How has mathematics gotten so abstract? • Hacker News Board

Re: How has mathematics gotten so abstract?

#161

Post by jjgreen » Tue, Sep 30, 2025, 8:51 PM UTC

prmph wrote: post #139

jjgreen wrote: post #25

The number 1 is what a cow, a fox, a stone ... have in common, oneness. Mathematics is abstraction, written down.

That's not obvious. - they are material objects - they are concepts I understand - they are sequences of letters - they are English words - ... Not sure why oneness is privileged as what they have in common, and their oneness is meaningless by itself. Oneness is a property that is only meaningful in relation to other concepts of objects.

A rock is not physically a material object, it is a region of space where the electrons, protons and neutrons are differently arranged, and that region is fuzzy, difficult to determine; but as physical beings, as monkeys, we recognise its oneness, that's necessary for our survival in this physical world, we see this blurred outline of a rock, we feel it's weight in our hand, we observe its practical difference from two rocks. Just as we recognise twoness in a pair of rocks, fish, apples, threeness in a triple of parrots, of carrots, we abstract those out into 1, 2, 3, ...

Re: How has mathematics gotten so abstract?

#162

Post by naasking » Tue, Sep 30, 2025, 8:51 PM UTC

goatlover wrote: post #148

abdullahkhalids wrote: post #87

Earlier quoted context omitted.

Mathematical proofs are checked by noisy finite computational machines (humans). Even computer proofs' inputs-outputs are interpreted by humans. Your uncertainty in a theorem is lower bounded by the inherent error rate of human brains.

Plenty of mathematical proofs have been proven true with 100% certainty. Complicated proofs that involve a lot of steps and checking can have errors. They can also be proven true if exhaustively checked.

> Plenty of mathematical proofs have been proven true with 100% certainty

Solipsists would like to have a word with you...

Re: How has mathematics gotten so abstract?

#163

Post by mrguyorama » Tue, Sep 30, 2025, 8:56 PM UTC

kannanvijayan wrote: post #74

empath75 wrote: post #68

Earlier quoted context omitted.

Quite the opposite, Plato, several hundred years before Euclid was already talking about geometry as abstract, and indeed the world of ideas and mathematics as being _more real_ than the physical world, and Euclid is very much in that tradition. I am going to quote from the _very beginning_ of the elements: Definition 1. A point is that which has no part. Definition 2. A line is breadthless length. Both of these two…

Another point to keep in mind is that a lot of mathematics that's not considered abstract _now_ was definitely considered "hopelessly" abstract at the time of its conception. The complex number system started being explored by the greeks long before any notion of the value of complex spaces existed, and could be mapped to something in reality.

Hell, 0 used to be considered too abstract!

Re: How has mathematics gotten so abstract?

#164

Post by drdeca » Tue, Sep 30, 2025, 9:09 PM UTC

btilly wrote: post #147

zozbot234 wrote: post #135

Earlier quoted context omitted.

> In what sense do they exist? In the sense that all statements of non-constructive "existence" are made, viz. "you can't prove that they don't exist in the general case", so you are allowed to work under the stronger assumption that they also exist constructively, without any contradiction resulting. That can certainly be useful in some applications.

Sure, we can choose to work in a set of axioms that says that there exists an oracle that can solve the Halting problem. But the fact that such systems don't create contradictions emphatically *DOES NOT* demonstrate the constructive existence of such an oracle. Doubly not given that in various usual constructivist systems, it is easily provable that nothing that exists can serve as such an oracle.

If such a system proved that the answer to some decidable question was x, when the actual answer was y, then the system would prove a contradiction. If the system doesn’t prove a contradiction, then that situation doesn’t happen, so you can trust its answers to decidable questions.

If the only questions you accept as meaningful are the decidable ones, then you can trust its answers for all the questions you accept as meaningful and for which it has answers.

Also, “provable that nothing that exists can serve as such an oracle” seems pretty presumptive about what things can exist? Shouldn’t that be more like, “nothing which can be given in such-and-such way (essentially, no computable procedure) can be such an oracle”?

Why treat it as axiomatic that nothing that isn’t Turing-computable can exist? It seems unlikely that any finite physical object can compute any deterministic non-Turing-computable function (because it seems like state spaces for bounded regions of space have bounded dimension), but that’s not something that should be a priori, I think.

I guess it wouldn’t really be verifiable if such a machine did exist, because we would have no way to confirm that it never errs? Ah, wait, no, maybe using the MIP* = RE result, maybe we could in principle use that to test it?

Re: How has mathematics gotten so abstract?

#165

Post by anthk » Tue, Sep 30, 2025, 9:13 PM UTC

stonemetal12 wrote: post #60

rob74 wrote: post #24

How has mathematics gotten so abstract? My understanding was that mathematics was abstract from the very beginning . Sure, you can say that two cows plus two more cows makes four cows, but that already is an abstraction - someone who has no knowledge of math might object that one cow is rarely exactly the same as another cow, so just assigning the value "1" to any cow you see is an oversimplification. Of course, simp…

Mathematics arose from ancient humans need to count and measure. Even the invention\discovery of Calculus was in service to physics. It has probably only been 300 years or so since Mathematics has been symbolic, before that it was more geometric and more attached to the physical world. Leibniz (late 1600s) helped to popularize negative numbers. At the time most mathematicians thought they were "absurd" and "fictitiou…

Archimedes did Calculus before Newton.

https://en.wikipedia.org/wiki/The_Method_of_Mechanical_Theor...

