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.

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)

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 ...

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)

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)

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)

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)

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 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)

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)

The idea of natural truth is the idea of truths that are the same for all rational beings with our biological form of life. The thought is that in regard to at least some issues, for example the ontological status of fish, there are natural truths, and that it is the task of philosophy in particular to discover such truths. In my essay I distinguish such truths from empirical truths such as, for example, that water nourishes plants or that there (...) are black swans, as well as from matters of taste. Angle reminds us of yet another sort of truth that is not amenable to the kind of treatment that I outline for natural truth. It is not, or at least seems not to be, a natural truth that one ought to rid... (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)

As Friedman has argued, Kant's argument for the ideality of space turns on the nondeductive character of geometrical reasoning in Euclid's system. Since geometry can be axiomatized, this argument fails. But ("pace" Russell) Leibniz's argument based on the unreality of constitutive relations is not thereby answered as well. I argue that what is needed in response to Leibniz is a properly post-Kantian conception of concepts as inferentially articulated. This conception, I suggest, is based on the same fundamental insight that underlies (...) the axiomatization of geometry. (shrink)

What is the place of philosophy in today's intellectual culture? What should its place be? My intent, ultimately, is to answer this second, and more interesting, question, to show that philosophy should be a truly global dialogue the aim of which is to discover what I shall refer to as natural truths about us and about the world in which we live our lives. But in order to show the place of philosophy in this way, I need to begin by (...) focusing on the place of Western philosophy, in particular analytic philosophy, in today's intellectual culture. If philosophy is to emerge as a truly global dialogue in the intellectual culture, we need explicitly to recognize that philosophy is not and cannot be a science but is... (shrink)