I used to say stuff like this, but honestly... don't. If you're mentoring a young impressionable mathematician, explain the trade-offs in pure vs applied work, and explain how to determine which pure math is likely to be helpful in their useful lifetime.

Besides, "pure" and "applied" distinction is hilariously subjective and stupid. IME a lot of theoretical CS is more-or-less accurately characterized "pure math" and also far more financially remunerative than the bullshitty pointless PDE hacking lots of "applied math" folks do. Pure vs. applied is a stupid perspective these days because it almost exclusively defines a delineation among communities of mathematicians during the late 20th century; "useful in the next 3 years to ad tech folks" vs. "long-term/foundational importance in science and engineering" is a more useful and relevant distinction.

But once people choose to focus on pure mathematics, don't shit on them for making that choice. They KNOW it's not a great financial commitment. It's like telling a committed humanist that "no one pays for poetry". Like, they get it already... (and besides, sometimes you're wrong and then you're that asshole that always doubted them.)

Plus, plenty of pure mathematicians make high five/low six figures working 12 months with free trips all over the world. Or 9 months without the trips but with 3 months of freetime every year. And in both cases with near perfect job security. They don't have any social currency in the startup $$$$$$ obsessed world, but I bet they spend a fuckload more time with their kids and enjoy a lot more sunsets than any of startup founders. And probably make more money than the 90% of "failure cases" in the startup world too...

This article is great case-in-point. All the profiled folks are sure to live very comfortable lives doing work they love patronized by lovers of the mathematical arts (or anyways live off of rich endowments one way or the other) without ever having to figure out how to nickle and dime customers.

And none of this is to say that building businesses isn't valuable, since that's more-or-less taken for granted by the current venue. But it takes all types.

I was not really running down pure math. Indeed, see my post below where I explained that some pure math is crucial to my startup.

I was saying that I, personally, find foundations as in the OP down in the basement, dark and too far from applications in any sense.

For making money as a full prof of math, first have to get there, and that usually takes over 10 years if make it at all. Yes, it's possible to play the academic game; heck one paper in math I published is pretty, surprising, etc. but I can't imagine that it will ever be useful directly or even indirectly even by several steps of indirection for the foreseeable future if ever. I've seen people publish such things and make some progress in an academic career; to me that's playing an academic game; I chose not to do that.

When I was a prof (I didn't want to be but did it for a while trying to help my wife in her illness), it seemed to me that I was getting paid by students, pizza parlor owners and workers, auto dealership owners and workers, farmers, etc., and for them I wanted my work and teaching to be useful -- to me, personal curiosity, art, etc. didn't count. I really wanted the department to be clinical, professional, practical, like law and medicine, i.e., welcome people from outside academics with real problems and then seek to solve those problems. When can't solve a problem, then maybe that will be a good research direction; if make good progress, then already have one application!

For applied math, determine that not by PDEs but by what can find that is useful. Early in my career around DC for mostly US national security, I found lots of such applied math. And my startup, while more focused, is basically some applied math. PDEs? I had very little to do with those; the one case was the Navier-Stokes equations, and they were so difficult to work with that the project wasn't making much progress and I was pleased to move on to other topics.

It's not easy to see what pure math is the more useful. For the pure math I do respect, especially for utility, it appears that in some vague, large sense the results are fundamental, important broadly, and, eventually, inescapably relevant, but that is a difficult judgment call. Functional analysis? Sure. Algebraic geometry? Less sure. Foundations? Slim chance.

I was not trying to give career advice to other people possibly interested in math but just commenting that I found the foundations as deep as in the OP just too far down in a dark basement. In math, that's an old remark about foundations -- "Drop what you are doing trying to be useful and come with me down into the dark basement and wrestle with really subtle issues down there." Standard, old remark in math.

You read lots of stuff between the lines I wrote, stuff not really there.

Who gets to decide what is and isn't useful? Number theory was the most useless of mathematics for a long time; interesting for sure, but not applicable in any sense. Then we discovered applications to crypto, and now the security and privacy of the entire digital world depends on number theory. You mention the questionable applicability of algebraic geometry, yet elliptic curves are incredibly important for efficient crypto.

Let's take an example from TCS: Quantum computers. Initially, they were only a theoretical exploration of the the additional power (if any) that Quantum mechanics can give in computing things. Then Shor came around and shook up our understanding of the world.

Many such examples abound in the world of mathematics. And lastly, even if a subfield provides no use to "common" humans, how should that matter? People should be free to study what they want.

I was giving my opinion for myself and not "deciding" for anyone else.

> Who gets to decide what is and isn't useful?

There is a recipe for rabbit stew that starts out "First catch a rabbit". Well my own recipe for applied math starts out "First find an application."

I don't define applied math as just mathematical physics or its connections with, say, mechanical or electrical engineering.

For me, applied math is math that has been applied. Sure, you mention number theory and cryptography; so, now at least in part they are applied math. There is also finite fields and error correcting codes -- I took a whole grad course on coding theory. I had a good background in abstract algebra and was torqued that the prof was sloppy with the math. But coding theory is important, and for me that makes abstract algebra applied math. Just how R. Hamming saw to use finite field theory I don't know -- curious insight. My undergraduate honors paper was on group representation theory, and that is applied math at least for its connections with molecular spectroscopy. In fact, my work was stimulated by some people from the chemistry department that came to the math department for help with group representation theory. IIRC, the quantum mechanics part was in part from E. Wigner.

> And lastly, even if a subfield provides no use to "common" humans, how should that matter?

