8 ms·

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 m

by 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!
  - sfpotter 3y ago
    It's different. Painters, sculptors, musicians, etc. are all doing something which is ultimately in service of making others happy and fulfilled. A mathematician or scientist working on something terribly narrow and abstruse is arguably only working on something in service of making themselves happy and fulfilled. Since they are frequently supported by taxpayer money, asking whether what they're working on is worth that expense is totally legitimate. The idea that we can't question the legitimacy or applicability of research because we can't predict how useful that research will be useful in 300 years is dumb. If you want to be taken seriously, it's better to spend some time dwelling on whether you are responsibly fulfilling your obligation to society. For many professors, I think this could easily be addressed by shifting the focus away from research back to where it belongs: teaching.
    - chongli 3y ago
      Since they are frequently supported by taxpayer money, asking whether what they're working on is worth that expense is totally legitimate. That's not the deal they signed up for. They were told that they were going into a purely theoretical field. To turn around and demand applications is called a bait-and-switch. For many professors, I think this could easily be addressed by shifting the focus away from research back to where it belongs: teaching. Same as above. The universities hired these professors on the basis of their research. If they want teaching staff they should hire teachers. The researchers will seek greener pastures elsewhere. As for taxpayers demanding applications? Vote for it! Give the money to DARPA or Johns Hopkins instead of Princeton IAS. But to demand that theoreticians come up with applications on the spot is disingenuous.
      - sfpotter 3y ago
        Most people don't have the luxury of tenure. Frankly, most people don't have the luxury of a single career that's well paying throughout their professional life. If funding dried up tomorrow and every number theory professor was out on their ear, that wouldn't be a bait and switch. That would be a uniquely nice and pleasant situation for them coming to an end. It's probably also good to contextualize this historically. The NSF has only existed since 1950. The high degree of funding for pure math let alone the rest of science and engineering could easily be a historical blip. I don't really care what happens to people who are currently employed doing some kind of useless research in academia. Indeed, they got hired for it. I'm simply pointing out that it's totally fair and reasonable for anyone to ask whether it's a good idea to employ people this way. Your earlier comment seems to suggest that you don't think it's OK for people to call this into question. I'm also not particularly worried about it. I'm not sure how much money IAS gets, but my guess is that DARPA and the DOE get a hell of a lot more, which (at least in my opinion ;-) ) is good.
- DrDroop 3y ago
  This is the reason I prefer the intuitionist view of mathematics vs the more common platonic view. Something as simple as the Pythagorean theorem is true because it has found so many ways to solve practical problems, the Sumerians and Egyptians basically worshiped that formula.
  - BalinKing 3y ago
    I really don't think that intuitionism has that much to do with practical application—rather, it's a view about what it means for something to be "true" in the first place. That is to say, whether an intuitionist accepts a piece of mathematics is not contingent on if it has any practical applications. I suppose one could argue that something proven in an intuitionistic manner would have a higher probability of being practically useful (since it could e.g. be extracted into an algorithm), but IMO any such correlation is pretty weak.
- Someone 3y ago
  > that the math done for its own sake is just not as beautiful to me as math that has applications. I think the reverse. In general, the math done for its own sake will be more beautiful. Think of it this way: mathematicians doing math to get some place will accept to take a long path that’s ugly and/or extremely dull, as long as it gets them there; mathematicians doing math for math’s sake will prefer to walk nicer paths. But of course, beauty is in the eye of the beholder; those who define ‘beautiful’ as ‘has direct application’ will disagree.
chongli 3y ago
decades of work in number theory Centuries of work. To Galois and at least all the way back to the Chinese Remainder Theorem, and to Euclid as well! Mathematicians have been fascinated by this stuff for a very, very long time.
- jacobolus 3y ago
  The Chinese Remainder Theorem had explicit applications to astronomical cycles and calendrical calculations right from the start. See https://people.math.harvard.edu/~knill/crt/lib/Kangsheng.pdf https://people.math.harvard.edu/~knill/crt/lib/Kangsheng.pdf The "Euclidean" algorithm was also directly relevant to both plane geometry and astronomical calculations. Many of the relevant ideas were worked on by Plato and his contemporaries at the Athenian Academy, and then in the next generation by Eudoxus and his followers. (Unfortunately most of the precursors to the Elements have been lost.)
  - thaumasiotes 3y ago
    Fun fact sort of about the Chinese Remainder Theorem: It appears to have been named after the fact that it appears in a medieval Chinese mathematical treatise. But it always reminds me of the fact that in old Chinese texts it is frequently necessary to determine a number given its remainders modulo 10 and 12. Why? Because that is the formal way to record dates! It has been for more than 3000 years. There is a Chinese cycle of 10 sequence markers, the "heavenly stems", and a separate cycle of 12, the "earthly branches". Because Chinese characters have no real ordering (there are ordering systems, but only specialists would bother learning or using them), the stems and branches are traditionally used when you need to give arbitrary names to things. (Examples of this survive in modern Chinese -- the two parties to a contract are 甲方 and 乙方, after the first two celestial stems 甲 and 乙 -- but modern Chinese are likely to use Western letters like A and B for their arbitrary names.) The stems and branches together form the cycle of sixty, from 甲子 (1-1, meaning one) up to 癸亥 (10-12, meaning sixty). The system is so old that instead of incrementing one place value at a time for a cycle of 120 the values of which can be easily calculated in your head, incrementing bumps both place values, so the value following 甲子 is 乙丑, 2-2, meaning two. What does 乙卯 refer to? Well, 乙 is 2 (of 10) and 卯 is 4 (of 12). A little work with the Chinese Remainder Theorem will get you there!
