The Case of Platonism in Mathematics. Philosophical discussions concerning the outside world typically take place within the framework of epistemology. These discussions center on the subject of how justified our views about the outside world are and what paths we can take to get there.

The Case of Platonism in Mathematics

Awareness of the outside world

We might be curious about how we might prove that the outside world doesn’t exist. Because external, and so objectively existing, entities may exist even if we assume that we can explain all we want to explain in terms of objects that depend on human interests. Similar to the Kantian noumenon, they might be completely unapproachable by human epistemic endeavors.

Naturally, it is impossible to establish such noumena through experience since, in either case, the universe would appear exactly the same. Nevertheless, it appears that we are unable to formulate a defense against the possibility of such cognition-transcending noumena.

Although there is some validity to this criticism, it is crucial to remember that we shouldn’t include completely idle wheels into our explanation of reality. If we acknowledge the existence of an entity like a “world behind the appearances,” it ought to accomplish some theoretical or explanatory job; yet, since the world remains the same whether this entity exists or not, it is unable to accomplish any such work.

A stronger argument is required to support a hypothesis than the assertion that, based on what we currently know and what we may learn in the future, such a thing might exist.

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Take mathematical Platonism as an example. For the purposes of argument, let’s imagine that we can demonstrate that whatever we wish to describe about mathematics can be stated (by using, example, a number of formalisms) without making reference to a world of necessarily existent objects outside of time and space.

Assume further that we can contend that platonic objects cannot potentially contribute to our understanding of mathematical facts (e.g., because entities outside of space and time cannot be causal relata and all of our epistemic functions entail causal processes).

The Platonism defender could still dig in his heels and concede that although we have accepted certain rules for manipulating marks on paper, we still believe that these abstract objects exist even though we do not know that 7 + 5 = 12. This is because we intuitively understand the identity relation between a set of abstract objects, a number and a pair of numbers connected by the addition function.

He could say this while still acknowledging that our capacity to learn mathematics is unaffected by the existence or nonexistence of these items, and that we would still be able to do so even if they were all to disappear.

I think we can all agree that there is no hope for this defense of Platonism. The mere suggestion of an outside world would not fair much better in the face of arguments that claim such a universe would not be necessary to explain how we perceive the world as it is.

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