Is there a way to do a PhD such that you get a broad view of a field or subfield as a whole?

Question

I will aim to keep this as general as possible, given the already narrow scope of the question.

I am looking for a way to design a PhD around the broader study of mathematics. I want to learn more about the relative relationships of different subjects within math, and am less interested in the arcane outer reaches of a particular subject. I understand that in order to do this successfully, I will probably need to find some lens through which to focus my study. I am curious if there are existing ways that others have managed to do this.

As an example, suppose you are interested in analysis. Rather than study whatever particular problems lie on the periphery of the field, you might focus your research on the particular approach of 19th century mathematicians like Cauchy and Weierstrass. In this way, you've chosen history to be the lens through which you are able to examine the whole (or at least a larger part) of analysis.

Maybe another way to do this is to develop alternative explanations for theorems and/or subjects? It bleeds a bit into pedagogy. Could you direct your PhD research on re-framing traditional explanations in a geometric/visual context? I'm thinking specifically of Visual Complex Analysis and Tristan Needham's idea of the amplitwist. Is it viable to have a thesis that is based on developing/reexamining alternative approaches to understanding math, while still maintaining a rigorous relationship with the mathematical content?

To be clear, I am not speaking specifically about history of math or math education. These are just the only examples I can think of. I am curious if anyone has tried to broaden the scope of their study through other lenses.

Answer 1

I think you can do research like this, but it usually wouldn't be in a math department.

Your two examples fit squarely within math history and math education, respectively. Philosophy of math is another topic where people take a broad view of math and look at it from different lenses. By contrast, to receive a PhD in math, one is generally expected to do new math research, i.e. develop new mathematics that has never been written down before.

There is some wiggle room here. The line between "new mathematics" and "new ways of viewing previously-existing mathematics" is somewhat blurry. But to get a PhD in math, it's expected that your contribution gives some new mathematical insight, in some sense.

One could argue that this segregation between researchers who "do" math and those who "think about" math (for lack of better terms) isn't a perfect state of affairs. We need big picture thinking as well as progress in specific subareas, and modern math academia tends to push people toward the latter.

Answer 2

While you might not be able to find a program like that, you might be able to find an advisor somewhere who meets your needs. But their focus might be more on the history of math or the philosophy of math. Such folks exist.

Doctoral programs, however, in the US, at least, include a breadth component in the initial coursework and the comprehensive exams that give a fairly broad view of the field. Most research, however, following comps is very narrow based as you note. But if you can find an advisor who has a broader view you might be happy enough.

Alternatively, you can get a "typical" doctorate and broaden your view, though you might need to wait to mid career to really do that.

At liberal arts colleges in the US, faculty often take a broader view and it is possibly more acceptable there than at strictly research universities. They need to earn a doctorate (mostly) somewhere, of course.

The path is uncommon, but it is there. You'll have to seek it out.

Let me note that mathematicians seek insight. Insight is what gives you the ability to come up with things that might be true (and significant) but aren't known to be true or false. Insight can be hard to come by and takes work to achieve. One strives for this in a doctoral program. My experience, however, is that mathematical insight isn't general, but can be specific to a subfield. One can, for example, have great insight into, say, abstract algebra, but little if any into topology. This can make your quest difficult. You want more than superficial knowledge of these areas and it can take effort for each of them independently of the others.

Answer 3

It is indeed possible to write a Ph.D. thesis providing alternative explanation for theorems -- but to write a good such Ph.D. thesis is much harder than writing one proving new theorems!

A good example is the 1950 thesis of John Tate. Tate's main results were the analytic continuation and functional equation for the Dedekind zeta and Hecke L-functions associated to number fields. (This includes the Riemann zeta function, as a special case.)

The results were already known, but Tate's methods were extremely innovative, developing an elegant approach in terms of adelic integration and Poisson summation.

Tate's work was enormously influential in the development of modern number theory; almost any working number theorist will be at least broadly familiar with Tate's thesis, and many of them will have read it. It is widely regarded as one of the best Ph.D. theses in math to have ever been written. Tate then went on to have a brilliant career, and advised many students who are now famous number theorists in their own right.

Most of us (including me!) have far more modest talents than Tate. So my advice is to set out and initially try to do something new. That is easier than having new insight into something old. However, if at any point you discover new insights into old methods, by all means pursue them!