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I have sort of a contrarian take on this, at least by traditional academic standards. I think applicability is extremely undervalued in modern mathematics. Tao

by j2kun 3y ago

I have sort of a contrarian take on this, at least by traditional academic standards. I think applicability is extremely undervalued in modern mathematics. Tao references algebraic geometry in the interview, but the people who do algebraic geometry in applied settings ("applied" is also being quite generous here, in my opinion) are doing math that is basically unrelated to algebraic geometry as practiced by theoretical mathematicians. Even in connections to computer science, the people studying algebraic circuit complexity-which a priori should benefit a lot from algebraic geometry-basically can't use any modern algebraic geometry theory because (to my understanding) the theoreiticians choose to work in a setting that is so elegant it rules out interesting algebraic circuits. And circuit complexity is still quite far removed from applications the way I think of it. I think the best and most beautiful mathematics is driven directly by applications, and made better for the pressure to apply to a real problem (not a problem that just mathematicians care about). A quote from Milind Tambe's 2018 AAAI keynote talk sums up why: > Deploying the algorithms in the field often gives us new insights into what might be wrong with our models and provides us new research directions. If you've never heard of Tambe's work, you should really check it out. He literally sends his grad students to do field work to study how their algorithms are performing. Too many mathematicians consider "applications" as "can be used in other, purely theoretical academic papers," and based on my research for my next book (pmfpbook.org) I find that large swaths of what is claimed to be applied math is actually not useful for solving the real world problems they claim to be useful for. A math thing will be tried in production, to much acclaim and fanfare, and then it will be quietly discarded in favor of simpler, more holistic, or less complicated solutions that end up performing better. But the academic publications praising it persist for decades, and nobody seems to get the message that it's not used. One thing Tao gets right is that the best math is interdisciplinary. Connecting ideas across different theories is one great way to demonstrate the power of a theory. I would add that, to be great a theory needs to _grounded_ in a practical setting, where it's not just mathematicians judging the math for its beauty but an external reality pushing back. For my aesthetic, a pretty theory is just not enough.

