Mathematics is hard for mathematicians to understand too • Hacker News Board

Re: Mathematics is hard for mathematicians to understand too

#41

Post by nkrisc » Wed, Dec 03, 2025, 1:38 PM UTC

bell-cot wrote: post #32

johngossman wrote: post #5

Mathematics is such an old field, older than anything except arguably philosophy, that it's too broad and deep for anyone to really understand everything. Even in graduate school I often took classes in things discovered by Gauss or Euler centuries before. A lot of the mathematical topics the HN crowd seems to like--things like the Collatz conjecture or Busy Beavers--are 60, 80 years old. So, you end up having to spe…

I'll argue for astronomy being the oldest. Minimal knowledge would help pre-humans navigate and keep track of the seasons. Birds are known to navigate by the stars.

I would argue that some form of mathematics is necessary for astronomy, for “astronomy” as defined as anything more than simply recognizing and following stars.

Re: Mathematics is hard for mathematicians to understand too

#42

Post by nkrisc » Wed, Dec 03, 2025, 1:40 PM UTC

pfortuny wrote: post #26

mkl wrote: post #20

Earlier quoted context omitted.

> That depends what you mean by "literature". If you want it to be written down That is what literature means: https://en.wiktionary.org/wiki/literature#Noun

Well, then poetry is not literature.

If it’s not written down, then that’s true.

Once someone writes it down, it is.

Re: Mathematics is hard for mathematicians to understand too

#43

Post by nkrisc » Wed, Dec 03, 2025, 1:42 PM UTC

worthless-trash wrote: post #31

BlackFingolfin wrote: post #8

Earlier quoted context omitted.

I find it strange to compare "math" with one programming language. Mathematics is a huge and diverse field, with many subcommunities and hence also differing notation. Your rant would be akin to this if the sides are reversed: "It's surprising how many different ways there are to describe the same thing. Eg: see all the notations for dictionaries (hash tables? associative arrays? maps?) or lists (vectors? arrays?). Y…

> You don't have "the manual" of programming languages. " Well, we kinda do when you can say "this python program" the problem with a lot of math is that you can't even tell which manual to look up.

Someone not educated in programming would not know that a given text is Python source code.

Re: Mathematics is hard for mathematicians to understand too

#44

Post by MangoToupe » Wed, Dec 03, 2025, 1:43 PM UTC

segfaultex wrote: post #23

MrDrDr wrote: post #19

I think this would be extremely valuable: “We need to focus far more energy on understanding and explaining the basic mental infrastructure of mathematics—with consequently less energy on the most recent results.” I’ve long thought that more of us could devout time to serious maths problems if they were written in a language we all understood. A little off topic perhaps, but out of curiosity - how many of us here hav…

Yeah, I don't want to be uncharitable, but I've noticed that a lot of stem fields make heavy use of esoteric language and syntax, and I suspect they do so as a means of gatekeeping. I understand that some degree of formalism is required to enable the sharing of knowledge amongst people across a variety of languages, but sometimes I'll read a white paper and think "wow, this could be written a LOT more simply". Statis…

> I suspect they do so as a means of gatekeeping.

What, as opposed to using ambiguous language and getting absolutely nothing done?

Re: Mathematics is hard for mathematicians to understand too

#45

Post by nyeah » Wed, Dec 03, 2025, 1:46 PM UTC

Davidzheng wrote: post #13

geomark wrote: post #2

I thought we were well past trying to understand mathematics. After all, John von Neumann long ago said "In mathematics we don't understand things. We just get used to them."

sorry but that is a dumb quote.

Yeah, I wonder how exactly he meant that. I doubt that Von Neumann believed in random plug-and-chug, which is what I'd probably mean if I said I had given up on understanding something. Possibly von N was being very careful and cautious about what "understanding" means.

For example there's a story that von Neumann told Shannon to call his information metric entropy, telling S "nobody really understands entropy anyway." But if you've engaged with Shannon to the point of telling him that quantity seems to be the entropy, you really do understand something about entropy.

So maybe v N's worry was about really undertanding math concepts fully and extremely clearly. Going way beyond the point where I'd say "oh I get it!"

Re: Mathematics is hard for mathematicians to understand too

#46

Post by adornKey » Wed, Dec 03, 2025, 1:47 PM UTC

johngossman wrote: post #5

Mathematics is such an old field, older than anything except arguably philosophy, that it's too broad and deep for anyone to really understand everything. Even in graduate school I often took classes in things discovered by Gauss or Euler centuries before. A lot of the mathematical topics the HN crowd seems to like--things like the Collatz conjecture or Busy Beavers--are 60, 80 years old. So, you end up having to spe…

The desire to hide all traces where a proof comes from is really a problem and having more context would often be very helpful. I think some modern authors/teachers are nowadays getting good at giving more context. But mostly you have to be thankful that the people from the minimalist era (Bourbaki, ...) at least gave precise consistent definitions for basic terminology.

Mathematics is old, but a lot of basic terminology is surprisingly young. Nowadays everyone agrees what an abelian group is. But if you look into some old books from 1900 you can find authors that used the word abelian for something completely different (e.g. orthogonal groups).

Reading a book that uses "abelian" to mean "orthogonal" is confusing, at least until you finally understand what is going on.

