existence of derived sequences for countable closed sets is proved in ATR0. This existence theorem is an intermediate step in a proof that a statement concerning topological comparability is equivalent to ATR0. In actuality, the full strength of ATR0 is used in proving the existence theorem. To show this, we will derive a statement known to be (...) equivalent to ATR0, using only RCA0 and the assertion that every countable closed set has a derived sequence. We will use three of the subsystems of second order arithmetic defined by H. Friedman , which can be roughly characterized by the strength of their set comprehension axioms. RCA0 includes comprehension for Δmath image definable sets, ACA0 includes comprehension for arithmetical sets, and ATR0 appends to ACA0 a comprehension scheme for sets defined by transfinite recursion on arithmetical formulas. MSC: 03F35, 54B99. (shrink)

In the 1920s, David Hilbert proposed a research program with the aim of providing mathematics with a secure foundation. This was to be accomplished by first formalizing logic and mathematics in their entirety, and then showing---using only so-called finitistic principles---that these formalizations are free of contradictions. ;In the area of logic, the Hilbert school accomplished major advances both in introducing new systems of logic, and in developing central metalogical notions, such as completeness and decidability. The analysis of unpublished material presented (...) in Chapter 2 shows that a completeness proof for propositional logic was found by Hilbert and his assistant Paul Bernays already in 1917--18, and that Bernays's contribution was much greater than is commonly acknowledged. Aside from logic, the main technical contribution of Hilbert's Program are the development of formal mathematical theories and proof-theoretical investigations thereof, in particular, consistency proofs. In this respect Wilhelm Ackermann's 1924 dissertation is a milestone both in the development of the Program and in proof theory in general. Ackermann gives a consistency proof for a second-order version of primitive recursive arithmetic which, surprisingly, explicitly uses a finitistic version of transfinite induction up to www . He also gave a faulty consistency proof for a system of second-order arithmetic based on Hilbert's &egr;-substitution method. Detailed analyses of both proofs in Chapter 3 shed light on the development of finitism and proof theory in the 1920s as practiced in Hilbert's school. ;In a series of papers, Charles Parsons has attempted to map out a notion of mathematical intuition which he also brings to bear on Hilbert's finitism. According to him, mathematical intuition fails to be able to underwrite the kind of intuitive knowledge Hilbert thought was attainable by the finitist. It is argued in Chapter 4 that the extent of finitistic knowledge which intuition can provide is broader than Parsons supposes. According to another influential analysis of finitism due to W. W. Tait, finitist reasoning coincides with primitive recursive reasoning. The acceptance of non-primitive recursive methods in Ackermann's dissertation presented in Chapter 3, together with additional textual evidence presented in Chapter 4, shows that this identification is untenable as far as Hilbert's conception of finitism is concerned. Tait's conception, however, differs from Hilbert's in important respects, yet it is also open to criticisms leading to the conclusion that finitism encompasses more than just primitive recursive reasoning. (shrink)

Lewis' concept of admissibility was introduced as an integral part of his famous Principal Principle; the principle that initial rational/reasonable belief should conform to objective chance unless there is evidence to the contrary. At that time Lewis offered only the rough and ready characterisation that evidence not to the contrary of such dependence is admissible. This, together with some sufficiency conditions, served well enough until it became clear that admissibility was central to debates on the viability of Humean Supervenience (...) and the analysis of objective chance. In response, Thau and Lewis refined the concept of admissibility in various ways. Since the mid 90's those who have employed the concept have, with minor variations and additions, followed the Thau/Lewis line. Yet, in the 30 years since its introduction what has been all too conspicuous by its absence is a full formal definition of admissibility and its degrees. Herein a family of definitions—all in terms of screening off by chance—that capture much that has been agreed about admissibility are proposed and evaluated; one of which is ultimately found to be serviceable as a definition schema for relative admissibility and its degrees. (shrink)

Two thoughts about the concept of number are incompatible: that any zero or more things have a number, and that any zero or more things have a number only if they are the members of some one set. It is Russell's paradox that shows the thoughts incompatible: the sets that are not members of themselves cannot be the members of any one set. The thought that any things have a number is Frege's; the thought that things have a number only (...) if they are the members of a set may be Cantor's and is in any case a commonplace of the usual contemporary presentations of the set theory that originated with Cantor and has become ZFC.In recent years a number of authors have examined Frege's accounts of arithmetic with a view to extracting an interesting subtheory from Frege's formal system, whose inconsistency, as is well known, was demonstrated by Russell. These accounts are contained in Frege's formal treatise Grundgesetze der Arithmetik and his earlier exoteric book Die Grundlagen der Arithmetik. We may describe the two central results of the recent re-evaluation of his work in the following way: Let Frege arithmetic be the result of adjoining to full axiomatic second-order logic a suitable formalization of the statement that the Fs and the Gs have the same number if and only if the F sand the Gs are equinumerous. (shrink)

