Wikipedia:Reference desk/Archives/Mathematics/2021 November 25

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November 25

Why are some structures part of the math canon?

How come some structures get a name and become object of study in academic math and part of school curricula? I get that trigonometry or linear algebra can be used to describe loads of real phenomena, so they get their special place. I also see that the Möbius strip not only looks cool but has some applications. However what about, for example, the Klein bottle that is physically infeasible in our space? So, what is the benchmark for inclusion? Real-life application, looking cool, being unexpected, maybe-useful-someday? --Bumptump (talk) 12:29, 25 November 2021 (UTC)

There are obviously a lot of school curricula, and each has their own standards of when to include something. For a Klein bottle one of the reasons is probably because it's a well known object in mathematics. It's a well known object because it's a good example for some questions in topology, so it's often discussed in that context. El sjaako (talk) 16:32, 25 November 2021 (UTC)

There is no international standards committee that decides these things and there's no systematic rational being applied. Historical accident and tradition play a role as well applicability and relevance to current areas of research. The Klein bottle is an example relevant to certain areas of topology, but why teach about topology rather than, say, category theory or group theory? 200 years ago a school curriculum would include Euclid's Elements, not just Euclidean geometry but Euclid's actual books; nowadays geometry in general is waning as a school subject. You could ask the same question about a lot of subjects and get similar answers. Why study the plays of William Shakespeare and not William Congreve? --RDBury (talk) 23:42, 25 November 2021 (UTC)

I'm not sure if comparing math with a subject like literature theory is meaningful. Art forms have not well-defined boundaries, and are often a matter of subjective preferences. Why not compare it with physics or computer science? When would mathematician dismiss something as 'not an object of their field'?--Bumptump (talk) 08:44, 27 November 2021 (UTC)

Your question appears to be based on the assumption that there is such a thing as "the maths canon", quod non. Things get a name because someone assigns a name to it. If enough people are interested in whatever it is, and discuss it amongst themselves, they need a name to refer to it. If there are many such references, the whatever will eventually get an entry under that name in encyclopedias, may become the subject of conferences, and find a place in textbooks and curricula. The driver of the process is being found of interest by a sufficiently large group of people.  --Lambiam 09:05, 27 November 2021 (UTC)