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.
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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? - 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? - Are there other criteria for forming
mathematics categories?
Thanks and regards!
Best Answer
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.