ducttapecrown 3y ago
"The people who do algebraic geometry in applied settings are doing math that is basically unrelated to algebraic geometry as practiced by theoretical mathematicians" seems like it is just a gatekeeping statement. It is basically related---it is algebraic geometry.
nerdponx 3y ago
An effect similar to this is what ultimately drove me away from pursuing an advanced degree in economics. I kept seeing what were supposedly great innovations in our understanding of economic theory, which were mathematically elegant and completely unrealistic with respect to how people actually behave. What's the value in a closed-form expression that doesn't describe any real-world phenomenon? I came to realize later that there is plenty of "good" economics research and modeling out there, but I had no desire to enter a field where models were not expected to be useful, in the sense of Box's aphorism.
- ysofunny 3y ago
  > What's the value in a closed-form expression that doesn't describe any real-world phenomenon? it keeps smart, driven, and potentially disruptive people busy chasing around closed forms while "real" power politics are kept firmly and rigorously under the control of incumbent powers. this may be a perspective too cynical, nonetheless I offer it as an answer to your question; still working on making this funny somehow as it hits close to why I couldn't make it through the academic-industrialized pipeline of institutional corporations (big universities) 'manufacturing' and 'selling' graduated professionals in the labor 'marketplace'
- BalinKing 3y ago
  This example feels categorically different to me, since economics is fundamentally about modeling the real world, whereas many would argue that the mathematics' ability to do so is more of a convenient byproduct. This reads to me like an indictment of economics in particular, rather than a general statement about the value of the real-world applicability of research.
ink_13 3y ago
The idea that academic work should be grounded in "applicability" is a lazy canard. Nobody was much convinced that decades of work in number theory was worth much of anything until applications in cryptography were discovered. We just don't know what we'll find. Academic research in every field is undertaken for its own sake. If you want to work on "applicability", go get a job in industry. Finance is still hiring quants, for example.
- j2kun 3y ago
  People are welcome to do whatever they like. I'm expressing an opinion, my value judgement, that the math done for its own sake is just not as beautiful to me as math that has applications. You call it laziness, and I call it a refined view built up over decades of study and repeatedly hearing mathematicians bullshit and be misinformed about how applicable their work is.
  - wadd1e 3y ago
    >that the math done for its own sake is just not as beautiful to me as math that has applications. The point that the person who you're replying to is making is that you dont know the distinction between maths done for its own sake and maths that has applications until potentially hundreds of years later.
  - chongli 3y ago
    If you want to read a more thorough defence of the value of pure mathematics research, I recommend that you read A Mathematician's Apology by G.H. Hardy. Hardy witnessed first-hand the beginnings of the applications of his field, number theory, to cryptography. At that time, the only use for cryptography was in warfare so as a committed pacifist you can well imagine how much this upset him. misinformed about how applicable their work is I think I know exactly what you're talking about and it's unfair. I've also seen it happen to scientists across the spectrum. A question from a journalist or other layperson -- about the applicability of their research -- to catch them off guard and put them on the spot, designed to humble them and make the questioner feel superior. You never hear this sort of question levelled at painters, sculptors, musicians, writers, game designers, or any other artist!
eachro 3y ago
Every year I see tons of CS students captivated by the beauty of algorithms in theoretical CS. I lose track of all the bright eyed undergraduates saying they love thinking about algorithms and would ideally like to spend their summer doing research on these sorts of problems. More often than not, I end up telling them they should focus their efforts on the systems side of things and chat with an OS or DB professor rather than a prof in TCS, but very few of them actually take this advice.
- gcanko 3y ago
  I’m curious why would it better to focus on the system side of things instead of algorithms?
  - priprimer 3y ago
    software algorithms need to be written once to be used by anyone (digital copies and all that) systems side doesn’t have this problematic quality
  - eachro 3y ago
    Because there are real problems to work on that benefit from the sort of problem solving with algorithms that students are hoping to do.
umutisik 3y ago
As a former mathematician, I completely agree. Academic math is good stuff but you end up making too many assumptions that end up being hard to reconcile with a working system. Personally, I stopped caring much about beauty because doing work guided by some beauty heuristics didn't make me happy. Doing work that is useful to many people does make me happy; and there ends up being beauty in it somehow.
somethingsome 3y ago
> I think the best and most beautiful mathematics is driven directly by applications, and made better for the pressure to apply to a real problem (not a problem that just mathematicians care about). That's the thing, in my opinion, when you study math, you start with applications, you work your way to solve complex applicative problems, and then, those problems start to be less interesting, in part because they become easy for you, and the other part is that most of the problem is just 'mechanical and technical' work. Not really interesting. But what you like in math, it's not the maths themselves, it's to be challenged by problems. So you start to work on abstracts problems, until they become easy, then you go one more step in the abstraction, to find challenging problems, etc.. So basically, I find it somewhat natural to be driven always to more abstract theory, with maybe the hope that one day they will be applied. It's far from you to solve too 'technical' problems in the sense that 'it's just an application of XYZ problem to solve, now, I don't want to implement it, as it will take me a lot of time, while I can find pleasure again in discovering new hard things' I'm also on the applicative side, but I understand the attraction of the abstract side, and sometimes, I allow myself to work with just a bit of higher abstraction. So with enough luck, some day, a mathematician, with the right level of abstraction at that moment, will discover an applicative problem that is of your interest, and dedicate to it some time, maybe a lot of time, and maybe even, his/her whole carreer, making it more abstract, elegant and practical.

by j2kun 3y ago

ducttapecrown 3y ago

"The people who do algebraic geometry in applied settings are doing math that is basically unrelated to algebraic geometry as practiced by theoretical mathematicians" seems like it is just a gatekeeping statement. It is basically related---it is algebraic geometry.

nerdponx 3y ago

An effect similar to this is what ultimately drove me away from pursuing an advanced degree in economics. I kept seeing what were supposedly great innovations in our understanding of economic theory, which were mathematically elegant and completely unrealistic with respect to how people actually behave. What's the value in a closed-form expression that doesn't describe any real-world phenomenon? I came to realize later that there is plenty of "good" economics research and modeling out there, but I had no desire to enter a field where models were not expected to be useful, in the sense of Box's aphorism.

