[jonbronson]

> But I ultimately think of mathematics as a just an invented tool whose only reason for existence is to solve concrete problems.

This might be the source of disconnect. I frequently encounter this perspective and worry there's a fundamental problem with how mathematics is taught if so many people walk away believing this. Whether or not humans ever mastered mathematics, what is and isn't mathematically true would not change. Humans can create notation and formalisms, but they do not invent the truths those mathematics represent.

[thwayunion]

> Humans can create notation and formalisms, but they do not invent the truths those mathematics represent.

The land represented by a map exists independently of humanity. Another intelligent species would have to come up with a roughly isomorphic representation if they wanted a similar tool.

Maps, to be clear, are just invented tools. They can be more or less right or wrong, but they are not the territory.

Moving up the meta stack does sort of confuse this initially if you don't stay grounded. I wonder if there is a field of meta-map-making and whether map makers sometimes confuse maps with territories when they start meta-map-making work.

> I frequently encounter this perspective and worry there's a fundamental problem with how mathematics is taught if so many people walk away believing this.

I didn't walk away (undergrad, PhD, PI, editorial committees, grant reviewer, ...). I'm about as far from walking away as is possible. But I suppose it is possible I'm an imposture :)

[VirusNewbie]

'roughly isomorphic' would be saying 'not isomorphic', so I'm not sure what you're trying to say. Actually I'm frustrated with most physicists/math folks misusing this term 'isomorphism' to mean "a bijection".

Which gets to the second point, if there is a true isomorphism between the map and the land, it doesn't matter that one isn't the other. That would mean that the land is constrained by the same axioms as the 'map', which gives some significance to them.

[whatshisface]

Any species would prove the exact same theorems given the same axioms, and since mathematicians only claim that their axioms imply their theorems, I think they are right to claim absolute truth.

[marcosdumay]

> I wonder if there is a field of meta-map-making

There certainly is, although I'm not sure it has a name. Kids gets introductions to it on schools, when they have classes about how to read a map in Geography.

[JumpCrisscross]

> Whether or not humans ever mastered mathematics, what is and isn't mathematically true would not change...[we] can create notation and formalisms, but they do not invent the truths those mathematics represent

This is a big philosophical question. (Kant would agree with you. Some of his detractors would not.)

Setting that aside, I agree with your criticism of the claim that mathematics only exists to solve concrete problems. Mathematical fiction, e.g. exploring how a system following nonsense rules might behave, is perfectly good math. It's interesting, potentially beautiful, first and foremost; it might also be useful, though that's of secondary concern. (It has an uncanny knack for being so [1].)

To say math must serve physical reality is to discard its artistic side, perhaps essence; that's disappointing, debilitating and reductive.

[1] https://aapt.scitation.org/doi/abs/10.1119/1.2402156?journal...

[thwayunion]

With respect to aesthetics, to each their own. I do find some mathematics beautiful. In fact, "I am bored" is a real problem and mathematics can be used to solve that problem by being a tool that tickles our brains in pleasant ways.

The "tool" and "problem" here are meant as comments on the metaphysical content of mathematics, not some sort of statement that mathematics is for engineering and that's all.

In particular: I'm commenting on the imbued/latent metaphysics of Wolfram's post, which goes beyond mere artistic appreciation. If his framing were "and look how pretty cellular automata are!" then I guess my reaction would be "yeah they are quite cool aren't they?"

I find Church-Rosser quite beautiful and also think Wolfram puts way too much metaphysical weight into the behavior of confluent rewrite systems. Similarly, some Psalms are beautiful and the story of Jesus is very nice but god does not actually exist. There's no contradiction there -- you can take the beauty and spit out the metaphysics.

[thebooktocome]

> Whether or not humans ever mastered mathematics, what is and isn't mathematically true would not change. Humans can create notation and formalisms, but they do not invent the truths those mathematics represent.

We quite literally have no way of ever knowing this. This proposition and its negation are both beyond the scope of human knowledge.

[whatshisface]

We do have a way of knowing this, it is as simple as saying that the finish line of a symbol game will stay the same given the initial symbols and the rules that can be used to move them around.

[ganzuul]

It's almost as if there is a phase diagram with impedance mismatches between systems of belief.

[mrob]

How is the Banach–Tarski paradox a truth that exists independent of humanity? It makes a physically implausible assumption (existence of infinitely small objects) and reaches a physically implausible conclusion (violation of conversation of mass). Mathematics is full of things like this. They all look like human inventions to me.

[BreakfastB0b]

Banach–Tarski relies upon the Axiom of Choice / Law of the Excluded Middle. Zermelo–Fraenkel set theory is independent of the Axiom of Choice and there's an entire field of Mathematics called Constructivist Mathematics which avoids including the Axiom of Choice / Law of the Excluded Middle.

I had a hard time grasping why the Axiom of Choice / Law of the Excluded Middle was so problematic until I heard it translated into a Computer Science context.

The Law of The Excluded Middle sounds very reasonable at first. For all propositions P, P ∨ ¬P. i.e. Every proposition is either true or false. Sounds fine right? But when viewed in the context of computer science via the Curry-Howard Isomorphism. A proposition is actually a program, and deciding the truth value of a proposition involves "running" that program. So The Law of the Excluded Middle is actually the Halting Problem! It's really saying that all possible programs terminate and yield true or false, but we know that some programs don't terminate, some propositions aren't true or false, but undecidable.

So circling back around to the Banach-Tarski paradox. I would be very skeptical of any paradoxes resulting from assuming the halting problem doesn't exist!

[ganzuul]

General Relativity does not conserve energy.

[vba616]

>they do not invent the truths those mathematics represent

I got the idea somewhere that because Principia Mathematica was doomed to failure, that means any two "islands" of math are not necessarily related to each other.

So I would think that hypothetical aliens in different circumstances could in fact have math that didn't intersect with ours at all.

If they did, wouldn't it be possible to build one system up from foundations?

[arka2147483647]

Most people are intrested in what they can do with mathematics, and therefore, for them, it is a tool.

Because of this, they are more likely to ’get’ mathemathics if it is presented to them as a tool, instead of as an abstract truth-of-everything.