Regarding the relation of Plato’s early and middle period dialogues, scholars have been divided to two opposing groups: unitarists and developmentalists. While developmentalists try to prove that there are some noticeable and even fundamental differences between Plato’s early and middle period dialogues, the unitarists assert that there is no essential difference in there. The main goal of this article is to suggest that some of Plato’s ontological as well as epistemological principles change, both radically and fundamentally, between the early and (...) middle period dialogues. Though this is a kind of strengthening the developmentalistic approach corresponding the relation of the early and middle period dialogues, based on the fact that what is to be proved here is a essential development in Plato’ ontology and his epistemology, by expanding the grounds of development to the ontological and epistemological principles, it hints to a more profound development. The fact that the bipolar and split knowledge and being of the early period dialogues give way to the tripartite and bound knowledge and benig of the middle period dialogues indicates the development of the notions of being and knowledge in Plato’s philosophy before the dialogues of the middle period. -/- Keywords Plato; early dialogues; middle dialogues; being; knowledge; development -/- Introduction The differences between two groups of the early and middle period dialogues have always been a matter of dispute. Whereas the developmentalists like Vlastos, Silverman (2002), Teloh (1981), Dancy (2004) and Rickless (2007) think that from the early to the middle dialogues Plato’s philosophy changes, at least in some essential points, the unitarists like Kahn, Cherniss and Shorey believe that there happens no development and the differences must be taken as natural, ignorable and even pedagogic. In his well-known article, Socrates contra Socrates in PLATO, Vlastos lists ten theses of difference between two groups of dialogues. The first group which includes Plato’s early dialogues he divides to 'elenctic' (Apology, Charmides, Crito, Euthyphro, Gorgias, Hippias Minor, Ion, Laches, Protagoras and Republic I) and 'transitional' (Euthydemus, Hippias Major, Lysis, Menexenus and Meno) dialogues. These transitional come after all the elenctic dialogues and before all the dialogues of the second group which compose Plato’s middle period dialogues, including (with Vlastos' chronological order) Cratylus, Phaedo, Symposium, Republic II-X, Phaedrus, Parmenides and Theaetetus (1991, 46-49). Vlastos asserts strictly that his list of differences are 'so diverse in content and method that they contrast as sharply with one another as with any third philosophy you care to mention, beginning with Aristotole’s' (ibid, 46). Vlastos’ list of differences between two Socrateses is considered a view breaking sharply between the early and the middle dialogues. Besides all ten differences between the two Socrateses in Vlastos’ list that can be supportive for our doctrine here, we intend to focus on some ontological as well as epistemological distinctions that have not completely been discussed hitherto. Contrary to the developmentalist theory of Vlastos, unitarian theory of Charles Kahn wishes to eliminate any substantial difference between the early and the middle dialogues. He distinguishes seven 'pre-middle' or 'threshold' dialogues including Laches, Charmides, Euthyphro, Protagoras, Meno, Lysis and Euthydemus from the other Socratic dialogues which he calls 'earliest group' (1996, 41). The threshold dialogues, Kahn thinks, 'embarke upon a sustained project' that is to reach to its climax in the middle period dialogues, namely Phaedo, Symposium and Republic. Believing in that there is no 'fundamental shift'between the early and middle dialogues (ibid, 40), Kahn thinks the Socratic dialogues are just the 'first stage' with a 'deliberate silence' towards the theories of later periods (ibid, 339). One of the reasons of the surprising fact that Plato gives no hint of the metaphysics and epistemology of the Forms in the early dialogues, he thinks, is the pedagogical advantages of aporia. He thinks that Plato 'obscurely', and mostly because of education, hinted to some doctrines and conceptions in his early dialogues and with the aim of clarifying them only in the later ones (ibid, 66). The seven threshold dialogues, Kahn asserts, 'had been designed from the first' to prepare the pupils and readers for the views expounded in the middle period dialogues (ibid, 59-60). To show that the differences of the two Socrateses of the early and middle period dialogue are in their onto-epistemological grounds and thence cannot be explained by a unitaristic view, we try to draw the ontological and epistemological principles of the early dialogues in the first part below in order to show, in the second part, that those principles have been developed in the middle period dialogues. -/- A. The Onto-Epistemologic Principles of the Early Dialogues Socratic dialogues are paradoxical about knowledge because while being knowledge-oriented always searching for knowledge, they deny it and even never discuss it directly. Three elements of Socrates’ way of searching Knowledge throughout the early dialogues, i.e. Socratic 'what is X?' question, his disavowal of knowledge and his elenctic method combined together produce something like a circle which works, more or less, in the same way in these dialogues. Though this circle is an embaressing inquiry always resulting in ignorance instead of knowledge, its motivation is surprisingly Socrates’ passionate enthusiasm for knowledge, an intensive love of wisdom. The starting point of this circle is Socrates’ confession of having no knowledge which might be explicitly asserted or presupposed, maybe because it was one of the famous characteristics of Socrates; a confession always paradoxically accompanied with his intense longing for knowledge (e.g., Gorgias 505e4-5). Every time Socrates encounters with someone who thinks he knows something (οἴεταί τι εἰδέναι) (Apology 21d5) and tries to examine him. This examination seems to be the simplest one asking just what it is that he knows. Socrates’ elenchus, therefore, is always connected with 'what is X?' (τίς ποτέ ἐστιν) question, a question that most of the early dialogues of Plato are concerned with; 'what is courage?' in the Laches, 'what is piety?' in Euthydemus, 'what is temperance?' in Charmides and 'what is beauty?' in Hippias Major. This question we call here 'Socratic question' and probably is the very question the historical Socrates used to employ, is tightly interrelated with both his disavowal and his elenchus. He disclaims knowledge because he cannot find the answer to the question himself and he rejects others’ since they cannot offer the correct answer too. Every interlocutor can claim he knows X, if and only if he can answer the Socratic question. Otherwise, he is more of an ignorant than of a knower of τί ποτέ ἐστι X. Knowing the answer to this question is, thus, knowledge’s criterion for Socrates. Having found out that he cannot answer what it is which he would claim to know, the interlocutor comes to the point Socrates was there at the beginning. The least advantage of this circle is that he becomes as wise as Socrates does about the subject, becoming aware that he does not know it. At the end of the circle they are both still at the beginning, not knowing what X is. So let us first take a glance at these three elements. Socratic disavowal of knowledge is strictly asserted in some passages. At Apology 21b4-5, Socrates says: -/- I do not know of myself being wise at all (οὔτε μέγα οὔτε σμικρὸν σύνοιδα ἐμαυτῷ σοφὸς ὤν) -/- Moreover, at 21d4-6: -/- None of us knows anything worthwhile (καλὸν κἀγαθὸν εἰδέναι) …. I do not know (οἶδα) neither do I think I know. -/- In Charmides Socrates speaks about a fear about his probable mistake of thinking that he knows (εἰδέναι) something when he does not (166d1-2). The somehow generalization of this disavowal can be seen in Gorgias. After calling his disavowal 'an account that is always the same' (509a4-5), Socrates continues (a5-7): -/- I say that I do not know (οἶδα) how these things are, but no one I have ever met, like now, who can say anything else without being absurd. -/- At the end of Hippias Major (304d7-8), Socrates affirms his disavowal of knowledge of 'what is X?' this time about the fine: -/- I do not know (οἶδα) what that is itself (αὐτὸ τοῦτο ὅτι ποτέ ἐστιν). Some other passages, however, made a number of scholars dubious about Socrates’ disavowal. Vlastos mentions Apology 29b6-7 as an evidence : 'that to do wrong and to disobey one’s master, both god and men, I know (οἶδα) to be evil and shameful'. He thinks that if we give this single text 'its full weight' it can suffice to show that Socrates does claim knowledge of a moral truth (1985, 7). Vlastos’ claim is not admittable since it would be too strange, I think, for a man like Socrates to violate his disavowal claim just after emphasizing it. We can see his claim just before the already mentioned passage: -/- And surely it is the most blameworthy ignorance to believe that one knows what one does not know. It is perhaps on this point and in this respect, gentlemen, that I differ from the majority of men and if I were to claim that I am wiser than anyone in anything it world be in this, that, as I have no adequate knowledge (οὐκ εἰδὼς ἱκανῶς) of things in the underworld, so I do not think I have (οὐκ εἰδέναι) (Apology 29b1-6) Vlastos tries to solve what he calls the 'paradox' of Socrates’ disavowal of knowledge distinguishing the 'certain' knowledge from 'elenctic' knowledge (1985, 11) and thinking that when Socrates avows knowledge, we must perceive it as an elenctic knowledge, a knowledge its content 'must be propositions he thinks elenctically justifiable' (ibid, 18). Irwin’s solution is the distinction of knowledge and belief. He approves that Socrates does not 'explicitly' make such a distinction, but still thinks that Socrates’ 'test for knowledge would make it reasonable for him to recognize true belief without knowledge, and his own claims are easily understood if they are claims to true belief alone' (1977, 40). While I agree with Vlastos up to a point, I strongly disagree with Irwin about the early dialogues. As I will try to show below, we are not permitted to consider any kind of distinction within knowledge in Socratic dialogues because only one category of knowledge is alluded to there and the distinction of knowledge and belief thoroughly belongs to the second Sorcates. Although no kind of distinction can be admittable here, I think Vlastos’ distinction can be accepted only if we regard it as a distinction between knowledge, which is unique and without any, even incomparable, rival, and a semi-idiomatic and ordinary one that is a necessary requirement for any argument, and thus, unavoidable even for someone who does not claim any kind of knowledge. An apparent evidence of this is Gorgias 505e6-506a4: -/- I go through the discussion as I think it is (ὡς ἄν μοι δοκῇ ἔχειν), if any of you do not agree with admissions I am making to myself, you must object and refute me. For I do not say what I say as I know (οὐδὲ γάρ τοι ἔγωγε εἰδὼς λέγω ἃ λέγω) but as searching jointly with you (ζητῶ κοινῇ μεθ᾽ ὑμῶν). -/- The minimal degree of knowledge everyone must have to take part in an argument, conduct it and use the phrase "I know" when it is necessary is what Socrates cannot deny. We can call it elenctic knowledge only if we agree that it is not the kind of knowledge that Socrates has always been searching, the one that can be accepted as the answer of Socratic 'what is X?' question. His disavowal of knowledge is applied only to the knowledge which can truly be the answer of Socratic question and pass the elenctic exam; a knowledge that, I believe, is never claimed by first Socrates. Socrates conducts his method of examining his interlocutors’ knowledge, our second element here, by almost the same method repeated in Socratic dialogues. That whether we are allowed to regard all the examinations of Socrates in the early dialogues as based on the same or not has been a matter of dispute. Vlastos himself (1994, 31) distinguished Euthydemas, Lysis, Menexenns and Hippias Major from the other Socratic dialogues because he thinks Socrates has lost his faith to elenchus in there. Irwin (1977, 38) distinguishes Apology and Crito where Socrates’ own convictions is present. Contrary to some scholars like Benson (2002, 107) who take elenchus in all the Socratic dialogues as a unique method, Michelle Carpenter and Ronald M. Polansky (2002, 89-100) argue pro the variety of methods of elenchus. Thomas C. Brickhouse and Nicholas D. Smith (2002, 145-160) even reject such a thing as Socratic elenchus which can gather all of the Socrates’ various arguments under such a heading. There can be no solution for the problem of elenchus, they think, is due to 'the simple reason that there is no such thing as 'the Socratic elenchos"'(p.147). Though we consider elenchus a somehow determined process and a part of Socratic circle in the early dialogues, all we assume is that whatever differences it might have in different dialogues, it has the same onto-epistemological principles and, thus, we are not going to take it necessarily as a unique method. This method sets out to prove that the interlocutors are as ignorant as Socrates himself is. That whether elenchus is constructive, capable of establishing doctrines as Vlastos (1994) or Brickhouse and Smith (1994, 20-21) believe or not is another issue to which this paper is not to claim anything. What is crucial for our discussion here is that elenchus does not reach to the very knowledge Socrates is looking for. This is strictly against Irwin who thinks that not only elenchus leads to positive results, 'it should even yield knowledge to match Socrates’ conditions'(1977, 68, cf. 48). He does not explain where and how they really yield to that kind of knowledge. He explains his elenctic method in his apology in the court (Apology 21-22), that how he used to examine wise men, politicians, poets and all those who had the highest reputation for their knowledge and every time found that they have no knowledge. If we accept, as I strongly do, that Socrates’ disavowal of his knowledge is not irony, it might seem more reasonable to agree that the process that has that disavowal as its first step cannot be irony as well. The key of the circle which can explain why Socrates disclaims knowledge and how he can reject others’ claim of having any kind of knowledge lies in the third element, Socratic question. In Hippias Major (287c1-2), Socrates asks: 'Is it not by Justice that just people are Just? (ἆρ᾽ οὐ δικαιοσύνῃ δίκαιοί εἰσιν οἱ δίκαιοι). He insists at 294b1 that they were searching for that by which (ᾧ) all beautiful things are beautiful (cf. b4-5, 8). At Euthyphro 6d10-11 it is said that the Socratic question is waiting for 'the form itself (αὐτὸ τὸ εἶδος) by which (ᾧ) all the pious actions are pious; and at 6d11-e1: -/- Through one form (που μιᾷ ἰδέᾳ) impious actions are impious and pious actions pious. -/- Since the X itself is that by which X is X, knowing 'X itself' is the only way of knowing X. It is probably because of this that Socrates makes the distinction between the ousia as a right answer to the question and effect as a wrong one: -/- I am afraid, Euthyphro, that when you were asked what piety is (τὸ ὅσιον ὅτι ποτ᾽ ἐστίν) you did not wish to make clear its nature itself (οὐσίαν … αὐτοῦ) to me, but you said some effect (πάθος) about it. (Euthp. 11a6-9) -/- 1. Knowledge of what X is Now it is time to look for the onto-epistemological principles of Socratic circle and its three elements. It cannot reasonably be expected from the first Socrates to present us explicitly and clearly formulated principles of his onto-epistemology when such explications cannot be found even in the second Socrates who has some obviously positive theoris. Since there must be some principles underlying this first systematic inqury of knowledge, we must seek to them and be satisfied with elicitation of the first Socratas’ principle. What will be drawn out as his principles cannot and must not, thus, be taken as very fix and dogmatic principles. Some very slightest principles and grounds suffice for our purposes here. The first principle that is the prima facie significance of the Socratic question and his implementation of all those elenctic arguments I call the principle of 'Knowledge of What X is' (KWX): -/- KWX To know X, it is required to know what X is. -/- Aristotle (Metaphysics 1078b23-25, 27-29, 987b4-8) takes Plato’s τί ἐστι question as seeking the definition of a thing that is the same as historical Socrates’ search but applying to a different field. That Plato’s 'what is X?' question was a search for definition became the prevailed understanding of this question up to now. I am going neither to discuss the answer of Socratic question nor to challenge taking it as definition. What I am to insist is that knowledge is attached firmly to the answer of the question: Knowledge of X is not anything but the knowledge of what X is. It entails, certainly, the priority of this knowledge to any other kind of knowledge about X but it also has something more fundamental about the relation of knowledge and Socratic question. Socrates’ exclusive focus on the answer of his question can authorize the consideration of such an essential role for KWX in his epistemolgoy. The total rejection of his interlocutors’ knowledge when they are unable to answer the question can be regarded as a strong evidence for it. Socrates’ elenctic method and his rejection of others’ knowledge in the early dialogues, which all end aporetically with no one accepted as knower and nothing as knowledge, prevent us from finding any positive evidence for this. We have to be content, therefore, of negative evidences which, I think, can be found wherever Socrates rejects his interlocutors’ knowledge when they are not able to give an acceptable answer to his 'what is X?' question. 2. Bipolar Epistemology Socrates’ rejection of his interlocutors’ knowledge has another epistemological principle as its basis. Let me call this principle Bipolar Epistemology' (BE): BE There is no third way besides knowledge and Ignorance. -/- BE says that about every object of knowledge there are only two subjective statuses: knowledge and Ignorance. Socrates’ disavowal, however, says nothing but that he is ignorant of knowledge of X because he does not know what X is. This means that BE is presupposed here. Socrates’ elenchus and his rejection of interlocutors’ having any kind of knowledge are the necessary results of the fact that he does not let any third way besides knowledge and ignorance. The first Socrates never let anyone partly know X or have a true opinion about it, as he would not let anyone know anything about X when he did not know what X is. -/- 3. Bipolar Ontology The principle of BE in first Socrates’ epistemology is parallel to another principle in his ontology. Plato’s bipolar distinction between being and not being is as strict and perfect as his distinction between knowledge and ignorance. This principle I shall call the principle of 'Bipolar Ontology' (BO): BO Being is and not being is not. BO is apparently the same with the well-known Parmenidean Principle of being and not being (cf. Diels-Kranz (DK) Fr. 2.2-5). Euthydemus’ statement against the possibility of false knowledge can be good evidence for this principle: -/- The things which are not surely do not exist (τὰ δὲ μὴ ὄντα … ἄλλο τι ἢ οὐκ ἔστιν). (Euthydemus 284b3-4) -/- He continues (b4-5): -/- There is nowhere for not being to be there (οὖν οὐδαμοῦ τά γε μὴ ὄντα ὄντα ἐστίν). -/- That BE does not let true opinion as a third option besides knowledge and ignorance seems to be related to BO’s rejecting a third option besides being and not being, which is itself the basis of the impossibility of false belief. Socrates’ elaborate discussion of the problem of false belief in Theatetus that leads to a more decisive discussion and finally to some solutions in Sophist can make our consideration of BO for the first group of dialogues authentive. -/- 4. &5. Split Knowledge and Split Being The fourth principle I shall call the principle of 'Split Knowledge' (SK): -/- SK Knowledge of X is separated from any other knowledge (of anything else) as if the whole knowledge is split to various parts. -/- This Principle is hinted and criticized as Socrates’ way of treating with knowledge in Hippias Major: -/- But Socrates, you do not contemplate the entireties of things, nor do people you have used to talk with (τὰ μὲν ὅλα τῶν πραγμάτων οὐ σκοπεῖς, οὐδ᾽ ἐκεῖνοι οἷς σὺ εἴωθας διαλέγεσθαι). (301b2-4) -/- Contrary to Rankin who regards the passage as 'antilogical, almost eristic in tone rather than presenting a serious philosophy of being' (1983, 54), I think it can be taken as serious. Hippias criticizes Sorcates that he does not contemplate (σκοπεῖς) the entireties of things (ὅλα τῶν πραγμάτων), a critique which Socrates is not its only subject but all those whom Socrates accustomed to talk with (ἐκεῖνοι οἷς σὺ εἴωθας διαλέγεσθαι). This last phrase, I think, extends this critic beyond this dialogue to other Socratic dialogues. Hippias’ use of the perfect tense of the verb ἔθω (to be accustomed) is a good evidence of this extension. We can get, thus, these ἐκεῖνοι as Socrates’ interlocutors in his other dialogues that hints that this criticism has the epistemological groundings of the previous dialogues as its subject. What ἐκεῖνοι οἷς σὺ εἴωθας διαλέγεσθαι points to is that Hippias does not have in mind Socrates’ way of treating things only in this dialogue, but he is also criticizing Socrates’ way throughout his dialogues. At 301b4-5, Socrates and his interlocutors’ way of beholding things is described as such: -/- You people knock (κρούετε) at the fine and each of the beings (ἕκαστον τῶν ὄντων) by taking each being cut up in pieces (κατατέμνοντες) in words (ἐν τοῖς λόγοις). This leads us to our next principle that is parallel to, and the ontological side of, SK, the principle of 'Split Being'(SB): -/- SB Everything (being) is separated from any other thing as if the whole being is split to many beings. -/- Socrates and his interlocutors and thus, as we saw, the Socratic dialogues are accused to regard everything as it is separated from all other things. This separation is en tois logois that, I think, can reasonably be taken as saying that Socratic dialogues (it refers of course to the dialogues before Hippias Major) cut up all things which have the same name/definition from all other things and try to understand them separately, without considering other things that are not in their logos. They, for example in Hippias Major, are cutting up to kalon and try to understand what it and all others inside this logos are by separating it from all other things. This is directed straightly against Socratic question and the way Socratic dialogues follow to find its answer, every time separating one logos. As Meyer (1995, 85) points out, every question 'presupposes that the X in question in a logos is something' and 'every answer to every question aims at unity' (84). The critique of Hippias Major is, therefore, at the same time a critique of SK and SB. It is also a critique of KWX because it is only based on KWX that Socratic dialogues could search for the answer of 'what is X' question supposing that knowing what X is is enough for the knowledge of X. SK and KWX are absolutely interdependent. What Socrates and all his previous interlocutors have neglected in Socratic dialogues, Hippias says, was 'continuous bodies of being' (διανεκῆ σώματα τῆς οὐσίας) (301b6). Either this theory actually belongs to the historical Hippias as it is being said or not, it is strictly criticizing SB. We have the same phrase with changing sūma to logos some lines later at 301e3-4: διανεκεῖ λόγῳ τῆς οὐσίας. Although this theory that can be observed as both an ontological and an epistemological theory is rejected by Socrates (301cff.), it is still against first Socrates’ onto-epistemological principles and might let us look at what is rejected as Socratic prineple. -/- 6. Knowledge is of Being The sixth principle that I think is presupposed by the first Socrates, is the 'Knowledge of Being' Principle (KB): -/- KB Knowledge is of Being. -/- This principle is obviously the source of the problem of false knowledge, a very important problem throughout Plato’s philosophy. We face this problem maybe for the first time in Euthydemus. In less than 20 Stephanus’ pages, 276-295, we are encountered with different interwoven problems about knowledge , all grounded in the problem of false opinion. Having challenged the obvious possibility of telling lies at 283e, at 284 Euthydemus discusses it saying that the man who speaks is speaking about 'one of those things that are (ἓν μὴν κἀκεῖνό γ᾽ ἐστὶν τῶν ὄντων)' (284a3) and thus speaks what is (λέγων τὸ ὄν) (a5). He must necessarily be saying truth when he is speaking what is because he who speaks what is (τὸ ὄν) and the things that are (τὰ ὄντα) speaks truth (τἀληθῆ λέγει) (a5-6). This is based on Parmenidean principle of the impossibility of being of not being which Euthydemus restates (284b3-4) and we mentioned discussing BO principle above. The things that are not are nowhere (οὐδαμοῦ) and there is no possibility for anyone to do (πράξειεν) anything with them because they must be made as being before anything else can be done, which is impossible (b5-7). The words, then, are of things that exist (εἰσὶν ἑκάστῳ τῶν ὄντων λόγοι) (285e9) and as they are (ὡς ἔστιν) (e10). The result is that no one can speak of things as they are not. It is this impossibility of false speach that is extended to thinking (δοξάζειν) at 286d1 and leads to the impossibility of false opinion (ψευδής … δόξα) (286d4). The general conclusion is asserted at 287a extending this impossibility to actions and making any kind of mistake. Not only knowledge is of being but speech, thought and action are of being simply because of the fact that nothing can be of not being. -/- B. First Socrates’ Principles in the Middle Period Dialogues Out of what were presupposed or criticized mostly in five dialogues, Euthyphro, Euthydemus, Hippias Major, Laches and Charmides, we tried to draw the first Socrates’ principles. Our inquiry here is directed to find out the fate of these principles in the three dialogues of the middle period, Meno, Phaedo and Republic. To do this, first we ought to check the situation of Socratic circle in these dialogues. The Socratic circle that was predominant in the early dialogues, does not look like a circle here anymore though they certainly have some features in common. Meno, Phaedo and Republic II-X are not committed to the principles of the circle and the whole circle in disrupted in them. The difference of the two Socrateses towards acquiring knowledge is obvious. The Socrates of Meno, Phaedo and Republic is evidently more self-confident that he can get to some truths during his arguments as he does. They are in their first appearance, as almost all other dialogues, committed to Socrates’ disavowal. All of them try to keep the shape of the Socratikoi Logoi genre, which is committed to the historical Socrates’ way of discussion; a dramatic personage who is to challenge his interlocutors, ask them and refuse the answers. Nevertheless, the fact is that what we have in common between two groups of dialogues is mostly a dramatic structure. Whereas the first group’s arguments are based on Socrates’ disavowal and lead to no positive results, the second group is decisively going to achieve some positive results. The aim of the first Socrates was to show others that they are ignorant of what they thought they knew. The new Socrates of Meno, on the opposite, makes so much efforts to show that the slave boy has within himself true opinions (ἀληθεῖς δόξαι) about the things that he does not know (οἶδε) (85c6-7). Despite his lack of knowledge, he has true opinions nevertheless. The way from these true opinions to knowledge, as Socrates states, is not so long. These true opinions are now like a dream but 'can become knowledge of the same things not less accurate than anyone’s' (οὐδενὸς ἧττον ἀκριβῶς ἐπιστήσεται περὶ τούτων) (c11-d1), if they be repeated by asking the same questions. The most outstanding text regarding Socrates’ disavowal is Meno 98b where he explicitly claims knowledge: -/- And indeed I also speak as (ὡς) one who does not know (εἰδὼς) but is guessing (εἰκάζων). However, [about the fact] that true opinion and knowledge are different, I do not altogether expect (δοκῶ) myself to be guessing (εἰκάζειν), but if I say about anything that I know (εἰδέναι) -which about few things I say- this is one of the things that I know (οἶδα). (98b1-5) This passage is very significant about Plato’s disavowal of knowledge. There can hardly be found, I think, anywhere else in Plato’s corpus where Socrates speaks about his knowledge of something as such. He says first that he speaks as someone who does not know. This ὡς οὐκ εἰδὼς λέγω comparing with what he used to say in the first group, οὐκ οἶδα, has this added ὡς. Socrates does not claim strongly anymore that he does not know anything but speaks only as someone who does not know. He needs this ὡς not only because he is going to accept that he does know some, though few, things immediately after this sentence, but also because he needs his previous disavowal to be loosened from Meno on. He does not merely say here that he knows something. It is then different from the examples mentioned before which could be taken as idiomatic or at least not emphatic. Socrates’ remarkable emphasis on distinguishing εἰδέναι from εἰκάζειν departs it from all other passages where he says only he knows something. Moreover, he claims definitely that he has knowledge about few (ὀλίγα) things. From the early to the middle dialogues, Socrates’ attitude to knowledge has totally renewed its face. He brings forth a new concept, true opinion, and he does not speak of knowledge as he used to before; the rough, perfect and unachievable knowledge of the first group has turned to something more smooth, realistic and achievable. Comparing with the early dialogues that did not set out from the first to reach positive results, the middle ones are extraordinarily and surprisingly positive and hence destroy the basis of the Socratic circle. The questions and answers are purposely directed to some specific new theories; most of them are not directly related to the topics or the main questions of the dialogues. They are suggested when Socrates draws the attention of the interlocutor away from the main question because of the necessity of another discussion. Even if the main question remains unanswered, we have still many positive theories, prominently of metaphysical type. These theories are so abundant and dominant in these dialogues, especially Phaedo and Republic, that one might think that they may appear to be arbitrarily sandwiched in there. This helps dialogues to keep their original Socratic structure while they are suggesting new theories. Hence, the Socratic question of 'what is X?', though is still used to launch the discussion, is loosened and is forgotten for most part of the dialogues. Meno that has first a differently formulated question, 'can virtue be taught?', leads finally to the Socratic question of 'What is virtue?'