This chapter contains sections titled: Abstract Introduction Mathematical Relativism: Does Everything Go In Mathematics? Conceptual, Structural and Logical Relativity in Mathematics Mathematical Relativism and Mathematical Objectivity Mathematical Relativism and the Ontology of Mathematics: Platonism Mathematical Relativism and the Ontology of Mathematics: Nominalism Conclusion: The Significance of Mathematical Relativism References.

We set out a doctrine about truth for the statements of mathematics—a doctrine which we think is a worthy competitor to realist views in the philosophy of mathematics—and argue that this doctrine, which we shall call ‘mathematical relativism’, withstands objections better than do other non-realist accounts.

Recent no-go theorems for absolute facts in single-world interpretations are widely considered the strongest arguments in favour of ‘quantum relativism’: interpretations according to which measurement results are observer-relative. In this paper, however, I show that relativist interpretations are themselves vulnerable to mathematically identical ‘revenge’ theorems, unless they assume a particularly radical form. To this end, a novel distinction between tame and feral varieties of quantum relativism is introduced, where a relativist interpretation counts as tame if and (...) only if it postulates i) absolute facts about relative facts (absolute facts of second order) and ii) the possibility of communication across perspectives. While at first sight preferable to feral views, tame relativism turns out to be subject to revenge problems: arguments against absolute measurement outcomes can be modified and turned against absolute facts about which outcome a measurement has relative to a particular observer. This observation is developed in detail for the case of the ‘Local Friendliness’ EWF theorem. Moreover, it is shown to generalise to absolute facts about the facts that obtain relative to sequences of specified reference systems of arbitrary length. This suggests that only the most radical types of relativist interpretations have a hope of being motivated with recourse to EWF scenarios. (shrink)

In §69.b of BT Heidegger attempts an existential genetic analysis of science, i.e. a phenomenology of the conceptual process of the constitution of the logical view of science (science seen as theory) starting from the Dasein. It attempts to do so by examining the special intentional-existential modification of (human) being-in-the-world, which is called the "mathematical projection of nature"; that is, by examining that special modification of our being, which places us in the state of experience that (...) presents the world to us as taught by, e.g., Galileo and Newton. The idea he puts forward is that modern mathematical physics and the corresponding experience of the real is the result of the mathematical projection of nature. Modern science, Heidegger says, stands out because it is mathematical. But we cannot get the answer to what this mathematicality consists in from mathematics, because mathematics is only a special manifestation of mathematicality. Here is what Heidegger means by "mathematicality" and, accordingly by "mathematical projection of nature." If one were to let two bodies of different weights fall simultaneously from the top of the tower of Pisa, Heidegger notes, both Galileo and an Aristotelian would see that the two bodies would reach the ground with as little time difference as possible, and not at exactly the same moment. Galileo, however, does something decisive: he claims that under conditions that would be impossible to ensure (if we removed absolutely all resistance to the motion of the bodies), the two bodies would reach the ground absolutely simultaneously. This is not something that he or the Aristotelian sees happening empirically. How, then, does Galileo insist so firmly on his position? How exactly does it happen that he has before him such a view of things? Heidegger's answer is that Galileo is mathematically projecting nature. In his Dialogues Galileo says "I conceive with my mind" (mente concipio) a body thrown to move on an infinitely extending horizontal plane, having eliminated from its path every possible obstacle. Finally, I argue that from the detailed interpretation of Heidegger's points of the mathematical and of the notion of mathematical projection of nature, we can arrive at the conclusions I had arrived at in my doctoral dissertation (2000) as regards the meaning and function of thought experiments in physics as seen from a Husserlian phenomenological point of view as intentional acts. In a nutshell, in thought experiments physics constitutes the special metaphysics (to use the Kantian term) of the primordial ontological region it investigates as well as attempts to make natural sense of the mathematical formalism in cases it proves necessary. (shrink)

The determinism-free will debate is perhaps as old as philosophy itself and has been engaged in from a great variety of points of view including those of scientific, theological, and logical character. This chapter focuses on two arguments from logic. First, there is an argument in support of determinism that dates back to Aristotle, if not farther. It rests on acceptance of the Law of Excluded Middle, according to which every proposition is either true or false, (...) no matter whether the proposition is about the past, present or future. In particular, the argument goes, whatever one does or does not do in the future is determined in the present by the truth or falsity of the corresponding proposition. The second argument coming from logic is much more modern and appeals to Gödel's incompleteness theorems to make the case against determinism and in favour of free will, insofar as that applies to the mathematical potentialities of human beings. The claim more precisely is that as a consequence of the incompleteness theorems, those potentialities cannot be exactly circumscribed by the output of any computing machine even allowing unlimited time and space for its work. The chapter concludes with some new considerations that may be in favour of a partial mechanist account of the mathematical mind. (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)

Multiverse Views in set theory advocate the claim that there are many universes of sets, no-one of which is canonical, and have risen to prominence over the last few years. One motivating factor is that such positions are often argued to account very elegantly for technical practice. While there is much discussion of the technical aspects of these views, in this paper I analyse a radical form of Multiversism on largely philosophical grounds. Of particular importance will be an account (...) of reference on the Multiversist conception, and the relativism that it implies. I argue that analysis of this central issue in the Philosophy of Mathematics indicates that Radical Multiversism must be algebraic, and cannot be viewed as an attempt to provide an account of reference without a softening of the position. (shrink)

The aim of this paper is to show that John Wyclif’s theory of space is at once an interpretation of the Platonic theory of place and a Neopythagorean conception of magnitudes and numbers. The result is an original form of mathematical atomism in which atoms are point-like entities with a particular situation in space. If the core of this view comes from Boethius’ De arithmetica, John Wyclif is also influenced by Robert Grosseteste’s metaphysics, which includes the (...) Boethian number theory within the Christian tale of the creation of the world ex nihilo. John Wyclif, however, adds some novelty to this theory concerning the epistemological status of this hypothetical description of the creation of the world out of atoms. First, according to Wyclif, whereas geometry is concerned with sensible and imaginable beings, arithmetic, which is purely intellectual, has access to the deep mathematical structure of the universe. He then suggests a subordination of geometry under arithmetic, which he considers the most solid basis for his metaphysics. As a result, with the attribution of numbers and units to every level of reality, it becomes possible to reform our natural imagination, so that it can imagine the atomic structure of matter and space. (shrink)

Multiverse Views in set theory advocate the claim that there are many universes of sets, no-one of which is canonical, and have risen to prominence over the last few years. One motivating factor is that such positions are often argued to account very elegantly for technical practice. While there is much discussion of the technical aspects of these views, in this paper I analyse a radical form of Multiversism on largely philosophical grounds. Of particular importance will be an account (...) of reference on the Multiversist conception, and the relativism that it implies. I argue that analysis of this central issue in the Philosophy of Mathematics indicates that Radical Multiversism must be algebraic, and cannot be viewed as an attempt to provide an account of reference without a softening of the position. (shrink)

