# Categories of mathematics

I am interested in understanding how mathematics is divided into many categories, such as what categories are particular cases of what, what categories do not or have little overlap with what. This is meaningful to me, because it helps me get a big picture and not mess up many categories.

1. Quoted from Arturo:

think if "Algebra", "Geometry", "Analysis", "topology, "Number Theory", etc. as 'first-level subjects'; then you have algebraic number theory, algebraic topology, analytic geometry, etc., as 'second-level subjects.' Now we have algebraic arithmetic geometry, a 'third level subject'

I was wondering what criterion is used to divide mathematics into the first level subjects?

My understanding for these subjects are:

The objects studied in algebra are sets with operators with some properties, and mapping between such sets. So algebra is dealing with general and abstract objects.

Geometry is, quoted from Wikipedia:

a branch of mathematics concerned with questions of shape, size, relative position of figures, and the properties of space.

To make these questions meaningful, is the space a general inner product space? Or must it be a particular one, an Euclidean space $\mathbb{E}^n$? In either case, geometry is dealing with some kind of topological vector space, which seems to be more concrete than algebra.

The subjects studied in analysis are derivatives and integrals of some mapping between some sets( others that I miss?). To make derivative concept meaningful, the domain and codomain of the mapping must be Banach spaces (?); to make integral concept meaningful, the domain and codomain of the mapping must be measure space and Banach space respectively(?).

Topology is about neighborhood of each element in a set, defined as a class of subsets that are closed under arbitrary union and finite intersection. This is also quite general and abstract.

Number theory is about properties of natural, integer, rational, real, complex, algebraic numbers, that are represented in terms of four specific operators $+, -, \times, \div$. This is quite concrete.

In summary, the first-level subjects algebra, geometry, analysis, topology and number theory seem not stand at the same level of abstraction or concreteness. Is there a criterion or reason for dividing mathematical topics into these first-level subjects?

2. There are also other categories of mathematics, such as set theory, category theory, logic and measure theory, which especially the first three seem quite general and each does not very much overlap with other categories of mathematics, including algebra, geometry, analysis, topology and number theory. So what kind of criterion is used to form these other categories?
3. Are there other criteria for forming mathematics categories?

Thanks and regards!

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I would suggest you to read the first couple of sections of first chapter of "Princeton Companion to Mathematics". – user17762 May 7 '11 at 22:34
Your view of analysis is severly limited. But overall, you are still suffering from a kind of reductionistic approach that is not very helpful. Mathematics is not a bunch of discrete subjects that are not allowed (or cannot) interact; it's a continuum where fields have large overlaps, where one field informs another, often in surprising ways, where deep connections lie between subjects that may seem (superficially) distinct. There are no sharp boundaries, no absolute categories. (And you are equivocating between the colloquial meaning of "category" and a formal meaning in 'Category Theory') – Arturo Magidin May 7 '11 at 22:41
@Arturo: (1) I admit my views are limited. So I am open to and wait for any correction. (2) When saying dividing mathematics into categories, I already knew there are probably not strict boundaries in many cases. But I also knew there are distinctions between the goals/subjects of a mathematical category and other categories used as tools for it to achieve its goals. I think the goals/subjects of different mathematical categories are likely to be different. Otherwise, there would be no different mathematical categories. – Tim May 7 '11 at 22:51
@Tim: People are free to do lots of things. But what you seem to be asking about is some sort of "library science" for mathematics, that decides on classifications, and determines in which section to put a particular idea, question, or object. And there is no such thing. – Arturo Magidin May 8 '11 at 1:51
I do think that it helps a lot, for those serious about pursuing mathematics (beyond studying math as a necessary evil to satisfy general ed requirements, or degree requirement in another field), that having some sort of map (albeit general and "fuzzy"), can really help to begin to make some sense out of the extensively broad (and intensely specialized) field of mathematics. But any organizational scheme, to be of help, requires at least grounding in fundamental math, lest it be incomprehensible, or too overwhelming! – amWhy May 8 '11 at 4:05

I first refer you here, to the math subject classification system of the American Mathematical Society. I also refer you Arxiv's Math subject classification system. These are the two major systems that I use and that I refer to when classifying or looking for mathematics. As for the categories - these are often made the way they are due to historical events or interpretations.

In reference to the distinctions between 'first' and 'second' level math, and so on: I think that Arturo was basing these on necessary prerequisites. For example, one can take a first class on Algebra, Geometry, Elementary Number Theory, Real Analysis, or Topology without having taken any of the others. Of course, one might argue that there are many interconnections and that one would benefit from knowing algebra before learning number theory, or topology before real analysis, etc. I think this is true, but that it misses the point: it's not necessary at first.

On the other hand, algebraic number theory, algebraic topology, analytic geometry, etc (to directly quote your quote of Arturo) all require multiple previous topics, i.e. some mixture of topology, number theory, algebra, geometry, analysis, etc.

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Thanks! (1) Things that could have been clear can be made ambiguous by historical reasons. So I feel hard to appreciate such role played by history, even though they are used by AMS and Arxiv. (2) what do you mean by "it's not necessary at first" at the end of second paragraph? – Tim May 7 '11 at 22:57
@Tim: I mean that one does not need the concepts of algebra, say, to learn number theory. Often people learn number theory first and then use examples from number theory to learn algebra (Fermat's Little Theorem and Euler's Theorem are manifestations of Lagrange's Theorem, for example). Others learn algebra first and then can associate number theory with well-known algebraic structures. So it doesn't really matter in what order you take the classes. – mixedmath May 7 '11 at 23:06
To a large extent, yes, you got it. It comes form a discussion I once had with a friend in grad school, when we were talking about algebra geometry vs. geometric algebra, algebraic geometry vs differential geometry, etc. He happens to be an arithmetic geometer. – Arturo Magidin May 8 '11 at 1:53
@Arturo: How about set theory, category theory, logic and measure theory? Are they at first level? – Tim May 9 '11 at 14:55
@Tim: Measure theory isn't; logic/set theory/category theory are foundational theories. To really study them you need a lot of baggage, but they are all useful in small doses to study other things. – Arturo Magidin May 9 '11 at 14:56

Based on the AMS classification scheme (linked in mixedmath's answer) the Mathematical Atlas has visual representations of overlaps, connections between fields, etc. I encourage you to explore the math-map link, and other links in right upper corner of the link I'm providing:

I found it helpful when I first encountered it. It elaborates a bit on the indexing/categories given by the AMS (American Mathematical Society).

Enjoy exploring! (It can be overwhelming to realize just how expansive the field of mathematics is, so take your time!)

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Thanks! I am more interested than how goals/subjects of different fields/categories are different/related to each other and criteria used to distinguish them, than a long list. By the way, are you related to the famous ID J.M.? – Tim May 8 '11 at 0:19
@Tim That's what I found helpful about the Mathematical Atlas: the links it provides to "mapping out" the territory, etc. and providing spatial representations of how "large" a field is, and where it "lies" in relation to other fields. And no, I'm not related to J.M. (J and M happen to be the initials to my middle and last name. I might change my ID to something more "fun"!) – amWhy May 8 '11 at 0:26
1. We're not related. 2. I consider myself more "notorious" than "famous". 3. Amy's a nice name. – J. M. May 8 '11 at 0:43
@Tim: I added - to my answer above - a direct link to the MathMap page available at the Mathematics Atlas: That page has a link for descriptions of fields/categories, including descriptions in "layman's terms". – amWhy May 8 '11 at 2:02
@Amy: That's really cool! I haven't seen that before. – mixedmath May 8 '11 at 3:41