Re: Mathematics is hard for mathematicians to understand too

#47

Post by locknitpicker » Wed, Dec 03, 2025, 1:52 PM UTC

segfaultex wrote: post #23

MrDrDr wrote: post #19

I think this would be extremely valuable: “We need to focus far more energy on understanding and explaining the basic mental infrastructure of mathematics—with consequently less energy on the most recent results.” I’ve long thought that more of us could devout time to serious maths problems if they were written in a language we all understood. A little off topic perhaps, but out of curiosity - how many of us here hav…

Yeah, I don't want to be uncharitable, but I've noticed that a lot of stem fields make heavy use of esoteric language and syntax, and I suspect they do so as a means of gatekeeping. I understand that some degree of formalism is required to enable the sharing of knowledge amongst people across a variety of languages, but sometimes I'll read a white paper and think "wow, this could be written a LOT more simply". Statis…

> Yeah, I don't want to be uncharitable, but I've noticed that a lot of stem fields make heavy use of esoteric language and syntax, and I suspect they do so as a means of gatekeeping.

I think you're confusing "I don't understand this" with "the man is keeping me down".

All fields develop specialized language and syntax because a) they handle specialized topics and words help communicate these specialized concepts in a concise and clear way, b) syntax is problem-specific for the same reason.

See for example tensor notation, or how some cultures have many specialized terms to refer to things like snow while communicating nuances.

> "wow, this could be written a LOT more simply"

That's fine. A big part of research is to digest findings. I mean, we still see things like novel proofs for the Pythagoras theorem. If you can express things clearer, why aren't you?

Re: Mathematics is hard for mathematicians to understand too

#48

Post by beng-nl » Wed, Dec 03, 2025, 1:54 PM UTC

segfaultex wrote: post #23

MrDrDr wrote: post #19

I think this would be extremely valuable: “We need to focus far more energy on understanding and explaining the basic mental infrastructure of mathematics—with consequently less energy on the most recent results.” I’ve long thought that more of us could devout time to serious maths problems if they were written in a language we all understood. A little off topic perhaps, but out of curiosity - how many of us here hav…

Yeah, I don't want to be uncharitable, but I've noticed that a lot of stem fields make heavy use of esoteric language and syntax, and I suspect they do so as a means of gatekeeping. I understand that some degree of formalism is required to enable the sharing of knowledge amongst people across a variety of languages, but sometimes I'll read a white paper and think "wow, this could be written a LOT more simply". Statis…

My opinion on this is that in mathematics the material can be presented in a very dry and formal way, often in service of rigor, which is not welcoming at all, and is in fact unnecessarily unwelcoming.

But I don’t believe it to be used as gatekeeping at all. At worst, hazing (“it was difficult for me as newcomer so it should be difficult to newcomers after me”) or intellectual status (“look at this textbook I wrote that takes great intellectual effort to penetrate”). Neither of which should be lauded in modern times.

I’m not much of a mathematician, but I’ve read some new and old textbooks, and I get the impression there is a trend towards presenting the material in a more welcoming way, not necessarily to the detriment of rigor.

Re: Mathematics is hard for mathematicians to understand too

#49

Post by otoburb » Wed, Dec 03, 2025, 1:57 PM UTC

adornKey wrote: post #46

johngossman wrote: post #5

Mathematics is such an old field, older than anything except arguably philosophy, that it's too broad and deep for anyone to really understand everything. Even in graduate school I often took classes in things discovered by Gauss or Euler centuries before. A lot of the mathematical topics the HN crowd seems to like--things like the Collatz conjecture or Busy Beavers--are 60, 80 years old. So, you end up having to spe…

The desire to hide all traces where a proof comes from is really a problem and having more context would often be very helpful. I think some modern authors/teachers are nowadays getting good at giving more context. But mostly you have to be thankful that the people from the minimalist era (Bourbaki, ...) at least gave precise consistent definitions for basic terminology. Mathematics is old, but a lot of basic termino…

>>[...] at least gave precise consistent definitions for basic terminology.

Hopefully interactive proof assistants like Lean or Rocq will help to mitigate at least this issue for anybody trying to learn a new (sub)field of mathematics.

Re: Mathematics is hard for mathematicians to understand too

#50

Post by TimPC » Wed, Dec 03, 2025, 2:04 PM UTC

beng-nl wrote: post #48

segfaultex wrote: post #23

Earlier quoted context omitted.

Yeah, I don't want to be uncharitable, but I've noticed that a lot of stem fields make heavy use of esoteric language and syntax, and I suspect they do so as a means of gatekeeping. I understand that some degree of formalism is required to enable the sharing of knowledge amongst people across a variety of languages, but sometimes I'll read a white paper and think "wow, this could be written a LOT more simply". Statis…

My opinion on this is that in mathematics the material can be presented in a very dry and formal way, often in service of rigor, which is not welcoming at all, and is in fact unnecessarily unwelcoming. But I don’t believe it to be used as gatekeeping at all. At worst, hazing (“it was difficult for me as newcomer so it should be difficult to newcomers after me”) or intellectual status (“look at this textbook I wrote t…

If it's actually in the service of rigor then it's not unnecessaryily unwelcoming. If it's only nominally in the service of rigor than maybe, but Mathematics absolutely needs extreme rigor.