The paper explores the issue of moral relativism in Nietzsche, and tries to argue that Nietzsche's attitude towards moral values does not support a radical relativism according to which since (i) every moral interpretation is relative to a judging perspective, and (ii) an absolute viewpoint is lacking, then (iii) every moral interpretation seems to be as true, valid or justified as the others. On the contrary, Nietzsche's perspectivism leaves space for a rank order among (...) values, whose establishment is considered by Nietzsche as the future task of the philosopher. (shrink)

In broad terms, realism, relativism and pluralism can be regarded as the theoretical articulations of the following insights. Realism embodies the sense that what is at stake in our beliefs is something serious, i.e. that there is a fact of the matter, independent from our desire, which is going to decide whether what we believe in is true or not. Relativism, on the other hand, incorporates the realization that our cognitive take on the world is always perspectival, (...) that there is no way to overcome the blind spot which enables the knower to have a world in view at all. Pluralism, finally, draws on the intuition that every human being and every human community cannot fully understand, let alone save themselves, without the help of others’ sense-making efforts. Against the background of Charles Taylor’s philosophy, the core of truth of the above insights will be discussed and arranged to develop an active view of toleration that not only urges us to put up with others, but encourages us to rely on the benefit of coming to terms with different outlooks and ways of life. (shrink)

It was shown in Dirac A117, 610; A118, 351, 1928) that the ultra-violet divergence in quantum electrodynamics is caused by a violation of the time-energy uncertainly relationship, due to the implicit assumption of infinitesimal time information. In Wheeler et al. it was shown that Einstein’s special theory of relativity and Maxwell’s field theory have mathematically equivalent dual versions. The dual versions arise from an identity relating observer time to proper time as a contact transformation on configuration space, which leaves (...) phase space invariant. The special theory has a dual version in the sense that, for any set of n particles, every observer has two unique sets of global variables \\) and \\) to describe the dynamics, where \ is the canonical center of mass. In the \\) variables, time is relative and the speed of light is unique, while in the \\) variables, time is unique and the speed of light is relative with no upper bound. In the Maxwell case, the two sets of particle wave equations are not equivalent. The dual version contains an additional longitudinal radiation term that appears instantaneously with acceleration, leading to the prediction that radiation from a cyclotron will not produce photoelectrons. A major outcome is the dual unification of Newtonian mechanics and classical electrodynamics with Einstein’s special theory of relativity, without the need for point particles, without a self-energy divergency and, without need for the problematic Lorentz–Dirac equation. The purpose of this paper is to introduce the dual theory of relativistic quantum mechanics. In our approach, we obtain three distinct dual relativistic wave equations that reduce to the Schrödinger equation when minimal coupling is turned off. We show that the dual Dirac equation provides a new formula for the anomalous magnetic moment of a charged particle. We can obtain the exact value for the electron g-factor and phenomenological values for the muon and proton g-factors. (shrink)

There is a family of metaethical views according to which (i) there are no objectively correct moral standards and (ii) whether a given moral claim is true depends in some way on moral standards accepted by either an individual (forms of subjectivism) or a community (forms of relativism). This chapter outlines the three most important versions of this type of theories: old-fashioned subjectivism and relativism, contextualism and new wave subjectivism and relativism. It also explores the (...) main advantages of these views and the key objections to them. (shrink)

Adopting a dispositional theory of value promises to deliver a lot of theoretical goodies. One recurring problem for dispositional theories of value, though, is a problem about nonconvergence. If being a value is being disposed to elicit response R in us, what should we say if it turns out that not everybody is disposed to have R to the same things? One horn of the problem here is a danger of the view collapsing into an error theory—of it turning (...) out, on account of the diversity of agents' relevant dispositions, that nothing is really a value, since nothing is disposed to elicit R in everybody. Alternatively, there is a danger of an objectionable fragmentation of value, according to which there is no such thing as a value simpliciter, but only valuesme and valuesyou, valuesus and valuesthem. I advocate a de se relativist version of a dispositional theory of value. If we go for this sort of de-se-ified dispositional theory, we get to keep our theoretical goodies, but we avoid the problem of nonconvergence that leads to a danger of either collapse into an error theory, or else talking-past, and a loss of common subject matter. (shrink)

Radical relativism was born with a promise: to account for certain phenomena that opposite views are unable to explain. One example is the phenomenon of “faultless disagreement”, according to which two people, while disagreeing, are not at fault in any substantive way. The phenomena of retraction and assessments of truth in cases of eavesdropping are others. All these phenomena have been claimed to pose serious problems for rival views and be best accounted for within a radical relativistic (...) framework. While “faultless disagreement” and the notion of disagreement in general has benefited from extensive discussion in current debates over semantic content, retraction has not been in the spotlight that much. In particular, very few things have been said about what retraction exactly amounts to and how to conceive of its normative profile. This will be the focus of our paper. (shrink)

Radical relativism was born with a promise: to account for certain phenomena that opposite views are unable to explain. One example is the phenomenon of “faultless disagreement”, according to which two people, while disagreeing, are not at fault in any substantive way. The phenomena of retraction and assessments of truth in cases of eavesdropping are others. All these phenomena have been claimed to pose serious problems for rival views and be best accounted for within a radical relativistic (...) framework. While “faultless disagreement” and the notion of disagreement in general has benefited from extensive discussion in current debates over semantic content, retraction has not been in the spotlight that much. In particular, very few things have been said about what retraction exactly amounts to and how to conceive of its normative profile. This will be the focus of our paper. (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)

I think that there are good reasons to adopt a relativist semantics for epistemic modal claims such as ``the treasure might be under the palm tree'', according to which such utterances determine a truth value relative to something finer-grained than just a world (or a <world, time> pair). Anyone who is inclined to relativise truth to more than just worlds and times faces a problem about assertion. It's easy to be puzzled about just what purpose would be served (...) by assertions of this kind, and how to understand what we'd be up to in our use of sentences like ``the treasure might be under the palm tree'', if they have such peculiar truth conditions. After providing a very quick argument to motivate a relativist view of epistemic modals, I bring out and attempt to resolve this problem in making sense of the role of assertions with relativist truth conditions. Solving this problem should be helpful in two ways: first, it eliminates an apparently forceful objection to relativism, and second, spelling out the relativist account of assertion and communication will help to make clear just what the relativist position is, exactly, and why it's interesting. (shrink)

