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!

Solution 1:

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.

Solution 2:

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:

http://www.math.niu.edu/~rusin/known-math/index/index.html

http://www.math.niu.edu/~rusin/known-math/index/mathmap.html

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!)