Conceptual engineers have made hay over the differences of their metaphilosophy from those of conceptual analysts. In this article, I argue that the differences are not as great as conceptual engineers have, perhaps rhetorically, made them seem. That is, conceptual analysts asking ‘What is X?’ questions can do much the same work that conceptual engineers can do with ‘What is X for?’ questions, at least if conceptual analysts self-understand their activity as a revisionary enterprise. I show this with a study (...) of Russell's metaphilosophy, which was just such a revisionary conception of conceptual analysis. (shrink)

Word problems in mathematics seem to constantly pose learning difficulties for all kinds of students. Recent work in math education (for example, [Lakoff, G. & Nuñez, R. E. (2000). Where mathematics comes from: How the embodied mind brings mathematics into being. New York: Basic Books]) suggests that the difficulties stem from an inability on the part of students to decipher the metaphorical properties of the language in which such problems are cast. A 2003 pilot study [Danesi, M. (2003a). Semiotica, 145, (...) 71–83] confirmed this hypothesis in an anecdotal way. This paper reviews the implications of that study and of a follow-up one that is described here as well, in the light of how the metaphorical analysis of word problems allows learners to overcome typical difficulties in word problem-solving by teaching them how to flesh out the underlying concepts and convert them into appropriate representations. (shrink)

In this interesting and engaging book, Shabel offers an interpretation of Kant's philosophy of mathematics as expressed in his critical writings. Shabel's analysis is based on the insight that Kant's philosophical standpoint on mathematics cannot be understood without an investigation into his perception of mathematical practice in the seventeenth and eighteenth centuries. She aims to illuminate Kant's theory of the construction of concepts in pure intuition—the basis for his conclusion that mathematical knowledge is synthetic a priori. She does this through (...) a contextualized interpretation of his notion of mathematical construction, which she argues can be approached by looking at Euclid's Elements and Christian Wolff's mathematical textbooks. The importance of the former for her interpretation is justified by the fact that nearly all of Kant's mathematical examples in the Critique are Euclidean propositions. The importance of the latter is revealed through the fact that Wolff's textbooks were not only widely read and representative of the state of elementary mathematics during Kant's time; Kant was also intimately familiar with them. During the thirty years prior to the publication of the Critique, he used the textbooks in the college-level introductory courses in mathematics and physics that he taught.In the introduction to her book, Shabel helpfully distinguishes her approach to Kant's philosophy of mathematics from that of previous commentators. She points out that most commentators assessed Kant's thoughts on mathematics in terms of the ‘supposedly devastating effects of the discovery of non-Euclidean geometry on his theory of space’ .1 Bertrand Russell, for example, criticized Kant for his lack of a proper …. (shrink)

This paper provides a definitive answer, based on considerations derived from first-order logic, to the question regarding the status of elementary geometry, whether elementary geometry can be reduced to algebra. The answer we arrive at is negative, and is based on a series of structural questions that can be asked only inside the geometric formal theory, as well as the consideration of reverse geometry, which is the art of finding minimal axiom systems strong enough to prove certain geometrical theorems, given (...) that there are no algebraic structures that one could associate with those minimal axiom systems. (shrink)

This paper sets out to adduce visual evidence for Kroneckerian finitism by making perspicuous some of the insights that buttress Kronecker’s conception of arithmetization as a process aiming at disclosing the arithmetical essence enshrined in analytical formulas, by spotting discrete patterns through algorithmic fine-tuning. In the light of a fairly tractable case study, it is argued that Kronecker’s main tenet in philosophy of mathematics is not so much an ontological as a methodological one, inasmuch as highly demanding requirements regarding mathematical (...) understanding prevail over mere preemptive reductionism to whole numbers. (shrink)

We shall address from a conceptual perspective the duality between algebra and geometry in the framework of the refoundation of algebraic geometry associated to Grothendieck’s theory of schemes. To do so, we shall revisit scheme theory from the standpoint provided by the problem of recovering a mathematical structure A from its representations \ into other similar structures B. This vantage point will allow us to analyze the relationship between the algebra-geometry duality and the structure-semiotics duality. Whereas in classical algebraic geometry (...) a certain kind of rings can be recovered by considering their representations with respect to a unique codomain B, Grothendieck’s theory of schemes permits to reconstruct general rings by considering representations with respect to a category of codomains. The strategy to reconstruct the object from its representations remains the same in both frameworks: the elements of the ring A can be realized—by means of what we shall generally call Gelfand transform—as quantities on a topological space that parameterizes the relevant representations of A. As we shall argue, important dualities in different areas of mathematics can be understood as particular cases of this general pattern. In the wake of Majid’s analysis of the Pontryagin duality, we shall propose a Kantian-oriented interpretation of this pattern. We shall use this conceptual framework to argue that Grothendieck’s notion of functor of points can be understood as a “relativization of the a priori” that generalizes the relativization already conveyed by the notion of domain extension to more general variations of the corresponding domains. (shrink)

Donald Coxeter died in 2003, at a ripe old age of 96. Though I had regularly seen him at mathematics talks in Toronto for over twenty years, I never felt rushed to seek him out. It seemed he would go on forever. His death left me regretting my missed opportunity and Siobhan Robert's excellent book makes me regret it even more. Like any good biography of an intellectual, King of Infinite Space contains personal details and mathematical achievements in some detail. (...) Thus, we learn of the traumatic effects on Coxeter of his parents' divorce, his search for a spouse, his vegetarianism, and his progressive politics. We also learn a fair bit about the kaleidoscopes he made while in Cambridge during his student days in order to study the symmetry properties of polyhedra. These involved mirrors that Coxeter had specially constructed for this purpose. Along the way we are treated to interesting tidbits, such as G.H. Hardy's detestation of mirrors . There are brief accounts of the brilliant notation Coxeter invented, known as Coxeter diagrams, and of course, the now famous Coxeter groups. Short appendices fill in a bit more detail. It is all very well done and thoroughly engrossing.Wittgenstein befriended Coxeter, who was part of the very small group to whom …. (shrink)