Dependencies of various of areas of mathematics

Question

Specifically, I want to learn category theory to improve my programming ability in Haskell, but it seems like that would be difficult without a knowledge of abstract algebra, and it seems that a solid understanding of linear algebra is useful to understand abstract algebra.

I'm therefore looking for something showing how these areas of maths depend on each other, but I think that a more general table or graph probably exists (or would be useful).

This question is the same as this one on Stack Overflow, but it seems like a good question for this site, and likely to get much better answers here.

Just as a side note: if you manage to understand this material: https://en.wikibooks.org/wiki/Haskell/Category_theory then you won't really find any benefits from learning more category theory. Learning all this mathematics just for the purpose of getting better at Haskell is not a good idea, since it won't work. — user2520938, Dec 19 '16 at 15:41
@nbro The bounty offered 3 days ago, which already attracted an answer, is asking for an altogether different question than the one asked and answered 2.5+ years ago. As such, it seems to me that posting a new question would have been a better solution. — Did, Dec 22 '16 at 20:08
@Did and of course, such a question would've been closed as way, way too broad. — Mathmo123, Dec 22 '16 at 20:20
@nbro "but it risked to be closed" Are you explaining that you offered this bounty as a deliberate and relatively elaborate attempt to get answers to an offtopic question or am I imagining things? — Did, Dec 22 '16 at 20:26
@nbro Sure, it might have been closed. But placing a bounty to ask a new question is not what bounties are intended for. However, since the bounty is already underway for three days, I think removing it now would do more harm than good, so I'll leave it. But if a similar situation arises in future, ask a new question. — Daniel Fischer, Dec 22 '16 at 20:28
@DanielFischer For the record, I find the "more harm than good" argument rather unconvincing in the present case. Concretely, it amounts to putting the preservation of a single answer from an OP who should have known better higher than the rules of the site (and allowing other answers to accumulate). The bounty could have been nullified and its author invited to repost their question separately, new question to which Han could have copy/pasted their answer, et voilà. Instead, we have a precedent. — Did, Dec 22 '16 at 20:44
@Did The problem is not the preservation of answers. The problem is the precedent of refunding a bounty that already is running some time (and has attracted an answer during that time). I don't know what will be the worse choice in the long run. But either way is not good. — Daniel Fischer, Dec 22 '16 at 20:58
@DanielFischer I do not know what "refunding a bounty" means exactly (my fault) but if it means that the user offering the bounty loses the points with no compensation and that the users adding answers to this 2.5+ years old question to get the bounty, do not get it, then so be it. They played, choosing to disregard the site's rules, and they lost, so, Es ist alles in Ordnung. (By the way, I feel I have posted what I had to say on this, so please do not deem necessary to continue the exchange, and thanks for your comments.) — Did, Dec 22 '16 at 21:17
@Did It means the user offering the bounty gets the points back. If the points vanished into thin air, there'd be no problem. — Daniel Fischer, Dec 22 '16 at 21:18
@Did: Worse. Suppose an answer has been posted as a reply to the implicit question as formulated in an elaborate comment accompanying the bounty. That comment will evaporate after the bounty has expired. So that leaves us with an answer to a question that is no longer there, in the end. Weird, huh? — Han de Bruijn, Dec 24 '16 at 11:00
@HandeBruijn If you are alluding to the fact that users seeing the bounty could also have detected that it was inappropriate and, thus, could have abstained from posting answers about the subject delineated by the bounty, I fully agree. Unfortunately, I see two such answers below... :-) — Did, Dec 24 '16 at 16:16

score 1 · Answer 1 · answered Jun 22 '14 at 20:17

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I can't give a fully authoritative answer to this, but I can try to give some indication. The key issue with category theory is that whilst you could probably understand the theory without too many pre-requisites, examples of categories come from all branches mathematics and in order to fully appreciate it, you'd need to have seen some of these examples before. You'd also need a good deal of experience in other disciplines, especially abstract algebra. As such at my university in England, category theory is not offered as an undergraduate course.

I'm going to assume that you have a basic high school proficiency in mathematics - as I have to start somewhere.

First thing you'll need is some elementary set theory - by which I mean a knowledge of what we mean by sets (unions, intersections etc), functions (injections, bijections, surjections), basic notions of countability etc. In fact this is one of the simplest examples of a category.

