Is it necessary for a Mathematician to learn Maths foundations?

Question

Is it necessary for a mathematician to learn Maths foundations/philosophy and Maths history?

Answer 1

Here is a simple answer reflecting my personal views on this:

I think you can get along fine with mathematics without concerning yourself with whether the number 5 really exists, and whether results are invented or discovered. Of course, this is a very simplified account of what mathematical philosophy is about. There may be problems within the field of mathematical philosophy that you find interesting, and in that case, by all means, study away. But I don't think you need it.

History is a bit different. Yes, you can learn math as well as the best without ever hearing about Euler (you will, of course, necessarily come across a few entries in the long list of things named after him, but apart form that you don't need to know anything about him).

That being said, a lot of theory, standard examples and results are tied to its place in history. For instance, graph theory started with Euler being asked to consider the problem of the bridges of Königsberg. Whenever you come across a mathematical concept that you find unintuitive and cumbersome, it might be because historically, it made sense, but how we think about mathematics has changed over the centuries. Knowing its place in history has, at least for me, made it easier to accept.

One example is the large load of trigonometric functions that we have ($\sin, \sec, \tan, \cos, \csc, \cot$ just to mention a few). Most of these are easily expressed in terms of the others ($\sec = 1/\cos$, for instance), so we don't really need that many in modern mathematics (you usually get by with $\sin, \cos$ and $\tan$, but even that list can be shortened by removing $\tan$ without too much actual trouble).

The reason that we have them (and have names for them) is that back before calculators, we had tables. You could buy a small booklet containing the values of these trigonometric functions. And that table would contain all of them, because there was room for it, and because if you really wanted $\sec 42^\circ$, then being told what $\cos 42^\circ$ is from the table would still leave you with a long calculation. Therefore you woud want a table which also contained $\sec$, if you were going to do a lot of trigonometry.

As for the logic foundation of mathematics, you don't need a firm grasp. Again, you can do (higher) math as well as the best of them without knowing about forcing, or Gödel's constructible universe. However, knowing the basics will probably help your appreciation of everything which is built upon it. Personally, I think that being aware of the Axiom of choice, its consequences, and the consequences of not assuming it is probably the most visible in higher areas, but there are others as well.

In general, I think it will help you as a mathematician to have a feel about many different fields and what they're about. You don't need to study them and become an expert, but before you know it, there is potential for you to find a connection from your specialisation field to some other, seemingly unrelated field, and being able to spot such a connection and work to prove it is where many of the big names of mathematics come from.