The most enlightening examination to date of the developments of Frege's thinking about his logic, this book introduces a new kind of logical language, one that ...

Danielle Macbeth offers a new account of mathematical practice as a mode of inquiry into objective truth, and argues that understanding the nature of mathematical practice provides us with the resources to develop a radically new conception of ourselves and our capacity for knowledge of objective truth.

Throughout its long history, mathematics has involved the use ofsystems of written signs, most notably, diagrams in Euclidean geometry and formulae in the symbolic language of arithmetic and algebra in the mathematics of Descartes, Euler, and others. Such systems of signs, I argue, enable one to embody chains of mathematical reasoning. I then show that, properly understood, Frege’s Begriffsschrift or concept-script similarly enables one to write mathematical reasoning. Much as a demonstration in Euclid or in early modern algebra does, a (...) proof in Frege’s concept-script shows how it goes. (shrink)

In Part III of his 1879 logic Frege proves a theorem in the theory of sequences on the basis of four definitions. He claims in Grundlagen that this proof, despite being strictly deductive, constitutes a real extension of our knowledge, that it is ampliative rather than merely explicative. Frege furthermore connects this idea of ampliative deductive proof to what he thinks of as a fruitful definition, one that draws new lines. My aim is to show that we can make good (...) sense of these claims if we read Frege’s notation diagrammatically, in particular, if we take that notation to have been designed to enable one to exhibit the (inferentially articulated) contents of concepts in a way that allows one to reason deductively on the basis of those contents. (shrink)

Prouver des théorèmes est une pratique mathématique qui semble clairement améliorer notre compréhension mathématique. Ainsi, prouver et reprouver des théorèmes en mathématiques, vise à apporter une meilleure compréhension. Cependant, comme il est bien connu, les preuves mathématiques totalement formalisées sont habituellement inintelligibles et, à ce titre, ne contribuent pas à notre compréhension mathématique. Comment, alors, comprendre la relation entre prouver des théorèmes et améliorer notre compréhension mathématique. J'avance ici que nous avons d'abord besoin d'une notion différente de preuve (formelle), qui (...) ne tienne pas la forme pour opposée au contenu. La pratique de la preuve algébrique au xviiie siècle fournit un exemple de preuve à la fois pleinement rigoureuse et porteuse de contenu, et dans ce cas, il est possible de voir comment une preuve mathématique apporte une compréhension mathématique. Il s'agira alors de mobiliser les enseignements de cet exemple pour étudier le type de raisonnement déductif à partir de concepts, qui a constitué la norme dans la pratique mathématique depuis le xixe siècle. (shrink)

In the late 1980s, I wrote up some notes on the play of pick-up basketball and sent them to Harold Garfinkel, who incorporated them into an un-published monograph in 1988. They were motivated by an interest in exhibiting the sense of "detail" for ethnomethodological studies. An edited version is presented below. They follow a front piece of recollection and discussion about Garfinkel's distinctive interests in matters of "detail," their tie to structure and structure's circumstantiality, and their place in EM studies.

François Viète is often regarded as the first modern mathematician on the grounds that he was the first to develop the literal notation, that is, the use of two sorts of letters, one for the unknown and the other for the known parameters of a problem. The fact that he achieved neither a modern conception of quantity nor a modern understanding of curves, both of which are explicit in Descartes’ Geometry, is to be explained on this view “by an incomplete (...) symbolization rather than by any obstacle intrinsic in the system.” Descartes’ Geometry provides only a “clearer expression” of themes already sounded in Viète’s work, one that perfects Viète’s literal calculus and gives it “its modern form”; it merely continues the “‘new’ and ‘pure’ algebra which Viète first established as the ‘general analytic art’.” It can seem, furthermore, that this must be right, that had there been some obstacle intrinsic to Viète’s system that barred the way to a modern conception of quantity and a modern understanding of curves, then Descartes’ Geometry would have had to have taken a very different form than it did. As it was, Descartes had only to improve Viète’s symbolism, free himself of the last vestiges of the ancient view of geometrical and arithmetical objects, and apply the new symbolism to the study of curves in order to achieve what Viète did not but could have. But what, really, is the status of this “could have”; what would it actually have taken for Viète to achieve Descartes’ results in the Geometry? The interest of the question lies in its potential to better our understanding of the nature of modern mathematics. (shrink)

