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November 1

What does it mean to be mathematically educated?

Since there does not seem to be anyone asking questions today, I shall propose this one (for which I myself do not know the answer). What does it mean to be mathematically educated? I should make it clear that I am not wondering about the ability required to do mathematics, but rather the breadth of knowledge necessary (within reasonable bounds!) that should allow one to claim that he/she "knows" mathematics. Succintly, for one particular aspect of the question - which fields of mathematics approximately encapsulate all ways of thinking within mathematics (fundamentally) and which such fields arise in "many areas"? By "all ways of thinking", I mean "sorts of thinking". Perhaps more concretely, thinking geometrically/topologically is an important skill, but so is thinking algebraically, or thinking like a number theorist. However, the possibility exists that these ways of thinking are actually the same, but in a different guise.

In simpler terms, consider the examples of finite group theory, field theory and ring theory. Connections underpinned by Galois theory show that certain finite degree field extensions may be analysed via finite group theory. But general field theory, such as the theory of totally transcendental extensions, must be tackled differently; at least in some respects. Is the thinking involved simply that of ring theory, lattice theory or category theory in a different guise? For an analytic example, consider set-theoretic topology and low-dimensional topology. Of course, set-theoretic topology deals with more abstract spaces than the latter (and thus bears deep relations to axiomatic set theory), but nevertheless, low-dimensional topology may have similarities with set-theoretic topology, at least in terms of the thinking involved (geometric intuition - "moving around a manifold" and using this to construct embeddings is an example (keeping in mind the algebraic topology)). Can one claim, in terms of ways of thinking, that one of these fields is a subfield of the later?

Are the fields of number theory (algebraic, analytic et cetera), analysis and algebra exhaustive in terms of encapsulating ways of thinking? Do other fields merely disguise "ways of thinking", but still have the same ways of thinking? Is familiarity with these fields enough to guarantee "mathematical education"? Note that I have of course not considered all fields within mathematics; rather I have given examples which may be modified to answer (or at least provide insight) into these (rather philosophical) questions. --PS T 12:08, 1 November 2009 (UTC)[reply]

This may have nothing to do with the question but to do number theory, specifically right now I'm reading a textbook in modular forms (no research yet), I have to know abstract algebra, linear algebra, complex analysis, at least undergrad real analysis, point-set topology, algebraic topology, hyperbolic geometry, differential geometry, number theory itself, and maybe stuff I don't know I need to know yet. I'm not saying I do know all these. But, I need to. My friends doing work in graph theory just need to know graph theory. My friends doing work in Moufang loops are just working with loops. I am not suggesting that someone must be a number theorist to be able to be called a mathematician. It's just that such a person needs to know a lot more just to work on one problem it seems to me. StatisticsMan (talk) 14:22, 1 November 2009 (UTC)[reply]

Number Theory is the most interdisciplinary field of maths I know. It includes bits of pretty much every other field of pure maths, although you don't need to learn about those fields in as much depth as a specialist would (if you did, nobody would be able to do number theory). --Tango (talk) 15:57, 1 November 2009 (UTC)[reply]

Trying to partition mathematics into different fields isn't particularly useful. All the commonly mentioned fields overlap in various ways. I don't think it is possible for anyone to know all of mathematics (it may have been a couple of hundred years ago, but so much more research has been done since then). I think the key feature of a good mathematical education is to develop "mathematical maturity". By learning and doing lots of maths (it doesn't really matter what maths) you get a feel for how the subject works and how to go about learning it. Once you have that you can learn any necessary maths to do whatever it is you are trying to do. --Tango (talk) 15:57, 1 November 2009 (UTC)[reply]

I agree that number theory is one of the most diverse fields of mathematics. It demonstrates that the better you understand the techniques used in fields prerequisite to number theory, the better your ability will be to research number theory (an enormous amount of field theory or algebraic topology may not be needed, but I feel that since a huge number of people have already approached number theory from a number-theoretic perspective, people should start attempting to develop new insights). I also partially agree with Tango that "mathematical maturity" is quite an important skill. However, my question is about the sorts of ways of thinking different fields encapsulate.

For example, in your mathematical experience, do you believe that fields like category theory, measure theory, or functional analysis, have any connections with number theory (of course they do!)? Would a number theorist be aided by appreciating such fields (that is, do such fields encapsulate different ways of thinking)? Or, are these fields redundant, in the sense that although interesting questions arise, and although they are wonderful subjects, can an algebraist have an intuitive feel for these fields (that is, do these fields have the same sorts of thinking to algebra, involved)?

I am of course not trying to partition different fields of mathematics, but rather trying to understand various points of view. Specialists abound in numerous disciplines, and many (if not all) have the qualities Tango suggests. But would not broader education help a specialist in his field (at least, aid him/her with new ways of thinking)? For instance, consider the famous Burnside problem. Analyzing the problem from a group-theoretic point of view is useful, but connecting the problem to graph theory may yield new insights (as researchers have already learnt). In effect, a group theorist should have some appreciation of graph theory (a quite well known link). But suppose someone else arrives with a topological viewpoint. It is quite possible that he may provide new insight, and this would imply that topology encapsulates different ways of thinking. Is this the case, in your experience; namely that although fields overlap in terms of content, they actually encapsulate the same ways of thinking? In the simplest terms possible, which fields encapsulate all sorts of thinking (or if there are none, why is this the case?)? --PS T 01:53, 2 November 2009 (UTC)[reply]