Abstract
THE CLOSE CONNECTION BETWEEN mathematics and philosophy has long been recognized by practitioners of both disciplines. The apparent timelessness of mathematical truth, the exactness and objective nature of its concepts, its applicability to the phenomena of the empirical world—explicating such facts presents philosophy with some of its subtlest problems. In this final chapter we discuss some of the attempts made by philosophers and mathematicians to explain the nature of mathematics. We begin with a brief presentation of the views of four major classical philosophers: Plato, Aristotle, Leibniz, and Kant. We conclude with a more detailed discussion of the three “schools” of mathematical philosophy which have emerged in the twentieth century: Logic ism, Formalism, and Intuitionism.
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Notes
On the other hand, empirical propositions containing mathematical terms such as 2 cats + 3 cats = 5 cats are true because they hold in the actual world, and, according to Leibniz, this is the case only because the actual world is the “best possible” one. Thus, despite the fact that 2 + 3 = 5 is true in all possible worlds, 2 cats + 3 cats = 5 cats could be false in some world.
In fact, it was the imprecision surrounding the concept of continuity that impelled him to embark on the program of critical analysis of mathematical concepts.
This idea of correspondence or functionality, taken by Dedekind as fundamental, is in fact the central concept of category theory (see Chapter 6).
Here and in the sequel we employ the logical operators introduced at the beginning of Appendix 2. Thus “∀” stands for “for every”, “∃” stands for “there exists”, “1” for “not”,” ⋀” for ‘and”, “⋁” for “or”, “→” for “implies” and “↔” for “is equivalent to”.
It is helpful to think of the extension of a concept as the class of all entities that fall under it, so that, for example, the extension of the concept red is the class of all red objects. However, it is by no means necessary to identify extensions with classes; all that needs to be known about extensions is that they are objects satisfying (1).
A numerical concept is one expressing equinumerosity with some given concept.
Thus at the time this was asserted Russell was what could be described as an “implicational logicist”.
One may get an idea of just how difficult this work is by quoting the following extract from a review of it in a 1911 number of the London magazine The Spectator: It is easy to picture the dismay of the innocent person who out of curiosity looks into the later part of the book. He would come upon whole pages without a single word of English below the headline; he would see, instead, scattered in wild profusion, disconnected Greek and Roman letters of every size interspersed with brackets and dots and inverted commas, with arrows and exclamation marks standing on their heads, and with even more fantastic signs for which he would with difficulty so much as find names.
The self-contradictory nature of the “paradoxical” entities we have described derives as much from the occurrence of negation in their definitions as it does from the circularity of those definitions.
The idea of stratifying classes into types had also occurred to Russell in connection with his analysis of classes as genuine pluralities, as opposed to unities. On this reckoning, one starts with individual objects (lowest type), pluralities of these comprise the entities of next highest type, pluralities of these pluralities the entities of next highest type, etc. Thus the evident distinction between individuals and pluralities is “projected upwards” to produce a hierarchy of types.
Here by a quantifier we mean a an expression of the form “for every” (∀) or “there exists” (∃).
Thus, if a domain of discourse D comprises entities a.b.c,⋯, then for every x in D, P(x) is construed to mean P(a) and P(b) and P(c) and⋯, and there exists x in D such that P(x) to mean P(a) or P(b) or P(c) or⋯ 13 In this connection one recalls his famous remark: one must be able to say at all times, instead of points, lines and planes—tables, chairs, and beer mugs.
Hilbert actually asserted that “no one will ever be able to expel us from the paradise that Cantor has created for us.” There is no question that “Cantor’s Paradise” furnishes the ideal site on which to build Hilbert’s hotel (see the previous chapter.)
It should be emphasized that Hilbert was not claiming that (classical) mathematics itself was meaningless, only that the formal system representing it was to be so regarded.
A sketch of Göder s arguments is given in Appendix 2.
This is not to say that Brouwer was primarily interested in logic, far from it: indeed, his distaste for formalization led him not to take very seriously subsequent codifications of intuitionistic logic.
Hermann Weyl said of nonconstructive existence proofs that “they inform the world that a treasure exists without disclosing its location.”
This is the assertion that, for any proposition p, either p or its negation p holds.
This is the assertion that, for any proposition p, p implies p.
And indeed may never have; as observed in Chapter 3, little if any progress has been made on the ancient problem of the existence of odd perfect numbers.
In fact a much deeper argument shows that is irrational, and is therefore the correct value of a.
Here by proof we are to understand a mathematical construction that establishes the assertion in question, not a derivation in some formal system. For example, a proof of 2 + 3 = 5 in this sense consists of successive constructions of 2, 3 and 5, followed by a construction that adds 2 and 3, finishing up with a construction that compares the result of this addition with 5.
Remarkably, it is also the logic of smooth infinitesimal analysis: see Appendix 3.
In a famous remark opposing intuitionism, Hilbert said “to deny the mathematician the use of the law of excluded middle would be to deny the boxer the use of his fists.” But with experience in using the refined apparatus of intuitionistic logic one comes to regard Hilbert’s simile as inappropriate. It would, perhaps, be more apposite to compare the mathematician’s frustration in being denied the use of the law of excluded middle with the frustration a nineteenth century surgeon might well have felt when denied the use of a butcher knife, or a twentieth century general the use of nuclear weapons.
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© 1999 Springer Science+Business Media Dordrecht
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Bell, J.L. (1999). The Philosophy of Mathematics. In: The Art of the Intelligible. The Western Ontario Series in Philosophy of Science, vol 63. Springer, Dordrecht. https://doi.org/10.1007/978-94-011-4209-0_12
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DOI: https://doi.org/10.1007/978-94-011-4209-0_12
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