Severin Schroeder’s book Wittgenstein on Mathematics is reviewed and at the same time critically discussed by concentrating on its main aim: to show the coherence of Wittgenstein’s mature philosophy of mathematics. Although Schroeder is dealing with Wittgenstein’s philosophy of mathematics in its entirety, he is mainly interested in the mature philosophy which he sees as dominated by two central ideas: that mathematics is essentially algorithmic, called the calculus view, and that the results of mathematical proofs are grammatical (...) propositions, called the grammar view. According to Schroeder there is an initial tension between these two ideas which he then resolves in two steps: he first clarifies the specific sense of »grammatical« with respect to mathematical propositions and, secondly, he emphasizes the application of mathematical propositions. The reviewer’s criticism of Schroeder’s approach is mainly concerned with this second point. Discussing several specific cases, the reviewer shows that Wittgenstein is much less oriented towards applicability than is assumed by Schroeder. (shrink)

It is plausible to think that some animals perceive the world as coloreddifferently from the way humans perceive it. I argue that the best way ofaccommodating this fact is to adopt perceiver-relativism, the view that colorpredicates express relations between objects and types of perceivers.Perceiver-relativism makes no claim as to the identity of color properties;it is compatible with both physicalism and dispositionalism. I arguehowever for a response-dependence version of it according to which an object counts as (...) red (for a type of perceiver) iff it standardly looks red to normal perceivers (of that type). Finally, I develop a notion of minimal realism on which this account counts as realist despite its subjectivist elements, in that it is committed to the objectivityof truth. (shrink)

According to Ibn Khaldūn, man is a social entity deeply influenced by the geo-economics-politics of the environment in which he lives. The effect is seen as so strong that nearly all of these structures in their relationship to human beings are dominated by it. In this system, we see human beings as a creature who is both able to adapt himself to the environment and able to evolve in this harmony. From the perspective of Ibn Khaldūn, man cannot (...) be evaluated or defined separately from the environmental conditions, the technique and architecture he has developed. Ibn Khaldūn sees the primary source of this change to be the desire to make life both more prosperous and bearable. Therefore, the most natural motivations that affect man’s desire to alter the natural world are the environmental challenges that he encounters. The challenges that are imposed upon man in the natural world are those that have led to human progress. From badāwa (nomadic society) to ḥaḍāra (sedentary society) and so to the ceaseless drive of human progress that has served to make life more endurable, every effort in this sense leads to a physical and moral loosening as the incessant struggle to survive that defines human existence is made more bearable. Physical slack by breaking the resistance of man to resist the environmental conditions; and moral looseness by making man more vulnerable to the temptation of desires, pleasures and welfare that are the cause of moral dilemmas. Ibn Khaldūn’s moral thought with its social, political, and other fields of application are all contexts that enable an understanding of the devaluation of money-hedonist morality in all ways. This article aims to analyze the social, political and economic dilemmas and challenges that the capitalist, consumption-based money-hedonist morality (obesity, social media addiction, consumption, and moral devaluation in the face of financial, virtual and imagined valuations, etc.) from the perspective of Ibn Khaldūn. Summary: Man is endowed with both good and evil. Ibn Khaldūn notes that man tends to misuse these two orientations with a higher inclination towards evil, and that social conditions have an effect on this. He sees another characteristic of man, that they develop habits; accordingly, man is the child of his habits. Habits are second nature to man and affects all of his perceptions, particularly notions concerning truth. At the same time, human beings are under the constant influence of both habits, culture, and climate and geography; and all of these together affect our ethics, psychology and characteristics. Thus, man is a historical, social being that is changed, transformed and determined by adapting to these cultural, social, political and ecological conditions. The concept of Money-Hedonism is one that we have created by combining the terms monetarism, an economic concept, and hedonism, a concept in moral philosophy; the resulting term we seek to produce essentially translates as the adoration of Money. As is well known, hedonism refers to a philosophy of life that has its origins in the ancient Greek world which sees the purpose of life as one in which the pursuit of pleasure is the highest good. Whereas, Monetarism is an American-based economic view developed by Milton Friedman in 1970, linking all the balance in the economy to the supply and demand of money. According to this, the demand for money is operation, precaution and motives of speculation of individuals the amount of money they want to keep in cash on them. The power of an exchange tool increases with the amount and quality of what it is achievable in its utilization. In this respect, money has been an unprecedented source of power in modern society as a means by which change can be achieved. The increase in the power of money as an effective means as manifesting power has led to a socio-cultural and socio-economic superstructure around this power. In the life of an individual that is part of such a culture and is unable to isolate himself from the economy in which he lives, the values that can be expressed with money are consequently formed. In this structure, since the measure of life is money, money determines the quality of human life and human pleasure to a far greater degree than at any time in history. Thus, those living in this age have developed a new moral philosophy of value formed by the combination of these two concepts (monetarism and hedonism), which we have defined as Money-Hedonism. The money-hedonist individual is one that is in love with money, the joy of having it, the pleasure of using it, who desires it with immeasurable passion and ambition and is willing to use all legitimate and illegitimate methods of obtaining it. The most prominent characteristics of this type of individual is egoism, a desire to possess, hatred and mistrust. Ibn Khaldūn argues that the inhabitants of a city that live a settled life and who are living in the city have become economically prosperous and absorbed by consumption and pleasure, and, consequently morally degenerate as a result of the abuse of their evil nature. As can be understood from Ibn Khaldūn’s explanations, the basic element that determines the transition of people from the badāwi way of life to ḥaḍāri (settled) is the change in the livelihoods and related needs. He perceives this change as progressing from necessity to luxury, and, with this change, ethical concern in the nature of people being replaced by aesthetic care. In this way, since aesthetic concern reveals the dissatisfaction of the human self, man always pursues what is perceived to be better and more beautiful, and therefore he easily tends to any malignancy that may disrupt his religious and worldly peace. The ḥaḍāra community of Ibn Khaldūn versus the badāwa represent a modern society with a much more heterogeneous population, with a significant increase in lifestyle diversity along with urbanization. However, the determinative effect of the relationship with nature or the way in which society engage in economic activities have not changed. While modern people are urbanized, they are also heterogeneously shaped by psycho-social necessities demanded by urbanization. Man has created a new socio-psychological condition that not only affects the socio-economic but the way of forming a relationship through the accelerated development of new communication methods, facilitating access to information, and the opening of new methods of communication field between institutions. While man’s connection to nature has become disconnected, he no longer perceives himself as human being in nature but as a human within the city, the individual urban life and the law define this perception. The rupture from nature has created an inevitable result of alienation to nature as well as the estrangement of man with his own nature. The liberal policies adopted with capitalism have encouraged competition and free entrepreneurship. In the time that Ibn Khaldūn lived, man who created himself with his struggle in the nature brings himself / herself into the economic structure of competition in the modern period. Unlike in more traditional societies of the past, the modern way of life not only expresses itself in our new way of life and moral values, but also by statistical values. The exploitative and mathematical world, which forms the current economic science, shapes the socio-economic and socio-psychological structure of modern man. It is understood that the life brought by luxury renders the dynamic, subjective, disciplined sense of morality dysfunctional by destroying the moral values of man, particularly by reducing their feeling of shame. As a result, while the world of value of money-hedonism is solid, materialistic and cold; man in trying to exist in this world of value carries a flexible, spiritual and emotional nature. As such, human beings are pushed into social anomaly situations such as a constant conflict, dilemma, suicide, insecurity and solitude in this world of value. (shrink)