After this, you'll need a first course in linear algebra, and a first, and possibly second course in abstract algebra covering the rudiments of group theory, ring theory, and perhaps module theory and field theory. This will give you a large bank of examples of categories, as well as some familiarity with some of the ideas that will be explored and their significance.

Other useful things to know about would be posets, which can often be found in a first course on logic. It'd also be handy to know some basic topology for some more examples.

I'd say the raw essentials are:
- Elementary set theory
- Linear algebra
- Group theory
- Ring theory

But the more experience you have in mathematics, the more accessible it will become.

answered Jun 22 '14 at 20:17

Mathmo123

23,018

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Have no experience with Haskell, at all, but advertising says it is a functional language. Therefore I would rather think of e.g. lambda calculus instead of set theory. – Han de Bruijn Jun 24 '14 at 19:25
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@HandeBruijn Yes, lambda calculus is important to it, but the type system is based on category theory. In fact, lots of it is based on high level areas of maths - I would definitely recommend it to a mathematician, as although most programmers find it quite hard to learn, I'd imagine it would by quite easy for someone with a maths degree. – rlms Jun 29 '14 at 19:25
Historically, there are only two mainstreams in mathematics: (1) Algebra and (2) Geometry. All the rest is a mixture and (sometimes too much of) a generalization. Personally, I find it a pity that pure Geometry has been absorbed by other formal mathematical disciplines a great deal. – Han de Bruijn Dec 21 '16 at 14:34
@nbro I'm really not sure what you're hoping to achieve. You've taken a very specific question and asked for an extremely broad answer, sufficiently broad that it is probably unanswerable. What makes you think that such a web actually exists? – Mathmo123 Dec 21 '16 at 20:20
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@nbro that's just the point. In order to have a graph, you need vertices and edges. Branches of mathematics are not well-defined. The traditional algebra and geometry each split up into hundreds of sub-disciplines. The vertices are also not well-defined. There are aspects of number theory for which high level algebraic geometry is essential. Other aspects don't use it at all. On the one hand, you seem to think that mathematics can be fit into neat boxes. On the other hand, you seem to underestimate just how interconnected mathematics is. And new connections are being found all the time. – Mathmo123 Dec 21 '16 at 20:26
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Not to mention the fact that you've taken a decent question and completely changed it to a new question. A question that, had you posted it separately, would've been closed as too broad. – Mathmo123 Dec 21 '16 at 20:28

Han de Bruijn · Answer 2 · 2016-12-24T11:27:07.237

It seems that user nbro essentially is looking for a Taxonomy of Mathematics :

Maybe a little bit more on the internet : let Google be your friend.

EDIT. Taxonomy is one of the best things we can do. Any decent table of contents in a book has this structure too. It's well-defined, and throughout centuries it has proved its power to classify subjects, mathematical and other. Furthermore, any document in plain text format can always be converted into a tree structure, given some programming effort. Indeed, trees are not always a perfect model of reality (whatever that might mean). It is well known that the good old hierarchical database has its limitations and it would be better to replace them by relational databases. But I have yet to see an example of the latter containing a decent overview of mathematics, while there are a few humble attempts of the hierarchical kind available.

Correction! Clearly I wasn't aware of possibilities like this : A Graph Map of Math.SE and this : Mathematics Subject Classification (according to the above comment by Watson) .

A document in plain text format like this one can always be converted into a tree with some programming effort. — Han de Bruijn, Dec 22 '16 at 19:25
@nbro: That may be true, but I think taxonomy nevertheless is one of the best things we can do. By the way, any decent table of contents in a book has this structure too. It's well-defined, and it has proved its power to classify throughout centuries. — Han de Bruijn, Dec 23 '16 at 13:26
@nbro: This has been an issue I've been involved with on a project basis ( called NKBW ) ten years ago or so. I do agree with your conclusion that the final outcome is rather unsatisfactory. But I think that outcome is more or less inevitable. I'm not very optimistic about the idea that mathematicians can have an complete overview of their own discipline. — Han de Bruijn, Dec 27 '16 at 10:40

score -1 · Answer 3 · answered Dec 22 '16 at 20:27

Not sure if this is what you're looking for, but the book Mathematics - Form and Function by Saunders Mac Lane contains several figures with "dependency trees" connecting different mathematical areas or different concepts within one area. See pages 149, 184, 306, 408, 423, and 425-428.

Dependencies of various of areas of mathematics

3 Answers3

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