. Phaedo is dedicated to the demonstration of the immortality of soul and the life after death without having a central Socratic question. The case of Republic is more complicated. The first book, on the one hand, has all the criteria of a Socratic circle: its 'what is justice?' question, Socrates’ strong disavowal (e.g. 337d-e), his rejection of all answers and coming back to the first point without finding out any answer. This Book, considered alone, is a perfect Socratic dialogue, as many scholars regard it as early and separated from other books. The books II-X are, on the other hand, far from implementing a Socratic circle. They have still the 'what is justice?' question as their incentive leading question, but they are, in most of the positive doctrines and methods that encompass the main parts, ignoring the question. Even these books that, I believe, are the farthest discussions from the Socratic circle are so cautious not to break the Socratic structure of the dialogue as long as it is possible. What is changed is not the structure of the dialogue but the ontological and epistemological grounds based on which new theories are suggested. -/- 1. Knowledge of the Good We can clearly see in the second group of dialogues that the KWX principle loses the place it had before in our first group. It is not, of course, rejected, but still we cannot say that it has the same situation. KWX that was based on Socratic question, as we discussed before, was the leading principle of the first Socrates’ epistemology and of the highest position. Other epistemological principles, SK directly and BE indirectly, were relying on Socratic question and therefore on KWX. Such a position does not belong to KWX from Meno on. What makes it different in the second Socrates is another principle that is needed it not only as its complementary principle but also as what is more fundamental. Plato, then, does not reject KWX in this period, but, it seems, he transcends to another more basic principle; a principle we shall call the principle of 'Knowledge of the Good' (KG): -/- KG Knowledge of X requires knowledge of the Good. -/- Whereas all the dialogues of our first group are free from any discuscon about KG, it bears a very important role in the second group so as becomes the superior principle of knowledge in Republic. Trying to solve the problem of teachability of virtue, Socrates says that it can be teachable only if it is a kind of knowledge because nothing can be taught to human beings but knowledge (ἐπιστήμην) (Meno 87c2). The dilemma will be, then, whether virtue is knowledge or not (c11-12) and since virtue is good, we can change the question to: whether is there anything good separate from knowledge (εἰ μέν τί ἐστιν ἀγαθὸν καὶ ἄλλο χωριζόμενον ἐπιστήμης) (d4-5). Therefore, the conclusion will be that if there is nothing good which knowledge does not encompass, virtue can be nothing but knowledge (d6-8). What let us discuss KG as an epistemological principle for the second Secrates is the relation he tries to establish between knowledge and the Good which, though is alongside with the mentioned thesis and the idea of virtue as knowledge, goes much deeper inside epistemology asking to regard the Good as the basis of knowledge. The effort of Phaedo cannot succeed in establishing the Good as the criterion of explanation and knowledge since, I think, it needs a far more complicated ontology of Republic where Socrates can finally announce KG. What is said in Republic is totally compatible with Phaedo 99d–e and the metaphor of watching an eclipse of the sun. In spite of the fact that we do not have adequate knowledge of the Idea of the Good, it is necessary for every kind of knowledge: 'If we do not know it, even if we know all other things, it is of no benefit to us without it' (505a6-7). The problem of our not having sufficient knowledge of the Idea of Good is tried to be solved by the same method of Phaedo 99d-e, that is to say, by looking at what is like instead of looking at thing itself (506d8-e4). It is this solution that leads to the comparison of the Good with sun in the allegory of Sun (508b12-13). What the Good is in the intelligible realm corresponds to what the sun is in the visible realm; as sun is not sight, but is its cause and is seen by it (b9-10), the Good is so regarding knowledge. It has, then, the same role for knowledge that the sun has for sight. Socrates draws our attention to the function of sun in our seeing. The eyes can see everything only in the light of the day being unable to see the same things in the gloom of night (508c4-6). Without the sun, our eyes are dimmed and blind as if they do not have clear vision any longer (c6-7). That the Good must have the same role about knowledge based on the analogy means that it must be considered as a required condition of any kind of knowledge: -/- The soul, then, thinks (νόει) in the same way: whenever it focuses on what is shined upon by truth and being, understands (ἐνόησέν), knows (ἔγνω) and apparently possesses understanding (νοῦν ἔχειν). (508d4-6) -/- Socrates does not use agathon in this paragraph and substitutes it with both aletheia and to on. He links them with the Idea of the Good when he is to assert the conclusion of the analogy: -/- That which gives truth to the objects of knowledge and the power of knowing to the knower, you must say, is the Idea of the Good: being the cause of knowledge and truth (αἰτίαν δ᾽ ἐπιστήμης οὖσαν καὶ ἀληθείας) so far as it is known (ὡς γιγνωσκομένης μὲν διανοοῦ). (508e1-4) Knowledge and truth are called goodlike (ἀγαθοειδῆ) since they are not the same as the Good but more honoured (508e6-509a5). KG, which had been implicitly contemplated and searched in Phaedo, is now explicitly being asserted in Republic. As what was quoted clearly proves, this principle is the very one which we can observe as the most fundamental principle of the second Socrates in Republic, corresponding to the role KWX had in the first Socrates. The Form of the Good in Republic, of which Santas speaks as 'the centerpiece of the canonical Platonism of the middle dialogues, the centerpiece of Plato’s metaphysics, epistemology, ethics and …' (1983, 256) much more can be said. Plato’s Cave allegory in Book VIII dedicates a similar role to the Idea of the Good. The Idea of the Good is there as the last thing to be seen in the knowable realm, something so important that its seeing equals to understanding the fact that it is the cause of all that is correct and beautiful (517b). Producing both light and its source in visible realm, it controls and provides truth and understanding in the intelligible realm (517c). -/- 2. Tripartite Epistemology BE is thoroughy rejected in the second Socrates and substituted by its opposite, the principle of 'Tripartite Epistemolgy' (TE): -/- TE Opinion is an epistemological status between knowledge and ignorance. BE of the first Socrates was denying any third way besides knowledge and ignorance which was the foundation of Socratic circle without which Socrates could not reject his interlocutors’ possessing any kind of knowledge. We cannot say, however, that the first Socrates had a third epistemological status in mind but rejected it. Such a status was unacceptable for him so that one can say that he would reject any kind of such status if suggested. There were only two possibilities about knowledge: either one knows something or he does not know it. TE, thus, was not the first Socrates’ discovery and, I think it is not the second Socrates’discovery. All we can see in our second group of dialouges is that he uses this principle as an already demonstrated one. Having examined the slave boy in Meno for the prupose of showing the working of recollection, it truns out that he has some opinions in him while he still does not know. Without trying to prove it, Socrates takes this as the distinction between knowledge and opinion: -/- So, he who does not know (οὐκ εἰδότι) about what he does not know (περὶ ὧν ἂν μὴ εἰδῇ) has within himself true opinions (ἀληθεῖς δόξαι) about the same things he did not know (περὶ τούτων ὧν οὐκ οἶδε) (85c6-7). -/- The same distinction is set between ὀρθὴν δόξαν and ἐπιστήμην at 97b5-6 ff. (also cf. 97b1-2: ὀρθῶς μὲν δοξάζων … ἐπιστάμενος). He connects then their difference to the myth of Daedalus, the statue that would run away and escape. So are true opinions, not willing to remain long in mind and thus not worthy until one ties them down by αἰτίας λογισμῷ (98a3-4). Socrates says that this tying down is anamnesis. We face, in Phaedo, the same relation is settled between knowledge as the process of being tied down and getting the capability to give an account, on the one hand, and anamnesis, on the other hand. When a man knows, he must be able to give an account of what he knows (Phaedo 76b5-6) and since not all people are able to give such an account, those who recollect, recollect what once they learned (76c4). Although the distinction between knowledge and opinion is not explicitly used in Phaedo, referring to the parallel link between knowledge plus account and anamnesis in Phaedo and Meno, one can say that those who cannot recollect are not able to give account and, thus, are in a state of opinion. What is said at Phaedo 84a, though not yet a definite distinction between knowledge and opinion, makes a distinction between their objects so as we can agree that it is presupposed. The soul of the philosopher, Socrates says, follows reason, stays with it forever and contemplates the divine, which is not the object of opinoin (ἀδόξαστον) (84a8, cf. Meno 98b2-5). The distinction of knowledge and belief in Republic has a significant difference with what we discussed in Meno and Phaedo since the distinction of Meno was based on anamnesis and thus more an epistemological distinction. Even in Phaedo that we do not have any elaborate discussion about the distinction, the only hint to the matter at 76 is bound to the theory of anamnesis. In addition to the relation of the distinction with this theory, there is another evidence that does not permit us to consider the distinction as an ontological distinction. Let’s see Meno 85c6-7: -/- So, he who does not know about what he does not know has within himself true opinions about the same things he did not know (τῷ οὐκ εἰδότι ἄρα περὶ ὧν ἂν μὴ εἰδῇ ἔνεισιν ἀληθεῖς δόξαι περὶ τούτων ὧν οὐκ οἶδε). (85c6-7) -/- This last sentence persists that the objects of knowledge and true opinion are the same. What Socrates is to say here is that whereas he does not know X he has true opinion about the same X. I think Socrates’ sentence that the slave boy οὐκ εἰδότι ἄρα περὶ ὧν ἂν μὴ εἰδῇ and his restatement of it by saying περὶ τούτων ὧν οὐκ οἶδε is because he wants to emphasize that the slave boy who does not know, has true opinion about the same thing. Socrates could say this just with using τούτων and there would be no necessity to bring περὶ ὧν ἂν μὴ εἰδῇ for οὐκ εἰδότι if he did not want to emphasize. -/- 3. Tripartite Ontology The distinction between knowledge and true opinion in Republic, on other side, has nothing to do with recollection, but is based on an ontological principle, 'Tripartite Ontology' (TO): -/- TO There are things that both are and are not. -/- This principle I confine, among our three dialogues of the second group, to Republic not extended to Meno and Phaedo, is obviously the opposite of BO. Speaking about the lovers of sights and sounds in the fifth book, Socrates distinguishes them from philosophers because their thought is unable to understand the nature of beautiful itself besides beautiful things (476a6-8) and hence they can only have opinions. The philosopher who, on the contrary, believes in beautiful itself and can distinguish it from beautiful things (476c9-d3), has knowledge because he knows, contrasting others who have opinion because they only opine (d5-6). Since those whose knowledge were degraded as opinion will complain about Socrates’ such calling their thought, he provides them the following argument (476e7-477b1): -/- - Does the person who knows, knows (γιγνώσκει) something (τὶ) or nothing (οὐδέν)? - He knows something (τί). - Something that is (ὂν) or is not (οὐκ ὄν)? - Something that is (ὂν) for how could something that is not be known (πῶς γὰρ ἂν μὴ ὄν γέ τι γνωσθείη)? - Then we have an adequate grasp of this: No matter how many ways we examine it, what completely is (παντελῶς ὂν) is completely knowable (παντελῶς γνωστόν) and what is in no way (μὴ ὂν δὲ μηδαμῇ) is in every way unknowable (πάντῃ ἄγνωστον). - A most adequate one. - Good. Now, if anything is such as to be and also not to be (ὡς εἶναί τε καὶ μὴ εἶναι), won’t it be intermediate (μεταξὺ) between what purely is (εἰλικρινῶς ὄντος) and what in no way is (μηδαμῇ ὄντος)? - Yes, it’s intermediate. - Then as knowledge (γνῶσις) is set over what is (τῷ ὄντι), while ignorance (ἀγνωσία) is of necessity set over what is not (μὴ ὄντι) mustn’t we find an intermediate between ignorance (ἀγνοίας) and knowledge (ἐπιστήμης) to be set over the intermediate, if there is such a thing? -/- From the third status of being we must reach to the third status of knowledge. The simple reading of this text can be an existential reading, taking the "is" of the mentioned sentences as existence. The problem is that when it is said that there is something that both is and is not reading "is" existentially, it sounds too bizarre to be acceptable. It cannot easily be understandable to have something as both existent and non-existent at the same time. This problem arose so many debates and led many scholars to reject the existential reading of "is" and suggest some other readings like predicative or veridical readings. I think though Plato’s complicated ontology of Republic cannot be correctly understood by a simple existential reading, this "is" cannot be free from existential sense of being and, thus, cannot be reduced to just a predicative or veridical sense of being. -/- 4. Bound Knowledges In addition to KG, the Good is also the basis of another principle in the second Socrates, namely the principle of 'Bound Knowledge' (BK): -/- BK Knowledge of everything is bound to the knowledge of the Good. -/- We distinguished BK from KG because we want to insist, in BK, on what had not been insisted upon in KG, that is, the binding role that the Good plays in the second Socrates, contrasting the absence of such a role in the first Socrates. Socrates remembers, in Phaedo, his wonderful keen on natural philosophers' wisdom when he was young. The origin of this enthusiasm was Socrates’ hope to know the cause of everything as they used to claim. When he was searching the matters of his interest on their basis, Socrates says, he became convinced he can get no acceptable answer from them and found himself blind even to the things he thought he knew before. One day he hears Anaxagoras’ theory that 'it is Mind that arranges and is the cause of everything (ὡς ἄρα νοῦς ἐστιν ὁ διακοσμῶν τε καὶ πάντων αἴτιος)' (Phd. 97c1-2 cf. DK, Fr.15.8-9, 11-12, 12-14) and thinks that he can finally find what he has always expected, i.e. something which can explain all things. What I intend to show here is that what makes Socrates hopeful is that Anaxagoras’ theory tries 1) to explain all things by one thing and 2) this explanation is understood by Socrates as if it is based on the concept of the Good. That Socrates was searching for one explanation for all things can be proved even from what he has been expecting from natural philosophers. The case is, nonetheless, more clearly asserted when he speaks about Anaxagoras’ theory. In addition to διακοσμῶν τε καὶ πάντων αἴτιος of 97c2 mentioned above, we have τὸ τὸν νοῦν εἶναι πάντων αἴτιον (c3-4) and τόν γε νοῦν κοσμοῦντα πάντα κοσμεῖν (c4-5) all emphasizing on the cause of all things (πάντα) which can clearly prove that one of the reasons which caused Socrates to embrace it delightfully was its claim to provide the cause of all things by one thing. Another reason was that Anaxagoras’ Mind, at least in Socrates’ view, was attempting to explain everything by the concept of the Good. This connection between Mind and Good belongs more to the essential relation they have in Socrates’ thinking than Anaxagoras’ theory because there are almost nothing about such a relation in Anaxagoras. The reason for Socrates’ reading can be that Mind is substantially compatible with Socrates’ idea of the relation between good and knowledge. Both the thesis 'no one does wrong willingly' and the theory of virtue as knowledge we pointed to above are evidences of this essential relation. Nobody who knows that something is bad can choose or do it as bad. The reason, when it is reason, that means when it is as it should be, when it is wise or when it knows, works only based on good-choosing. In this context, when Socrates hears that Mind is considered as the cause of everything, it sounds to him like this: good should be regarded as the basis of the explanation of all things. We see him, thus, passing from the former to the latter without any proof. This is done in the second sentence after introducing Mind: -/- I thought that if this were so, the arranging Mind would arrange all things and put each thing in the way that was Best (ὅπῃ ἂν βέλτιστα ἔχῃ). If one then wished to find the cause of each thing by which it either perishes or exists, one needs to find what is the best way (βέλτιστον αὐτῷ ἐστιν) for it to be, or to be acted upon, or to act. On these premises then it befitted a man to investigate only, about this and other things, what is the most excellent (ἄριστον) and best (βέλτιστον). The same man must inevitably also know what is worse (χεῖρον), for that is part of the same knowledge. (97c4-d5) This passage is a good evidence of Socrates’ leap from Anaxagoras’ Mind to his own concept of the Good that can explain why Socrates found Anaxagoras theory after his own heart (97d7). Mind is welcomed because of its capability for explanation on the basis of good to 'explain why it is so of necessity, saying which is better (ἄμεινον), and that it was better (ἄμεινον) to be so' (97e1-3). What Socrates thought he had found in Anaxagoras can indicate what he had been expecting from natural scientists before. Socrates could not be satisfied with their explanations because they were unable to explain how it is the best for everything to be as it is. It can probably be said, then, that it was the lack of the unifying Good in their explanation that had disappointed him. We must insist that we are discussing what Socrates thought that Anaxagoras’ theory of Mind should have been, not about Anaxagoras’ actual way of using Mind. Phaedo 97c-98b, is not about what Socrates found in Anaxagoras but what he thought he could find in it. On the contrary, it should also be noted that it was not this that was dashed at 98b, but Anaxagoras’ actual way of using Mind. It was Anaxagoras’ fault not to find out how to use such an excellent thesis (98b8-c2, cf. 98e-99b). Socrates gives an example to show how not believing in 'good' as the basis of explanation makes people be wanderers between different unreal explanations of a thing. His words δέον συνδεῖν (binding that binds together) as a description for the Good we chose as the name of BK principle: They do not believe that the truly good and binding binds and holds them together (ὡς ἀληθῶς τὸ ἀγαθὸν καὶ δέον συνδεῖν καὶ συνέχειν οὐδὲν οἴονται). (99c5-6) -/- Having in mind Plato’s well-known analogy of the sun and the Good at Republic 508-509, we can dare to say that his warning of the danger of seeing the truth directly like one watching an eclipse of the sun in Phaedo (99d-e) is more about the difficulty of so-called good-based explanation than its insufficiency, a difficulty which is precisely confirmed in Republic (504e-505a, 506d-e). Moreover, BK is asserted in a more explicit way in the Republic, where the Good is considered not only as a condition for the knowledge of X, as was noted above discussing KG, but also as what binds all the objects of knowledge and also the soul in its knowing them. At Republic VI, 508e1-3, when Socrates says that the Form of the Good 'gives truth to the things known and the power to know to the knower (τοῦτο τοίνυν τὸ τὴν ἀλήθειαν παρέχον τοῖς γιγνωσκομένοις καὶ τῷ γιγνώσκοντι τὴν δύναμιν ἀποδιδὸν)', he wants to set the Good at the highest point of his epistemological structure by which all the elements of this structure are bound. This binding aspect of the Good is by no means a simple binding of all knowledges or all the objects of knowledge, but the most complicated kind of binding as it is expected from the author of the Republic. The kind of unity the Good gives to the different knowledges of different things is comparable with the unity which each Form gives to its participants in Republic: as all the participants of a Form are united by referring to the ideas, all different kinds of knowledge are united by referring to the Good. If we observe Aristotle's assertion that for Plato and the believers of Forms, the causative relation of the One with the Forms is the same as that of the Forms with particulars (e.g. Metaphysics 988a10-11, 988b4), that is to say the One is the essence (e.g., ibid, 988a10-11: τοῦ τί ἐστὶν, 988b4-6: τὸ τί ἢν εἶναί) of the Forms besides his statement that for them One is the Good (e.g., ibid, 988b11-13) the relation between the Good and unity may become more understandable. Since the quiddity of the Good (τί ποτ᾽ ἐστὶ τἀγαθὸν) is more than discussion (506d8-e2), we cannot await Socrates to tell us how this binding role is played. All we can expect is to hear from him an analogy by which this unifying role is envisaged, the sun. The kind of unity that the Good gives to the knowledge and its objects in the intelligible realm is comparable to the unity that the sun gives to the sight and its objects in the visible realm (508b-c). The allegory of Line (Republic VI, 509d-511), like that of the Sun, tries to bind all various kinds of knowledges. The hierarchical model of the Line which encompasses all kinds of knowledge from imagination to understanding can clearly be considered as Plato’s effort to bind all kinds of knowledges by a certain unhypothetical principle. The method of hypothesis starts, in the first subsection of the intelligible realm, with a hypothesis that is not directed firstly to a principle but a conclusion (510b4-6). It proceeds, in the other subsection, to a 'principle which is not a hypothesis' (b7) and is called the 'unhypothetical principle of all things' (ἀνυποθέτου ἐπὶ τὴν τοῦ παντὸς ἀρχὴν) (511b6-7). This παντὸς must refer not only to the objects of the intelligible realm but to the sensible objects as well. Plato does posit, therefore, an epistemological principle for all things, a principle that all things are, epistemologically, bound and, thus, unified by. -/- 5. Bound Beings The ontological aspect of BK we shall call the principle of 'Bound Beings' (BB): -/- BB Being of all things are bound by the Good. -/- We saw in our principle of Split Being (SB) how the first Socrates was criticized because of his approach to split being and separate each thing from other things. The principle of Bound Beings intends to make the things more related, a duty which is done again by the Good. In the allegory of Sun, there are two paragraphs that evidently and deliberately extend the binding role of the Good to the ontological scene: -/- You will say that the sun not only makes the visible things have the ability of being seen but also coming to be, growth and nourishment. (509b2-4) -/- This clearly intends to remind the ontological role the sun plays in bringing to being all the sensible things in order to display how its counterpart has the same role in the intelligible realm (b6-10): -/- Not only the objects of knowledge (γιγνωσκομένοις) owe their being known (γιγνώσκεσθαι) to the Good, but also their existence (τὸ εἶναί) and their being (οὐσίαν) are due to it, though the Good is not being but superior to it in rank and power. That the Good is here represented as responsible for being of things in addition to their being known means, in my opinion, that Plato wants to posit BB in addition to BK. The allegory of Cave at the very beginning of the seventh Book (514aff.) can be taken as another evidence. The role of the Good, one might say, is confined to the intelligible realm because it is asserted that the role the Good plays in this realm is corresponding to that of the sun in the visible realm. The fact that Plato wants to observe the Good also as the ontological cause of the sensible things is obvious from his saying, in the allegory of Cave, that the Form of the Good 'produces light and its source (τὸν τούτου κύριον) [i.e. the sun] in the visible realm' (517c3). We can conclude, then, that the ontological function of the Good is not confined to the intelligible realm in which it is the lord and provides truth and understanding (c3-4) because it is also responsible to produce τὸν κύριον of light. 6. Proportionality of Being and Knowledge Insofaras BE and BO principles of the first Socrates gave way to TE and TO principles of the second Socrates, we cannot expect him to preserve KB in the same way as it was in the first Socrates. The new tripartite ontology and epistemology necessitates some modifications in KB which results in the principle of 'Proportionality of Being and Knowledge' (PBK): -/- PBK To every class of being there is a proportionate category of knowledge. -/- This principle, of course, does not entail the refutation of KB and thus is not kind of rejecting PBK but only a more complicated version of it. Based on PBK, we can still agree that knowledge is of being (KB) but the issue is that since none of the concepts of knowledge and being in the second Socrates are as simple as they were for the first Socrates, we need a more complicated principle for their relation here. Although from Meno 97a where the distinction of knowledge and true opinion is drawn out in the second group of the dialogues, we can expect a new relation, it is articulated in its most complete way in Republic and specifically in the allegory of Line. All the beings are divided there hierarchically to four classes, to each of them belongs a class of knowledge: imagination to images, belief to the sensible things (more correctly: the things of which they[in previous class] were images (ᾧ τοῦτο ἔοικεν)), thought to mathematical objects (?) and understanding to the Forms and the first principle. The degree of clarity that each of the classes of knowledge shares in (σαφηνείας ἡγησάμενος μετέχειν) is proportionate to the degree that its object shares in truth (ἀληθείας μετέχει). (511e2-4) -/- Conclusion From the six onto-epistemilogical principles of the first Socrates, four principles turn to their opposite in the middle period dialogues. While bipolar epistemology and ontology of the early dialogues give place to tripartite epistemology in the middle period dialogues and tripartite ontology in Republic, the split knowledge and being of the first Socrates are inclined to be substituted by bound knowledges and bound beings in the second Socrates and specifically in Republic. Not all our system of principles in this article is necessarily determinative. Either they are rightly formulated or not, our result would not be vulnerable if we accept that 1) making the distinction of knowledge and belief, 2) accepting the being of not being and 3) trying to bind both being and knowledge by the concept of the Good happens only in middle period dialogues, having been absent in the early ones. These are the favourite results all those somehow arbitrary and even oversimplified principles were to illustrate; that there is kind of a development in the epistemological as well as the ontological grounds of Plato’s philosophy. -/- Notes . (shrink)