In this paper, I explore a range of existent and possible ameliorative semantic theories of gender terms: invariantism, according to which gender terms are not context-sensitive, contextualism, according to which the meaning of gender terms is established in the context of use, and relativism, according to which the meaning of gender terms is established in the context of assessment. I show that none of these views is adequate with respect to the plight of (...) trans people to use their term of choice to self-identify and be referred to accordingly. I then consider an invariantist view based on self-identification and explore some of its challenges. (shrink)

It has sometimes been argued that Ludwig Wittgenstein’s philosophy is marked by relativism. According to this view people view general concepts in relative terms due to different considerations and perceptions. More specifically, there is a tendency among some philosophers to assign the Wittgensteinian concept of form of life to the idea of culture. The argument goes like this: as language games and forms of life develop within a particular culture and cultures are different they are no (...) absolute phenomena but rather relative depending on the frame within which they developed. I would like to take up this thought and examine whether we can find any relativistic position in Wittgenstein’s writings. The question is whether the existence of a multitude of forms of life points to any sort of relativism. Many philosophers have speculated over the significant concept of “form of life”. There has been an ongoing debate over the question whether the concept is to be understood in the singular or in the plural. The discussion here revolves around the question of whether there is only one prevailing form of life to which others can be subsumed. Or else, is there a variety of coexisting forms of life of equal value? (shrink)

The deviation of mathematical proof—proof in mathematical practice—from the ideal of formal proof—proof in formal logic—has led many philosophers of mathematics to reconsider the commonly accepted view according to which the notion of formal proof provides an accurate descriptive account of mathematical proof. This, in turn, has motivated a search for alternative accounts of mathematical proof purporting to be more faithful to the reality of mathematical practice. Yet, in order to develop and (...) evaluate such alternative accounts, it appears as a necessary prerequisite to first possess a clear picture of what the deviation of mathematical proof from formal proof consists in. The present work aims to contribute building such a picture by investigating the relation between the elementary steps of deduction constituting the two types of proofs—mathematical inference and logical inference. Many claims have been made in the literature regarding the relation between mathematical inference and logical inference, most of them stating that the former is lacking properties that are constitutive of the latter. Such differentiating claims are, however, usually put forward without a clear conception of the properties occurring in them, and are generally considered to be immediately justified by our direct acquaintance, or phenomenological experience, with the two types of inferences. The present study purports to advance our understanding of the relation between mathematical inference and logical inference by developing a detailed philosophical analysis of the differentiating claims, that is, an analysis of the meaning of the differentiating claims—through the properties that occur in them—as well as the reasons that support them. To this end, we provide at the outset a representative list of the different properties of logical inference that have occurred in the differentiating claims, and we notice that they all boil down to the three properties of formality, generality, and mechanicality. For each one of these properties, our analysis proceeds in two steps: we first provide precise conceptual characterizations of the different ways logical inference has been said to be formal, general, and mechanical, in the philosophical and logical literature on formal proof; we then examine why mathematical inference does not appear to be formal, general, and mechanical, for the different variations of these notions identified. Our study results in a precise conceptual apparatus for expressing and discussing the properties differentiating mathematical inference from logical inference, and provides a first inventory of the various reasons supporting the observations of those differences. The differentiating claims constitute thus a set of data that any philosophical account of mathematical inference and proof purporting to be more faithful to mathematical practice ought to be able to accommodate and explain. (shrink)

Charles Parsons examines the notion of object, with the aim to navigate between nominalism, denying that distinctively mathematical objects exist, and forms of Platonism that postulate a transcendent realm of such objects. He introduces the central mathematical notion of structure and defends a version of the structuralist view of mathematical objects, according to which their existence is relative to a structure and they have no more of a 'nature' than that confers on them. Parsons (...) also analyzes the concept of intuition and presents a conception of it distantly inspired by that of Kant, which describes a basic kind of access to abstract objects and an element of a first conception of the infinite. (shrink)

Approaches to the interpretation of physical theories provide accounts of how physical meaning accrues to the mathematical structure of a theory. According to many standard approaches to interpretation, meaning relations are captured by maps from the mathematical structure of the theory to statements expressing its empirical content. In this article I argue that while such accounts adequately address meaning relations when exact models are available or perturbation theory converges, they do not fare as well for models that (...) give rise to divergent perturbative expansions. Since truncations of divergent perturbative expansions often play a critical role in establishing the empirical adequacy of a theory, this is a serious deficiency. I show how to augment state-space semantics, a view developed by Beth and van Fraassen, to capture perturbatively evaluated observables even in cases where perturbation theory is divergent. This new semantics establishes a sense in which the calculations that underwrite the empirical adequacy of a theory are both meaningful and true, but requires departure from the assumption that physical meaning is captured entirely by the exact models of a theory. (shrink)

When the traditional distinction between a mathematical concept and a mathematical intuition is tested against examples taken from the real history of mathematics one can observe the following interesting phenomena. First, there are multiple examples where concepts and intuitions do not well fit together; some of these examples can be described as “poorly conceptualised intuitions” while some others can be described as “poorly intuited concepts”. Second, the historical development of mathematics involves two kinds of corresponding processes: poorly conceptualised (...) intuitions are further conceptualised while poorly intuited concepts are further intuited. In this paper I study this latter process in mathematics during the twentieth century and, more specifically, show the role of set theory and category theory in this process. I use this material for defending the following claims: (1) mathematical intuitions are subject to historical development just like mathematical concepts; (2) mathematical intuitions continue to play their traditional role in today's mathematics and will plausibly do so in the foreseeable future. This second claim implies that the popular view, according to which modern mathematical concepts, unlike their more traditional predecessors, cannot be directly intuited, is not justified. (shrink)

