Abstract
In this paper I argue that constructivism in mathematics faces a dilemma. In particular, I maintain that constructivism is unable to explain (1) the application of mathematics to nature and (2) the intersubjectivity of mathematics unless (3) it is conjoined with two theses that reduce it to a form of mathematical Platonism. The paper is divided into five sections. In the first section of the paper, I explain the difference between mathematical constructivism and mathematical Platonism and I outline my argument. In the second, I argue that the best explanation of how mathematics applies to nature for a constructivist is a thesis I call Copernicanism. In the third, I argue that the best explanation of how mathematics can be intersubjective for a constructivist is a thesis I call Ideality. In the fourth, I argue that once constructivism is conjoined with these two theses, it collapses into a form of mathematical Platonism. In the fifth, I confront some objections.
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Notes
The classic exposition of this problem for MP is Benacerraf (1965).
However, MC does presuppose some special faculty of the mind that is capable of engaging in synthetic, constructive procedures, and some assert that this is simply substituting one mysterious faculty for another. I am sympathetic to such worries, but I shelve them for now: they will become more pressing in Sect. 4.
This can be illustrated by means of a well-known example. Suppose you want to demonstrate that there are two irrational numbers, a and b, such that \(a^b\) is rational. You might begin by observing that (PREMISE) either \(\sqrt{2}^{\sqrt{2}}\) is rational or it is not. If it is rational, then let \(a=b=\sqrt{2}\) and you are done. If it is not, then let \(a=\sqrt{2}^{\sqrt{2}}\) and let \(b=\sqrt{2}\) and you are done. Because these two cases are exhaustive, you might take the proof to be complete even if we do not know which case is correct. But constructivists would not accept it. The problem is that the most obvious way to prove PREMISE is by appeal to the principle that all real numbers are rational or not (and the claim that \(\sqrt{2}^{\sqrt{2}}\) is a real number). But the reals are an infinite set, so if this is how one went about proving PREMISE, it would involve an appeal to the uLEM.
Two things are worth pointing out here. One is that if it could be proved that \(\sqrt{2}^{\sqrt{2}}\) belongs to a finite set (not necessarily of cardinality 1) each of whose members is either rational or irrational, this would suffice for MC (pace Bridges and Palmgren who assert that MC requires us to “decide whether \(\sqrt{2}^{\sqrt{2}}\) is rational or irrational” (Bridges and Palmgren 2018, section 1)). The other is that the Gelfond–Schneider theorem, proved in the first half of the twentieth century, shows that \(\sqrt{2}^{\sqrt{2}}\) is transcendental and, thus, irrational.
The PoA also is raised for MP. However, the PoA is not a problem for MN. For example, on Mill’s famous account, addition is merely the abstract function of gathering objects together. MN, by way of contrast with both MC and MP, faces the problem of universality (PoU): if mathematical truths are generalized from experience, how can they have strict (rather than merely inductive) universality? The PoU also brings with it another problem, one related to the PoA, the problem of projectability (PoP): how can the projection of mathematics onto previously unobserved objects be explained? However, such issues are beyond the scope of the present investigation.
Steiner recently has argued that the application of mathematics in modern physics is especially problematic. His argument is based on the fact that modern physics frequently moves immediately from mathematical possibility to physical reality, presupposing what Steiner calls a Pythagorean (and I would call a Platonic) faith in mathematical formalism (Steiner 1998).
Mancosu (2018, section 1).
See Friedman (1992).
In brief, Kant argued that (1) because mathematical truths are (a) universal, (b) necessary, and (c) non-analytic, they must be conditions of the possibility of experience; (2) conditions of the possibility of experience must be understood as laws that govern the functioning of the mind; therefore (3) mathematical truths must be understood as laws that govern the functioning of the mind. He then further argued that (4) because of (a), (b), and (c), mathematical truths cannot apply to mind-independent things as they are in themselves; (5) if mathematical truths cannot apply to mind-independent things as they are in themselves, then space and time are only mental constructs (i.e., they cannot be both mental constructs and things in themselves); therefore (6) space and time are only mental constructs. See Guyer (1987).
The name ‘Copernicanism’ is taken from Kant’s characterization of his idea that the only way to explain our knowledge of objects of experience is by means of a “Copernicanism revolution” in which we try out the hypothesis that objects conform to the mind (rather than the other way around) and see how far we get.
Beilby (2002).
The Pythagoreans guarded epistemic access to a truth about \(\sqrt{2}\), but I doubt that any of them would have conflated Kp with p in the way suggested in the sentence to which this note is appended.
There are difficult textual debates about whether the God solution or the counterfactual solution is logically prior on Berkeley’s account. I cannot pursue such debates here.
Niekus (2010).
These kinds of questions are familiar: they come up in metaethics in discussions of cultural relativism, and they also come up in social epistemology in discussions of communities of knowers and their respective epistemic authorities.
Steiner (1998)).
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Kahn, S. A Dilemma for Mathematical Constructivism. Axiomathes 31, 63–72 (2021). https://doi.org/10.1007/s10516-020-09475-x
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DOI: https://doi.org/10.1007/s10516-020-09475-x