umutisik 3y ago
This assumes that the parts of number theory that ended up being useful could not have been developed after people realized you could do public key cryptography with primes. If the work is undertaken for its own sake, there should not be a need to argue about how it will be useful in the future.
- feoren 3y ago
  No, we could not wait to start doing number theory until after we discover that it's useful for cryptography. It took thousands of years to get to the point where we understood it well enough to use it. Its use would never occur to us if we had not discovered it beforehand. That's completely the wrong causal direction. What completely undiscovered branch of mathematics do you think we should explore based on an immediate need that we have right now? Not so easy, is it?
  - umutisik 3y ago
    What specific, useful things in cryptography would never have happened if we had not been studying number theory for thousands of years? Even if there are some examples, would we be significantly behind in useful capability if we didn't have those specific results? It's more efficient to work backwards from the problems you have and build out the math. That's what they did with a lot of linear algebra and functional analysis when quantum mechanics came about. I am not saying discovery-based exploration would never work; I am saying it's inefficient if the goal is technological progress.
    - feoren 3y ago
      It just feels like asking which bits of a human wouldn't have been possible without having evolved for billions of years. It's an interconnected body of work that made cryptography possible at all. So ... all of it? I know it sounds like I'm copping out of the question, and maybe I am, because it's a really complicated question you're asking. I just don't know how you're imagining humanity came up with the ideas for: - Diffie-Hellman key exchange without a deep understanding of quotient groups (and their properties, and proofs of their properties), large (co)prime numbers, and computability - Quotient groups and its applicability to this problem without a deep understanding of group theory, equivalence classes, isomorphisms, etc. - Large (co)prime numbers without work by Euler, calculations of GCD, proofs of infinite primes, understanding their densities and occurrence on the number line, etc. - Computability without tons of work by Turing, von Neumann, Church, Babbage, Goedel, etc. relying on ideas on recursion, set theory, etc. - Ideas on recursion and set theory without work on the fundamental axioms of mathematics, Peano arithmetic, etc. - Group theory without modular arithmetic, polynomials and their roots, combinatorics, etc. - Polynomials and their roots without a huge body of work going back to 2000 BC - Calculations of GCD without work by Euclid Most of these generalized abstractions came about by thinking about the more specific problems: e.g. Group Theory only exists at all because people were thinking about equivalence classes, roots of polynomials, the Chinese remainder theorem, modular arithmetic, etc. Nobody would have thought of the "big idea" without first struggling with the smaller ideas that it ended up encompassing. You can't just take out half of these pillars of thought and assume the rest would have just happened anyway.
PH95VuimJjqBqy 3y ago
> 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. There are plenty of examples of this, Bezier Curves is one such.
- jacobolus 3y ago
  "Bézier curves" were separately invented by De Castlejau at Citroën (1959) and Bézier at Renault (early 60s) and were explicitly developed for their application to computer-aided car modeling right from the start.
  - PH95VuimJjqBqy 3y ago
    that's not the history as I understand it but I also don't claim to be an expert.
    - jacobolus 3y ago
      You can read about it here, §3 https://web.archive.org/web/20170830031522id_/http://www.farinhansford.com/gerald/papers/history.pdf https://web.archive.org/web/20170830031522id_/http://www.far... Of course both De Castlejau and Bézier were building on other tools developed previously: Most importantly, Bernstein polynomials, which were developed by Bernstein in service of a constructive proof of the theorem that every continuous function can be approximated arbitrarily well by a polynomial. You can read about that history here https://doi.org/10.1016/j.cagd.2012.03.001 https://doi.org/10.1016/j.cagd.2012.03.001 https://www.researchgate.net/profile/James-Peters-3/post/Since_the_Bernstein_basis_is_more_stable_than_the_1_x_x2_x3_basis_why_do_we_use_the_second_one/attachment/59d62e4cc49f478072e9ef35/AS%3A273573887840257%401442236505726/download/Farouki-on-Berstein-basis-2012.pdf https://www.researchgate.net/profile/James-Peters-3/post/Sin...
      - PH95VuimJjqBqy 3y ago
        From memory, I recall reading that there was work done by a french mathematician that didn't seem to have much applicability until the advent of 3d modeling. I'm probably wrong here, it's mostly from memory some 20 years ago. But having said that, I think the overarching point is still valid, it's often the case that someone does something to explore and an application for it is discovered, sometimes posthumously.
sfpotter 3y ago
This is simply wrong. There is loads of academic research born out of industrial collaborations which is absolutely focused on how useful it is. Applied research exists in academia. On the flip side, if you think the only place you can do something "applicable" is by working as a quant... again, wrong! If anything, the idea that research is free to disregard applicability (read: "relevance to anyone outside academia") is what's a lazy canard. :-) People who are wrestling with whether their research is applicable are struggling to figure out how to connect it to the rest of society, which is valuable. It's responsible of them to do so.
- ducttapecrown 3y ago
  You didn't engage with the argument of the comment above you at all, you engaged with imagined arguments: that there is no academic research that is focused on research, and that the only place to do something applicable is finance. The above comment said nothing of the sort, it only claimed that research did not have to be motivated by applications, and provided evidence: such research in the past has been incredibly applicable.
  - sfpotter 3y ago
    'If you want to work on "applicability", go get a job in industry.'