Few have entertained the idea that Georg Cantor, the creator of set theory, might have influenced Edmund Husserl, the founder of the phenomenological movement. Yet an exchange of ideas took place between them when Cantor was at the height of his creative powers and Husserl in the throes of an intellectual struggle during which his ideas were particularly malleable and changed considerably and definitively. Here their writings are examined to show how Husserl's and Cantor's ideas (...) overlapped and crisscrossed in the areas of philosophy and mathematics, arithmetization, abstraction, consciousness and pure logic, psychologism, metaphysical idealism, new numbers, and sets and manifolds. (shrink)

Sophists’ apologia. -/- Sophists were the first paid teachers ever. These ancient Greek enlighteners taught wisdom. Protagoras, Antiphon, Prodicus, Hippias, Lykophron are most famous ones. Sophists views and concerns made a unified encyclopedic system aimed at teaching common wisdom, virtue, management and public speaking. Of the contemporary “enlighters”, Deil Carnegy’s educational work seems to be the most similar to sophism. Sophists were the first intellectuals – their trade was to sell knowledge. They introduced a new type of teacher-student relationship (...) – the mutually beneficial communication on equal terms. They taught pupils how to think independently and how to persuade others, which was inseparably connected with the rise of democracy in the most advanced Greek polices. Sophists were the first to proclaim that all people are naturally equal. They put forward the idea of natural rights and social contract, working out the fundamentals of the present-day law. In addition, they developed the basics of philology, psychology, logic, and gave a scientific explanation to the origins of religions. They embodied definite positive ideals of their epoch. Sophistry flourished in the second half of the Vth through he first third of the IVth centuries B. C. The period coincides with the heyday of the entire Greek history – the so called Greek Miracle epoch. Sophists expressed some present-day concepts of Ancient Greek ideology and mentality, characteristic of its Golden Age times. The Ancient Athens of the times of Periclus were similar to the Florence of Renaissance as to the type of social structure. That is why (?) the art of eloquence (public speaking) was highly respected in Florence. The two cities had a lot of features of the present-day capitalist society, which was due to their role as the main centres of industry of their “worlds-economies”. During the subsequent Hellenic period, the Old Order was restored, based on the feudal rent recipients’ rule. The degradation of society resulted in the decline of sophistry. Sophistry represents the highest point in the evolution of Greek philosophy. It is also the most prominent school as to the novelty of its ideas. That was owing to the common propensity for innovation typical of Periclus’ Golden Age, which in its turn resulted from the establishment of the individualistic and rationalistic thinking similar to present-day mentality. Among other aspects, close attention was paid to ethnic issues; some humanistic ideas such as Master and Servant ethics and Enlightment as the primary condition of man’s freedom, received scientific grounding. The genre of philosophical dialogue was developed, as well as the rationalistic scientific gnoceology, aimed at consistently opposing any religion. Dialectics, the most outstanding achievement of the Antiquity, was also worked out by the sophists. Zeno of Helleas was one of the sophists. Democrites’ ideas were, too, quite close to dialectics. Socrates’ contemporaries regarded him ironically, as shown, for example, in the Aristophanes’ comedy “Clouds”. Nevertheless, this scholar can still be referred to as a sophist, although of somewhat weird personality. Socrates’ popularity, which came later, should be attributed to the fact that he was the first philosopher ever to have been sentenced and executed by law, rather than to any valuable scientific contributions. Sophistry as such, rather than Socrates, marks the line between pre-Socratic and post- Socratic schools. Plato, with the exception of his latest works, is considered to belong to the same philosophical school. Specifically, there are valid reasons to believe that he was paid fees by his disciples. In Plato’s Dialogues, all points of view will be proven and then disproved, so that an integral philosophical system is hard to observe. Plato’s “Idealism” is but one of the many possible theories. His ideal philosophy is a high-minded argument of wise men. As late as towards the end of his life, Plato betrayed the ideals of sophistry, turning into the world’s first proponent of totalitarianism. Sophistry is Greek classical literature. Later on, philosophy, as well as the entire Greek civilization, started to fall into decline. Over 500 subsequent years, Hellenistic-Roman scholars never came up with anything novel. The entire process of antique philosophical evolution can be subdivided into the following six stages: Phoenician (IX–VIIth centuries B. C.), archaic (VIth – the first half of IVth century B. C., classic (the epoch of sophistry: the second half of the Vth through the first decades of the IIIrd centuries B. C.), late antique (middle of the IIIrd – first decades of the VI th centuries B.C.) The fact that early in the VIth century B. C. sophistry was rejected in Athens, has a lot to do with the consequences of their defeat in Peloponnesian war, a social catastrophe. Ever since, the prevalent part of the criticism of sophistry is a criticism of sophists’ immoral ways of life and personalities, rather than the essentials of their philosophy. For instance, Aristotle’s criticism primarily concerns their pragmatism and tendency to make a profit out of teaching. However, Aristotle himself taught his disciples along similar lines, showing them how to “be able to prove both opposites”. It was Aristotle who summed up sophists’ century-long elaborations. The traditional modern interpretation of sophists as shown in Plato’s dialogues appears to be groundless. The truth of the matter is that Plato regards sophists as more serious philosophers than Socrates was. It becomes especially clear in the dialogue titled “Protagoras”, which shows the latter as gaining an undoubted victory over Socrates. Generally speaking, Plato’s works never regard sophists as weak or worthless opponents. A common opinion as to the essence of sophists’, in particular, Protagoras’s philosophy, has hitherto not been worked out. Many researchers renounce the very existence of any specific philosophical position in sophistry. The term “relativism” itself now has a bad name. However, it should be borne in mind that Protagoras doctrine emerged as a result of an attempt to get out of the ontological dead-end to which the entire Greek natural philosophy had come. To this purpose, he employed Anaxagoras’s idea about everything comprising everything, whereupon he built his physical-ontological theory, and Zeno of Helleas’s concept that all objects are such and different at the same time in the same respect. According to Protagoras’s philosophy, every object does not only seem, but is different from every other object, including human beings. Therefore, every person is his or her own measure of all other entities. The objective reality is the Ego’s relations with the outer world. Every object’s essence is defined through its interaction with ourselves. Nothing remains stagnant – every existence already contains non-existence, and vice-versa. However, the relativity of being is not to be regarded as absolute. That means, that relatively stable and therefore relatively cognizable entities do exist, among them the words we say. Doubt is not considered absolute by relativists either, which differentiates them from skeptics and agnostics. They dare to believe. Uncertainty is Godly – here, Protagoras becomes very close to the concept of apathetic theology. Gorgius shared Protagoras’s views and tried to prove them applying the same method as Zeno applied when upholding Parmenidis’s philosophy. He argued that if one accepts the opposite point of view and agrees that nothing can exist and not exist at the same time, one is bound to come to an the absurd conclusion that nothing is cognizable and nothing exists. To prove that there is a universal contradiction within the Existence itself, Protagoras and his followers in their turn put forward a number of special logical arguments, called rules of contraries, or sophisms. The most well-known of them is the Euathle’s sophism, which proves that an object can be such and different at the same time in the same respect. Generally speaking, all relationships of any entity are extraneous. The idea that all entities, including concepts, exist and do not exist at the same time, was also accepted by Plato. It should be stressed, that Plato’s philosophy as a whole is relativists’ greatest achievement. In his dispute against relativism, Aristotle put forward the main concepts of his ontology: the existence falls into the existence in possibility and the existence in reality, or substance and accidence – the ever changing “sublunary” world and the stable world of divine heavenly bodies. This division made his system self-contradictory to the point of absurd. Aristotle quite realized, that the direct disproof of relativism is hardly possible, which fact makes his criticism a mere tautology. It is very indicative in this respect that materialists, idealists and even skeptical agnostics applied the same arguments do disprove relativism. That demonstrates that the inner contradiction between any forms of “absolutism” and relativism is the most fundamental philosophical problem. Relativist ideas were repeatedly renewed throughout history, provided the social conditions were favourable. In variations, the concepts were advanced by Tzuan Tsi and ancient Chinese sophists, apostle Paul, Nicolas of Cusa, Galileo, Rene Descartes, G. Berkley, D. Hume, Ch. S. Pierce, E. Mach, A. Bogdanov, F. de Saussure, N. Bore, P. Feuerabend, U. Quain and others. However, no consistent relativist ontological system was developed after Protagoras. Meanwhile, a valid relativist theory could provide a clue to a fuller and more adequate general physical concept of the world, and to a deeper understanding of quantum mechanics. -/- Relativism as an ontological system of beliefs. -/- The relativist principle of relativity of all that exists is fundamental for the entire system of philosophy. Absolutely everything exists, but, at the same time, no existence is absolute. Anything is possible, but those entities only exist for us with which we interact this way or another, i. e., reality is interaction, “I interact – hence, I exist”. However, an overly close relationship between entities leads to non-existence – in other words, there is a certain existential optimum. For all of us, information, or perceptible heterogeneities, is real. Each of us is the center of a definite system of interactions with the world. If God is an endless Possibility, then the World is but the man’s dialogue with God. However, the ties that bind us with the world are of some definite kind, which predetermines our reality. The principle of relativity constrains itself, as any relativity is relative. On the one hand, that calls forth the Evil – caused by our own weakness. On the other hand, it makes our world relatively stable and cognizable. Generally speaking, the world is what we are. Everybody has an own Universe, but since people are fairly similar, our universes can be regarded as one Universe with common laws and regulations. It is possible for us to change the degree of objects’ existence in relation to ourselves and one another. It is equally important, though, to remain ourselves (keep following our “warrior’s path”). Since everything exists and no existence is absolute, the process of cognition should involve a criticism of everything but never a complete rebuttal of anything. The criterion of verity for any theory then is in its capacity to build itself upon a bigger number of diverse statements than in any other theory. Theoretically, all philosophies should ideally be complementary. The question of what in fact happened is meaningless unless we specify the “we”, the “where” and the “when”. The past can only exist as part of the present. When we change, our past undergoes corresponding qualitative objective changes, all of which are irreversible. The contemporary physics regards all objects of the Universe as existing for one another, which results in absurd conclusions. In fact, only what we interact with, exists. The so called “speed of light in vacuum” c is the maximum speed of direct interaction and direct exchange of signals. Specifically, at the oncoming speed c with a significant blue parallax the gamma-quantum is unable to interact with the matter. When this speed is reached, the gravitational mass of objects in relation to one another becomes equal to zero (imaginary), whereas their inertial mass equals infinity. Further acceleration will bring us into an entirely different Universe. That means that the Universe is an open system, which fact destroys all present-day cosmological models. Anything is possible in this world. Our reality is magic. Every bit of space contains a potentially limitless number of bits of other, however virtual, universes, as well as an infinite number of particles, which are still virtual for us (as postulated by quantum mechanics). In other words, there exists an infinity of different realities. With all the above in mind, the different levels of entities’ existence, in particular, of Life and Reason, should be taken into consideration. For example, to which extent a live system exists in relation to the inanimate forms of being is very indefinite, as what life is has to be determined by living creatures. An even greater compass of levels of existence can be brought about through interaction – that is the core idea of Godel’s theorem. As a result, the range of forms of existence tends towards infinity – the evolution of matter through super-intelligence will bring us up to God’s heights. A more generalized mathematical model of the physical world, including a universal operator of the co-existence of objects, can be developed based on the above considerations. The bonds between objects become weaker owing to 1) a high speed, 2) acceleration, 3) rotation, 4) strong gravity, 5) strong fields of other kinds, 6) the difference of levels of existence , 7) phase shifts. Yet, there always is an existential optimum. Such mathematical model is to operate certain non-transitive algebras. Relativist ontology assumes that the notion of “temperature” is relative, thus making it possible to ignore the Second Law of Thermodynamics. In its turn, that will allow for phase shifts. A perpetual motion machine of the second type will become a reality. Rotation could presumably be used to increase inertial and decrease gravitational mass, slowing down the time. An experimental research could shed light on the mentioned effects. (shrink)

Cantor’s theory of cardinal numbers offers a way to generalize arithmetic from finite sets to infinite sets using the notion of one-to-one association between two sets. As is well known, all countable infinite sets have the same ‘size’ in this account, namely that of the cardinality of the natural numbers. However, throughout the history of reflections on infinity another powerful intuition has played a major role: if a collectionAis properly included in a collectionBthen the ‘size’ ofAshould be (...) less than the ‘size’ ofB(part–whole principle). This second intuition was not developed mathematically in a satisfactory way until quite recently. In this article I begin by reviewing the contributions of some thinkers who argued in favor of the assignment of different sizes to infinite collections of natural numbers (Thabit ibn Qurra, Grosseteste, Maignan, Bolzano). Then, I review some recent mathematical developments that generalize the part–whole principle to infinite sets in a coherent fashion (Katz, Benci, Di Nasso, Forti). Finally, I show how these new developments are important for a proper evaluation of a number of positions in philosophy of mathematics which argue either for the inevitability of the Cantorian notion of infinite number (Gödel) or for the rational nature of the Cantorian generalization as opposed to that, based on the part–whole principle, envisaged by Bolzano (Kitcher). (shrink)

Those inquiring into the nature of mind have long been interested in the foundations of mathematics, and conversely this branch of knowledge is distinctive in that our access to it is purely through thought. A better understanding of mathematical thought should clarify the conceptual foundations of mathematics, and a deeper grasp of the latter should in turn illuminate the powers of mind through which mathematics is made available to us. The link between conceptions of mind and of mathematics (...) has been a central theme running through the great competing philosophies of mathematics of the twentieth century, though each has refashioned the connection and its import in distinctive ways. The present collection will be of interest to students of both mathematics and of mind. Contents include: "Introduction" by Alexander George; "What is Mathematics About?" by Michael Dummett; "The Advantages of Honest Toil over Theft" by George Boolos; "The Law of Excluded Middle and the Axiom of Choice" by W.W. Tait; "Mechanical Procedures and Mathematical Experience" by Wilfried Sieg; "Mathematical Intuition and Objectivity" by Daniel Isaacson; "Intuition and Number" by Charles Parsons; and "Hilbert's Axiomatic Method and the Laws of Thought" by Michael Hallett. (shrink)

We address the following questions in this paper: (1) Which set or number existence axioms are needed to prove the theorems of ‘ordinary’ mathematics? (2) How should Frege’s theory of numbers be adapted so that it works in a modal setting, so that the fact that equivalence classes of equinumerous properties vary from world to world won’t give rise to different numbers at different worlds? (3) Can one reconstruct Frege’s theory of numbers in a non-modal setting (...) without mathematical primitives such as “the number of Fs” ( $$\#F$$ ) or mathematical axioms such as Hume’s Principle? Our answer to question (1) is ‘None’. Our answer to question (2) begins by defining ‘x numbers G’ as: x encodes all and only the properties F such that being-actually-F is equinumerous to G with respect to discernible objects. We answer (3) by showing that the mere existence of discernible objects allows one to reconstruct Frege’s derivation of the Dedekind-Peano axioms in a non-modal setting. (shrink)

