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"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?

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    Where is the quote from? The characterization of intuitionism is pretty bad, cf. SEP "traditional intuitionist views... affirm the existence of mathematical objects but maintain that these objects depend on or are constituted by mathematicians and their activities". And "platonism" as currently used is very broad, with no consensus on what "existence" even means, see e.g. varieties of light platonism where no "realm" is needed at all.
    – Conifold
    Jul 10 at 21:03
  • the quote is from philosophy.stackexchange.
    – Sayaman
    Jul 10 at 21:14
  • Were you asking about "Platonists" who subscribe to Plato's specific metaphysics, or mathematical platonists more generally? Modern mathematical platonists may have little in common with Plato, all that's really required is belief in the existence of mathematical "abstract objects", so that the truth or falsity of our statements about math depends on what is true or false about these abstract objects.
    – Hypnosifl
    Jul 11 at 21:12
  • @Sayaman: You should include a link to the post, for context.
    – CriglCragl
    Jul 14 at 10:12
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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.

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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.

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  • Still the question is where the ideas themselves reside. Plato was cler that they dont exist in the world around us. They might be contained in physical processes but is not what Plato thought. And by the way, an antenna is not needed ti hear the sounds really. A brain is needed for that. Jul 11 at 1:07
  • @DescheleSchilder That depends what you mean by “hear”. Is an analogue circuit that transcribes speech into phonemes not “hearing” that speech? And yet there is no brain.
    – wizzwizz4
    Jul 11 at 18:37

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