by ink_13 3y ago

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!
- sfpotter 3y ago
  It's different. Painters, sculptors, musicians, etc. are all doing something which is ultimately in service of making others happy and fulfilled. A mathematician or scientist working on something terribly narrow and abstruse is arguably only working on something in service of making themselves happy and fulfilled. Since they are frequently supported by taxpayer money, asking whether what they're working on is worth that expense is totally legitimate. The idea that we can't question the legitimacy or applicability of research because we can't predict how useful that research will be useful in 300 years is dumb. If you want to be taken seriously, it's better to spend some time dwelling on whether you are responsibly fulfilling your obligation to society. For many professors, I think this could easily be addressed by shifting the focus away from research back to where it belongs: teaching.
  - chongli 3y ago
    Since they are frequently supported by taxpayer money, asking whether what they're working on is worth that expense is totally legitimate. That's not the deal they signed up for. They were told that they were going into a purely theoretical field. To turn around and demand applications is called a bait-and-switch. For many professors, I think this could easily be addressed by shifting the focus away from research back to where it belongs: teaching. Same as above. The universities hired these professors on the basis of their research. If they want teaching staff they should hire teachers. The researchers will seek greener pastures elsewhere. As for taxpayers demanding applications? Vote for it! Give the money to DARPA or Johns Hopkins instead of Princeton IAS. But to demand that theoreticians come up with applications on the spot is disingenuous.
    - sfpotter 3y ago
      Most people don't have the luxury of tenure. Frankly, most people don't have the luxury of a single career that's well paying throughout their professional life. If funding dried up tomorrow and every number theory professor was out on their ear, that wouldn't be a bait and switch. That would be a uniquely nice and pleasant situation for them coming to an end. It's probably also good to contextualize this historically. The NSF has only existed since 1950. The high degree of funding for pure math let alone the rest of science and engineering could easily be a historical blip. I don't really care what happens to people who are currently employed doing some kind of useless research in academia. Indeed, they got hired for it. I'm simply pointing out that it's totally fair and reasonable for anyone to ask whether it's a good idea to employ people this way. Your earlier comment seems to suggest that you don't think it's OK for people to call this into question. I'm also not particularly worried about it. I'm not sure how much money IAS gets, but my guess is that DARPA and the DOE get a hell of a lot more, which (at least in my opinion ;-) ) is good.
DrDroop 3y ago
This is the reason I prefer the intuitionist view of mathematics vs the more common platonic view. Something as simple as the Pythagorean theorem is true because it has found so many ways to solve practical problems, the Sumerians and Egyptians basically worshiped that formula.
- BalinKing 3y ago
  I really don't think that intuitionism has that much to do with practical application—rather, it's a view about what it means for something to be "true" in the first place. That is to say, whether an intuitionist accepts a piece of mathematics is not contingent on if it has any practical applications. I suppose one could argue that something proven in an intuitionistic manner would have a higher probability of being practically useful (since it could e.g. be extracted into an algorithm), but IMO any such correlation is pretty weak.
Someone 3y ago
> that the math done for its own sake is just not as beautiful to me as math that has applications. I think the reverse. In general, the math done for its own sake will be more beautiful. Think of it this way: mathematicians doing math to get some place will accept to take a long path that’s ugly and/or extremely dull, as long as it gets them there; mathematicians doing math for math’s sake will prefer to walk nicer paths. But of course, beauty is in the eye of the beholder; those who define ‘beautiful’ as ‘has direct application’ will disagree.