Re: How has mathematics gotten so abstract?

#166

Post by falcor84 » Tue, Sep 30, 2025, 9:14 PM UTC

Ar-Curunir wrote: post #125

falcor84 wrote: post #36

I found it a bit ironic that the author introduced C code there as an aid, but didn't incorporate it into their argument. As I see it, code is exactly the bridge between abstract math and the empirical world - the process of writing code to implement your mathematical structure and then seeing if it gives you the output you expect (or better yet, with Lean, if it proves your proposition) essentially makes math a natu…

No, the correctness of your implementation is a mathematical statement about a computation running a particular computational environment, and can be reasoned about from first principles without ever invoking a computer. Whether your computation gives reasonable outputs on certain inputs says nothing (in general) about the original mathematics.

While mathematics "can" be reasoned about from first principles, the history of math is chock-full of examples of professional mathematicians convinced by unsound and wrong arguments. I prefer the clarity of performing math experiments and validating proofs on a computer.

Re: How has mathematics gotten so abstract?

#167

Post by drdeca » Tue, Sep 30, 2025, 9:20 PM UTC

abdullahkhalids wrote: post #87

pdpi wrote: post #21

Earlier quoted context omitted.

> Pure mathematics is regarded as an abstract science, which it is by definition. I'd argue that, by definition, mathemtatics is not, and cannot be, a science. Mathematics deals with provable truths, science cannot prove truth and must deal falsifiability instead.

Mathematical proofs are checked by noisy finite computational machines (humans). Even computer proofs' inputs-outputs are interpreted by humans. Your uncertainty in a theorem is lower bounded by the inherent error rate of human brains.

This may be, but not, I think, in a way that is particularly worth modeling?

When we try to model something probabilistically, it is usually not a great idea to model the probability that we made an error in our probability calculations as part of our calculations of the probability.

Ultimately, we must act. It does no good to suppose that “perhaps all of our beliefs are incoherent and we are utterly incapable of reason”.

Re: How has mathematics gotten so abstract?

#168

Post by mensetmanusman » Tue, Sep 30, 2025, 9:21 PM UTC

btilly wrote: post #89

thadt wrote: post #2

This reminds of of that one time when I was on a date with a girl from the history department who somehow bemusedly sat through my entire mini-lecture on comparing infinite sets. Twenty years and three kids later, she'll still occasionally look me straight in the eye and declare "my infinity is bigger than your infinity."

I'm curious. Did either of you ever notice the implicit philosophical assumptions that you have to make to come to the conclusion that one infinity can be larger than another? Despite the fact that this was actively debated for decades, modern math courses seldom acknowledge the fact that they are making unprovable intellectual leaps along the way.

Don’t worry, we have only decided that there are two sizes of Infinitis- normal ones and really big ones.

Re: How has mathematics gotten so abstract?

#169

Post by btilly » Tue, Sep 30, 2025, 9:28 PM UTC

zozbot234 wrote: post #157

btilly wrote: post #154

Earlier quoted context omitted.

That's like asserting the existence of a bank account in my name with a billion dollars in it that I know nothing about. I won't ever be able to find a contradiction from that claim, because I have no way to find that bank account if it exists. But that argument also won't convince me that the bank account exists.

That argument ought to convince you that there's a mere "possible world" where that bank account turns out to exist. Sometimes we are implicitly interested in these special-cased "possible worlds", even though they'll involve conditions that we aren't quite sure about. Non-constructive existence is nothing more than a handy way of talking about such things, compared to the constructively correct "it's not the case th…

It would be weird for a constructivist to be interested in a possible world that they don't believe exists.

Theoretically possible? Sure. But the kinds of questions that lead you there are generally in opposition to the kinds of principles that lead someone to prefer constructivism.

Re: How has mathematics gotten so abstract?

#170

Post by btilly » Tue, Sep 30, 2025, 9:42 PM UTC

drdeca wrote: post #164

btilly wrote: post #147

Earlier quoted context omitted.

Sure, we can choose to work in a set of axioms that says that there exists an oracle that can solve the Halting problem. But the fact that such systems don't create contradictions emphatically *DOES NOT* demonstrate the constructive existence of such an oracle. Doubly not given that in various usual constructivist systems, it is easily provable that nothing that exists can serve as such an oracle.

If such a system proved that the answer to some decidable question was x, when the actual answer was y, then the system would prove a contradiction. If the system doesn’t prove a contradiction, then that situation doesn’t happen, so you can trust its answers to decidable questions. If the only questions you accept as meaningful are the decidable ones, then you can trust its answers for all the questions you accept as…

You're literally talking about how I should regard the hypothetical answers that might be produced by something that I think doesn't exist. There's a pretty clear case of putting the cart before the horse here.

On being presumptive about what things can exist, that's the whole point of constructivism. Things only exist when you can construct them.

We start with things that everyone accepts, like the natural numbers. We add to that all of the mathematical entities that can be constructed from those things. This provides us with a closed and countable universe of possible mathematical entities. We have a pretty clear notion of what it means for something in this universe to exist. We cannot be convinced of the existence of anything that is outside of the universe without making extra philosophical assumptions. Philosophical assumptions of exactly the kind that constructivists do not like.

This constructible universe includes a model of computation that fits Turing machines. But it does not contain the ability to describe or run any procedure that can't fit onto a Turing machine.

Therefore an oracle to decide the Halting problem does not exist within the constructible universe. And so your ability to imagine such an oracle, won't convince a constructivist to accept its existence.