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Axiomatic Method and Category Theory

Part of the Synthese Library Book Series (volume 364)

DOI: 10.1007/978-3-319-00404-4

ISBN: 978-3-319-00403-7

Publisher: Springer International Publishing - Springer Journals

Aug 28, 2013

1 - 10 of 10 Chapters

Available

Introduction

Rodin, Andrei

Pages: 1 –12

[The main motivation of writing this book is to develop the view on mathematics described in the above epigraphs. Some 200 years ago this view used to be by far more common and easier to justify than today. It is sufficient to say that it made part of Kant’s view on mathematics, and that Kant’s...
Available

Euclid: Doing and Showing

Rodin, Andrei

Pages: 15 –37

[Reading older mathematical texts always involves a hermeneutical dilemma: in order to make sense of the mathematical content of a given old text one wants to interpret it in modern terms; in order to see the difference between the modern mathematical thinking and older ways of mathematical...
Available

Hilbert: Making It Formal

Rodin, Andrei

Pages: 39 –72

[In the standard textbooks Hilbert’s philosophy of mathematics is commonly labelled formalism and under this title distinguished from Brouwer’s intuitionism, on the one hand, and Russell’s logicism, on the other hand. However, as Hintikka (1997a) rightly remarks, this popular name is very...
Available

Formal Axiomatic Method and the Twentieth Century Mathematics

Rodin, Andrei

Pages: 73 –97

[The Formal Axiomatic Method has been proposed by Hilbert about a century ago and it is appropriate to ask how it performed during the past century. It appears to me that its impact is somewhat controversial. On the one hand, during this time period the Formal Axiomatic Method was and still...
Available

Lawvere: Pursuit of Objectivity

Rodin, Andrei

Pages: 99 –143

[The author’s attitude to foundations of mathematics, which he describes as commonsensical, assumes a permanent interaction between the foundations and the current mathematical practice (“applications”). It is opposed in this sense to the “Bolzano-Frege-Peano-Russell tradition” (making part of...
Available

Identity in Classical and Constructive Mathematics

Rodin, Andrei

Pages: 149 –173

[Changing objects (of any nature) pose a difficulty for the metaphysically-minded logician known as the Paradox of Change. Suppose a green apple becomes red. If A denotes the apple when green, and B when it is red then A = B (it is the same thing) but the properties of A and B are different: they...
Available

Identity Through Change, Category Theory and Homotopy Theory

Rodin, Andrei

Pages: 175 –209

[The replacement of the equivalence xEy by the identity x = y discussed by Frege (Sect. 6.5) allows for an interpretation, which differs from Frege’s. Namely, equivalence E can be understood as an invertible transformation (rather than relation), which turns x into y and vice versa; then the...
Available

How Mathematical Concepts Get Their Bodies

Rodin, Andrei

Pages: 215 –234

[By Kant’s popular word “thoughts without content are empty, intuitions without concepts are blind” Critique of Pure Reason (B75). In Sect. 2.3 we have seen how this general claim applies to Euclid’s geometry, that is, how exactly concepts and intuitions work here closely together. In this...
Available

Categories Versus Structures

Rodin, Andrei

Pages: 235 –263

[In this Chapter I shall discuss a notion, which have been already used throughout this book, namely, the notion of mathematical structure. I shall also discuss a philosophical view known as structuralism and analyze its relationships with the category theory. According to a popular opinion the...
Available

New Axiomatic Method (Instead of Conclusion)

Rodin, Andrei

Pages: 265 –272

[In the following long promised presentation of the New Axiomatic Method I shall use as a guide Lawvere’s description of Axiomatic Method as “unification and concentration” (Lawvere 2003, p. 213) and generalize upon some examples of axiomatic thinking due to Lawvere and Voevodsky. I begin with...

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