In what realm do mathematics objects exist according to Platonists?

Question

"Intuitionists" believe that mathematics is just a creation of the human mind. In that sense you can argue that mathematics is invented by humans. Any mathematical object exists only in our mind and doesn't as such have an existence.

"Platonists", on the other hand, argue that any mathematical object exists and we can only "see" them through our mind. Hence in some sense Platonists would vote that mathematics was discovered.

If this is what Platonists believe, then where do they think that these objects exist? If it's not inside our physical realm then in what realm do these objects exist and do they move inside of it?

Answer 1

They exist in the metaphysical world of forms. Plato's basic geometrical objects like the cubes, tetrads, octahaeders,etc. live there. The are eternal and static. All other mathematical structures kive there too and they can only be approximated in our wordly domain. It is impossible to construct a perfect material cube. Only math can describe them perfectly but this description is not the idea itself. From the Platonic ideas many others can be constructed. The five Platonic bodies serve as atoms for the other forms. After Plato many other forms were developed serving as other atoms. The forms derived from the basic forms belong to this ideal world too. So there are changing forms (ideas) and non-changing eternal ones. For the changing ones time has to exist. The non-changing ones are static and eternal.

Answer 2

Interpretation of the Platonic forms is a big wrangle and may be undertaken in many ways. But the simplest answer to your question "in what realm" might just be to say everywhere or in every "realm."

Thus the "squareness" of a square, to use the old trope, is not "in" this or that particular square thing. In fact, any particular square will not be perfectly square and will not stay square forever. Yet we may see "squareness" reflected in this or that thing and so attach the geometrical predicate. To Plato this is not mere immanence, for it indicates that "squareness" is more real than any things or brain states dimly reflecting it.

The view goes back to Parmenides that what does not change cannot, of course, be itself perceived by our changing sensory input, but is for that very reason more real than shifting phenomena, thus cannot depend upon those phenomena and so exists prior to and independent of any such phenomena.

These are not easy adjustments for modern minds, but not a few notable mathematicians and physicists, Goedel and Penrose, for example, are Platonists in this sense. They simply cannot feel they are "making things up" mathematically or arriving at solutions by way of observations. They feel they are struggling beyond perceptions to discover something objectively "out there."

Perhaps another way to think about it is to consider gravity. Where is it? Everywhere. What is it? It is not this or that material object in which we see "gravity" reflected. It is basically a mathematical form of the universe that we do come to think of as very real, though it would be problematic to say it is "in" the universe.

We are also thrown off by the word mathematical "object," which suggests perceptible things. Gravity or squareness or oneness can be "objects" of our consideration unseen but for, as we say, the practiced "mind's eye," just as a radio antenna is needed to hear the voices "really" in the airwaves all around you. The mind "senses" not a thing but a limit or necessity that we "know must be the case" and so must "be."

To Pythagoreans like Plato such was the "real" necessity of numbers and forms, without which no "ideas" could even begin to take shape out the "radio static" of unformed sense data.