Danielle Macbeth

Diagrammatic reasoning in Frege's Begriffsschrift

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.

Précis of Realizing Reason: A Narrative of Truth and Knowing

In the seventeenth century, Descartes introduced a radically new form of mathematical practice, one that he came to think was the work of pure reason. Kant saw that it was not: Descartes’ mathematical practice no less than ancient diagrammatic practice constitutively involves paper-and-pencil reasoning, not, to be sure, images or drawn figures but written signs nonetheless, in particular, equations in the symbolic language of arithmetic and algebra. While the ancient paradigm of knowing as perception, whether with one’s bodily eyes or with the eyes of the mind, was indeed superseded with the appearance of Descartes’ Geometry in 1637, it was not pure reason but only the understanding that was now to be regarded as the power of knowing. In 1637 Descartes fundamentally transformed the practice of mathematics. Fifty years later with the publication of his Principia (1687), Newton transformed the practice of physics. It was left to Kant, nearly a century on, to transform the practice of philosophy in the Critique of Pure Reason (1781/1787). Over the course of the nineteenth century, mathematical practice was again transformed to become, as it remains today, a practice of deductive reasoning directly from concepts. And in the twentieth century the practice of fundamental physics was again transformed as well. Philosophy has not had its revolution but remains merely Kantian. And it does so because although Frege in fact saw (pace Kant) that deductive reasoning can nonetheless be ampliative, and thereby that pure reason had been realized as a power of knowing, Frege’s logic to show this was systematically misunderstood. As a result, the potential of Frege’s Begriffsschrift to revolutionize philosophical practice has, for over a century, lain dormant. Realizing Reason: A Narrative of Truth and Knowing aims to complete what Frege began, to catalyze the needed revolution in philosophy, thereby ushering in a new, profoundly post-Kantian era in philosophy. A central aim of Realizing Reason is to trace developments over the course of the intellectual history of the West that have culminated, so I argue, in the

STRIVING FOR TRUTH IN THE PRACTICE OF MATHEMATICS: KANT AND FREGE

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. ! e account that is sketched draws fi rst 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 (e.g., the ten digits of arithmetic or the points, lines, angles, and areas of Euclidean geometry) that combine into wholes (numerals or drawn Euclidean fi gures) that are themselves parts of larger wholes (the array of written numerals in a calculation or the diagram of a Euclidean demonstration). Because wholes such as numerals and Euclidean fi gures 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 Begriff sschrift 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.

Logical form, mathematical practice, and Frege's Begriffsschrift

Abstract For over a century we have been reading Freges Begriffsschrift notation as a variant of standard notation. But Freges notation can also be read differently, in a way enabling us to understand how reasoning in Begriffsschrift is at once continuous with and a significant advance beyond earlier mathematical practices of reasoning within systems of signs. It is this second reading that I outline here, beginning with two preliminary claims. First, I show that one does not reason in specially devised systems of signs of mathematics as one reasons in natural language; the signs are not abbreviations of words. Then I argue that even given a system of signs within which to reason in mathematics, there are two ways one can read expressions involving those signs, either mathematically or mechanically. These two lessons are then applied to a reading of Freges proof of Theorem 133 in Part III of his 1879 logic, a proof that Frege claims is at once strictly deductive and ampliative, a real extension of our knowledge. In closing, I clarify what this might mean, and how it might be possible.

The Place of Philosophy

What should be the place of philosophy in todays intellectual culture? This exploration begins with Western philosophy, especially analytic philosophy, and aims to show that we are at a singular historical moment: it is now clear, as it could not have been hitherto, that the conversation of philosophy is inherently global.

Responses to Brassier, Redding, and Wolfsdorf

Realizing Reason aims to catalyze a revolution in philosophy by recovering the extraordinary and hitherto unrecognized achievements of Frege in his 1879 logic Begriffsschrift. Following in the age-old tradition of the self-conscious development of systems of written marks within which to formulate the contents of mathematical concepts and to reason, Frege’s mathematical language was devised for the newly established mathematical practice of deductive reasoning from the contents of defined concepts. This language, adequately understood, provides us with all the resources we need to move the practice of philosophy beyond Kant into what I think of as its properly Hegelian phase. And it enables us to understand one thing, at least, that this whole twenty-five-hundred-year intellectual adventure has been about, the realization of reason, pure reason, as a power of knowing. My critics convey a good sense of the overall shape of my narrative, as well as some of the details. They also raise important questions. I begin with a worry Redding raises following Rorty, then turn to a concern Wolfsdorf has regarding my reading of the ancient Greeks, which will lead in turn to issues, raised one way or another by all three of my critics, regarding where my narrative ends up. After next responding to another of Wolfsdorf ’s concerns, this time regarding what I suggest is an analogy between the Gibsonian picture of affordances and our cognitive involvements in the world, I end with some reflections on Brassier’s comments on history and nature.

Frege and the Aristotelian Model of Science

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.