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I will explain why I am asking this question. Let's say, there are mathematical truths and truths about our physical reality. But, there is no way we can establish the truth of any statement about our Physical Reality.

Consider a mathematical statement "There are infinitely many primes", I can give a formal proof to establish the truth of this statement.

Consider a statement about our Physical Reality "Sun Exists", I can give many Physical evidences that correspond to the Existence of sun, but still, there is no way I can establish the truth of the statement, like the Mathematical statement.

So, what is this distinction that sets apart Mathematical Truths from Physical truths? Or Is there anything such as Physical truths? Even if so, Mathematical Truths are "kind of" different from truths about the Physical Reality.

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    Your question is a little too broad for our site, the nature of mathematical vs physical existence is a very vast subject, see SEP, Platonism in the Philosophy of Mathematics for a general introduction. – Conifold Feb 29 at 0:53
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    Physical truths - nobody has ever seen what matter 'is'. We only perceive what our individual senses perceive. Nobody has been able to go outside their bodies and understand what matter 'is'. – Swami Vishwananda Feb 29 at 5:11
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    The problem of the "nature" of mathematical objects is on the table since Plato and is still lacking a clear answer. See Abstract Objects: numbers are the paradigmatic examples of them. If there are really abstract objects, where they live? how we know them (how we "interact" with them, due to the fact that they are not physical?) – Mauro ALLEGRANZA Mar 1 at 10:46
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The reason for this difference is due to relativity. By relativity I mean "the state of being judged in comparison with other things and not by itself."

Physical reality is based on physical world and we know that it is ever changing. Therefore we cannot establish physical truths. Unfortunately we are not even aware that if there is another, that reality is based on this Physical reality. You would realize its seriousness if you could think about the 'reality of "one"'--the first natural number or 'the element of mathematics'. That means, Mathematics doesn't verify even the reality of one. Mathematics, since it is a subject, it doesn't need to verify the truth about a statement beyond its limits. If it did so, the subject would change and would merge in Physical reality and would grope for another reality.

As I said earlier, because of relativity 'Mathematical reality' seems 'rigid and Mathematical Truths have its own entity'. But the first one itself is baseless as for now. So Mathematics doesn't/can't “exist” in any sort of “Reality” that is different from our Physical Reality.

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    What do you mean by relativity here? Is it a stand-in term for someone or something? In which case, for the former - whom? And if the latter - what? – Mozibur Ullah Feb 29 at 16:49
  • In otherwise unqualified contexts, "relativity" usually refers to Einstein's theories of relativity. However, since those don't appear to be related to this topic, it's unclear what you might be referring to. – Nat Feb 29 at 21:37
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    @Mozibur Ullah: Comparison happens in persons only. So it is a term for someone. "In which case, for the former - whom?"~To those who see different realities. – SonOfThought Mar 1 at 6:56
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    @Nat:I used it as a commonly used word. I didn't wish to invite Einstein here and make it complicated. – SonOfThought Mar 1 at 6:59
  • @SonOfThought: comparison happens with things, place and ideas too. – Mozibur Ullah Mar 2 at 17:17
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Mathematics largely deals with formal systems. For example, much of modern mathematics can be constructed in terms of set theory. Once the formal system is established, mathematical "truths" are derived based on the formal rules of the system. If we consider chess as another example of a formal system, then a true statement is loosely equivalent to a sequence of valid moves. Similarly, a true mathematical statement is equivalent to "valid moves" in set theory or another formal system.

On the other hand there is no formal system for physical reality. So we can not establish "truth" through applying formal rules. Instead we generally use observations from physical reality in the form of experiments, etc. to obtain empirical, rather than formal, "truth". This is a crucial distinction in epistemology.

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The proof of physical truth is your own consciousness. Your direct awareness of some phenomena cannot be proved by any external or material means. In this sense, your direct conscious experience of Sun in undemonstrable (and all people know this). At the same time you have no doubt that you are experiencing the Sun. And you don't need any proof because you know it - your consciousness is good for it self; it is complete and it doesn't needs any backup proof from outside. Consciousness is self-referential.

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    Welcome to SE Philosophy! Thanks for your contribution. Please take a quick moment to take the tour or find help. You can perform searches here or seek additional clarification at the meta site. – J D Mar 11 at 15:58
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Math is based on logic and logic is based on physical experience. Therefore there is no difference between physical and mathematical "truths" other than the level of abstraction.

You could of course make up alternative logics that are not based on experience, but these wouldnt lead to any useable "truth". If mathematically I define bananas to be the same as apples, then say "bananas equal apples", I would have reached a mathematical "truth", but no one would care.

The question of why physics and math are so consistent often pops up and I would answer that : 1)logic is based on experience (A=A because that's what we usually observe)

2)math is based on logic

3)physics is based on math, logic and experience

4) it all works out because nature is consistent with itself

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What you call "physical reality" was called by Kant noumenon or thing-in-itself. We can't know it, we don't know it, we have no access to it. Its counterpart is phenomenon, what we can perceive and cognize, which is subjective for every individual. Reality is just a structure of ideas, based on perception, that exists in each individual mind.

Therefore, a “Reality” that is different from our Physical Reality is not possible. The only possible reality is not unique, it is subjective, and is an attempt to model the noumenon.

Mathematics just express facts about such subjective reality. Therefore, it exists on the same realm as objects (objects being a subjective model of part of such reality).

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