Some Musings on Mathematics

Although this might apply for early number theory, how does it fit with more abstract mathematics that doesn't deal with numbers, let alone units, at all?

However, I completely agree math (as in the human practice) is a collaborative and inventive process. The directions in which it progresses is often determined by what mathematicians consider "interesting", and not some purely analytical and sterile reasoning.

> Pure Calculus is motivated by applications in ballistics and weather forecasting,

> Pure Combinatorics is motivated by analysis of computer networks and data processing,

> Pure Statistics is motivated by life assurance, insurance and gambling,

> Pure Linear Algebra is motivated by optimization problems and Google's Page Rank algorithm.

Math is really sweet, math allows you to make so much, just look around you. I see it as the most powerful, low level, oldest (probably) API in the world. And it's mostly free and open source ! Math gives you tools and tells you in what context they work and don't work. Then it's up to you using it to make or understand something cool. Actually, math purposely tries to abstract itself as much as possible from reality in order to give you a robust framework to work with.

I think this quote of David Hilbert's response upon hearing that one of his students had dropped out to study poetry made me understand why pure math and applied math were two distinct fields: "Good, he did not have enough imagination to become a mathematician" [1]

Like honestly, who cares about whether or not all simply connected closed 3-manifold are homeomorphic to a 3-sphere. But the understanding it brings us about the behavior of manifolds in particular contexts is very real. Whether it's useful or not isn't a pure mathematician's problem though :D (but the truth is that it probably is, just that somebody else will make use of that)

[1] http://www.amazon.com/The-Universal-Book-Mathematics-Abracad... pp. 151 (according to wikipedia, I haven't read the book)

I see a few holes with this argument:

1) Who is this "we"? I'm sure not all non-"non-mathematicians" agree with this sentiment.

2) If a mathematician is still looking for an answer to a question, they don't yet know if an answer exists or not (see: Godel, halting problem). Perhaps this should be rephrased "because there might be an answer and it would be unbearable not to know it if it did exist."

3) The argument is circular (calling not knowing bad doesn't answer why knowing is good).

I think of (pure) mathematics as exploring the structures generated by simple rules. You start with some system of axioms, maybe those of group theory or linear algebra, and you see where it takes you. Often, there are richer, but closely related structures available by adding additional axioms or constraints. For example, add commutativity to group theory and you get abelian groups. Add metrics to vector spaces and you get topology (sorta).

This is useful because the world is full of complex systems that emerge from simple rules. Therefore, when we observe that some system in the real world displays the characteristics of a known mathematical structure, we inherit a bunch of free knowledge about that system.

In practice, most math falls on some spectrum between the above definition and "applied math". Historically speaking, it's a pretty modern idea (although its pedigree begins with Euclid). Before the mid-nineteenth century, mathematicians had indeed been chasing puzzles like "how to find the roots of polynomials" and "can you square a circle using constructions?" for several centuries. And puzzles are certainly not dead, as the millenium prize clearly shows. Number theory also doesn't play nice with this definition. I intentionally ignored that - if I start thinking about the ontology of numbers I risk losing quite a bit of sleep :p.