One of the most intuitive views about the metaphysics of laws of nature is Tim Maudlin's idea of a Fundamental Law of Temporal Evolution. So-called FLOTEs are primitive elements of the universe that produce later states from earlier states. While FLOTEs are at home in traditional Newtonian and non-relativistic quantum mechanical theories (not to mention our pre-theoretic conception of the world), I consider here whether they can be made to work with relativity. In particular, shifting to relativistic spacetimes poses two (...) threats to FLOTEs. First, the lack of a privileged spacelike hypersurface in relativistic spacetime makes it unclear how to understand what produces what. A survey of several conceptions of the nomic production relation compatible with relativity reveals all of them to be lacking. Second, relativity motivates a four-dimensional block universe conception of time, according to which all events that will ever occur already exist. On such a view, it's unclear what work there is to be done by FLOTEs. I consider how a proponent of FLOTEs might respond, but conclude that these combined threats seriously undermine the prospects of a productive conception of laws. In short, if spacetime is relativistic, laws are not productive. (shrink)

An ab-initio foundation for relativistic spacetime is given, which is a conservative extension of Zermelo’s set theory with urelemente. Primitive entities are worldlines rather than spacetime points. Spacetime points are sets of intersecting worldlines. By the proper axioms, they form a manifold. Entities known in differential geometry, up to a metric, are defined and have the usual properties. A set-realistic point of view is adopted. The intended ontology is a set-theoretical hierarchy with a broad base of the empty (...) set and urelemente. Sets generated from the empty set are mathematically interpreted, all other sets are physically interpreted. (shrink)

Almost no systematic theorizing is generality-free. Scientists test general hypotheses; set theorists prove theorems about every set; metaphysicians espouse theses about all things of any kind. But do we ever succeed in theorizing about absolutely everything? Not according to generality relativism, which J.P. Studd defends in this book.

Leaving aside here the question of the author of the Essai de logique, I show that, if Mariotte insisted on the specificity of physics, he also sought a certain inspiration in mathematics as to the way in which to lay out the propositions in a proof. To do so, I start off from the ontological distinction made in the Essai among three types of possibles; next we will show that the three types of propositions correspond to three types of (...) knowledge, and, correlatively, that the main problem of physics is that it is impossible to establish in a certain fashion universal sensory propositions; I next characterize the notion of principles of experience; finally, I examine a bit systematically the relationship between mathematics and physics. (shrink)

Epistemic possibilities are relative to bodies of information, or perspectives. To claim that something is epistemically possible is typically to claim that it is possible relative one’s own current perspective. We generally do this by using bare, unqualified epistemic possibility (EP) sentences, ones that don’t mention our perspective. The fact that epistemic possibilities are relative to perspectives suggests that these bare EP sentences fall short of fully expressing propositions, contrary to what both contextualists and relativists take for granted. Although they (...) rightly reject propositional invariantism, the implausible view that a bare EP sentence expresses the same classical (absolutely true or absolutely false) proposition in any context, they maintain that a change in perspective shifts either the sentence’s propositional content (to a proposition involving a different perspective) or its truth-value (the same perspectivally neutral proposition now evaluated from a different perspective). I deny that the semantic contents of bare EP sentences shift at all. But I also deny that these contents have truth-values. Rather, according to the radical invariantism I defend, these contents are not full-fledged propositions but merely propositional radicals. Only explicitly relativized EP sentences manage fully to express propositions, and these perspective-involving propositions are the only EP propositions there are. Nevertheless, bare EP sentences are perfectly capable of being used to assert EP propositions, because utterances of them implicitly allude to the relevant perspective. Various problem cases challenge radical invariantism to explain pragmatically which perspective is read into the utterance of a given bare EP sentence. Unlike contextualism and relativism, it can do this without having to resort to any semantic bells and whistles.. (shrink)

Elkana's paper “Two-Tier Thinking” contains the thesis that became the foundation of all his later work. This thesis is best summarized by the author himself:The thesis of this paper is that this distinction [between realists and relativists] is not a logical necessity but a historical situation in Western scientific culture. It is claimed here that the distinction is spurious: every problem has a realist and a relativist dimension, and the two views can be, and are actually being, held simultaneously. (...) Once a frame of reference has been selected, in it realism prevails. With respect to selection of an appropriate framework the approach has to be relativist since there is no absolute, external-to-all framework which would fit absolute realism. (shrink)

According to the math tea argument, there must be real numbers that we cannot describe or define, because there are uncountably many real numbers, but only countably many definitions. And yet, the existence of pointwise-definable models of set theory, in which every individual is definable without parameters, challenges this conclusion. In this article, I introduce a flexible new method for constructing pointwise-definable models of arithmetic and set theory, showing furthermore that every countable model of Zermelo-Fraenkel ZF (...) set theory and of Peano arithmetic PA has a pointwise-definable end extension. In the arithmetic case, I use the universal algorithm and its Σ n generalizations to build a progressively elementary tower making any desired individual a n definable at each stage n, while preserving these definitions through to the limit model, which can thus be arranged to be pointwise definable. A similar method works in set theory, and one can moreover achieve V = L in the extension or indeed any other suitable theory holding in an inner model of the original model, thereby fulfilling the resurrection phenomenon. For example, every countable model of ZF with an inner model with a measurable cardinal has an end extension to a pointwise-definable model of ZFC + V = L[μ]. (shrink)

In this paper, I argue that religious belief is epistemically equivalent to mathematical belief. Abstract beliefs don't fall under ‘naive’, evidence-based analyses of rationality. Rather, their epistemic permissibility depends, I suggest, on four criteria: predictability, applicability, consistency, and immediate acceptability of the fundamental axioms. The paper examines to what extent mathematics meets these criteria, juxtaposing the results with the case of religion. My argument is directed against a widespread view according to which belief in mathematics is (...) clearly rationally acceptable whereas belief in religion is not. The paper also aims to make some of the implications of contemporary mathematics available to philosophers working in different fields. (shrink)

How should one conceive of the method of mathematics, if one takes a naturalist stance? Mathematical knowledge is regarded as the paradigm of certain knowledge, since mathematics is based on the axiomatic method. Natural science is deeply mathematized, and science is crucial for any naturalist perspective. But mathematics seems to provide a counterexample both to methodological and ontological naturalism. To face this problem, some naturalists try to naturalize mathematics relying on Darwinism. But several difficulties arise when one tries to (...) naturalize in this way the traditional view of mathematics, according to which mathematical knowledge is certain and the method of mathematics is the axiomatic method. This paper suggests that, in order to naturalize mathematics through Darwinism, it is better to take the method of mathematics not to be the axiomatic method. (shrink)

This chapter contains sections titled: Abstract Introduction Classical Invariantism and the Ho ‐ Hum View Relativism and Contexualism: Clarifications and Distinctions Relativism and Contexualism: A Quick Look at Some Sample Views Flexibility and Disagreement, Charity and Error: A Common Motivating Idea, and a Common Objection Conclusion References.