My starting point is two themes from Peirce: his familiar pragmatist conception of meaning focused on what follows from an application of a term rather than on what is the case if it is correctly applied, and his less familiar and rather startling claim that even purely deductive, logical reasoning is not merely formal but instead constructive or diagrammatic — and hence experimental, and fallible. My aim is to show, using Frege’s two-dimensional logical language as a paradigm of a “constructive” (...) logic in Peirce’s sense, that taking this second theme into account in one’s interpretation of the first yield a very different, and arguably more fruitful, conception of meaning than is usually ascribed to Peirce, not only a different conception of the role of inference in meaning than is found in, say, Brandom following Sellars, but also a very different understanding of the role of pragmatics in semantics than is standard in social practice theories. (shrink)

Empirical knowledge is at once an exercise of freedom and rationally constrained by how things are. But if the reality on which empirical thought aims to bear is outside the sphere of the conceptual then, while it can exert a causal constraint on knowing, it cannot exert a rational constraint. Empirical reality both must and, so it seems, cannot have rational bearing on empirical thought. I consider the related ways Kant and Sellars try to avoid this antinomy, arguing that understanding (...) the relationship between their views reveals how unsatisfactory both are, and what might be said instead. (shrink)

The expression ‘Oh’ in natural conversation is a signal topic in the development of the Epistemic Program. This article attempts to bring into view a sense of place for this simple expression in the early literature, beginning with ‘Oh’ as a ‘change-of-state token’ and through its subsequent treatments in the production of assessments. It reviews them with an interest in two allied developments. One is the rendering of ‘Oh’ as an expression that ‘indexes’ epistemic structure. The other, pursued in the (...) detail of transcript in Part 2, is how, as of this rendering, the literature manages its tasks of ‘animating transcript’, or how we portray ordinary talk as social action. We think these two moves are closely connected within the EP. And we think they yield a very different ‘vocabulary of motives’, different from the natural language studies of conversation analysis. Our discussions address in turn the central phrases of our title. (shrink)

My aim is to understand the practice of mathematics in a way that sheds light on the fact that it is at once a priori and capable of extending our knowledge. The account that is sketched draws first on the idea, derived from Kant, that a calculation or demonstration can yield new knowledge in virtue of the fact that the system of signs it employs involves primitive parts that combine into wholes that are themselves parts of larger wholes. Because wholes (...) such as numerals and Euclidean figures both have parts and are parts of larger wholes, their parts can be recombined into new wholes in ways that enable extensions of our knowledge. I show that sentences of Frege 's Begriffsschrift can also be read as involving three such levels of articulation; because they have these three levels, we can understand in essentially the same way how a proof from concepts alone can extend our knowledge. (shrink)

Although profoundly influential for essentially the whole of philosophy’s twenty-five hundred year history, the model of a science that is outlined in Aristotle’s Posterior Analytics has recently been abandoned on grounds that developments in mathematics and logic over the last century or so have rendered it obsolete. Nor has anything emerged to take its place. As things stand we have not even the outlines of an adequate understanding of the rationality of mathematics as a scientific practice. It seems reasonable, in (...) light of this lacuna, to return again to Frege—who was at once one of the last great defenders of the model and a key figure in the very developments that have been taken to spell its demise—in hopes of finding a way forward. What we find when we do is that although Frege remains true to the spirit of the model, he also modifies it in very fundamental ways. So modified, I will suggest, the model continues to provide a viable and compelling image of scientific rationality by showing, in broad outline, how we achieve, and maintain, cognitive control in our mathematical investigations. (shrink)

In conversation analysis, through Sacks, Schegloff, Jefferson, and others, the conceptual architecture is joined at the hip to a technical architecture of transcripts, sequence, and turn productions. That the conceptual was to be found and demonstrated in the material detail of temporal productions was central to CA’s extraordinary innovations. As with CA, an Epistemic CA has the task of giving evidence of its conceptual order in actual materials, and thus animating the materials to show them. The task and relationship are (...) emblematically reflexive: we shall find the expression ‘Oh’ indexing ‘changes of state’ or ‘inapposite inquiries’, for example, as of the account-able animations of turns and sequences of turns. Our shared attachments to sequential analysis deliver the expectation that we shall see how Epistemic order is achieved on actual occasions, through actual materials, rendered as transcript. The discussion turns to how the Epistemic Program engages and acquits this analytic expectation. (shrink)