chongli 3y ago

decades of work in number theory Centuries of work. To Galois and at least all the way back to the Chinese Remainder Theorem, and to Euclid as well! Mathematicians have been fascinated by this stuff for a very, very long time.

jacobolus 3y ago
The Chinese Remainder Theorem had explicit applications to astronomical cycles and calendrical calculations right from the start. See https://people.math.harvard.edu/~knill/crt/lib/Kangsheng.pdf https://people.math.harvard.edu/~knill/crt/lib/Kangsheng.pdf The "Euclidean" algorithm was also directly relevant to both plane geometry and astronomical calculations. Many of the relevant ideas were worked on by Plato and his contemporaries at the Athenian Academy, and then in the next generation by Eudoxus and his followers. (Unfortunately most of the precursors to the Elements have been lost.)
- thaumasiotes 3y ago
  Fun fact sort of about the Chinese Remainder Theorem: It appears to have been named after the fact that it appears in a medieval Chinese mathematical treatise. But it always reminds me of the fact that in old Chinese texts it is frequently necessary to determine a number given its remainders modulo 10 and 12. Why? Because that is the formal way to record dates! It has been for more than 3000 years. There is a Chinese cycle of 10 sequence markers, the "heavenly stems", and a separate cycle of 12, the "earthly branches". Because Chinese characters have no real ordering (there are ordering systems, but only specialists would bother learning or using them), the stems and branches are traditionally used when you need to give arbitrary names to things. (Examples of this survive in modern Chinese -- the two parties to a contract are 甲方 and 乙方, after the first two celestial stems 甲 and 乙 -- but modern Chinese are likely to use Western letters like A and B for their arbitrary names.) The stems and branches together form the cycle of sixty, from 甲子 (1-1, meaning one) up to 癸亥 (10-12, meaning sixty). The system is so old that instead of incrementing one place value at a time for a cycle of 120 the values of which can be easily calculated in your head, incrementing bumps both place values, so the value following 甲子 is 乙丑, 2-2, meaning two. What does 乙卯 refer to? Well, 乙 is 2 (of 10) and 卯 is 4 (of 12). A little work with the Chinese Remainder Theorem will get you there!

umutisik 3y ago

This assumes that the parts of number theory that ended up being useful could not have been developed after people realized you could do public key cryptography with primes. If the work is undertaken for its own sake, there should not be a need to argue about how it will be useful in the future.

