4 ms·
> Illustrated Curiously, Category Theory appears to be the most “illustrated” mathematical theory by its very nature (because of the wide use of diagrams in an
by Koshkin 5y ago
> Illustrated
Curiously, Category Theory appears to be the most “illustrated” mathematical theory by its very nature (because of the wide use of diagrams in any text on it) and, at the same time, the most misunderstood one (or, often, not understood at all) - because its conceptual content and the patterns it considers are so deep that no amount of illustration could convey that depth, to the point that even drawing diagrams, however complex, is a skill comparable to being able to write numbers on a piece of paper - which is just as far from what we think what the understanding of the subject is.
- pfortuny 5y agoThe moment limits appear, illustrations stop. Not to speak about “derived” concepts…
- fn-mote 5y agoI guess you are talking about limits like in analysis, but even then I don't think this is really true. I'm not really sure if you intend to distinguish between limits and colimits; I'll assume not, but I would certainly agree that infinite (inverse) limits are harder to understand with diagrams. On a simple level, limits in category theory don't need to involve infinite things at all, you can take the limit over a small category like (x --> x <-- x). With regard to derived concepts, one can draw pictures of "projective resolutions" or draw level by level pictures of the nerve of the category you want to work with. So I think you can in fact make useful pictures even in this case.
- deleted 5y ago[deleted]
- fn-mote 5y ago> conceptual content and the patterns it considers are so deep that no amount of illustration could convey that depth My experience is different - for me category theory abstracts / generalizes / encodes ideas that we understand well in a particular setting. Intuition comes from particular examples and is then "categorified" to make more general arguments. Of course some times it goes the other way ("hey you really need to do X because the category you are working with doesn't have property Y"). And sometimes the imported understanding is not quite accurate, like always there are issues. I don't think that impacts the main line of this argument, though it may be an example of the depth that category theory brings.
- pvg 5y agoPlane geometry, arguably the OG mathematical theory, would probably take issue with who gets to be called 'most illustrated'.