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Platonism in the Philosophy of Mathematics

Platonism about mathematics is the idea that abstract mathematical objects have their own existence independent of our experience of them, similar to how Electrons have their own independent existence.

The premier argument for mathematical structures having their own existence, is derived from Frege: The language of mathematics refers to and quantifies over these objects, and many theorems are true, but a sentence cannot be true unless its sub-sentences are true.

Despite this, philosophers have laid out a number of issues with the concept, and these objects said to be epistemologically inaccessible, and metaphysically contentious.

1. What is Mathematical Platonism?

Mathematical platonism can be described as the union of the following three ideas:

Existence: There are mathematical objects.

Abstractness: Mathematical objects are abstract.

Independence: Mathematical objects are independent of intelligent agents.

The first two claims are clear enough to be useful, where existence can be formalized as ∃xMx\exists x \mathcal{M}x, with Mx\mathcal{M}x being the predicate xx 'is a mathematical object, which holds only for the objects of Pure Mathematics.

Abstractness, states that all mathematical objects are abstract, in the sense of being non-spatiotemporal, and not necessarily causally important.

Independence lacks clarity in comparison, due to the question of the meaning of ascribing independence to an object.

The obvious fix is simply the counterfactual that, were there intelligent agents, with different language or thought than we possess, there would still be mathematical objects.

1.1 Historical remarks

Platonism as it stands here is distinguished from the perspective of Plato, with his work not being considered critical for defending the argument, having grown beyond his theory of abstract and eternal forms.

Not only is this no longer Plato's formalism, but it is a purely metaphysical perspective, lacking substantive epistemological content.

Lastly, the description of mathematical platonism presented here does not claim that all of the truths of pure mathematics are necessary, as many philosophers who are platonists reject this claim.

1.2 Philosophical Significance

Platonism in this regard is philosophically significant, in that it challenges the physicalist idea that reality is exhausted by the physical, because it entails a sector of reality that is non-physical and contains objects which are removed from the causal, spatio-temporal hierarchy observed in physical science.

It also presents issues to naturalistic theories of knowledge, for mathematics is knowledge we posses, which is abstract, and this would be hard for naturalistic theories to account for.

These consequences are not unique to this form of platonism, but it is unusually well equipped to make such challenges, due to its ubiquitous success, and the fact that few contemporary analytic philosophers will rebuke fundamental claims made by a discipline with the track record mathematics possesses.

Should if it is revealed via philosophical analysis, that mathematics does lead to these consequences, the option of rejecting mathematics is simply untenable.

For instance, in cases where the consequences presented by the tenets of theologies, the parts relevant to the unacceptable consequences are rejected.

1.3 Object Realism

If we define object realism to be the belief that abstract mathematical objects exist, then it can be considered a conjunction of Existence and Abstractness.

It stands in opposition to nominalism, which maintains that there are no abstract objects.

Referenced in

Platonism in the Philosophy of Mathematics