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Comment by whatshisface | Hacker News Reader

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.

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Oh, dear. Truth being the operating word… There is no truth in a set of axioms we cannot even conceive properly (any infinite set has properties beyond what seems reasonable, even “just” the Natural numbers). From that comes arithmetic, the “most elementary” form of mathematics which cannot be proved consistent…

We (I am a working mathematician) do not understand our objects, we can just make do. Only finite graph theory has a chance of being “real”. And it stops being finite very soon.

And we certainly should be honest enough to admit that our “science” says very little about the “real” world, where truth lies.

Maths is just a tool. Funny, exciting and even in some sense beautiful. But “truth” does it not contain. Except, I insist, in very specific finite constructions.

Statements hold but they are not “true” because they do not relate to the real world (otherwise, Frodo reaching Mount Doom would also be “true”).

There are no continuous functions out there. Bolzano’s theorem is not “true”.

I would contend that A -> B can be true even if A is not true or more relevantly to this discussion if A is unknown. That's math's version of objective truth, where "A" is filled by our various axioms and rules of inference.

In ~300 BCE, Euclid decides to work with morphisms not objects.

How can you explain appealing to these “unreal objects” (real numbers, set theory, arithmetic) * does* help science? (Effectiveness maybe)

I see you are also a non realist about science.

But even the methodological naturalist (one who takes natural empirical science to be the best method but not an ontology) must wonder how we are uncovering and putting more precision to more and more of the world.

I don’t think we can currently explain why this made up tool “works”.

The enduring appeal of both triangles and Frodo is that they relate a truth that is hard to see, because it’s diffuse and abstract.

But that “abstract truth” is why your pizza doesn’t flop if you fold it or Amazon can’t make a good LOTR.

> From that comes arithmetic, the “most elementary” form of mathematics which cannot be proved consistent…

This is kind of wrong. Are you familiar with Godel's thoughts on this?

...are you?

Very much so.

So lording esoteric knowledge over the anonymous rabble is more productive than just explaining what is the issue?

ok fair enough. There is no issue with proving arithmetic consistency on a finite number of symbols. Incompleteness entirely relies on the unbounded induction step.

Any species could prove the same theorems given the same axioms, but (besides the fact that they might not choose the same axioms) I'm not sure if they would prove the same subset of theorems that we have proven/will prove. Perhaps they'd have different ideas about what is interesting.

Human mathematicians are already fanning out into other systems of deduction (constructive mathematics being a great example), and given enough time the mathematicians of each galaxy will eventually discover the other galaxy's mathematics, even if it perhaps happens in a different order.

Surely intergalactic mathematicians already know that the only time is now? =)

> As Prigogine explains, determinism is fundamentally a denial of the arrow of time. With no arrow of time, there is no longer a privileged moment known as the "present," which follows a determined "past" and precedes an undetermined "future." All of time is simply given, with the future as determined or as undetermined as the past. With irreversibility, the arrow of time is reintroduced to physics.

This is either an extremely obvious and boring observation or the basis for a metaphysical trip, depending on how pre-disposed you are to "mathematical spiritualism" :)

Nothing can be objectively interesting, only objectively true. Just because math is objectively true does not mean you've been robbed of your license to decide if you think it's interesting. :)

Agreed. Similarly, just because I don't "get" church doesn't mean I can prove God doesn't exist. And it certainly doesn't mean I should stand in the way of others enjoying the experience of going to church regardless of their beliefs. It just means I don't "get" it.

Religion is slightly different though, they claim actual direct truth (not mere truth of implication given certain assumptions) which makes their claims more interesting but prevents them from claiming automatic objective truth. The Formal Gospel would go, "If God so loved the world that he gave his only begotten son, ..." ;)

That's why I used words like "spirituality" and "faith" instead of "religion" and "dogma".

I'm describing a shared social/psychological phenomenon, not a shared epistemological status.

Read my posts less as works of the philosophy of mathematics and more as anthropological musings.

That begs the question. Would any other species pick the exact same axioms? Why would they have the exact same theorems? Are you suggesting there is only one way to think logically?

Any species would face a selective pressure towards theorems which help them understand the world around them (if they have any motivation to prove theorems at all), and they will similarly face a pressure towards choosing the smallest/simplest set of axioms which allows all those theorems to be proven (and new ones to be discovered).

In fact, if we assume that neural networks are the only sorts of intelligence that can occur naturally in the universe and be sophisticated enough for arbitrary abstract calculation[0], then we might be able to infer things about the sorts of concepts they will develop and in what order. For example, having the concept of finite sums would likely occur before having the concept of infinite sums.

[0] I know that cellular automata can emulate a universal Turing machine, but I can't imagine a situation existing in nature where the cells evolve into an arrangement that produces a Turing machine, much less a machine running a program of instructions that lead to it generating mathematical theorems.

> Any species would prove the exact same theorems given the same axioms

Assume there exists at least one species....

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