Over the past half-century, formal, machine-executable proofs have been developed for an impressive range of mathematical theorems. Formalists argue that such proofs should be seen as providing the fully worked out proofs of which mathematicians’ proofs are sketches. Nonformalists argue that this conception of the relationship of formal to informal proofs cannot explain the fact that formal proofs lack essential virtues enjoyed by mathematicians’ proofs, the fact, for example, that formal proofs are not convincing and lack the explanatory power of (...) their informal counterparts. Formal proofs do not yield mathematical insight or understanding, and they are not fruitful in the way that informal proofs can be, for instance, in suggesting novel approaches to old problems. And yet, there is a clear sense in which the formalist is right: Formal proofs do seem to make explicit what is otherwise taken for granted in the mathematician’s reasoning. Very recent work uncovers the source of the dilemma by showing that rigor does not entail formalism insofar as rigor, unlike formalism, is compatible with contentfulness. Mathematical proofs, were they to be recast in a Leibnizian language, at once a characteristica and a calculus, would be, or could be made to be, completely gap-free and hence machine checkable. Because as so recast they would be grounded in mathematical ideas, they would not be machine executable. It is on just this basis that a compelling account can be given of the fact that formal proofs are mostly irrelevant to mathematicians in their practice. (shrink)

In lieu of an abstract, here is a brief excerpt of the content:'Is* and Ought' in Context: Maclntyre's Mistake1 Murray MacBeth (1)What drives Hume to the conclusion that morality must be understood in terms of, explained and justified by reference to, the place of the passions and desires in human life is his initial assumption that either morality is the work of reason or it is the work of the passions and his own apparently conclusive arguments that it cannot be (...) the work ofreason. (2)At the same time as they [Diderot, Hume, Kierkegaard and Kant] agree largely on the character of morality, they agree also upon what a rationaljustification ofmorality would have to be. Its key premises would characterise some feature or features of human nature; and the rules of morality would then be explained and justified as being those rules which a being possessing just such a nature could be expected to accept. (3)[Hume] treats this ground for the justification of the rules of property and this explanation ofthem as holding for all times andplaces, prosperous as well as unprosperous, ever since the rules were first artificially contrived. (4)It isjust suchreasoning which Hume advanced both to explain and to justify the rules ofjustice, conceived as he conceived them, and the obligations imposed by those rules, understood as he understood them. One characteristic common to these four quotations from Alasdair Maclntyre's influential books After Virtue and Whose Justice? Which Rationality? is a yoking which is neither explained nor justified in the surrounding context, of the concepts of explanation and justification. Hume is said to be concerned both to explain and to justify (1) "morality," (2) "the rules of morality," (3) "the rules of property," and (4) "the rules ofjustice," and it seems that in each case he used the same argument to do both. My aim in this paper is to argue that Maclntyre altogether fails to dojustice to the distinctionbetweenjustification and Volume XVII Number 2 41 MURRAY MACBETH explanation, a distinction which is of the utmost importance for the interpretation ofHume—and notjust ofHume's moral philosophy. I take as my target, however, not either ofthe books from which I drew my opening quotations, but a much earlier article ofMaclntyre's, "Hume on 'Is' and 'Ought'," in which the ideas expressed in the above quotations are foreshadowed. An examination ofaspects ofthis article will enable me to show, if I am right, how Maclntyre goes startlingly astray. That 1959 article set in motion a lively discussion of the interpretation of Hume: no less than six articles, by Atkinson, Flew, Hudson, and Hunter, all contributions to the debate sparked off by Maclntyre, are reprinted in either or both of Chappell (1968) and Hudson (1969).6 I am in broad agreement with the criticisms made of his article by Atkinson and Flew; but I think it instructive to identify, as they failed to do, Maclntyre's fundamental mistake, which, I shall argue at the end ofthis paper, resulted from the very kind ofconfusion against which Hume warned his readers, and warned them in precisely that passage with which Maclntyre was primarily concerned, the famous paragraph about "is" and "ought" at the end of the opening section of book 3 of Hume's Treatise. My identification of Maclntyre's mistake involves putting that "is"-"ought" paragraph in context: in both, ambitiously on my part, the broad context of the Treatise as a whole, and also the more immediate context of the opening sections of book 3. Maclntyre wrote his article less as a contribution to an ongoing discussion of Hume than as a contribution to an ongoing discussion of ethical naturalism. Mid-twentieth century anti-naturalists, as Maclntyre pointed out, often invoked the "is"-"ought" paragraph as putting forward a view which was direct a ancestor oftheir own. Hare, for instance, appeals in his article "Universalisability" to what he calls "Hume's Law,"7 with a footnote referring the reader to this paragraph of the Treatise; the "law" in question is, roughly, that "ought"-judgements cannot be deduced from "is"-judgements. Maclntyre argued that such an interpretation of Hume was mistaken; and a crucial part ofhis evidence was the claim that, "ifthe currentinterpretation ofHume'sviews on "is'and'ought... (shrink)