A Mathematical Review by John Corcoran, SUNY/Buffalo -/- Macbeth, Danielle Diagrammatic reasoning in Frege's Begriffsschrift. Synthese 186 (2012), no. 1, 289–314. ABSTRACT This review begins with two quotations from the paper: its abstract and the first paragraph of the conclusion. The point of the quotations is to make clear by the “give-them-enough-rope” strategy how murky, incompetent, and badly written the paper is. I know I am asking a lot, but I have to ask you to read the quoted passages—aloud (...) if possible. Don’t miss the silly attempt to recycle Kant’s quip “Concepts without intuitions are empty; intuitions without concepts are blind”. What the paper was aiming at includes the absurdity: “Proofs without definitions are empty; definitions without proofs are, if not blind, then dumb.” But the author even bollixed this. The editor didn’t even notice. The copy-editor missed it. And the author’s proof-reading did not catch it. In order not to torment you I will quote the sentence as it appears: “In a slogan: proofs without definitions are empty, merely the aimless manipulation of signs according to rules; and definitions without proofs are, if no blind, then dumb.”[sic] The rest of my review discusses the paper’s astounding misattribution to contemporary logicians of the information-theoretic approach. This approach was cruelly trashed by Quine in his 1970 Philosophy of Logic, and thereafter ignored by every text I know of. The paper under review attributes generally to modern philosophers and logicians views that were never espoused by any of the prominent logicians—such as Hilbert, Gödel, Tarski, Church, and Quine—apparently in an attempt to distance them from Frege: the focus of the article. On page 310 we find the following paragraph. “In our logics it is assumed that inference potential is given by truth-conditions. Hence, we think, deduction can be nothing more than a matter of making explicit information that is already contained in one’s premises. If the deduction is valid then the information contained in the conclusion must be contained already in the premises; if that information is not contained already in the premises […], then the argument cannot be valid.” Although the paper is meticulous in citing supporting literature for less questionable points, no references are given for this. In fact, the view that deduction is the making explicit of information that is only implicit in premises has not been espoused by any standard symbolic logic books. It has only recently been articulated by a small number of philosophical logicians from a younger generation, for example, in the prize-winning essay by J. Sagüillo, Methodological practice and complementary concepts of logical consequence: Tarski’s model-theoretic consequence and Corcoran’s information-theoretic consequence, History and Philosophy of Logic, 30 (2009), pp. 21–48. The paper omits definitions of key terms including ‘ampliative’, ‘explicatory’, ‘inference potential’, ‘truth-condition’, and ‘information’. The definition of prime number on page 292 is as follows: “To say that a number is prime is to say that it is not divisible without remainder by another number”. This would make one be the only prime number. The paper being reviewed had the benefit of two anonymous referees who contributed “very helpful comments on an earlier draft”. Could these anonymous referees have read the paper? -/- J. Corcoran, U of Buffalo, SUNY -/- PS By the way, if anyone has a paper that has been turned down by other journals, any journal that would publish something like this might be worth trying. (shrink)

I defend a unified fictionalism about modality and mathematics. First, I defend each view separately against internal objections. Then, I attempt a unified fictionalism by giving an analysis of truth in fiction which is neither modal nor platonistic. Finally, I explore the prospects for nominalistic unified fictionalism. ;In the first chapter, I defend modal fictionalism: the view that statements about possible worlds are best understood as claims about the content of a fiction, the 'many-worlds story'. I address (...) the Brock-Rosen objection that the fictionalist is a modal realist despite himself. I argue that Noonan's solution, the best response so far, is inadequate, and then offer an alternative fictionalist translation scheme for modal statements. ;In the second chapter, I defend mathematical fictionalism. Mathematical fictionalism is the view that mathematics is best understood as a fiction whose value is quite independent of its truth-value. Field argues for fictionalism in this sense by constructing nominalistic alternatives to standard physical theories. The aim is to establish that mathematical entities are 'dispensable for the purposes of science', and so to undermine the only reason we might have for believing in the literal truth of standard mathematics. I address an objection that the nominalization of current science would not suffice to establish the 'dispensability' of mathematical entities in the epistemologically relevant sense. I argue that there is a plausible interpretation of the indispensability argument according to which successful nominalization would establish that it is unreasonable to believe in mathematical entities. ;The most important challenge for any attempt at a unified fictionalism is to provide an account of truth in fiction that is neither implicitly modal nor implicitly platonistic. To this end, I offer in the third chapter an analysis of truth in a story which is based on a modified version of Walton's account of truth in fiction. ;A second challenge is to show that the timelessness and necessity of modal truth are compatible with the view that fictions are contingent and temporally restricted entities. In the final chapter, I argue this problem constitutes a genuine obstacle for a unified nominalistic fictionalism. (shrink)

In this paper, I assume, perhaps controversially, that translation into a language of formal logic is not the method by which mathematicians assess mathematical reasoning. Instead, I argue that the actual practice of analyzing, evaluating and critiquing mathematical reasoning resembles, and perhaps equates with, the practice of informal logic or argumentation theory. It doesn’t matter whether the reasoning is a full-fledged mathematical proof or merely some non-deductive mathematical justification: in either case, the methodology of assessment (...) overlaps to a large extent with argument assessment in non-mathematical contexts. I demonstrate this claim by considering the assessment of axiomatic or deductive proofs, probabilistic evidence, computer-aided proofs, and the acceptance of axioms. I also consider Jody Azzouni’s ‘derivation indicator’ view of proofs because it places derivations—which may be thought to invoke formal logic—at the center of mathematical justificatory practice. However, when the notion of ‘derivation’ at work in Azzouni’s view is clarified, it is seen to accord with, rather than to count against, the informal logical view I support. Finally, I pose several open questions for the development of a theory of mathematical argument. (shrink)

In the past century the received view of definition in mathematics has been the stipulative conception, according to which a definition merely stipulates the meaning of a term in other terms which are supposed to be already well known. The stipulative conception has been so absolutely dominant and accepted as unproblematic that the nature of definition has not been much discussed, yet it is inadequate. This paper examines its shortcomings and proposes an alternative, the heuristic conception.