In lieu of an abstract, here is a brief excerpt of the content:Reviewed by:Sextus Empiricus: Against the Grammarians (Adversus Mathematicos I)George A. KennedyD[avid] L. Blank, trans. Sextus Empiricus: Against the Grammarians (Adversus Mathematicos I). With an introduction and commentary. Oxford: Clarendon Press, 1998. lvi + 436 pp. Cloth, $105. (Clarendon Later Ancient Philosophers).Sextus was a Greek physician whose "empirical" medical studies seem to have led him to an enthusiastic commitment to what he calls "Pyrrhonian" skepticism, though it perhaps has rather (...) little debt to Pyrrho, who lived in the fourth century B.C. When and where Sextus himself lived and worked is unknown: certainly during the time of the Roman Empire, probably in the second, possibly in the third century, maybe in Alexandria, or Rome, or elsewhere. His extensive writings are a major source for the skeptical tradition in ancient philosophy--a kind of anti-foundationalism that has some appeal in our own age--but their importance goes beyond that, since Sextus explicates in detail the dogmas and theories he seeks to counteract. Against the Grammarians, as Blank notes, is second only to the textbook attributed to Dionysius Thrax for information on the teaching of grammar and is important for knowledge of Epicureanism as well.Blank's introduction, translation (the Greek text is not included), and line-by-line commentary (the first ever published) is a work of great learning and, I would think, of great patience. The volume as a whole greatly facilitates understanding of Sextus, his sources, and the traditions to which he subscribed or which he resisted. Blank has adopted "expertise" as a translation for the somewhat troublesome word techne, and "expert" for technites. This seems a reasonable rendering of Sextus' usage. What he attacks is not grammar (or, in subsequent books, rhetoric and the other arts) as practiced in ordinary life, but the claims to expert, scientific knowledge on the part of doctrinaire teachers of the arts, as well as some physicians and philosophers.This leads us, however, to a general problem in Sextus' writings. At the beginning of Against the Grammarians (chapters 5-7) (and elsewhere) he claims to have discovered difficult problems in the claims of specialists in philosophy and the arts which lead him to suspension of judgment. He will attempt "without any spirit of competitiveness to select and put the effective arguments against the liberal studies." Implicitly, an equal amount might be said in their defense. In what follows, however, he repeatedly claims to have "demolished" dogmas, leading to what has been called "negative dogmatism," the dogmatic denial of dogma. Blank devotes a section of his introduction to this problem, concluding that Sextus' intent throughout is in fact "equipollence." That is probably what Sextus himself would claim. Assuming it is, Sextus must, however, be charged with the fault of expressing himself poorly, for he repeatedly exults at his conclusive refutations of the claims of others.Throughout the work Blank strives, often with considerable success, to provide a sympathetic explanation and logical justification for Sextus' arguments, [End Page 166] many of which may initially strike a reader as blatant sophistries. Sextus was no fool, but he had blind spots and obsessions--infinite regress being one--and he often fails to take into account lines of argument obvious to a modern reader. Blank's objectivity does not extend to blanket approval of all of Sextus' arguments; indeed, he criticizes and even rejects the validity of arguments on a number of occasions. His treatment can be described as generally even-handed, but giving his author the benefit of the doubt more often than I would be inclined to do.Blank's extended introduction considers what is known about Sextus' life and works; the history of attacks on the arts, which go back to the fifth century b.c.; evidence for empiricist attacks on grammar; Against the Grammarians in relation to the other writings of Sextus; and negative dogmatism; then a few concluding remarks on the translation. Both here and in the commentary he convincingly demonstrates that Sextus' primary source is a lost Epicurean treatise--something like the writings of Zeno of Sidon, the teacher of Philodemus, though Blank stops short of claiming Zeno as the source. This Epicurean work contained a detailed attack... (shrink)

The paper outlines a novel version of mathematical structuralism related to invariants. The main objective here is twofold: first, to present a formal theory of structures based on the structuralist methodology underlying work with invariants. Second, to show that the resulting framework allows one to model several typical operations in modern mathematical practice: the comparison of invariants in terms of their distinctive power, the bundling of incomparable invariants to increase their collective strength, as well as a heuristic (...) principle related to the search for new invariants. (shrink)

Salmon and Soames argue against nominalism about numbers and sentence types. They employ (respectively) higher-order and first-order logic to model certain natural language inferences and claim that the natural language conclusions carry commitment to abstract objects, partially because their renderings in those formal systems seem to do that. I argue that this strategy fails because the nominalist can accept those natural language consequences, provide them with plausible and non-committing truth conditions and account for the inferences made without committing themselves (...) to abstract objects. I sketch a modal account of higher-order quantification, on which instead of ranging over sets, higher order quantifiers are used to make (logical) possibility claims about which predicate tokens can be introduced. This approach provides an alternative account of truth conditions for natural language sentences which seem to employ higher-order quantification, thus allowing the nominalist to evade Salmon’s argument. I also show how the nominalist can account for the occurrence of apparently singular abstract terms in certain true statements. I argue that the nominalist can achieve this by, first, dividing singular terms into real singular terms (referring to concrete objects) and only apparent singular terms (called onomatoids), introduced for the sake of brevity and simplicity, and then providing an account of nominalistically acceptable truth conditions of sentences containing onomatoids. I develop such an account in terms of modally interpreted abstraction principles and argue that applying this strategy to Soames’s argument allows the nominalists to defend themselves. (shrink)

This paper consists of two parts and has two aims. The first is to elaborate on the historical-philosophical background of Cantor’s notions of infinity in the context of Spinoza’s influence on him. To achieve this aim, in the first part I compare Spinoza’s and Cantor’s conceptions of actual infinity along with their mathematical implications. Explaining the metaphysical, conceptual, and methodological aspects of Cantor’s expansion of the orthodox finitist conception of number of his time, I discuss how (...) he adopts Spinoza’s motifs to overcome the challenges on his way to proposing his transfinite infinites. To achieve the second aim, i.e., to investigate the philosophical foundations of Cantor’s transfinite infinites, I follow a three-step outline analyzing Cantor’s three principles along with their metaphysical and epistemological-linguistic aspects. I analyze the strengths and weak sides of Lakoff and Núñez’s Basic Metaphor of Infinity and Pantsar’s Process → Object Metaphor, then compare their interpretations with mine. I conclude that the metaphors can be improved when their source domains are limited to the target domains they apply to and when supported by a thorough analysis of the historical-philosophical background of their target domains. (shrink)

The Common Cause Principle was introduced by HansReichenbach, in The Direction of Time, which was publishedposthumously in 1956. Suppose that two events A and Bare positively correlated: p(A∩B)>p(A)p(B)p(A∩B)>p(A)p(B)p(A\textbackslashcap B)>p(A)p(B). Suppose,moreover, that neither event is a cause of the other. Then,Reichenbach’s Common Cause Principle (RCCP) states that Aand B will have a common cause that renders them conditionallyindependent. Reichenbach incorporated his RCCP into a new probablistictheory of causation, and used it to describe a (purported)macrostatistical temporal asymmetry in analogy with (...) the second law ofthermodynamics. The principle is significant because it posits aconnection between causal structure and probabilistic correlations,thus facilitating causal inference from observed correlations.However, RCCP has been controversial, and a number of counterexampleshave been proposed. RCCP is often seen as an antecedent of theCausal Markov Condition, which plays a central role in causalmodeling and causal inference. RCCP has also been taken to captureassumptions about the behavior of classical systems that appear to beviolated in quantum mechanics. (shrink)

In this paper, the authors discuss Frege's theory of "logical objects" and the recent attempts to rehabilitate it. We show that the 'eta' relation George Boolos deployed on Frege's behalf is similar, if not identical, to the encoding mode of predication that underlies the theory of abstract objects. Whereas Boolos accepted unrestricted Comprehension for Properties and used the 'eta' relation to assert the existence of logical objects under certain highly restricted conditions, the theory of abstract objects uses unrestricted Comprehension for (...) Logical Objects and banishes encoding formulas from Comprehension for Properties. The relative mathematical and philosophical strengths of the two theories are discussed. Along the way, new results in the theory of abstract objects are described, involving: the theory of extensions, the theory of directions and shapes, and the theory of truth values. (shrink)

The fragmentary nature ofGSmakes it difficult to read as it stands, and for this reason, I have rearranged the material slightly so that it falls into four primary, reasonably coherent, parts. Their titles are: ‘The nature of mathematical objects’, ‘Thirteen propositions of Euclid 1’, ‘The philosophy of parallel lines’ and ‘On the algebra of geometrical figures’.GSactually starts with ‘Thirteen propositions of Euclid 1’. The justification for the reversal of order in the translation is to have Hegel's philosophical basis for (...) geometry available first, to provide a context for his detailed discussion of geometrical propositions. The reader can easily reconstruct the original order ofGSas follows. Interchange the first and second parts and also the third and fourth parts. Next, move the passages marked a …. a', b …. b',c….c'to the places in the text markeda-a',b-b'andc-c'. For the reader wishing to locate the original German passages inDok., the page and paragraph for each of these passages are given at the start of that passage.In addition, subtitles for sections in these parts have been added which do not appear in the original. They are included in the translation to help orient the reader. I have inserted in some places words not in the original in order to clarify the meaning of the text. These insertions as well as my comments on the text are also enclosed within square brackets. Further clarifications can be found in the appended endnotes. (shrink)

In lieu of an abstract, here is a brief excerpt of the content:Journal of the History of Ideas 62.2 (2001) 193-210 [Access article in PDF] Zeno Against Mathematical Physics Trish Glazebrook Galileo wrote in The Assayer that the universe "is written in the language of mathematics," and therein both established and articulated a foundational belief for the modern physicist. 1 That physical reality can be interpreted mathematically is an assumption so fundamental to modern physics that chaos and super-strings are (...) examples of physical theories developed on the basis of success in mathematics. The mathematics came first, then the physical theory. Quantum theory is an awkward and similar case in point. Despite the fact that there is much agreement among physicists about the mathematical theory, its physical interpretation remains a matter of controversy. There are several interpretations, all of which challenge our everyday assumptions about reality. This led Bohr in 1935 to call for "a radical revision of our attitude towards the problem of physical reality." 2 Arthur Fine, originally a staunch defender of realist interpretations of quantum theory, saw the success of Bohr's Copenhagen interpretation as the end of Einsteinian realism, and subsequently Fine declared that "realism is well and truly dead." 3 Mathematical operators simply do not correspond with real entities in any conceivable way.Ilkka Niiniluoto has argued convincingly, however, that realism is alive and well in quantum theory. 4 The problem is that Bohr is right: quantum theory can be considered to concern the real only upon radical revision of ordinary conceptions of reality. Quantum theory does not describe the real in any meaningful sense of "the real." Rather, it is mathematical projection: ideal rather than real. [End Page 193]Aristotle did not hold that the physicist deciphers nature mathematically. He distinguished the physicist from the mathematician at Physics 2.2 by analogy to the definitions of "snub nose" and "curved." 5 The definition of "snub nose" requires the mention of a nose, that is, of matter which is inseparable from motion, whereas "curved" and other mathematical concepts like " 'odd' and 'even', 'straight'... 'number,' 'line,' and 'figure,' " which are separable from matter and from motion, are investigated by the mathematician. 6 Aristotle claims that physics is different from mathematics because the mathematician investigates physical lines, but not qua physical, and the physicist investigates mathematical lines "but qua physical, not qua mathematical." 7A peculiar consequence of my argument is that it establishes a common principle between two thinkers whose beliefs are otherwise so fundamentally in opposition. Zeno's paradoxes are commonly read as a denial of motion. For Aristotle, that move is axiomatic to the physicist. At 185a15 he considers arguments against this assumption outside the realm of objections to which the physicist must reply. If I am right that Zeno's paradoxes of motion challenge the very notion of mathematical physics, then he and Aristotle agree, contra Galileo, that the universe is not written in the language of mathematics. They simply come to this point from opposite directions, Zeno from an Eleatic thesis that physical reality is illusion, Aristotle from the natural scientist's pragmatic commitment to a truth about nature.I intend to use Zeno's paradoxes of motion to argue that the application of mathematical concepts to the physical world results in paradox. Zeno's arguments are limited within mathematics to geometry. This paper is accordingly a beginning from which I hope more work can be done later. Furthermore, Zeno's paradoxes are standardly read as directed against anyone who disagrees with Parmenides' thesis that there is but one thing which is motionless and indivisible. The paradoxes achieve this end by means of a reductio against the possibility of dividing space and time. That there are four paradoxes is accordingly explicable by the fact that there are four possibilities for the divisibility of space and time. The Achilles argues against the possibility that space and time are both infinitely divisible; the Dichotomy, against the infinite divisibility of space and finite divisibility of time; the Arrow, against the infinite divisibility of time and the finite divisibility of space; and finally, the Stadium shows that time and space cannot both... (shrink)