ysofunny 3y ago
> What's the value in a closed-form expression that doesn't describe any real-world phenomenon? it keeps smart, driven, and potentially disruptive people busy chasing around closed forms while "real" power politics are kept firmly and rigorously under the control of incumbent powers. this may be a perspective too cynical, nonetheless I offer it as an answer to your question; still working on making this funny somehow as it hits close to why I couldn't make it through the academic-industrialized pipeline of institutional corporations (big universities) 'manufacturing' and 'selling' graduated professionals in the labor 'marketplace'
BalinKing 3y ago
This example feels categorically different to me, since economics is fundamentally about modeling the real world, whereas many would argue that the mathematics' ability to do so is more of a convenient byproduct. This reads to me like an indictment of economics in particular, rather than a general statement about the value of the real-world applicability of research.

ink_13 3y ago

The idea that academic work should be grounded in "applicability" is a lazy canard. Nobody was much convinced that decades of work in number theory was worth much of anything until applications in cryptography were discovered. We just don't know what we'll find. Academic research in every field is undertaken for its own sake. If you want to work on "applicability", go get a job in industry. Finance is still hiring quants, for example.

j2kun 3y ago
People are welcome to do whatever they like. I'm expressing an opinion, my value judgement, that the math done for its own sake is just not as beautiful to me as math that has applications. You call it laziness, and I call it a refined view built up over decades of study and repeatedly hearing mathematicians bullshit and be misinformed about how applicable their work is.
- wadd1e 3y ago
  >that the math done for its own sake is just not as beautiful to me as math that has applications. The point that the person who you're replying to is making is that you dont know the distinction between maths done for its own sake and maths that has applications until potentially hundreds of years later.
- chongli 3y ago
  If you want to read a more thorough defence of the value of pure mathematics research, I recommend that you read A Mathematician's Apology by G.H. Hardy. Hardy witnessed first-hand the beginnings of the applications of his field, number theory, to cryptography. At that time, the only use for cryptography was in warfare so as a committed pacifist you can well imagine how much this upset him. misinformed about how applicable their work is I think I know exactly what you're talking about and it's unfair. I've also seen it happen to scientists across the spectrum. A question from a journalist or other layperson -- about the applicability of their research -- to catch them off guard and put them on the spot, designed to humble them and make the questioner feel superior. You never hear this sort of question levelled at painters, sculptors, musicians, writers, game designers, or any other artist!

eachro 3y ago

Every year I see tons of CS students captivated by the beauty of algorithms in theoretical CS. I lose track of all the bright eyed undergraduates saying they love thinking about algorithms and would ideally like to spend their summer doing research on these sorts of problems. More often than not, I end up telling them they should focus their efforts on the systems side of things and chat with an OS or DB professor rather than a prof in TCS, but very few of them actually take this advice.

gcanko 3y ago
I’m curious why would it better to focus on the system side of things instead of algorithms?
- priprimer 3y ago
  software algorithms need to be written once to be used by anyone (digital copies and all that) systems side doesn’t have this problematic quality
- eachro 3y ago
  Because there are real problems to work on that benefit from the sort of problem solving with algorithms that students are hoping to do.

umutisik 3y ago

As a former mathematician, I completely agree. Academic math is good stuff but you end up making too many assumptions that end up being hard to reconcile with a working system. Personally, I stopped caring much about beauty because doing work guided by some beauty heuristics didn't make me happy. Doing work that is useful to many people does make me happy; and there ends up being beauty in it somehow.

somethingsome 3y ago

> I think the best and most beautiful mathematics is driven directly by applications, and made better for the pressure to apply to a real problem (not a problem that just mathematicians care about). That's the thing, in my opinion, when you study math, you start with applications, you work your way to solve complex applicative problems, and then, those problems start to be less interesting, in part because they become easy for you, and the other part is that most of the problem is just 'mechanical and technical' work. Not really interesting. But what you like in math, it's not the maths themselves, it's to be challenged by problems. So you start to work on abstracts problems, until they become easy, then you go one more step in the abstraction, to find challenging problems, etc.. So basically, I find it somewhat natural to be driven always to more abstract theory, with maybe the hope that one day they will be applied. It's far from you to solve too 'technical' problems in the sense that 'it's just an application of XYZ problem to solve, now, I don't want to implement it, as it will take me a lot of time, while I can find pleasure again in discovering new hard things' I'm also on the applicative side, but I understand the attraction of the abstract side, and sometimes, I allow myself to work with just a bit of higher abstraction. So with enough luck, some day, a mathematician, with the right level of abstraction at that moment, will discover an applicative problem that is of your interest, and dedicate to it some time, maybe a lot of time, and maybe even, his/her whole carreer, making it more abstract, elegant and practical.