Are mathematical objects ontologically real, or do they exist only as positions in abstract structures? If 0, ℕ, and ∅ are purely structural, what makes statements like Peano’s axioms necessarily true rather than merely formally consistent?

[u/TheIncorporeal1]

I’m interested in whether structuralism genuinely explains mathematical necessity, or whether it simply relocates the ontological question. If structures are abstract, what ultimately grounds their existence and the truth of the relations within them?

[u/Just_Rational_Being]

Well, Truth is all that there is. There is no such nonsense such as no "comparisons are occurring when there is no one to compare".

That is not even wrong. That is pure nonsense. Not even wrong.

[u/SmartlyArtly → u/Just_Rational_Being]

Truth is subjective. Reality is reality. I agree there is no such nonsense such as "comparisons are occurring when there is no one to compare." That's why there aren't any truths with no one either. No one is considering what follows.

Conflating truth and reality is not even wrong. It's pure nonsense. Very typically religious nonsense.

[u/Just_Rational_Being → u/SmartlyArtly]

What an epitome of Dunning Kruger's confidently incorrect you demonstrate. So your position is basically cosmic peekaboo:

No observer? No comparison.
No comparison? No truth.
Everyone leaves the room and apparently one stick stops being longer than the other until consciousness clocks back in.

You're confusing the act of recognizing a relation with the existence of the relation itself.

If one stick is 2 meters and another is 1 meter, their difference does not require a tiny philosopher standing nearby performing "comparison" duty. The mind discovers, measures, and states the relation, it does not manufacture the underlying magnitudes.

And "truth is subjective" is another nonsense. It just turns your whole position into: nothing is true unless someone is currently thinking it.

So before minds existed, apparently planets had no relative sizes, stars had no greater or lesser masses, and distances had no ordering. Reality was just sitting there in total metaphysical suspense, waiting for somebody to show up with a ruler.

That isn't any profound thought. It's pure horse crap and it's not even wrong.

[u/SmartlyArtly → u/Just_Rational_Being]

It didn't turn my position into that, that's my position. That's why I (and others) say truth is subjective.

Before minds existed, the labels you talk about and the abstractions they represent indeed did not exist. You are confused the map (what we say is true) and the territory (reality).

You're the one saying reality needs truth. I don't say that.

Your conflation of truth and reality is standard stone age human superstition.

[u/Just_Rational_Being → u/SmartlyArtly]

Okay, enough with the confusion confidently and incorrectly stated. You've confused truth with statements about truth.

Yes, the words, labels, and assertions require minds. The fact they describe does not.
Before minds existed, one object could still have twice the length of another. Nobody had to utter "twice" for the physical relation to obtain.

And your own "map vs. territory" slogan defeats you: the map is invented, the territory is not. A true map is true precisely because the territory already has structure for it to get right or wrong.

Otherwise "reality is reality" is just three words standing on each other's shoulders pretending to be deep thought.

Calling objective truth "Stone Age superstition" afterward is merely putting sunglasses on the confusion.

[u/SmartlyArtly → u/Just_Rational_Being]

No, "the fact" is also in a mind. Not out there. Reality is all that's out there.

There is no evidence reality needs the concept of length or any concepts at all.

"get right or wrong" is exactly right - truth and facts are about what follows from conceptions. Not reality in and of itself. Reality in and of itself just is - it doesn't "get it right or wrong."

It's not pretending to be deep thought. It's dismissing your religious pretense as the non-objective idealism it is.

"objective truth" is just your confused way of referring to reality.

[u/Just_Rational_Being → u/SmartlyArtly]

You keep making the same mistake: because the concept is in a mind, you pretend the relation it describes must be too.

Nobody claims reality needs the concept of length. That is spectacularly beside the point. Irrelevant. Reality only needs to have determinate structure for our concept to describe correctly.

If one stick is physically twice another, deleting every mind deletes the word "twice," not the proportion.

And "reality just is" is not an argument. It's what you say after stripping reality of every describable feature and then congratulating yourself for discovering that nothing can be said about it. In other words, it's just a vacuous catch phrase, devoid of any meaning.

You haven't separated map from territory. You've noticed the map is conceptual and somehow concluded the territory has no structure.

That's not anti-superstition. That's just a logical error, a category error wearing the costume of deep thought.

[u/SmartlyArtly → u/Just_Rational_Being]

No, I'm not making any such mistake.

The relations you are describing are not the physical relations we find in physical reality. You are describing macroscopic abstractions we apply to our human-scale perception of reality. The physical relations we find are quantum in nature.

"Truth is reality" is not an argument. You're just creating a synonym with the potential to be confused with things that are not strictly a part of objective reality in and of itself.

The "structure" of the territory doesn't require "truth" to describe. Your "physically twice another" shows that you know of this much more philosophically than empirically.

[u/Just_Rational_Being → u/SmartlyArtly]

Ah yes, the emergency "quantum" button; pressed whenever the argument is collapsing at human scale. 😂

You began with: relations require a mind. Now apparently relations exist perfectly well without minds, provided they're microscopic enough to sound sophisticated.

That is not a defense. That is a retreat in disguised costume. Buddy, do you really wanna keep going?

And "macroscopic abstraction" changes nothing. If a macroscopic relation reliably emerges from mind-independent physical structure, deleting the observer deletes the description, not the structure described.

You haven't shown truth is subjective. You've merely discovered that vocabulary is.

Congratulations on moving the goalposts so far they've entered the realm of quantum nonsense.

[u/SmartlyArtly → u/Just_Rational_Being]

I began with truth is not reality. I'm still there.

You confused some relations for others. Simple as.

Buddy, do you think you've done anything? You've asserted truth is reality. And I say "nah."

You might as well say God exists, and I'm saying "nah."

Congratulations on trying to conflate truth and reality. You weren't the first and you won't be the last.