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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,

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.

graycat 9y ago
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.
- Ar-Curunir 9y ago
  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.
  - graycat 9y ago
    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.

graycat 9y ago

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.

Ar-Curunir 9y ago
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.
- graycat 9y ago
  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.