feoren 3y ago
No, we could not wait to start doing number theory until after we discover that it's useful for cryptography. It took thousands of years to get to the point where we understood it well enough to use it. Its use would never occur to us if we had not discovered it beforehand. That's completely the wrong causal direction. What completely undiscovered branch of mathematics do you think we should explore based on an immediate need that we have right now? Not so easy, is it?
- umutisik 3y ago
  What specific, useful things in cryptography would never have happened if we had not been studying number theory for thousands of years? Even if there are some examples, would we be significantly behind in useful capability if we didn't have those specific results? It's more efficient to work backwards from the problems you have and build out the math. That's what they did with a lot of linear algebra and functional analysis when quantum mechanics came about. I am not saying discovery-based exploration would never work; I am saying it's inefficient if the goal is technological progress.
  - feoren 3y ago
    It just feels like asking which bits of a human wouldn't have been possible without having evolved for billions of years. It's an interconnected body of work that made cryptography possible at all. So ... all of it? I know it sounds like I'm copping out of the question, and maybe I am, because it's a really complicated question you're asking. I just don't know how you're imagining humanity came up with the ideas for: - Diffie-Hellman key exchange without a deep understanding of quotient groups (and their properties, and proofs of their properties), large (co)prime numbers, and computability - Quotient groups and its applicability to this problem without a deep understanding of group theory, equivalence classes, isomorphisms, etc. - Large (co)prime numbers without work by Euler, calculations of GCD, proofs of infinite primes, understanding their densities and occurrence on the number line, etc. - Computability without tons of work by Turing, von Neumann, Church, Babbage, Goedel, etc. relying on ideas on recursion, set theory, etc. - Ideas on recursion and set theory without work on the fundamental axioms of mathematics, Peano arithmetic, etc. - Group theory without modular arithmetic, polynomials and their roots, combinatorics, etc. - Polynomials and their roots without a huge body of work going back to 2000 BC - Calculations of GCD without work by Euclid Most of these generalized abstractions came about by thinking about the more specific problems: e.g. Group Theory only exists at all because people were thinking about equivalence classes, roots of polynomials, the Chinese remainder theorem, modular arithmetic, etc. Nobody would have thought of the "big idea" without first struggling with the smaller ideas that it ended up encompassing. You can't just take out half of these pillars of thought and assume the rest would have just happened anyway.

PH95VuimJjqBqy 3y ago

> 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. There are plenty of examples of this, Bezier Curves is one such.

jacobolus 3y ago
"Bézier curves" were separately invented by De Castlejau at Citroën (1959) and Bézier at Renault (early 60s) and were explicitly developed for their application to computer-aided car modeling right from the start.
- PH95VuimJjqBqy 3y ago
  that's not the history as I understand it but I also don't claim to be an expert.
  - jacobolus 3y ago
    You can read about it here, §3 https://web.archive.org/web/20170830031522id_/http://www.farinhansford.com/gerald/papers/history.pdf https://web.archive.org/web/20170830031522id_/http://www.far... Of course both De Castlejau and Bézier were building on other tools developed previously: Most importantly, Bernstein polynomials, which were developed by Bernstein in service of a constructive proof of the theorem that every continuous function can be approximated arbitrarily well by a polynomial. You can read about that history here https://doi.org/10.1016/j.cagd.2012.03.001 https://doi.org/10.1016/j.cagd.2012.03.001 https://www.researchgate.net/profile/James-Peters-3/post/Since_the_Bernstein_basis_is_more_stable_than_the_1_x_x2_x3_basis_why_do_we_use_the_second_one/attachment/59d62e4cc49f478072e9ef35/AS%3A273573887840257%401442236505726/download/Farouki-on-Berstein-basis-2012.pdf https://www.researchgate.net/profile/James-Peters-3/post/Sin...
    - PH95VuimJjqBqy 3y ago
      From memory, I recall reading that there was work done by a french mathematician that didn't seem to have much applicability until the advent of 3d modeling. I'm probably wrong here, it's mostly from memory some 20 years ago. But having said that, I think the overarching point is still valid, it's often the case that someone does something to explore and an application for it is discovered, sometimes posthumously.

sfpotter 3y ago

This is simply wrong. There is loads of academic research born out of industrial collaborations which is absolutely focused on how useful it is. Applied research exists in academia. On the flip side, if you think the only place you can do something "applicable" is by working as a quant... again, wrong! If anything, the idea that research is free to disregard applicability (read: "relevance to anyone outside academia") is what's a lazy canard. :-) People who are wrestling with whether their research is applicable are struggling to figure out how to connect it to the rest of society, which is valuable. It's responsible of them to do so.

ducttapecrown 3y ago
You didn't engage with the argument of the comment above you at all, you engaged with imagined arguments: that there is no academic research that is focused on research, and that the only place to do something applicable is finance. The above comment said nothing of the sort, it only claimed that research did not have to be motivated by applications, and provided evidence: such research in the past has been incredibly applicable.
- sfpotter 3y ago
  'If you want to work on "applicability", go get a job in industry.'