In the paper I describe John MacFarlane’s version of relativism about truth. I begin by discussing Twardowski’s (1900) and Kokoszynska’s (1948; 1951) arguments against relativism. They think — just as Haack does (see 2011) — that sentences may be relatively true, if they are incomplete, but once they are completed they become true (or false) absolutely. MacFarlane distinguishes between nonindexical contextualism (which was anticipated by Kokoszynska (sic!)) and relativism which requires the introduction of the context (...) of assessment. According to him only the view which postulates double-indexed (to contexts of utterance and to contexts of assessments) truth is able to explain disagreement in subjective domains and contradicting intuitions about the truth-value of future-contingents. (shrink)

Over the past 50 years, Nelson algebras have been extensively studied by distinguished scholars as the algebraic counterpart of Nelson's constructive logic with strong negation. Despite these studies, a comprehensive survey of the topic is currently lacking, and the theory of Nelson algebras remains largely unknown to most logicians. This paper aims to fill this gap by focussing on the essential developments in the field over the past two decades. Additionally, we explore generalisations of Nelson algebras, such as N4-lattices (...) class='Hi'>which correspond to the paraconsistent version of Nelson's logic, as well as their applications to other areas of interest to logicians, such as duality and rough set theory. A general representation theorem states that each Nelson algebra is isomorphic to a subalgebra of a rough set-based Nelson algebra induced by a quasiorder. Furthermore, a formula is a theorem of Nelson logic if and only if it is valid in every finite Nelson algebra induced by a quasiorder. (shrink)

There is another sort of ‘defense’ of relativism that I mention in conclusion. Sometimes one finds the view that one is rightly punished for a crime only if they admit committing it, and that it was a crime — something wrongly done: ‘punishment conditional on confession’ is the rule proposed. It might seem that this would give impunity to a criminal hardy enough to deny the fact, or the evil, of her deed; so it would, unless it was (...) also understood that if a person was proved by accepted standards to be the perpetrator of an action agreed to be wrong, the opportunity to admit guilt would extend from open questioning through beating to exquisite torture until confession was extracted (cf. the Chinese customs reflected in the Judge Dee stories of Robert Van Gulik). More often, one finds the view that a person is not refuted unless he admits this, and a view is not refuted — and so is not properly dismissed — unless its proponents concede defeat. A similar view is that all really telling philosophical criticism must be internal or else non-neutral (a view we discussed above).If one can in principle produce a conceptual system from a list of all propositions by following the recipe: for any proposition P include P, or else include not-P, in S, then (providing one adds no constraints concerning consistency) there will be no proposition that will appear in all conceptual systems. If what it is for a proposition to be neutral is for it to appear in all conceptual systems, then no proposition is neutral. But of course, the same recipe that produces this result produces the result that the proposition (P) Some proposition is both true and false appears in some conceptual system which (since the denial of (P) appears in less than all systems and hence is not neutral) is (allegedly) none the worse for that. And the proposition (P1) Conceptual relativism is false appears in some conceptual systems which (since the denial of (P1) appears in less than all systems and hence is not neutral) is none the worse for that. But then what remains of the claim that conceptual relativism is true?Sometimes refuted is used in a person-independent manner: Proposition P is (and always was) refuted by the fact that F uses P is refuted in such a way that it does not entail There is someone who refuted P. Sometimes, of course, it is so used that P is refuted does have that entailment. Even in the latter case, refutation is an epistemological notion, and Person S is refuted does not entail S knows that S is refuted and is compatible with S believes that S is not refuted. The recommendation that we add a psychological component of ‘admits to’ to the epistemological notion ‘being refuted’ seems to have no other ground than the arbitrary one that if we do we shall then be able better to defend relativism.While the recipe for constructing conceptual systems suggested above may seem useful to a relativist — it gives one clear content to ‘neutral proposition’ — it has what must surely seem a defect to the relativist, namely that given it (even with consistency constraints added) there will be no translational incommensurability. For if P is contained in system S, then not-P is expressible in S, and conversely. And the recipe yields the result that for any proposition P, and any system S, S contains either P or else not-P.This, of course, brings out an ambiguity in ‘contains’. In one sense, a system ‘contains (1)’ whatever can be expressed in it. In another, it ‘contains (2)’ whatever, according to it, is true. If it is a complete system without suspense of judgment it contains (1) twice the propositions it contains (2). Further, a system contains (1) lots of propositions it (or its proponent) does not regard as true, and so the fact that a proposition P appears in system S does not mean that P favors S. One often at least can state the conditions of S's falsehood in S, and even prove in S that S is false, where “in” is used in the sense of “contains (1).”Sophisticated relativism, then, seems no more plausible, no more free from inconsistency, and no less open to devastating critique than are its less cultured relatives. Relativism, at least in the varieties discussed here, seems to live up to its unsavory reputation. Since not even translation incommensurability seems to entail lack of cognitive competitiveness, it seems that it will have to be admitted that, regarding apparent disagreement on basic matters — including fundamental religious disagreement — the appearance is also the reality. People do uneliminably disagree, and short of conversion or (what will seem from within the religious perspective in question) some other ‘failure in faith’ (e.g., coming to reject all religious views) will continue to do so. Whatever can be done in terms of common understanding and common humanitarian endeavors must be done without denying the many basic disagreements that there are. And whatever notion of inter-faith dialogue is other than espitemically defective will have to face that fact; for significant dialogue based on self-deception or on falsehood seems not a terribly promising enterprise. (shrink)

Structuralism is a view about the subject matter of mathematics according to which what matters are structural relationships in abstraction from the intrinsic nature of the related objects. Mathematics is seen as the free exploration of structural possibilities, primarily through creative concept formation, postulation, and deduction. The items making up any particular system exemplifying the structure in question are of no importance; all that matters is that they satisfy certain general conditions—typically spelled out in axioms defining the (...) structure or structures of interest—characteristic of the branch of mathematics in question. Thus, in the basic case of arithmetic, the famous “axioms” of Richard Dedekind (taken over by Giuseppe Peano, as he acknowledged) were conditions in a definition of a “simply infinite system”, with an initial item, each item having a unique next one, no two with the same next one, and all items finitely many steps from the initial one. (The latter condition is guaranteed by the axiom of mathematical induction.) All such systems are structurally identical, and, in a sense to be made more precise, the shared structure is what mathematics investigates. (In other cases, multiple structures are allowed, as in abstract algebra with its many groups, rings, fields, and so forth.) This structuralist view of arithmetic thus contrasts with the absolutist view, associated with Gottlob Frege and Bertrand Russell, that natural numbers must in fact be certain definite objects, namely classes of equinumerous concepts or classes. (shrink)

In this article I ask how moral relativism applies to the analysis of responsibility for mass crime. The focus is on the critical reading of two influential relativist attempts to offer a theoretically consistent response to the challenges imposed by extreme criminal practices. First, I explore Gilbert Harman’s analytical effort to conceptualize the reach of moral discourse. According to Harman, mass crime creates a contextually specific relationship to which moral judgments do not apply any more. Second, I (...) analyze the inability thesis, which claims that the agents of mass crime are not able to distinguish between right and wrong. Richard Arneson, Michael Zimmerman and Geoffrey Scarre do not deny the moral wrongness of crime. However, having introduced the claim of authenticity as a specific feature of the inability thesis, they maintain that killers are not responsible. I argue that these positions do not hold. The relativist failure to properly conceptualize responsibility for mass crime follows from the mistaken view of moral autonomy, which then leads to the erroneous explanation of the establishment, authority and justification of moral judgments. (shrink)