The book reconstructs the history of Western ethics. The approach chosen focuses the endless dialectic of moral codes, or different kinds of ethos, moral doctrines that are preached in order to bring about a reform of existing ethos, and ethical theories that have taken shape in the context of controversies about the ethos and moral doctrines as means of justifying or reforming moral doctrines. Such dialectic is what is meant here by the phrase ‘moral traditions’, taken as a name for (...) threads of moral discourse, made in turn of other interwoven threads, including transmission of shared codes, appeals to reform of prevailing custom, rational argument about the justification of some precept on the basis of some shared general teaching or principle, and rational argument about the ultimate basis for principles and justification of authoritative teaching. That is, the approach adopted to the reading of ethical texts depends on a firm belief in the fact that philosophers hardly created any ethical doctrine out of nothing. The main point this book tries to highlight is how different philosophical theories emerged and followed each other as a result of attempts at accounting for what was going on in the world of moral traditions. Changes were propelled by controversies between different schools, and highly abstract arguments were the unintended effects of moves made by controversialists forced to transform (and occasionally to turn upside down) their own doctrines in order to face the challenge posed by other arguments. This is the reason why the book examines not only texts that already enjoy pride of place in the history of philosophy (Aristotle, Kant, Hegel), but also other texts usually treated in the histories of religions (the Bible, the Talmud, the Quran), and others considered to be much less philosophical (Plutarchus, Pufendorf). -/- 1. Plato and a response to ethical scepticism. Two different traditions of morality in VI-V century Greece are described. The birth of philosophical questioning of traditional morality and temporal and spatial variation of custom is described within the context of the v century crisis, the demise of traditional aristocratic and tyrannical rule and the birth of democracy. Two conflicting answers to the challenge are reconstructed, namely conventionalist or immoralist theories formulated by the Sophists and the eudemonist and intellectualist Socratic theory. Plato’s own reformulation of Socrates answer to the Sophists is reconstructed. His psychological views, his classification of the four cardinal virtues and his political theory are described as parts of a unitary system, culminating in an extremely realist moral ontology identifying the idea of the good with the essence of the (moral and extra-moral) world itself. -/- 2. Aristotle and the invention of practical philosophy. Aristotle’s invention of practical philosophy as a field separated from first philosophy is shown to be an implication of his break with Plato moral ultra-realism. Aristotle’s agenda in his moral works is arguably dependent on a polemical intention, namely dismantling Socratic intellectualism. The semi-inductive or virtuously circular method of practical philosophy is illustrated, starting with the received opinions of the better and wiser individuals and trying dialectically to sift what is left of mistake and inconsistence in such opinions, finally trying to correct mistakes and make the overall practical science more consistent. The chapter illustrates then the relationship of individual ethics, or ‘monastics’, with the art and the science of the pater familias, or ‘economics’, and the science of the ruler and citizen, or politics. The nature of virtues, or better, excellences of character, is discussed, highlighting the basic role of hexis, or ‘disposition’. Prudence, or better practical wisdom, is the focus of the chapter. Its relationship with bouleusis or deliberation is examined, and its autonomous status vis-à-vis theoretical knowledge is stressed. -/- 3. Diogenes and philosophy as a way of life. The chapter provides an overview of Hellenistic ethics, which almost amounts to Hellenistic schools of philosophy, in so far as ‘philosophy’ became in these centuries primarily the name for a way of life. The typical character of the Cynical movement is highlighted, that of a school of life, not a school aimed at providing any kind of intellectual training was to be provided. -/- 4. Epicurus and ethics as self-care. The peculiar character of the Epicurean school is described, a combination of a science of well-being aiming, more than at pleasure as in the popular view, at reduction of useless suffering, of unnecessary needs, and at a balanced selection of pleasures of the best and most durable kind. -/- 5. Epictetus and ethics as therapy of the passions. The various phases of Stoicism are described, and the shifting place given to ethics in the Stoic system of idea, culminating in the paradoxical view of ethics, its impossibility in principle notwithstanding, as the only truly significant and necessary part of philosophy. Cicero is treated, showing how his own synthesis of various Hellenistic trends is as a truly philosophical enterprise, deserving serious consideration after one or two centuries when he was confined to the role of literate. Epictetus is chosen as the best example of what the Stoic tradition could yield, an art of living based on sophisticated introspection, in turn aimed at making a kind of cognitive therapy possible, dismantling obnoxious passions at their root by systematically correcting false representations. -/- 6. Philo and the reconciliation of Torah and Platonism. A reconstruction of basic ideas from a few books of the Hebrew Bible is provided, starting with the Prophetic tradition and the focus on God’s mercy as the source of motivation and standard for human behaviour. Then a comparative analysis is undertaken of a parallel tradition, namely the three codifications of the Torā (Law or, better, Instruction), highlighting how a core of moral ideas may be recognized as a basis and preamble of codification of civil law, cultural practice, and regulation of ritual purity. The importance of Leviticus is stressed as the turning point when emphasis mercy, typical of the Prophetic tradition, is combined with the legal tradition, yielding the change in sensibility of the Second Temple time. Philo of Alexandria is described as one of the three leading figures – together with Rabbi Hillel and Rabbi Yeshua – at the apex the emergence of the mentioned new sensibility, gradually including mercy as an essential part of justice and establishing the starting point of both the Rabbinic and the Christian tradition. This consists precisely in the precept of one’s neighbour’s love or of the golden rule, two eventually identical precepts whose meaning is arguably more sober and sensible than the long-lasting Christian tradition deriving from John and Augustine has made us believe and no novelty vis-à-vis so-called Ancient-Testament teachings. -/- 7. Augustine and Christianity as Neo-Platonism. The first section examines the moral doctrines of so-called 'ethical Treaties' from the Talmud, a group of treaties, among which the best known is Pirqé Avot, that were left out of the six "orders" of the canon as they did not fit in any of the six groups of issues ritual or legal on which the division was based. According to Maimonides, their peculiar theme is provided by the Deòt, 'opinions', i.e., mental dispositions, that is, the translation of the Greek term hexeis and Arab akhlak (in turn providing in this language the name for ethics as such). The three topics I reconstruct are: i) the notion of Torah: The Torah is understood as the world order itself, or as the ‘Wisdom’ that existed even before creation and was ‘the tool by which the world was built’; however, the Torah is an earthly and human entity, as it was "received" by humans, and from that very moment belongs to them; ii) the relationship between love of God and love of neighbour; the treaties require us to study and practice the Torah ‘for its own sake’, that is, require us to act out of love, not out of fear or hope of reward; iii) the idea of sanctification of daily life: having disappeared with the destruction of the Temple the possibility of any conflict between liturgical service and everyday life, the latter is assumed to be in itself divine service: to give food to the poor has the same value as sacrifices in the Temple, and as an implication, the insistence become recurrent on the goodness of created things in themselves along with a polemic against ascetic currents. The conclusions drawn are: i) the moral teachings of the Talmud and those of Yeshua are, rather than similar, virtually identical; one may safely say that the precept of love and the golden rule are central ones for all Talmudic rabbis, that mercy plays an indispensable role alongside with justice, and the latter is not a different thing from one’s neighbour’s love; ii) a peculiarity of Talmud rabbis facing Yeshua is the idea of study as worship, and knowledge as a source of justice; but this is an idea of Judaism after the Temple's destruction that cannot be attributed to the Pharisees of Yeshua’s time; iii) the relation of study and practice in the Talmud parallels that between faith and deeds in Paul's epistles, that is, respectively faith or learning are a necessary and sufficient condition to be recognized as righteous , but deeds are the inevitable effect of either faith or learning. The sayings ascribed to Yeshua are examined first, yielding the conclusion that a close equivalent may be found for every saying in Talmudic literature and yet the whole is ascribed to one rabbi, with rather consistent stress on God’s mercy and unconditional forgiving as the mark of true imitation of God. Thus, Yeshua’s teaching is pure Judaism. The third section describes briefly the galaxy of Gnostic currents and Manichaeism, trying to sketch the profile of moral teachings resulting from an encounter of Asian spiritual traditions, Hellenistic lore and sparks of teachings from apocalyptic Jewish currents. The last section summarizes the turbulent history of several encounters between Christian currents and Hellenistic philosophical schools. The first one was with late Cynicism. Recent, rather controversial, literature discovered the jargon and a few topics from the Stoic-Cynical popular philosophy in a few books from the New Testament itself. This, far from proving that Rabbi Yeshua had been influenced by cynic preachers, is a proof of the necessity felt by Christian preachers to translate the original ‘Christian’ teaching into a Greek lexicon deeply impregnated with cynic terminology. The second was with Platonism, yielding the mild and temperate moral teaching of Clemens of Alexandria, teaching the sanctity of nature and the human body, the joy of moderate fruition of ‘natural’ kinds of pleasures, and the beauty of the marital life – in short, the opposite of the standard picture of Medieval Christianity. Ambrose of Milan brings about a different kind of synthesis, namely with Middle Platonism, where Stoic themes prevail. The most shocking case is Augustine, where an early Manichean education is overcome in a former phase by a synthesis of Plotinian Neo-Platonism and Christian preaching, yielding a sustained polemic with the Manicheans and rather optimistic views on life and Creation, the body and sexuality, and Hebrew-Judaic tradition not far from Clemens of Alexandria. In a later phase, occasioned by controversy on the opposite front, with such Christian currents as the Pelagians and Donatists, Augustine comes back to heavily anti-Judaic and world-denying ascetic attitudes where is earlier Manichean upbringing seems to emerge again. The tragedy of medieval Christianity will be the later Augustine’s overwhelming influence. A final section is dedicated to the monastic tradition where a curious mixture of world-denying asceticism with an astonishingly penetrating ‘science’ of introspection emerges. -/- 8. Al Farabi and the reconciliation of Islam and Platonism. The Qurān and the qadit, that is, collections of sayings ascribed to the prophet Muhammad contain a wealth of precepts and catalogues of virtues mirroring moral contents from the Jewish and the Christian traditions, among them the basic notion of imitation of the Deity, where mercy is assumed to be its basic trait. An important tradition within Islam, namely Sufism, stressed to the utmost degree the role of mercy, turning Islam into a mystic doctrine centred on retreat from the world, abandonment to God’s will, and a peaceful and fraternal attitude to our fellow-beings. A secular literary tradition originating in the Sassanid Empire, the literature of advice to the Prince had for a time a widespread influence in the newly constituted Arabic-Islamic commonwealth. In a later phase, the legacy of Hellenic philosophy made its way into the intellectual elite of the Islamic world. The first important legacy was Platonic, and the Republic became the main text for Islamic Platonism. Al-Farabi wrote an enjoyable remake of the Platonic Republic, arguing that the citizen’s virtues were the basis on which the political building needs to rest. The secular lawgiver is enlightened by the light of reason in his everyday practice of governance, but room is made for the Prophet as the voice of revealed truth that confirms and completes the rational truth that philosophy may afford. In a second phase also Aristotelian and Stoic influence were assimilated. Miskawayh’s treatise was the masterwork at the time Arabic philosophy reached its Zenith. It is a treatment of the soul’s diseases and their remedies, combining the Aristotelian doctrine of the golden mean with the Stoic doctrine of the passions and elements of Galenic medicine. Towards the eleventh-twelve centuries a war raged among theologians and philosophers, finally won by the former with disappearance of philosophy as such. The newly established mainstream, yet, was no kind of intolerant fanaticism. It drew from the work of mystic theologian Al-Ghazālī, the best heir of Sufism, teaching a tolerant and peaceful attitude to our fellow-beings and a passive attitude to destiny as an expression of the Divine will. -/- 9. Moshe ben Maimon and the reconciliation of Torah and Aristotelian ethics. The encounter between the Talmudic tradition and Hellenic philosophy had taken place a first time in Alexandria at the time of Philo but the two traditions had parted way again. In fact, the kind of Platonic Judaism founded by Philo survived only within Christianity, in the fourth Gospel and then in writings by Clemens of Alexandria. The Talmudic literature had absorbed just less striking traits from the Hellenic Philosophy, namely an idea of ethics as care of the self and a role for the education of character as propaedeutic to theoretical knowledge. In the Arabic-Islamic world, a second round started when Jewish authors writing in Arabic undertook the task to prove the full compatibility of the tradition deriving from Torah and Platonic philosophy. The culmination of this attempt is provided by Moshe ben Maimon who tried to use Aristotelian ethics as a language into which the teaching from the Pirké Avot could be translated. -/- 10. Thomas Aquinas and the reconciliation of Christian morality and Aristotelian ethics. In the first section the fresh start is described of philosophical ethics in Latin Europe at the turn of the Millennium. In a first phase, Anselm and Peter Abelard built on a Platonic and Augustinian legacy. In this phase. remarkable innovations are introduced, including Abelard’s claim of the obliging character of erring consciousness. In a second phase, the rediscovery of Nicomachean Ethics thanks to Latin translation of Arabic versions gives birth to a new wave of ethical studies, recovering the very idea of Aristotelian practical philosophy, with the potential implication of full legitimization of natural morality, i.e. ethics without Revelation. Aquinas’s ethics is a theological ethics out of which it would be vain to try to extract a self-contained philosophical ethics. His treatment of topics in philosophical ethics, yet, does not boil down to repetition of Aristotelian arguments but is rather a creative reshaping of such arguments. For example, he introduces also in the practical philosophy a first principle parallel to the principle of non-contradiction; and he also carries out a synthesis of Aristotelianism, Stoicism, and neo-Platonism. Even though it is essentially moral theology, Aquinas’s doctrine - unlike Augustine – grants full citizenship to "natural" morality, firstly by rejecting the claim that the corruption of human nature due to the original sin is so radical as to leave "pure nature" incapable of moral goodness. The doctrine is presented in a more sophisticated formulation in a few of the Quaestiones, such as De Malo and De Veritate, in the Summa contra Gentes and in the commentaries to Aristotle than in the famous Summa Theologica, but the latter work includes the only or the largest exposition of some decisive part of the theory. Thus, the Summa Theologica should be read for what it is more than criticized for not being what it was not meant to be. It was not meant to be the brilliant synthesis of all that Reason he had been able to produce with what Revelation had added about which the Neo-Thomists used to dream, but rather a manual for the training of preachers and confessors, where theoretical claims are not too demanding and a few serious tensions are left. Besides a jump between the Prima Secundae and the Secunda Secundae, being the former an essay in virtue theory and the latter a handbook for confessors, the most serious tension is perhaps the one between the ethics of right reason presented in most of Ia-IIae and the eudemonistic ethics developed in quaestiones 1-5 of the same part; the alternative ethical theory which also may be found in Augustine, the Stoic view of a cosmic reason eventually coincident with the moral law, was believed by Anselm (followed by John Duns Scotus and William of Ockham) to be incompatible with eudemonism. It is questionable whether Thomas could reduce the tension proving that it is just an apparent tension, in so far as the right reason and bliss derived from knowledge of God tend to coincide, but this is just a conjecture. Thomas’s ethics is a virtue ethic, not a law-based one, and moral judgment focuses on the virtues, particularly charity, not on the commandments and even less on absolute prohibitions; Thomas, however, would not have considered a drastic alternative between law and the virtues such as the one which has been advanced in late twentieth-century philosophy to be justified. Nonetheless, when it discusses ‘special’ virtues, it ceases to be an ethics of virtue and becomes a disappointing and often contradictory discussion of legal and illegal acts. Such a discussion takes most of the time ‘reasonable’ middle positions on controversial issues but not the alternative approach that Aristotelianism would have made possible; even when some occasional Aristotelian claim shows up, such as money’s barrenness as a reason against usury, this seems to be made by an author who apparently ignores the Aristotelian Thomas of the Prima Secundae. It is an ethic of human autonomy which recognizes the binding character of the individual conscience and, potentially, even a duty to disobey unjust laws. It is true that what Thomas writes in his discussion of death penalty and persecution of heretics is simply disgusting, and yet we should blame Thomas the man, not Thomist ethical theory. Finally, Thomas’s ethics is not ethical ‘absolutism’, as both traditionalist Catholic theologians and secularist enemies of dogmatic morality tend to believe. It is instead an ethic of prudential judgment. Exceptions to this approach – or better results of logical fallacies – are such claims as the absolute character of negative precepts and of the existence of "intrinsically evil acts", claims that simply cease to make sense in the light of Thomas’s own distinction between human act and natural act, carrying consequences Thomas did not live long enough to draw or was not consistent enough to dare to draw. -/- 11. Francisco de Vitoria and casuistry. The first topic illustrated is the discussion about the notion of pura natura, namely the human condition after the original sin and before divine revelation. The core notion was already there in Aquinas but was later developed by Caietanus (Thomas de Vio) with a number of interesting implications: firstly a view of human nature as such much more positive than Augustine’s and his followers’ view, including both Calvinists and Jansenists; secondly, the implication that the philosopher’s morality, as opposed to the Christian’s morality, is a quite respectable kind of morality; thirdly, that theocracy and prosecution of the unfaithful lacked justification, with more far-fetching consequences than Aquinas himself had dared to draw. The second topic is casuistry, a new name for a comparatively older way of thinking, which was already there in late Stoicism and Cicero as well as in the Talmudic literature. This is an approach aimed at yielding probable enough solutions for doubtful cases even in absence of completely safe staring points. The genre developed from medieval reference books for confessors and became by the sixteenth-century a sophisticated tool-box for dealing with various fields of applied ethics. Francisco Vitoria, the main authority of Baroque Scholasticism, was a controversial figure, among the proponents of the new discipline of casuistry and a consistent proponent of more separation between the religious and the civil authority, stricter limits to the legitimacy of war, innate rights of non-Christian nations in the New World based on the notion of pura natura, providing a standard of ‘natural’ goodness, previous to revelation, on which the Indian nations were judged to live in a naturally good condition, provided with the institutions of family, justice, and religion had accordingly a right to full respect by Christian sovereigns. Bartolomé de Las Casas, arguing along similar lines, was the leader of a historical battle in defence of the rights of the Indios. -/- 12. Michel de Montaigne and the art of living. A short description of one of the Renaissance Phyrronism, one of the classical philosophies revived in the fifteenth and sixteenth century. Montaigne combines the sceptical epistemology with an understanding of ethics – indeed of philosophy as a whole – in terms of an art of living inspired by two basic ideas, sagesse, that is, practical wisdom as the only kind of rationality available after theology, science, and tradition have declared bankrupt, and an aesthetic ideal of self-transformation through the art of writing and introspection. -/- 13. Pierre Nicole and neo-Augustinianism. The chapter illustrates first the vicissitudes of Augustinianism newly born after the late medieval triumph of Thomist Aristotelianism in the alternative Protestant and Catholic versions processed by the Calvinists and the Jansenists. Then it illustrates the doctrines of Pierre Nicole, Blaise Pascal’s best disciple, on the impossibility of introspection, on the ubiquity of self-deception, on the incurably evil character of the passion of love, and on the two moralities, the one of the man of the world, the morality of honesty which is indispensable for granting peace and order but devoid of any true value for eternal salvation, since the same external behaviour may be prompted by opposite motivations, and the morality of charity, the only true one but also useless to society. The third topic is Pierre Malebranche’s view of self-deception as a ubiquitous phenomenon accounting for human action and responsibility and his reformulated theory of self-love, no less ubiquitous than for Nicole and yet not as incurably evil, given the distinction between morally indifferent and even necessary amour de soi and vicious amour proper, a distinction on which the whole of Rousseau’s moral and political theory rests. -/- 14. Samuel von Pufendorf and the new science of morality. The chapter is dedicated to the discovery of the idea of a ‘new moral science’. Hugo Grotius is discussed first highlighting the real character of his project, much closer to the Thomist idea of a natural law accessible to non-corrupted human reason when human beings are living in a state of pura natura. Then Hobbes’s combination of scepticism, voluntarism, and conventionalism is described and both the continuity with Grotius’s new science, in the search for non-revealed rational morality, and the break with him, in the adoption of voluntarism and refusal of an intellectualist view of natural law, are illustrated. Pufendorf’s work is illustrated as a synthesis of the two previous attempts and the – up to Schneewind underestimated – paradigmatic example of the new science of morality. -/- 15. Richard Cumberland and consequentialist voluntarism. The chapter gives an overview of eighteenth-century Anglican ethics, noticing how the Cambridge tradition gave special weight to natural theology as opposed to positive or revealed theology – and how two Cambridge fellows, John Gay and Thomas Brown, elaborated on Cumberland’s (and Malebranche’s, as well as Leibniz’s) strategy for finding a third way between intellectualist view and voluntarist view of the laws of nature. The result of their elaboration was a kind of a rational-choice account of the origins of natural laws, where a law-giver God chooses among a number possible sets of laws on a maximizing criterion, and God’s maximandum is happiness for his creatures. The chapter notices also how such a solution aimed at solving at once the problem of evil and that of the foundation of moral obligation by proving how God’s choice was justified as far as it was the one minimising the amount of suffering in the world. 16. Richard Price and intuitionism. The chapter describes first the doctrines of the Cambridge Platonists, an example of hyper-rationalist reaction to Calvinism. Secondly it describes the sophisticated and universally ignored – from Sidgwick to Anscombe –version of what was later labelled ‘ethical intuitionism’ – showing how it escapes familiar objections and misrepresentations of intuitionism – from Mill to Rawls – in grounding its argument on transcendental arguments while carefully avoiding recourse to introspection and psychological evidence, which has been taken as a too easy target by critics of intuitionism. Thirdly, the chapter discusses Whewell’s ‘philosophy of morality’, as opposed to ‘systematic morality’, not unlike Kant’s distinction between a pure and an empirical moral philosophy. Whewell worked out a systematization of traditional normative ethics as a first step before its rational justification; he believed that the point in the philosophy of morality is justifying a few rational truths about the structure of morality such as to rule hedonism, eudemonism, and consequentialism; yet a system of positive morality cannot be derived solely from such rational truths but requires consideration of the ongoing dialectics between idea and fact in historically given moralities. Whewell’s intuitionism turns out closer to Kantian ethics than commentators have made us believe until now, and quite different from what Sidgwick meant by intuitionism. -/- 17. Adam Smith and the morality of role-switch. The chapter describes first, Hutcheson’s attempt at basing the ‘new science’ of natural law on different bedrock than Pufendorf and the English Platonists, namely a moral sense, a faculty whose existence is assumed to be proved on an empirical basis. The second step is a reconstruction of Hume’s rejoinder in terms of a new science of man including morality on an ‘experimental’ basis, that is, a ‘Newtonian’ hypothetical-deductive approach, distinguished from Hutcheson’s allegedly uncritical descriptive approach to human nature. The third step is a reconstruction of Adam Smith theory of morality understood as emerging from a spontaneous interplay of exchanges of situations (the most basic meaning of the word ‘sympathy’ as construed by this author). -/- 18. Jeremy Bentham and utilitarianism. The bulk of the chapter is dedicated to the doctrines of Jeremy Bentham. The Enlightenment spirit that suggested the idea of a new morality, free from religion, is illustrated. The notion of utility is illustrated as well as the subsequent formulations of the principle of utility. The idea of felicific calculus is discussed, showing how its inner difficulties prompted several reformulations of the principle of utility in order to avoid undesirable implications of proposed formulations. The role of the thesis of spontaneous convergence between interest and duty is discussed, showing how it left numberless questions open, and the distinction between the virtues of prudence, justice and beneficence is described as a way out of the deadlocks of classical utilitarianism. After Bentham, Mill’s reformed utilitarianism is reconstructed, showing how it is a kind of mixed system – as closer to common sense as it gives up Bentham’s claims to consistency and simplicity – resulting as unintended consequences from the controversies in which Mill was too keen to engage. The hidden agenda of the controversy between Utilitarianism and Intuitionism, going beyond the image of the battle between Prejudice and Reason, is described, showing how both competing philosophical outlooks turn out to be more research programs than self-contained doctrinal bodies, and such programs appear to be implemented, and indeed radically transformed while in progress thanks to their enemies no less than to their supporters. -/- 19. Immanuel Kant, practical reason and judgment. The chapter argues that Kant took from Moses Mendelssohn the idea of a distinction between geometry of morals and a practical ethic. He was drastically misunderstood by his followers precisely on this point. He had learned from the sceptics and the Jansenists the lesson that men are prompted to act by deceptive ends, and he was aware that human actions are also empirical phenomena, where laws like the laws of Nature may be detected. His practical ethics made room for judgment as a holistic procedure for assessing the saliency of relevant moral qualities in one given situation; this procedure yields the same results as the geometry of morals does for abstract cases but does so immediately and without balancing conflicting duties with each other, since what makes for the salient quality of a situation is perceived from the very beginning. Kant's practical philosophy is richer than the received image, making room for an ‘empirical moral philosophy’ or moral anthropology including treatment of commerce, needs, value and price, happiness and well-being; the overall social theory and philosophy of history is less different from Adam Smith's than the received image makes believe; the paradox of happiness is central to Kant's philosophy; a distinction between happiness and well-being is clearly drawn; the distinction plays a basic role in establishing a link between the economic and the moral life. -/- 20. Georg Wilhelm Friedrich Hegel and the critique of abstract universalism. The chapter describes first the Romantic movement and the implications some of its concerns, rescuing passions, community, tradition, the individual as the bearer of a unique destiny, and the longing for organic unity between the individual, mankind and nature. Hegel’s contribution is discussed then, highlighting how, on the one hand, he shared a number of these concerns and on the other he had more rationalist leanings. The notion of morality is the pivotal point of the reconstruction, highlighting how Hegel construes this notion as a key to his own diagnosis of the malaise of modernity – the separation of individual and Gemeinschaft – and how his attack on Kant turns around this very idea. -/- 21. Friedrich Nietzsche against Christianity and the Enlightenment. The chapter illustrates first the idea of deconstruction of the back-world of values. Nietzsche claims to be legitimate heir of the French moralists, Leibniz, Kant and Hegel, in so far as he allegedly carries out to its deepest implications their discovery of what lies behind traditional naïve belief in the existence of an objective realm of values just waiting for description by the philosopher. The two exemplars form which genealogy draws inspiration are the classical philologist’s historical reconstructions of lost meanings and the chemist’s decomposition of elements. Then the genealogy of the notions of good and evil carried out in the first dissertation of Genealogy of morals is illustrated with its paradoxical conclusion that will to power is in fact the only ‘genuine’ kind of goodness. The third point illustrated is the dialectic of ascetism and self-realization with its ambiguous outcome. The suggested interpretation of such outcome is that Nietzsche’s normative ethics is a kind of virtue ethics taking an aesthetic ideal as a normative standard -/- 22. George Edward Moore and ideal utilitarianism. The chapter discusses the ideas of common sense and common-sense morality in Sidgwick. I argue that, far from aiming at overcoming common-sense morality, Sidgwick aimed purposely at grounding a consist code of morality by methods allegedly taken from the natural sciences to reach, also in ethics, the same kind of “mature” knowledge as in the natural sciences. His whole polemics with intuitionism was vitiated by the a priori assumption that the widespread ethos of the educated part of humankind, not the theories of the intuitionist philosophers, was what was worth considering as the expression of intuitionist ethics. In spite of a naïve positivist starting-point, Sidgwick was encouraged by his own approach in exploring the fruitfulness of coherentist methods for normative ethics. Thus, Sidgwick left an ambivalent legacy to twentieth-century ethics: the dogmatic idea of a “new” morality of a consequentialist kind, and the fruitful idea that we can argue rationally in normative ethics albeit without shared foundations. Then it reconstructs the background of ideas, concerns and intentions out of which Moore’s early essays, the preliminary version, and then the final version of Principia Ethica originated. I stress the role of religious concerns, as well as that of the Idealist legacy. I argue that PE is more a patchwork of rather diverging contributions than a unitary work, not to say the paradigm of a new school in Ethics. I add a comparison with Rashdall’s almost contemporary ethical work, suggesting that the latter defends the same general claims in a different way, one that manages to pave decisive objections in a more plausible way. I end by suggesting that the emergence of Analytic Ethics was a more ambiguous phenomenon than the received view would make us believe, and that the wheat (or some other gluten-free grain) of this tradition, that is, what logic can do for philosophy, should be separated from the chaff, that is, the confused and mutually incompatible legacies of Utilitarianism and Idealism. -/- 23. Edmund Husserl and the a-priori of action. The chapter illustrates first the idea of phenomenology and the Husserl’s project of a phenomenological ethic as illustrated in his 1908-1914 lectures. The key idea is dismissing psychology and trying to ground a new science of the a priori of action, within which a more restricted field of inquiry will be the science of right actions. Then the chapter illustrates the criticism of modern moral philosophy carried out in the 1920 lectures, where the main target is naturalism, understood in the Kantian meaning of primacy of common sense. The third point illustrate is Adolph Reinach’s theory of social acts as a key the grounding of norms, a view that basically sketches the very ideas ‘discovered’ later by Clarence I. Lewis, John Searle, Karl-Otto Apel and Jürgen Habermas. A final section is dedicated to Nicolai Hartman, who always refused to define himself a phenomenologist and yet developed a more articulated and detailed theory of ‘values’ – with surprising affinities with George E. Moore - than philosophers such as Max Scheler who claimed to Husserl’s legitimate heirs. -/- 24. Bertrand Russell and non-cognitivism. The chapter reconstructs first the discussion after Moore. The naturalistic-fallacy argument was widely accepted but twisted to prove instead that the intuitive character of the definition of ‘good’, its non-cognitive meaning, in a first phase identified with ‘emotive’ meaning. Alfred Julius Ayer is indicated as a typical proponent of such non-cognitivist meta-ethics. More detailed discussion is dedicated to Bertrand Russell’s ethics, a more nuanced and sophisticated doctrine, arguing that non-cognitivism does not condemn morality to arbitrariness and that the project of a rational normative ethics is still possible, heading finally to the justification of a kind of non-hedonist utilitarianism. Stevenson’s theory, another moderate version of emotivism is discussed at some length, showing how the author comes close to the discovery of the role of a pragmatic dimension of language as a basis for ethical argument. Last of all, the discussion from the Forties about Hume’s law is described, mentioning Karl Popper’s argument and Richard Hare’s early non-cognitivist but non-emotivist doctrine named prescriptivism. -/- 25. Elizabeth Anscombe and the revival of virtue ethics. The chapter discusses the three theses defended by Anscombe in 'Modern Moral Philosophy'. I argue that: a) her answer to the question "why should I be moral?" requires a solution of the problem of theodicy, and ignores any attempts to save the moral point of view without recourse to divine retribution; b) her notion of divine law is an odd one more neo-Augustinian than Biblical or Scholastic; c) her image of Kantian ethics and intuitionism is the impoverished image manufactured by consequentialist opponents for polemical purposes and that she seems strangely accept it; d) the difficulty of identifying the "relevant descriptions" of acts is not an argument in favour of an ethics of virtue and against consequentialism or Kantian ethics, and indeed the role of judgment in the latter is a response to the difficulties raised by the case of judgment concerning future action. A short look at further developments in the neo-naturalist current is given trough a reconstruction of Philippa Foot’s and Peter Geach’s critiques to the naturalist-fallacy argument and Alasdair MacIntyre’s grand reconstruction of the origins and allegedly unavoidable failure of the Enlightenment project of an autonomous ethic. -/- 26. Richard Hare and neo-Utilitarianism. The chapter addresses the issue of the complex process of self-transformation Utilitarianism underwent after Sidgwick’s and Moore’s fatal criticism and the unexpected Phoenix-like process of rebirth of a doctrine definitely refuted. A glimpse at this uproarious process is given through two examples. The first is Roy Harrod Wittgensteinian transformation of Utilitarianism in pure normative ethics depurated from hedonism as well as from whatsoever theory of the good. This results in preference utilitarianism combined with a ‘Kantian’ version of rule utilitarianism. The second is Richard Hare’s two-level preference utilitarianism where act utilitarianism plays the function of eventual rational justification of moral judgments and rule-utilitarianism that of action-guiding practical device. -/- 27. Hans-Georg Gadamer and rehabilitation of practical philosophy. The post-war rediscovery of ethics by many German thinkers and its culmination in the Sixties in the movement named ‘Rehabilitation of practical philosophy’ is described. Among the actors of such rehabilitation there were a few of Heidegger’s most brilliant disciples, and Hans-Georg Gadamer is chosen as a paradigmatic example. His reading of Aristotle’s lesson I reconstructed, starting with Heidegger’s lesson but then subtly subverting its outcome thanks to the recovery of the central role of the notion of phronesis. -/- 28. Karl-Otto Apel and the revival of Kantian ethics. Parallel to the neo-Aristotelian trend, there was in the Rehabilitation of practical philosophy a Kantian current. This started, instead than the rehabilitation of prudence, with the discovery of the pragmatic dimension of language carried out by Charles Peirce and the Oxford linguistic philosophy. On this basis, Karl-Otto Apel singled out as the decisive proponent of the linguistic and Kantian turn in German-speaking ethics, worked out the performative-contradiction argument while claiming that this was able to provide a new rational and universal basis for normative ethics. An examination of his argument is some detail is offered, followed by a more cursory reconstruction of Jürgen Habermas’s elaboration on Apel’s theory. -/- 29. John Rawls and public ethics as applied ethics. Rawls’s distinction between a “political” and a “metaphysical” approach to one central part of ethical theory, namely the theory of justice, is interpreted as a formulation of the same basic idea at the root of both the principles approach and neo-casuistry, both discussed in the following chapter, namely that it is possible to reach an agreement concerning positive moral judgments even though the discussion is still open – and in Rawls’ view never will be close – on the basic criteria for judgment. -/- 30. Beauchamp and Childress and bioethics as applied ethics. The chapter presents the revolution of applied ethics while stressing its methodological novelty, exemplified primarily by Beauchamp and Childress principles approach and then by Jonsen and Toulmin’s new casuistry. -/- . (shrink)