The discovery of social phenomena by a discipline whose roots are abidingly psychological has been a singular development in American educational research. Formulations of situatedness are emblematic of this rethinking, and depending on our understanding of it, we have in situatedness the possibility of a distinctive set of analytic commitments. This paper discusses these possibilities and their development in the educational research literature, in the particulars of the Brown, Collins and Duguid (1989) publication Situated Cognition. In the end, situatedness is (...) understood as a unifying formulation. And while it opens a huge domain of topics for inquiry, it also proves, as its inclusiveness, of no use whatsoever as a parsing device. Situated cognition, for example, could not be, nor has ever been, otherwise. (shrink)

Russell’s theory of descriptions in “On Denoting” has long been hailed as a paradigm of the sort of analysis that is constitutiue of philosophical understanding. It is not the only model of logical analysis available to us, however. On Frege’s quite different view, analysis provides not a reduction of some problematic notion to other, unproblematic ones -- as Russell’s analysis does -- but instead a deeper, clearer articulation of the very notion with which we began. This difference, I suggest, is (...) grounded in their two very different conceptions of the nature of language / thought; and it grounds in turn two very different conceptions of the nature of philosophical understanding. (shrink)

After sketching familiar pragmatist arguments that seem to show that relations of reference and meaning shed no light on the role of language in our claims to knowledge, an alternative conception (inspired by Kripke's work on proper names and Sellars' conception of concepts and causal laws) is outlined. Neither relations of reference nor meanings are given; instead both essentially involve commitments that are different in kind from the sorts of propositional commitments made in judgment. If so, the pragmatist is mistaken (...) in concluding that meaning is a philosopher's fiction and reference nothing more than a technical notion of formal semantics. (shrink)

It is a familiar fact that different systems of notation can function in radically different ways. Consider, to take a very simple example, the difference between the sign-designs ‘twenty-three’, ‘XXIII’, and ‘23’. The first is an expression of English tracing the sounds a speaker makes in uttering the words ‘twenty’ and ‘three’. The second is a Roman numeral that uses signs for collections of things—‘X’ for ten things and ‘I’ for one thing—to present by addition the idea of ten and (...) ten and one and one and one, that is, twenty-three, things. Instead of tracing the sounds a speaker makes in speaking in some natural language, the Roman numeral ‘XXIII’ directly represents a collection of things. The Arabic numeral ‘23’ is different again. Like the Roman numeral ‘XXIII’ it in some way represents a number directly; unlike the Roman numeral it is not immediately additive. The numeral ‘23’ is not to be read as designating two and three things. The Arabic numeration system functions as a notational system in a way that is different both from the notational system of a written natural language such as English and from that of the Roman numeration system. (shrink)

The following pages are focused on a better understanding of Richard Rorty‟s reflections on the problem of truth, emphasizing the idea that one of the keys to a better understanding of Rorty is to consider the fact that there are three very different kinds of discourse present in his philosophical endeavors. In order to support this unprecedented claim I bring forward the case of one of Plato‟s dialogues concerned with the definition of knowledge, Theaetetus. The similarities between Plato‟s dialogue and (...) Rorty are the core of my argumentation. (shrink)

Reading Rorty can be hard, not because his ideas are especially difficult but because they often are, or at least seem on the face of it to be, quite straightforward and yet somehow wrong. But this is very unlikely: Rorty is not a thinker to get straightforward things wrong. We need, then, a plan for reading Rorty. The plan I suggest finds Rorty engaged, at different points, in three fundamentally different sorts of discourse. In this regard Rorty's reflections on truth (...) in particular bear, we see, a remarkable, and remarkably suggestive, similarity to Plato's reflections on knowledge in his dialogue Theaetetus. As we also see, reading Rorty as engaged at various points in the three sorts of discourse outlined enable us to make good sense of claims that have seemed to many readers, even otherwise sympathetic ones (such as McDowell), to be quite wrongheaded. (shrink)