If the work is being paid for, then commonly it matters a LOT to some of the people paying for it.

> People should be free to study what they want.

Of course. For some years, I studied violin. I made some progress but set violin aside to have more time for my startup. Ah, but, no one paid me for studying violiin!

People misread my post: First, the only math I was criticizing was foundations as deep as in the OP. Second, my criticism was only for my own personal values and opinion. Sure, maybe work in Ramsey theory will be as important in applications as the Riemann integral -- I doubt it, but maybe.

I spent some months in that dark basement; others are welcome to do that if they wish; I wish I hadn't and wouldn't do it again.

Defining "applied math" as mathematics that has found applications is disingenous, IMO; what is and isn't applied can change rapidly in the course of a few years. A number theorist can go from pure to applied because someone else found an application of work done by other people in his field?

Taking an example from TCS: Probabilistically checkable proofs. They allow you to encode a proof for a statement so that you checking the proof requires looking at in only a few locations. Back when they were conceived, in the early 90s, all constructions were efficient only asymptotically, with galactic constants. Only in the past 2 years have constructions of PCPs been realized that are sufficiently efficient for (some) applications to checking program executions.

PCPs were for years what most theorists would consider a core theoretical object with not many hopes of finding application, and indeed 99% of research in the field focused on "negative" uses of PCPs for finding hardness of approximation results.

> If the work is being paid for, then commonly it matters a LOT to some of the people paying for it.

What makes an industry with a lot of money in it, like online advertising, worth more than study of something completely theoretical?

> Defining "applied math" as mathematics that has found applications is disingenuous, IMO; what is and isn't applied can change rapidly in the course of a few years.

Not "disingenuous" at all: I want to apply math, especially to making money. To make money is the main reason I studied math; I wanted math to help my career, to make money, to support a family. If not math, then maybe physics, some part of engineering, etc.

> what is and isn't applied can change rapidly in the course of a few years.

My experience is that that doesn't happen very often. But again, once again, over again, yet again, I am not, Not, NOT, N.O.T. -- clear enough -- running down pure math. NOT doing that. I just wrote that foundations as in the OP was too far from applications for me. I have been totally, overwhelmingly, crystal clear about this point in this thread, and there was nothing in my first post that ran down pure math. Again, yet again, ..., I was talking about my view for me of foundations such as in the OP. You misread my first post and didn't read my other posts.

If someone found some applications for PCPs, then good for them, and that makes PCPs applied math.

Better techniques for proofs of correctness obviously would be good work in computer science with plenty of valuable applications.

> What makes an industry with a lot of money in it, like online advertising, worth more than study of something completely theoretical?

Nothing, but most sources of financial investment want a financial return. That's much of how our economy works.

To be blunt and frank, Congress votes money for math mostly for US national security and because of the role of math in WWII and off and on since then. If some people at the NIH tells Congress that they like math, too, then that will help. But the remark at the beginning of the movie on Nash is basically correct: "Mathematics won WWII". Congress can be slow on the uptake, but they tend to believe that math did win WWII and had a big role in GPS, Keyhole, stealth, design of the core of fission and fusion bombs, etc. Congress takes US national security quite seriously. The other biggie for Congress is the NIH -- if only because most people in Congress have gray hair and want research that will help them if they get sick.

It remains, if you want a job in math, especially connected with computing, then those jobs are still where I started my career (and should have stayed there) within 100 miles of the Washington Monument.

I'm on my way down but haven't seen the darkness yet :)

Maybe because I didn't start out with money making in mind (not to say I don't want to make money. I do. I do.)

Also how long did the whole process take for you? ... If you make up your mind in advance, as I have, that you're going to be in there for 10 years give or take, (in 8-hour days; so 20 calendar years if you spend 4 hours a day), and try to make a living on the side, does it still feel as dark?

One way and another, starting in my senior year in college, I spent a year or so in that dark basement.

The most intense time was in a course in axiomatic set theory in an NSF summer math program at Vanderbilt.

The next fall, I was in a course in real analysis, and the prof started with foundations. After the first test he wanted to "see me". The exercises on the test were all trivial except one, and I got it only in the last minute or so so wrote quickly. I used little omega for the ordinal of the natural numbers without defining it. I told him that from the course I'd taken the previous summer I thought that that was standard notation. Apparently he didn't know that. After my explanation, he saw that my solution was correct and one step shorter than his. Then I asked him what he wanted to "see me" about, and he said "Now, nothing.". Gads. I got my Ph.D. later from a better program at another university.

Occasionally later I touched on that material.

That was all.

No way did I spend or want to spend 10 years in that basement. And no way did I want to do original research with that material.

Some of the pure math I studied I liked and still like a lot, and some of it is a crucial pillar of the crucial applied math core of my startup.

Intellectual pursuits have value outside of money, you know.

You speak like someone who knows the price of everything and the value of nothing.

You seriously misread my post.

A lot of "pursuits have value outside of money"; e.g., for some years I pursued violin.

But if a person is being paid to do research in mathematics, then usually in some sense commonly the people paying will want to know if the work is or will soon become useful. In fact, there was the David Report that severely criticized Federally funded math research that seemed to have no intended connection with applications. The theme of that criticism was if the math is being pursued just as art, then fund it like art.

I was giving my personal opinion and not trying to change the opinion of anyone else. Read my other responses here.

that's very uncharitable. This person has a PhD in mathematics and has done some studying in the field. This is their own personal opinion of the matter after a firsthand experience.