It is widely agreed by philosophers that the so-called “Frege-Russell definition of natural number” is actually an assertion concerning the nature of the numbers and that it cannot be regarded as a definition in the ordinary mathematical sense. On the basis of the reasoning in this paper it is clear that the Frege-Russell definition contradicts the following three principles (taken together): (1) each number is the same entity in each possible world, (2) each number exists in each possible (...) world, (3) some entities existing in the actual world do not exist in every possible world. Since these principles seem to be true, the paper is a refutation of the Frege-Russell definition. The paper does more. It shows that the contradictory of the Frege-Russell definition follows even when principles 2 and 3 are replaced by one considerably weaker principle. The ideas contained in the paper are related to two earlier objections to the definition. The first, sometimes attributed to the mathematician, C. S. Keyser, is that existence of the numbers as defined implies the existence of infinitely many particulars in each possible world. The second is, in effect, an idea which is said to have led Whitehead to reject the definition of number to which he had subscribed in Principia Mathematica. Whitehead is supposed to have said that he could not believe that the number two changes every “time twins are born”. The mathematician H. Jeffreys expressed similar ideas [Philos. of Sci. 5 (1938), 434–451]. One of the merits of the author’s work is that it refutes the Frege-Russell definition without the need to take sides on controversial points presupposed by the Keyser and Whitehead objections. The objections made by the author are therefore not to be identified with the Keyser and Whitehead objections. Even if the author’s work is to be regarded as a refinement and integration of previous ideas, it is nevertheless a contribution—not only because the basic points are well worth repeating but also because the refinements are logically significant improvements and because the author has stated them clearly and concisely in the idiom of contemporary philosophy. (shrink)

The concept of a Mannigfaltigkeit in Husserl has been given various interpretations, due to its shifting role in his works. Many authors have been misled by this term, placing it in the context of Husserl’s early period in Halle, while writing the Philosophy of Arithmetic, as a friend and colleague of Georg Cantor.Yet at the time, Husserl distanced himself explicitly from Cantor’s definition and rather took Bernhard Riemann as example, having studied and lectured extensively on Riemann’s theories (...) of space. Husserl’s Mannigfaltigkeitslehre would then not be a Cantorian set-theory, but come rather closer to topology. Then, in the Prolegomena, Husserl introduces the idea of a pure Mannigfaltigkeitslehre as a meta-theoretical enterprise which studies the relations among theories, e.g. how to derive or found one upon another. When Husserl announces that in fact the best example of such a pure theory of manifolds is what is actually practiced in mathematics, this sounds slightly misleading. The pure theory of theories cannot simply be the mathematics underlying topology, but should rather be considered as a mathesis universalis. Indeed, while this might not have been fully clear yet in 1900/1901, Husserl will explicitly tie together the notions of pure theory of manifolds and mathesis universalis. The mathesis universalis in this sense is formal, a priori and analytic, as theory of theory in general. It is an analysis of the highest categories of meaning and their correlative categories of objects. In my paper I try to understand the development of the notion of Mannigfaltigkeit in Husserl’s thought from its mathematical beginnings to its later central philosophical role, taking into account the mathematical background and context of Husserl’s own development. (shrink)

A natural realist approach to the philosophy of arithmetic is defended by way of considering and arguing against contemporary attempts to solve Paul Benacerraf's dilemma . The first horn of the dilemma concerns the existence of abstract mathematical objects, which seems necessitated by a desire for a unified semantics. Benacerraf adopts an extensional semantics whereby the reference of terms for natural numbers must be abstract objects. The second horn concerns a desirable causal constraint on knowledge, according (...) to which "for X to know that S is true requires some causal relation to obtain between X and the referents of the names, predicates, and quantifiers of S". Within the philosophical tradition, Benacerraf's dilemma crystallizes the tension between realists and empiricists . It is shown that natural realism and naturalism meet. ;Both horns of the dilemma are amended.objects are conceived along conceptualist lines such that their existence is not mind-independent. The causal constraint is refined by drawing upon the work of Mark Steiner . Natural realism is the notion that the truth-values of arithmetical statements are recognition-transcendent. Natural realism explains why one and only one arithmetic is applicable. It is in some sense discovered, which is captured by the idea that arithmetical statements have truth-values even if one is not able to adduce what they are. ;It is argued that with proper emendation, like that of Philip Kitcher , empiricism as envisioned by J. S. Mill is defensible. Furthermore, the epistemology of arithmetical knowledge is divided into two tiers. The first principles of arithmetic are acquired by causal interaction with physical objects. As Kitcher has shown, empiricism functions at the first-tier. The a priori is revised such that arithmetical knowledge generated from the first-tier is considered non-empirical. ;It is argued that pragmatists' indispensability argument, once qualified, allows that arithmetic's applicability is one reason to consider natural realism for that domain . Furthermore, this thesis provides a case where aspects of Hilary Putnam's early and later views are rendered consistent. By way of utilizing both Putnam's writing in favour of realism , and those that break with them , it is suggested that he should not have abandoned his earlier view. In a nutshell, the realization of knowledge can depend upon agents' values, methods, and so on, without forcing the abandonment of realism . Putnam's early view sets the criteria for when one can be a realist about a given domain. (shrink)

Una obra frecuentemente consultada por Jorge Luis Borges fue Matemáticas e imaginación, de E. Kasner y J. Newman, en la que se discute la teoría de los conjuntos , propuesta por el matemático Georg Cantor , y mediante la cual se crea la aritmética transifinita y se establece un sistema epistémico para representar los diversos niveles del infinito. Así, Cantor le asigna a estas infinitudes la primera letra del alfabeto hebreo, el Aleph, seguido de un determinado número, (...) dependiendo del nivel de infinitud . Borges, de esta manera, teje varias de sus narraciones en las que se trata el tema del infinito y del absoluto; un ejemplo de ello es la colección de relatos bajo el título El Aleph, que abre con el cuento "El inmortal" y cierra con el quele da el título a la colección. Este ensayo tiene el propósito de estudiar "El inmortal" bajo la óptica cantoriana, para hablar de un absoluto en particular: el yo, y sugerir que no es posible establecer un vocabulario final, o una definición definitiva sobre el tema en cuestión. Esta imposibilidad, propone Borges, está dada en parte por la finitud lingüística, mientras que por otro lado la falibilidad de la memoria juega también un papel crucialen todo intento de definición. Sin embargo, como buen ironista, Borges, a través de "El inmortal", es capaz de proveer una redescripción del tema en cuestión mediante un lenguaje transfinito, sin pretender establecer un vocabulario final sobre el tema sino, al contrario, tratando de resolver ciertas paradojas a la vez que revela otras, promoviendo de esta manera el permanente diálogo entre las distintas disciplinas. Aunque este ensayose enfoca en el análisis de "El inmortal", a fin de desarrollar el tema propuesto, también estudia otros relatos contenidos en El Aleph, y utiliza un acercamiento teórico que se enmarca en la filosofía de la lengua. Jorge Luis Borges often consulted Mathematics and Imagination, by E. Kasner and J. Newman, where set theory is addressed . This theory was proposed by Georg Cantor and by it transfinite arithmetic is established and an epistemic system is created to represent different levels of the infinite. This way Cantor labels the different levels of the infinite by assigning to each the first letter of the Hebrew alphabet, the Aleph, followed by a number, depending on level of the infinite he is referring to . Following these ideas, Borges weaves several narratives in which the infinite and the absolute are discussed. An example of such narratives is the collection of stories compiled under the title The Aleph, which opens with "The Immortal" and closes with the story that gives the collection its title. The objective of this paper is to study "The Immortal" under the Cantorian lens to discuss one particular absolute, the self, and to suggest that it is impossible to establish a final vocabulary, or a definite definition, about this topic. This impossibility, Borges proposes, is in part due to the apparent finitude of language while at the same time the fallible attributes of human memory also is crucial when it comes to defining anything. However, as the ironist Borges is, he is capable of providing through "The Immortal" a re-description of these issues by means of a transfinite language that resolves some paradoxes while at the same time reveals others. By this way of writing, I propose, Borges fosters the continuation of the dialogue among the different disciplines. Though I will center my analysis on "The Immortal", to develop these ideas I also revisit other stories contained in The Aleph departing from a theoretical approach rooted in the philosophy of language. (shrink)