This thesis attempts to argue against an influential interpretation of Kant's philosophy of mathematics according to which the role of pure intuition is primarily logical. Kant's appeal to pure intuition, and consequently his belief in the synthetic character of mathematics, is, on this view, a result of the limitations of the logical resources available in his time. In contrast to this, a reading is presented of the development of Kant's philosophy of mathematics which emphasises a much (...) richer philosophical role for intuition, beyond that of filling in deductive gaps in mathematical arguments. This role is largely determined by two factors: on the one hand, the epistemological gap in a traditional account of mathematical certainty, and on the other hand, the metaphysical worry raised by the followers of Leibniz and Wolff regarding the status of mathematics as a science consisting of truths. ;By appreciating how Kant's philosophy of mathematics developed against the background of what he saw as a conflict between the 'metaphysicians' and the 'mathematicians', we can see how the doctrine of pure intuition was required to fill the gaps in the account of the method and certainty of mathematics which Kant presents in the Prize Essay of 1764. This conflict takes its sharpest form between the monadists who uphold the metaphysical necessity of simple elements, and the geometers who uphold the infinite divisibility of space. In the Prize Essay Kant indirectly addresses this conflict by comparing the two disciplines. He asserts that the method of attaining certainty in mathematics is different from the method of metaphysics, and that as a result of this difference--primarily the different roles of definitions in each--mathematics is capable of a higher degree of certainty than metaphysics. He does not, however, explain this difference. Such an explanation seems particularly important in light of Kant's wish to secure geometry against the challenge of metaphysics. This thesis attempts to show that it is to provide such an explanation, to provide a philosophical grounding for the Prize Essay account, that Kant invokes the doctrine of pure intuition in his later account of mathematical knowledge. (shrink)

In lieu of an abstract, here is a brief excerpt of the content:Reviewed by:Husserl and Mathematics by Mirja HartimoAndrea StaitiMirja Hartimo. Husserl and Mathematics. Cambridge: Cambridge University Press, 2021. Pp. 214. Hardback, $99.99.Mirja Hartimo has written the first book-length study of Husserl's evolving views on mathematics that takes his intellectual context into full consideration. Most importantly, Hartimo's historically informed approach to the topic benefits from her extensive knowledge of Husserl's library. Throughout the book, she provides references to texts and articles (...) that Husserl read, marked, and annotated, although regrettably, very few of these sources are quoted explicitly in his published works. The overall picture that emerges from Hartimo's book shows that Husserl continued to read and engage with mathematics throughout his career, and, even though extensive discussions of mathematical issues are not frequent in his later works, some of the ideas he develops, for instance, in The Crisis of the European Sciences and Transcendental Phenomenology, can be read against the backdrop of the developments of mathematics in his time.Chapter 1 sets the method for the rest of the book. Hartimo emphasizes the importance of a methodological device that takes center stage in Husserl's work from the 1920s onward: Besinnung. Hartimo claims that Besinnung is a method to reactivate the "goals and purposes" (21) that originally motivated scientists engaging in a particular line of research. According to Husserl, as specialized research progresses and becomes standard practice, such original goals and purposes might be blurred by other concerns. When applied to mathematics, Besinnung thus amounts to a reflection "on what mathematicians should do, on what the genuine goal of their activities should be" (21–22). Hartimo argues that this kind of approach to mathematics comes close to Penelope Maddy's naturalism, which takes its departure from what mathematicians are actually doing and then proceeds to inquire into the ontological status of mathematical objects and theories. Hartimo identifies particularly in David Hilbert, a colleague of Husserl's at Göttingen, a vivid embodiment of what Husserl considered to be exemplary work oriented toward the ideal goal of mathematics.In chapters 2 and 3, Hartimo expands upon Husserl's conception of what the true goal of mathematics should be. She does so, first, by addressing the classical topic of Husserl's psychologism in his first (and only) work entirely devoted to mathematics, The Philosophy of Arithmetic, and Frege's well-known critical review of it. In Hartimo's rendition, Husserl criticizes Frege for his attempt to provide formal definitions of basic notions, such as equality. For Husserl, such basic notions are intuitively available and require no definition—rather, they should be put to work to clarify vague notions. Frege, in return, attacks what he takes to be Husserl's wishy-washy psychological account of numbers, which revolves around the concrete act of collecting items and abstracting from their specific contents. Ultimately, Hartimo argues, Husserl takes Frege's criticism to heart and in his later Prolegomena no longer pursues a psychological foundation of number, but rather emphasizes the distinction between the subjective and the objective sides of mathematics, with philosophers studying the former and mathematicians focusing on the latter. While Hartimo's verdict on the philosophical yield of the debate is in line with the standard interpretation of Frege's review as prompting Husserl's antipsychologistic turn, she makes a remark that might be worthy of further discussion: "While for Husserl the debate was about logicism, Frege shifts the debate to be about psychologism and here commentators have followed Frege's example" (50). It would be interesting to imagine what a Fregean reply to Husserl's view about the uselessness of formal definitions for basic notions like equality might look like. In any case, Hartimo's reconstruction makes it clear that, however motivated, Husserl's rejection of psychologism does not amount to an acceptance of logicism.In chapter 3, Hartimo puts forward two of her central claims: (1) Husserl was a structuralist about mathematics; and (2) for Husserl, the goal of modern mathematics is encapsulated in the concept of definiteness. According to (1), mathematics is about the formal structures of systems, which can be isolated and compared. Definiteness, in turn, is the property of systems... (shrink)

Though my title speaks of Kant’s mathematical realism, I want in this essay to explore Kant’s relation to a famous mathematical anti-realist. Specifically, I want to discuss Kant’s influence on L. E. J. Brouwer, the 20th-century Dutch mathematician who built a contemporary philosophy of mathematics on constructivist themes which were quite explicitly Kantian. Brouwer’s theory is perhaps most notable for its belief that constructivism requires us to abandon the traditional logic of mathematical reasoning in favor of (...) different canon of reasoning, called intuitionistic logic. Brouwer thought that classical logic is intrinsically bound up with a nonconstructive view of mathematics. This means that, according to Brouwer, when we do mathematics we must give up bivalence, we must no longer use such familiar logical laws as excluded middle, and we must sometimes forebear from the classic method of reductio ad absurdum. All of these are intuitionistically invalid classical principles. (shrink)

Relativists make room for the possibility of “faultless disagreement” by positing the existence of subjective propositions, i.e. propositions true from some points of view and not others. We discuss whether the adoption of this position with respect to a certain domain of discourse is compatible with a realist attitude towards the matters arising in that domain. At first glance, the combination of relativism and realism leads to an unattractive metaphysical picture on which reality comprises incoherent facts. We (...) will sketch the contours of a realist-relativist position called “subjectivism”, which avoids this result by giving up the assumption that the points of view of different subjects are all metaphysically “on a par”. (shrink)