Hacker News Reader

Comment by chr1 | Hacker News Reader

Does anyone know example of math that is not beautiful? As far as i can tell everything that explains something more complex in terms of simpler rules is called beautiful, but that is all the math! So it's not very surprising that math in physics is beautiful too.

View on HN

The Heegner numbers are a good gateway into a brutally ugly and weird corner of analysis, with lots of almost-numbers and almost-facts: https://en.wikipedia.org/wiki/Heegner_number

In higher (n-)category theory, there is not a single canonical way to describe higher categories, leading to lots of careful use of "strict" and "weak" qualifiers, as well as many different non-equivalent flavors of higher category: https://ncatlab.org/nlab/show/semi-strict+infinity-category

Speaking of categories, the number of finite categories, or other finite abstract algebraic objects, is not a beautiful combinatoric theorem, but instead a nasty study of computer-aided searches: https://www.mta.ca/uploadedFiles/Community/Bios/Geoff_Cruttw...

The classification of finite simple groups. From an outsider's perspective, it's ream upon ream of brutal case analysis. Some will certainly find beauty in the thousands of pages; the historicity nature of the ~150y effort is undeniable and awe-inspiring.

Classification theorems, in general, are frequently beautiful in their structure. But they often involve delving deep into rabbit holes, in order to prove the non-existence of the impossible.

Agreed. Few people ever read and follow those proofs. Another related issue is proofs by computer e.g. the four coloring problem. No human will ever understand every step the computer is doing in these proofs but they can understand the program doing the proof and why it must be correct. If we insist on proofs like Euclid's proof that there are infinite primes or sqrt(2) is irrational then these computer proofs are clearly different. If we must use a computer is the proof somehow "ugly"? I don't think so but maybe I'm wrong.

Good point, proofs that require brute force enumeration of all possibilities are not nice, but the physicists who use the results to describe some symmetries still think the results are beautiful.

The representation theory of Kac-Moody algebras. There are some issues with integration theories that need clearing up, should we really be ditching the Riemann integral for the Lebesgue integral so quickly without a closer look. Recently someone wrote a good book on linear algebra that puts of "ugly" determinants for as long as possible. Much of the problem is making what we already know cleaner and better. Some theories like set theory don't have a unique set of axioms and have lots of loose ends. General topology is a closed subject but there are still potentially better ways to formulate convergence.

My point is the stuff we already know so well still could use some polishing and everybody should be learning newer and better approaches to classical subjects.

Reply on news.ycombinator.com