The article is devoted to the comparison of two types of proofs in mathematical practice, the methodological differences of which go back to the difference in the understanding of the nature of mathematics by Descartes and Leibniz. In modern philosophy of mathematics, we talk about conceptual and formal proofs in connection with the so-called Hilbert Thesis, according to which every proof can be transformed into a logical conclusion in a suitable formal system. The analysis of the arguments (...) of the proponents and opponents of the Thesis, “conceptualists” and “formalists”, is presented respectively by the two main antagonists – Y. Rav and J. Azzouni. The focus is on the possibility of reproducing the proof of “interesting” mathematical theorems in the form of a strict logical conclusion, in principle feasible by a mechanical procedure. The argument of conceptualists is based on pointing out the importance of other aspects of the proof besides the logical conclusion, namely, in introducing new concepts, methods, and establishing connections between different sections of meaningful mathematics, which is often illustrated by the case of proving Fermat’s Last Theorem (Y. Rav). Formalists say that a conceptual proof “points” to the formal logical structure of the proof (J. Azzouni). The article shows that the disagreement is based on the assumption of asymmetry of mutual translation of syntactic and semantic structures of the language, as a result of which the formal proof loses important semantic factors of proof. In favor of a formal proof, the program of univalent foundations of mathematics In. Vojevodski, according to which the future of mathematical proofs is associated with the availability of computer verification programs. In favor of conceptual proofs, it is stated (A. Pelc) that the number of steps in the supposed formal logical conclusion when proving an “interesting” theorem exceeds the cognitive abilities of a person. The latter circumstance leads the controversy beyond the actual topic of mathematical proof into the epistemological sphere of discussions of “mentalists” and “mechanists” on the question of the supposed superiority of human intelligence over the machine, initiated by R. Penrose in his interpretation of the Second Theorem of Goedel, among whose supporters, as it turned out, was Goedel himself. (shrink)

We study a transfinite iteration of the ordinal Hessenberg natural sum obtained by taking suprema at limit stages. We show that such an iterated natural sum differs from the more usual transfinite ordinal sum only for a finite number of iteration steps. The iterated natural sum of a sequence of ordinals can be obtained as a mixed sum (in an order‐theoretical sense) of the ordinals in the sequence; in fact, it is the largest mixed sum which satisfies (...) a finiteness condition. We introduce other infinite natural sums which are invariant under permutations and show that all the sums under consideration coincide in the countable case. (shrink)

In lieu of an abstract, here is a brief excerpt of the content:98 COMMENTS ON TWEYMAN AND DAVIS Tweyman contends that in Parts X and XI of the Dialogues Philo sets aside his Pyrrhonian or skeptical approach to theology, which consists in falsifying or casting doubt on the hypotheses of Cleanthes, and instead argues for a thesis of his own, viz. what we might call the "indifference thesis" that the original source of all things is morally indifferent. Davis counters (...) with an alternative interpretation of these two Parts of the Dialogues, arguing that Philo's approach to Cleanthes' arguments is consistent throughout the Dialogues and that Philo's aim is always the same, viz., to show that if cleanthes remains true to his principles and accepts their logical consequences, it is Cleanthes who is the Pyrrhonian and Philo who is the moderate skeptic. My position is that neither Tweyman nor Davis have given a proper analysis of Parts X and XI. I would agree with Davis that Tweyman' s position does seem to have an initial implausibility to it for at least two reasons. First of all, in Part II Philo commits himself to the "mystical incomprehensibility thesis" that the attributes of God are perfect but incomprehensible; and the mystical incomprehensibility thesis is incompatible with the indifference thesis. It cannot be the case both that God has perfect but incomprehensible attributes and that we also know he is morally indifferent. Moreover, the mystical incomprehensi2 bility thesis is reiterated in Part VI and more importantly, in Part X: "None but we mystics, as you were pleased to call us, can account for this strange mixture of phenomena, by deriving it from attributes, infinitely perfect, but incomprehensible" 99 (D 199). Since Philo appears to stick to the mystical incomprehensibility thesis, Tweyman has a problem explaining how Philo can hold one position on the attributes of God in Part X (the mystical incomprehensibility thesis) and then supposedly reverse himself in Part XI to argue for another (the indifference thesis). Second, Tweyman' s position seems to contradict the strategic position outlined by Philo in Part II: "You seem not to apprehend, replied Philo, that I argue with Cleanthes in his own way; and by showing him the dangerous consequences of his tenets, hope at last to reduce him to our opinion." (D 145) The strategy is to show Cleanthes that any position but that of the mystics has dangerous, i.e., skeptical or theologically unwelcome consequences, and so should be abandoned. Philo's strategy is not to defend his own position, but rather to reveal the weaknesses in Cleanthes' position. Thus far I would agree that Davis' interpretation is more consistent with both Philo's strategic position and with his mystical incomprehensibility thesis. Yet Davis does not directly take up the important point about Parts X and XI which Tweyman raises: that Philo changes his way of arguing in Parts X and XI from that which he had been employing previously in the first eight sections, i.e., that he changes from a Pyrrhonian to a non-Pyrrhonian way of arguing. This is how Tweyman puts it: In the first eight sections, Philo argues against Cleanthes' hypothesis by advancing his own hypotheses. His effort to advance hypotheses must be assessed in light of the fact that (by his own admission) all such hypotheses are based on insufficient 100 data, and, therefore, none is, strictly speaking, acceptable. Their use is not to establish truths about the nature of God, but to establish the conclusion which we find at the end of Part VIII that 'a total suspense of judgment is here our only reasonable resource.' The design of the world is compatible with, and could have arisen from, an indefinite number of designing principles. On the other hand, we have seen that in dealing with Cleanthes' hypotheses in Parts X and XI we are able to proceed more scientifically, and, in this manner, eliminate all but one of the hypotheses which can be introduced to explain the design of the world. (Tweyman 84-85) Now Tweyman does seem to have a point here. It is true that Philo takes his arguments in Parts X and XI to have a conclusiveness which he did not attribute to his... (shrink)

Frege proposed that his Context Principle—which says that a word has meaning only in the context of a proposition—can be used to explain reference, both in general and to mathematical objects in particular. I develop a version of this proposal and outline answers to some important challenges that the resulting account of reference faces. Then I show how this account can be applied to arithmetic to yield an explanation of our reference to the natural numbers and of (...) their metaphysical status. (shrink)

In the thesis some combinatorial statements are analysed from the reverse mathematics point of view. Reverse mathematics is a research program, which dates back to the Seventies, interested in finding the exact strength, measured in terms of set-existence axioms, of theorems from ordinary non set-theoretic mathematics. After a brief introduction to the subject, an on-line (incremental) algorithm to transitively reorient infinite pseudo-transitive oriented graphs is defined. This implies that a theorem of Ghouila-Houri is provable in RCA_0 and hence (...) is computably true. Interval graphs and interval orders are the common theme of the second part of the thesis. A chapter is devoted to analyse the relative strength of different characterisations of countable interval graphs and to study the interplay between countable interval graphs and countable interval orders. In this context the theme of unique orderability of interval graphs arises, which is studied in the following chapter. The last chapter about interval orders inspects the strength of some statements involving the dimension of countable interval orders. The third part is devoted to the analysis of two theorems proved by Rival and Sands in 1980. The first principle states that each infinite graph contains an infinite subgraph such that each vertex of the graph is adjacent either to none, or to one or to infinitely many vertices of the subgraph. This statement, restricted to countable graphs, is proved to be equivalent to ACA_0 and hence to be stronger than Ramsey's theorem for pairs, despite the similarity of partial order with finite width contains an infinite chain such that each point of the poset is comparable either to none or to infinitely many points of the chain. For each k less than or equal to 3, the latter principle restricted to countable poset of width k is proved to be equivalent to ADS. Some complementary results are presented in the thesis. (shrink)

I present a hypothetical account of how the ancients might have come to introduce mathematical objects in order to describe patterns, and I explain how working with patterns can generate information about the mathematical realm. The ancients might have started using what I call templates, i.e. concrete devices, like blueprints or drawings, to represent how things are shaped or structured, and this could have evolved into representing the abstract patterns that concrete things might fit. In this way, they (...) might have come to believe that written constructions and computations could provide information about the mathematical realm, for by their very nature, patterns should be structurally analogous to their templates and in positing that they are, one simply projects onto structures and features already attributed to templates. By reflecting on systems of dots representing cardinalities, the ancients could generate a body of results that then evolved into a systematic theory of numbers, but the approach fails when there is no direct connection between the computations and the patterns they are supposed to concern, e.g. those concerning trigonometric functions or transfinite ordinal numbers. In these cases, we forge a connection between proofs and patterns by positing that the premises of the proofs state uncontroversial features of the patterns, i.e. the premises constitute implicit definitions of the patterns. I explain how this position does not entail that mathematics is analytic or a priori. (shrink)

The object lesson concerns the passage from the foundational aims for which various branches of modern logic were originally developed to the discovery of areas and problems for which logical methods are effective tools. The main point stressed here is that this passage did not consist of successive refinements, a gradual evolution by adaptation as it were, but required radical changes of direction, to be compared to evolution by migration. These conflicts are illustrated by reference to set theory, (...) model theory, recursion theory, and proof theory. At the end there is a brief autobiographical note, including the touchy point to what extent the original aims of logical foundations are adequate for the broad question of the heroic tradition in the philosophy of mathematics concerned with the nature of the latter or, in modern jargon, with the architecture of mathematics and our intuitive resonances to it. (shrink)

During the first months of 1887, while completing the drafts of his Mitteilungen zur Lehre vom Transfiniten, Georg Cantor maintained a continuous correspondence with Benno Kerry. Their exchange essentially concerned two main topics in the philosophy of mathematics, namely, (a) the concept of natural number and (b) the infinitesimals. Cantor's and Kerry's positions turned out to be irreconcilable, mostly because of Kerry's irremediably psychologistic outlook, according to Cantor at least. In this study, I will examine and (...) reconstruct the main points in the discussion around (a) and (b) and stress some interesting aspects of the philosophical and mathematical thought of Benno Kerry. (shrink)

Humans are always interested in distinguishing natural and artificial entities although there is no sharp demarcation between the two categories. Surprisingly, things do not improve when the second type of entities is restricted to the arguably more constrained realm of physical technical artifacts. This paper helps to clarify the relationship between natural entities and technical artifacts by developing a conceptual landscape within which to analyze these notions. The framework is developed by studying three definitions of (...) class='Hi'>technical artifact which arise from different perspectives. All these perspectives share two intuitions: that technical artifacts are physical objects that exist by human intervention; and that technical artifacts are entities to be contrasted to natural entities. Yet the perspectives are different in the way they spell out these intuitions: the relevant human intervention may range from intentional selection to intentional production; and the contrast between technical artifacts and natural entities may be introduced by a constitution relation or by defining properties that set technical artifacts apart. The three perspectives are compared and their similarities and dissimilarities are explored with the help of ontological analysis. (shrink)

Paolo Mancosu provides an original investigation of historical and systematic aspects of the notions of abstraction and infinity and their interaction. A familiar way of introducing concepts in mathematics rests on so-called definitions by abstraction. An example of this is Hume's Principle, which introduces the concept of number by stating that two concepts have the same number if and only if the objects falling under each one of them can be put in one-one correspondence. This principle is at the (...) core of neo-logicism. In the first two chapters of the book, Mancosu provides a historical analysis of the mathematical uses and foundational discussion of definitions by abstraction up to Frege, Peano, and Russell. Chapter one shows that abstraction principles were quite widespread in the mathematical practice that preceded Frege's discussion of them and the second chapter provides the first contextual analysis of Frege's discussion of abstraction principles in section 64 of the Grundlagen. In the second part of the book, Mancosu discusses a novel approach to measuring the size of infinite sets known as the theory of numerosities and shows how this new development leads to deep mathematical, historical, and philosophical problems. The final chapter of the book explore how this theory of numerosities can be exploited to provide surprisingly novel perspectives on neo-logicism. (shrink)

This paper begins an axiomatic development of naive set theoryin a paraconsistent logic. Results divide into two sorts. There is classical recapture, where the main theorems of ordinal and Peano arithmetic are proved, showing that naive set theory can provide a foundation for standard mathematics. Then there are major extensions, including proofs of the famous paradoxes and the axiom of choice (in the form of the well-ordering principle). At the end I indicate how later developments of cardinal numbers will (...) lead to Cantor’s theorem, the existence of large cardinals, and a counterexample to the continuum hypothesis. (shrink)

This paper is concerned with the influence that the set theory of Georg Cantor bore upon the mathematical logic of Bertrand Russell. In some respects the influence is positive, and stems directly from Cantor's writings or through intermediary figures such as Peano; but in various ways negative influence is evident, for Russell adopted alternative views about the form and foundations of set theory. After an opening biographical section, six sections compare and contrast their views on matters (...) of common interest; irrational numbers, infinitesimals, cardinal and ordinal numbers, the axiom of infinity, the paradoxes, and the axioms of choice. Two further sections compare the two men over more general questions: the role of logic and the philosophy of mathematics. In a final section I draw some conclusions. (shrink)

This article concerns the ongoing neo-logicist program in the philosophy of mathematics. The enterprise began life, in something close to its present form, with Crispin Wright’s seminal [1983]. It was bolstered when Bob Hale [1987] joined the fray on Wright’s behalf and it continues through many extensions, objections, and replies to objections . The overall plan is to develop branches of established mathematics using abstraction principles in the form: Formula where a and b are variables of a given type , (...) Σ is a higher-order operator denoting a function from items of the given type to objects in the range of the first-order variables, and E is an equivalence relation over items of the given type. In what follows, I will sometimes omit the initial quantifiers.Frege [1884, 1893] himself employed three abstraction principles. One of them, used for illustration, comes from geometry: "The direction of l1 is identical to the direction of l2 if and only if l1 is parallel to l2."Call this the direction principle. The second was dubbed N= in [Wright, 1983] and is now called Hume’s principle: Formula where F≈G is an abbreviation of the second-order statement that there is a one-to-one relation mapping the F’s onto the G’s. In words, states that the number of F is identical to the number of G if and only if F is equinumerous with G. Unlike the direction-principle, the relevant variables, F, G here are second-order. Georg Cantor …. (shrink)

Contemporary philosophers of mathematics are deadlocked between two alternative ontologies for numbers: Platonism and nominalism. According to contemporary mathematical Platonism, numbers are real abstract objects, i.e. particulars which are nonetheless “wholly nonphysical, nonmental, nonspatial, nontemporal, and noncausal.” While this view does justice to intuitions about numbers and mathematical semantics, it leaves unclear how we could ever learn anything by mathematical inquiry. Mathematical nominalism, by contrast, holds that numbers do not exist extra-mentally, (...) which raises difficulties about how mathematical statements could be true or false. Both theories, moreover, leave inexplicable how mathematics could have such a close relationship with natural science, since neither abstract nor mental objects can influence concrete physical objects. The Thomist understanding of quantity as an accident inhering in concrete substances breaks this deadlock by granting numbers a foundation in extra-mental reality which explains why numeric expressions are relevant in natural science and how we come to know the truth or falsity of mathematical statements. Further, this kind of moderate realism captures the semantic advantages of the Platonists and the ontological parsimony of the nominalists (since for Thomists quantity is not a separate, free-standing ontological addition). Despite these advantages, the Thomistic understanding of number has been out of favor since the work of Cantor, who argued that actual infinities (regarded by Aristotelians as impossible) are required for modern developments in mathematics. This paper frees philosophers of mathematics to embrace the advantages of Thomistic realism by showing that Cantor’s understanding of actual and potential infinity is not the same as Aquinas’s, and that potential infinities (in Aquinas and Aristotle’s sense) can do all the work Cantor claims for actual infinities in modern mathematics. (shrink)

Structuralism, the view that mathematics is the science of structures, can be characterized as a philosophical response to a general structural turn in modern mathematics. Structuralists aim to understand the ontological, epistemological, and semantical implications of this structural approach in mathematics. Theories of structuralism began to develop following the publication of Paul Benacerraf’s paper ‘What numbers could not be’ in 1965. These theories include non-eliminative approaches, formulated in a background ontology of sui generis structures, such as Stewart Shapiro’s ante (...) rem structuralism and Michael Resnik’s pattern structuralism. In contrast, there are also eliminativist accounts of structuralism, such as Geoffrey Hellman’s modal structuralism, which avoids sui generis structures. These research projects have guided a more systematic focus on philosophical topics related to mathematical structuralism, including the identity criteria for objects in structures, dependence relations between objects and structures, and also, more recently, structural abstraction principles. Parallel to these developments are approaches that describe mathematical structure in category-theoretic terms. Category-theoretic approaches have been further developed using tools from homotopy type theory. Here we find a strong relationship between mathematical structuralism and the univalent foundations project, an approach to the foundations of mathematics based on higher category theory. (shrink)

I reply to two comments to my paper “Subjectivity and transcendental illusions in the Anthropocene,” by Johannes Schick and Melentie Pandilovski. Schick expands on the possibility that technical objects become “other” in a Levinasian sense, making use of Simondon’s three-layered structure of technical objects. His proposal is to free technical objects and install a different relationship between humankind and technology. I see two major difficulties in Schick's proposal. These difficulties are based on a number of features of (...) current digital technology which make it difficult to enter the proposed ethical relationship with it. A first cluster of difficulties consists of the phenomena of blackboxing, the intimate interwovenness of inventing technologies and profit on all levels of the technical object, and the ownership of and control over technologies. A second cluster revolves around the impossibility of a symmetrical relationship with the hyperobject because of current technology’s hyperobject-like nature. Next I discuss Pandilovski’s comments, where I point out that phenomenology is more encompassing than the study of having conscious experiences, and that phenomenology is essentially a method, rather than a collection of results. (shrink)