How has mathematics gotten so abstract? • Hacker News Board

Re: How has mathematics gotten so abstract?

#181

Post by griffzhowl » Wed, Oct 01, 2025, 1:14 AM UTC

empath75 wrote: post #68

nkrisc wrote: post #56

Earlier quoted context omitted.

> The next paragraph about how mathematics was closely coupled to reality for most of history and only recently with our understanding of infinite sets became too abstract is not really at all accurate of the history of mathematics. Euclid's Elements is 2300 years old and is presented in a completely abstract way. I may be off-base as an outsider to mathematics, but Euclid’s Elements, per my understanding, is very mu…

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…

Plato was only about a generation before Euclid. Their lives might have even overlapped, or nearly so: Plato died in 347BC and Euclid's dates aren't known but the Elements is generally dated ~300BC

Re: How has mathematics gotten so abstract?

#182

Post by griffzhowl » Wed, Oct 01, 2025, 1:31 AM 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.

I don't think we can say the Greeks were exploring complex numbers. There's something about Diophantus finding a way to combine two right-angled triangles to produce a third triangle whose hypotenuse is the product of the hypotenuses of the first two triangles. He finds an identity that's equivalent to complex multiplication, but this is because complex multiplication has a straighforward geometric interpretation in the plane that corresponds to this way of combining triangles.

There's a nice (brief) discussion in section 20.2 of Stillwell's Mathematics and its History

Re: How has mathematics gotten so abstract?

#183

Post by griffzhowl » Wed, Oct 01, 2025, 1:58 AM UTC

susam wrote: post #134

This article explores a particular kind of abstractness in mathematics, especially the construction of numbers and the cardinalities of infinite sets. It is all very interesting indeed. However, the kind of abstractness I most enjoy in mathematics is found in algebraic structures such as groups and rings, or even simpler structures like magmas and monoids. These structures avoid relying on specific types of numbers o…

When I was learning me a Haskell I had a great time when I realised that as long as my type was a monoid I could freely chain the operations together purely because of associativity

Re: How has mathematics gotten so abstract?

#184

Post by griffzhowl » Wed, Oct 01, 2025, 2:13 AM UTC

m_dupont wrote: post #137

trinsic2 wrote: post #136

>Next, consider the time needed for Achilles to reach the yellow dot; once again, by the time he gets there, the turtle will have moved forward a tiny bit. This process can be continued indefinitely; the gap keeps getting smaller but never goes to zero, so we must conclude that Achilles can’t possibly win the race. Am i daft, eventually (Very soon) Achilles would over take the turtles position regardless of how far i…

you're not, the proof is a famous error known as zenos paradox. Its only an apparent paradox, and indeed it's been disproven by observing that things do in fact move

I like the humourous way of putting it, but of course Zeno and his contemporaries knew that things moved - that's exactly why this seemed to be a paradox. Seemingly secure reasoning results in a conclusion that's obviously false.

To resolve the paradox, you have to show what's wrong with the reasoning, not just observe the obviously false conclusion.

Re: How has mathematics gotten so abstract?

#185

Post by griffzhowl » Wed, Oct 01, 2025, 2:25 AM UTC

naasking wrote: post #162

goatlover wrote: post #148

Earlier quoted context omitted.

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...

Tell them to mind their own business ;

Re: How has mathematics gotten so abstract?

#186

Post by aristofun » Wed, Oct 01, 2025, 2:57 AM UTC

The_suffocated wrote: post #94

aristofun wrote: post #42

How has blog posts authors gotten so uneducated or/and clickbaiting? Math in its core has always been abstract. It’s the whole point.

> Math in its core has always been abstract. It’s the whole point. I don't think so. E.g. there may be some abstractions in numerical linear algebra, but the subject matter has always been quite concrete.

It is not a matter of what you think it is a logical fact, part of the definition if you will.

What you call concrete - were the origins of math as we know it. Geometry, astronomy, metaphysics etc they all had in common fundamental abstract thing that we call math today.

Saying “math got abstract” is like saying “a tree got wooden”. Because when it was a seed - it wasn’t yet a tree in a full sense.

Re: How has mathematics gotten so abstract?

#187

Post by mcmoor » Wed, Oct 01, 2025, 3:57 AM UTC

pdpi wrote: post #21

Tazerenix wrote: post #13

>Today, mathematics is regarded as an abstract science. Pure mathematics is regarded as an abstract science, which it is by definition . Arnol'd argued vehemently and much more convincingly for the viewpoint that all mathematics is (and must be) linked to the natural sciences. >On forums such as Stack Exchange, trained mathematicians may sneer at newcomers who ask for intuitive explanations of mathematical constructs…

> 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.

It isn't, that's why it's in own section in STEM, and rightfully so. It's a higher tool that without it, science would come to a screeching halt.

Re: How has mathematics gotten so abstract?

#188

Post by drdeca » Wed, Oct 01, 2025, 4:43 AM UTC

btilly wrote: post #170

drdeca wrote: post #164

Earlier quoted context omitted.

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.…

> Things only exist when you can construct them.

This is exactly what I’m saying is presumptive! If constructivism is to earn the merit of being less presumptive by virtue of not assuming the existence of various things, it should also not assume the non-existence of those things.

Which, I think many visions of constructivism do earn this merit, but not your description of it.

Re: How has mathematics gotten so abstract?

#189

Post by nextaccountic » Wed, Oct 01, 2025, 4:57 AM UTC

edanm wrote: post #114

btilly wrote: post #104

Earlier quoted context omitted.

When you say "given ZFC", you're assuming a lot. Including a notion of mathematical existence which bears little relation to any concept that most lay people have of what mathematical existence might mean. In particular, you have made sufficient assumptions to prove that almost all real numbers that exist can never be specified in any possible finite description. In what sense do they exist? You also wind up with wei…

> When you say "given ZFC", you're assuming a lot. Errr, I'm just assuming the axioms of ZFC. That's literally all I'm doing. > In what sense do [numbers that can't be finitely specified] exist? In the sense that we can describe rules that lead to them, and describe how to work with them. I understand that you're trying to tie the notion of "existence" to constructability, and that's fine. That's one way to play the…

> Errr, I'm just assuming the axioms of ZFC. That's literally all I'm doing.

ZFC (and its underlying classical logic) is precisely the problem here though

Re: How has mathematics gotten so abstract?

#190

Post by btilly » Wed, Oct 01, 2025, 5:34 AM UTC

zozbot234 wrote: post #172

btilly wrote: post #170

Earlier quoted context omitted.

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.…

You can think that something doesn't exist in the general case, while still allowing that it might exist in unspecified narrow cases where additional constraints could apply. For example, there might be algorithms that can decide the halting problem for some non-Turing complete class of programs. Being able to talk in full generality about how such special cases might work is the whole point of non-constructive reaso…

Well yes. We can certainly make a function that acts something like that oracle in some special cases. But my point was to give an example of something that cannot be constructively created. The oracle that I described cannot exist within the universe of constructable things.