Mathematics and the Method of Abstraction
I provide a brief history and some philosophical reflections on the development of the method of abstraction in mathematics during the late nineteenth and early twentieth century.
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Contents 1 Cross-References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
Abstract
I provide a brief history and some philosophical reflections on the development of the method of abstraction in mathematics during the late nineteenth and early twentieth century. Keywords
Abstraction · Frege · von Neumann · Cantor · Locke · Berkeley · Russell · Number It is a familiar thought that mathematics derives from abstraction. We observe three rocks, three stars, three French hens and from these respective observations we are able to derive the concept of the number 3; or we observe a line in the sand and from this observed line, thick and wiggly as it may be, we are able to derive the concept of a line, perfectly indivisible in extent, absolutely straight, and extending infinitely in either direction; or we observe a car in motion, the rising temperature of the air, or the increasing volume of the water in the tub and from these sundry observations we are able to derive the concept of the rate of change of one quantity with respect to another.
This article is based upon an address I gave to a general audience in 2013, under the auspices of the Humboldt Foundation. K. Fine (*) New York University, New York, NY, USA e-mail: [email protected] © Springer Nature Switzerland AG 2020 B. Sriraman (ed.), Handbook of the History and Philosophy of Mathematical Practice, https://doi.org/10.1007/978-3-030-19071-2_91-1
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K. Fine
Abstraction may be a familiar idea but it is also vague. What exactly is abstraction? From what do we abstract? To what do we abstract? And how do we get from the one to the other? These are questions which have been discussed since the beginning of mathematics and philosophy. But what I want to consider is the discussion of the questions in a particular period and by a particular group of people. The period was the end of the nineteenth century and the beginning of the twentieth century. And the people in question were mathematicians who had a strong interest in foundational issues. They were not only concerned to do mathematics but also to put the subject on a firm foundation. The period was, I believe, a time when discussion of abstraction came to fruition. And this, certainly, was no accident. For this was a time when many of the most significant advances in mathematics depended upon extending its ontology. Mathematicians could no longer be content with the familiar numbers of arithmetic or the familiar points of Euclidean geometry; they also had to traffic in complex numbers, points at infinity, transfinite numbers, and the like. The question therefore arose as to what might justify these extensions of the familiar ontology; and to many mathematicians it seemed that the answer lay in the method of abstractio
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