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)

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)

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.

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 ...

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)

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)

On 15 April 1630, in a letter to Mersenne, Descartes announced that on his view God creates the truths of mathematics. Descartes returned to the theme in subsequent letters and some of his Replies but nowhere is the view systematically developed and defended. It is not clear why Descartes came to espouse the creation doctrine, nor even what exactly it is. Some have argued that his motivation was theological, that God creates the eternal truths, including the truths of logic, because (...) and insofar as God is omnipotent andthe creator of all things. I develop and defend a different reading according to which Descartes was led to espouse the creation doctrine by a fundamental shift in his understanding of the correct mode of inquiry in metaphysics and mathematics: by 1630, the God-created truths came to play the role in inquiry that until then, in the Rules for the Direction of the Mind, had been played by images. (shrink)

The Cerdanya valley in the eastern Pyrenees has a physical unity into which a political frontier has been imposed to divide it. The social and cultural repercussions of this Franco-Spanish border have created obstacles to marriage which are not due to topography. Choice of month of marriage is under cultural control and the study of seasonality in marriages recorded in the registers of all the Cerdan parishes on both sides of the border demonstrated differences over time and between French and (...) Spanish sectors. It is suggested that these changes demonstrate the process of distancing of the two populations. Cluster and correspondence analysis showed progressive differentiation of the seasonality patterns of the French and Spanish Cerdans despite the geographic unity of the valley. Sociocultural factors are presumed responsible. (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)

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)

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)

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)

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)

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)

In his Locke Lectures Brandom proposes to extend what he calls the project of analysis to encompass various relationships between meaning and use. As the traditional project of analysis sought to clarify various logical relations between vocabularies so Brandom’s extended project seeks to clarify various pragmatically mediated semantic relations between vocabularies. The point of the exercise in both cases is to achieve what Brandom thinks of as algebraic understanding. Because the pragmatist critique of the traditional project of analysis was precisely (...) to deny that such understanding is appropriate to the case of natural language, the very idea of an analytic pragmatism is called into question by that critique. My aim is to clarify the prospects for Brandom’s project, or at least something in the vicinity of that project, through a comparison of it with what I will suggest we can think of as Kant’s analytic pragmatism as developed by Peirce. (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)

1. In “Empiricism and the Philosophy of Mind,” Sellars argues that the notion of “self-authenticating nonverbal episodes” that would provide a foundation for empirical knowledge is a myth; nothing merely causal, not already in conceptual shape, could possibly play the justificatory role required of such a foundation. Rorty takes Quine, in “Two Dogmas,” to make the complementary point that the notion of analytic claims true by virtue of meaning, of self-authenticating verbal episodes that might provide a foundation for another sort (...) for knowledge, is again a myth. A third moment in the dismantling of the myth of a foundation—this time for the contentfulness of our thoughts rather than for the truth of our beliefs—is due to Rorty himself. As he argues in “Realism and Reference”, the notion of reference as a nonintentional or “external” word-world relation that would ground our thoughts’ representational bearing on things, and so explain how thoughts can so much as purport to be true, again involves illicit appeal to the idea that independent of what we take it as an object can have cognitive significance. An external relation of reference cannot serve as the unmoved mover of the contentfulness or aboutness of thought. Nor, if Quine is right about the breakdown of the analytic/synthetic distinction, can meanings or word-word relations play this role. (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)