> Mathematical truths are defined to be true

Perhaps a nitpick, but: axioms are defined to be true. Mathematics, roughly speaking, is the business of exploring the consequences of chosen sets of axioms. Sometimes published mathematical theorems turn out to be wrong.

I don't follow what you're taking issue with. Naturally by "mathematical truths" I meant axioms and statements that are reducible to axioms - there isn't really anything else for it to mean, after all. On the "sometimes published theorems are wrong" part you've lost me entirely.

> by "mathematical truths" I meant axioms and statements that are reducible to axioms

That distinction is important, and it seemed to me you'd skimmed over it.

Axioms are directly defined to be true for the purposes of mathematical inquiry, and as such can't really be wrong. At worst, a set of axioms can be inconsistent.

Emergent properties are different; they're not directly defined to be true. This is of practical consequence, as mathematicians sometimes get them wrong.

Do you think? I'd say that axioms were just taken; there's no commitment to their truth (since there is no need for it, and mathematics likes economy).

Axiom are just taken but over time we picked the most useful ones. This is why they map well to the physical world.

The baffling thing is how few and simple axioms we need for them to be useful.

In math "true" is essentially just a name for a symbol. Saying that an axiom is true is like saying that 3 comes after 2 - it doesn't imply any kind of truth in the usual "consistent with reality" sense.

It also doesn't make sense to define axioms as true. What if later it turns out that the axioms were a mutual contradiction?

Truth is tricky. If I state "The sky is blue", I can look out of the window, the sky is blue, my statement corresponds to the state of the world, my statement is true, right? But in mathematics there is no window, everything is internal, abstract, so what does "truth" mean when there is no state of the world with which to compare? More importantly, would the mathematics be any different if the axioms are declared to be true? No, it wouldn't make any difference at all. So let's leave "truth" to the physicists and philosophers.

That's the first error of the objectivists. Mistaking "blue" as a property of the sky.

Color happens in your head.

That is why trichromacy, tetrachromacy, pentachromacy, dodecachromacy etc. is a thing.

This is perhaps not all that relevant to your point, but "the sky is blue" so often being used as an example of an obvious truth is amusing to me, because very often the sky is not all that blue at all.

If I look outside right now, the sky is probably better described as cyan; quite obviously a different colour from the blue exercise mat that happens to be on my floor.

The sky can also be described by many of the other colours depending on weather or the time of day.

It's almost as if defining anything as a "mathematical truth" is definitionally wrong and anyone having done so should be called out as a liar.

Only the person who responded to the person who made such a statement were more polite about it, hence this "discussion".

This might be a good time to look up the definition of axiom: "a statement accepted as true as the basis for argument or inference".

Yes, but I think that's "accepted" in the sense of "until we run into a contradiction", at which point you have to start over with new axioms. I.e. it turned out your axioms weren't all true.

I think that in the definition you gave "accepted" should be replaced by "assumed".

"John can dig a hole ten feet deep in an hour. How long would it take him to dig a hole five feet deep?"

The first sentence of that problem is an axiom. It is absolutely 100% true, within the context of that story problem. It could contradict another axiom, but it can't turn out to be false, because it's not a statement about the real world.

Thank you, I took the same issue you did when reading this.

There are not "truths" in mathematics, it's literally a system of logic with definitions at the bottom (axioms) and then a following of the results based upon the agreed upon logic.

The argument echos Kant...which is too big a sea to boil in a comment. So at geostationary orbit, Kant's metaphysics says we cannot have absolute knowledge of the empirical world, only knowledge mediated by the a priori conditions of human experience, space and time. However, we can have absolute knowledge of morals because such knowledge is not dependent on the aforementioned a priori conditions of human experience.

Going further, in Critique of Pure Reason Kant uses mathematics as an example of synthetic a priori knowledge. This is the lever he uses to overturn the traditional halving of all knowledge into either analytic and a priori or synthetic and a posteriori (empirical). The halving was the basis for Hume's radical skepticism and it's predecessors.

Maths is what happens when you take nonsense very seriously.

For some odd reason a subset of all the possible non-sense maps really well to the real world.

I think the terms used in Mathematics are a tad more precise than those in natural language e.g. 1 is fairly clear, "moral" is not as clear.

The reason Maths works is because it is based on intuitions (counting, space) that are presumably shadows of the physical world cast on the evolving brain via coarse perception. The effectiveness of Mathematics in Physics is probably not unreasonable at all.

Clear to whom? The meaning of "1" is conventional/socially constructed.

For the sake of argument I can conveniently forget what "1" means, and then you can try and explain it to me like I am from another planet.

Semantics are a non-trivial matter.

To be extremely precise: 1 is the successor of 0. Anywhere that there is a natural numbers object, there's a semantics for this statement. What is socially constructed here is the choice of topos which hosts the natural numbers object!

SUCC(0) is no explanation of 1, until you explain 0.

Note how you are reaching for complex/abstract ideas (like a "topos" and "natural numbers") to explain simple/intuitive ones (like 0 and 1).

"toposes" and "sets" (of "natural numbers") are socially constructed in the subculture of Mathematicians.

Outside of that shared experience, they are pretty meaningless.

I guess you are on the "discovered" side, and I am on the "invented" side of Mathematics ;)

Humans are what, about 99.9% similar to each other on average? So that gives me 3 nines belief that yes, if I can understand, then you can understand; we're just humans after all. I think that you're being sarcastic; once we define categories, Cartesian closed categories, and elementary topoi, then categorical set theory is a straightforward sequence of diagram schemata. But I understand your point!

You're super-close to a deep realization: all words are only sensical to certain subcultures. IOW there's no absolute meaning to any word. In that perspective, we're both right; "1" is meaningless and 1 is categorical, and it's just a question of pointers vs. names.

Another quick and deep corollary is that reality is socially constructed; whenever a quorum of humans is mutually intelligible during a conversation, then they are agreeing on the local nature of reality. Humans can't construct global maps of reality, though, since they can't observe the Universe all at once. Indeed global maps of reality are forbidden by the Kochen-Specker Theorem.

Edit: To clarify for the audience, I have no appeal to higher categories here, and the definitions of category theory are not some sort of categorization or classification process, but axiomatic definitions akin to set theory.

Heh, how did I know you are going to appeal to infinity-categories at some point?

"defining categories" and their respective categorisation rules is the process of classification. They don't account for the classifiers themselves.

You are super-close to a deeper realisation even.

If any notation is meaningful (even one that uses symbols like ∞), then it's Turing-recognisable.

Type-0 Chomsky grammar. In formal languages syntax is semantics.

In so far as understanding (comprehension?) goes, you could say that I subscribe to the axiom of unrestricted comprehension. It's rather un-Mathematical doing so.