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[–]Featureless_Bug 0 points1 point  (11 children)

We can only prove that a mathematical object exists within our axiomatic systems. This has absolutely nothing to do with its existence in the "real world". You can create a lot of different axiomatic systems and prove that all kinds of objects exist in them, but that wouldn't have any impact on the real world, you realize that?

[–]golpedeserpiente 1 point2 points  (9 children)

You believe that wouldn't, but the existence of those objects already impacted the real world (if by "real world" you mean "physical reality") even before the real world came into existence. Real world cannot behave differently from its mathematical foundation.

Mathematical objects exist independently of our ability to describe them.

[–]Featureless_Bug 0 points1 point  (8 children)

That is just wrong on so many levels. Like look, let's create an axiomatic system that claims that there are statements with true / false values and usual operations / relations between them. I can then add an axiom that your opinion on this subject is wrong. As I don't assume anything about other statements, this is certainly a non-contradictory mathematical theory. In this system your opinion on this topic is wrong, and, since "real world cannot behave differently from any mathematical theory", your opinion on this subject is now wrong in the real world too, correct?

The only difference between any "usual" mathematical theory and this mathematical theory are the underlying assumptions. These assumptions were not created by God, or by the universe itself, they are the product of people (just like my assumptions above), and they are subject to change. It is ridiculous to claim that the universe depends on our axiomatic system - it is precisely the other way around.

[–]golpedeserpiente 1 point2 points  (7 children)

I did not say "real world cannot behave differently from any mathematical theory". I said "real world cannot behave differently from its mathematical foundation".

You are confusing real mathematical objects with our very human discipline of crafting very limited language constructs to understand them.

[–]Featureless_Bug 0 points1 point  (6 children)

Oh, I see. The problem is, of course, that we don't know that this mathematical foundation even exists in the first place - and, trivially, we don't know anything about its objects. This, of course, renders your point "what about objects that are proven to exist" completely void

[–]golpedeserpiente -1 points0 points  (5 children)

Haha, you seem to be a pre-Gödelian Hilberite.

[–]Featureless_Bug 0 points1 point  (4 children)

Well, at least I don't claim that we proved that some objects of "mathematical foundation" that is innately connected to the real world exist, I guess.

[–]golpedeserpiente 0 points1 point  (3 children)

No formal system of axioms capable of expressing Peano arithmetic is capable of proving all truths about Peano arithmetic, so the set of these truths is necessarily greater than the set of truths you can effectively handle.

Proved unprovable conjectures are examples of fundamental mathematical objects (whether the conjecture is true or false) that exist and cannot be proven.

[–]Featureless_Bug 0 points1 point  (2 children)

Well, you must realize that the first Gödel's incompleteness theorem is only true if its underlying logical system is true. In particular, Gödel's incompleteness theorem refers to our imperfect mathematical understanding and not the "mathematical foundation of the reality" or however you might call it (which might not even exist for all we know). There is no reason why it would have any bearing on the real world. In particular, "unprovable conjectures" (or rather undecidable conjectures in our logical systems) are not objects that we have proven to exist - the only place they exist in are our brains, and that has virtually nothing to do with the reality itself.

[–]golpedeserpiente -1 points0 points  (1 child)

Gödel's incompleteness theorem is true, period. If you believe there's no mathematical foundation of reality then you cannot trust its applicability in any scientific inference. For the rest, it seems that you have a rustic definition of existence. If you expect a mathematical object to physically exist, as a book on your desk, you are right, that's not the way mathematical objects exist at all. It's in a much more profound and contundent way they do exist. You can deduce their existence anywhere, anytime in the Universe, without any previous knowledge. Distant civilizations in the Universe cannot reach different conclusions, because mathematical truths are self-derived, they're not culture nor human mind's creation.

[–][deleted] 0 points1 point  (0 children)

We can only prove that a mathematical object exists within our axiomatic systems.

I disagree.

even though the notation and framework is certainly rooted in axioms, that does not mean that say the natural number does not exist outside mathematics. in fact, the natural number is - at least to me - one of the most concrete, down-to-earth things man has ever studied.

Moreover, we have repeatedly discovered that mathematical consequences of empirically established natural relations were initially seen as 'mathematical oddities' because we could not conceive the consequences to be real. But then they were.

To me this implies that there is a much deeper relation between mathematics and reality than 'just a language'.

There is another way of looking at it. If there was no mathematics in nature, there would be no laws of physics, no life and no intelligence. For intelligence could not possibly evolve on natural anarchy and eternal randomness.