I read this essay while I was still working in pure mathematics. I’m not sure whether I agreed with it then (I think I saw it more as a cheer for the home team than something that I analyzed critically). Skimming through it now, I am very skeptical of his main claim, which is that math is so interconnected that the very applied end of math benefits from the very pure end of math. His argument breaks down for at least four reasons:
(1) He gives examples of problem P1 in field F1 being related to problem P2 in field F2, then of problem P2′ in field F2 being related to problem P3 in field F3, and concludes therefore that F1 and F3 are related. There is no reason why this should be true.
(2) Just because two problems P1 and P2 are distantly related to each other doesn’t mean that the most efficient use of resources for solving problem P2 is to work on problem P1. The obvious counterargument is that the point is not to solve a specific problem, but to advance the field of mathematics as a whole. But one always has the choice to work on problems that are more or less centrally related to mathematics as a whole, and to the parts of mathematics that have any chance of actually being applied.
(3) Gowers’ main example of a pure math problem tying in to applied mathematics is the relationship between partial differential equations and various problems in discrete mathematics. However the typical application here is that partial differential equations are used as a solution method to the discrete problem—we turn the discrete problem into its continuous analog, which is much better understood, and then bound the difference between the continuous analog and the original discrete problem. Thus in his example it is actually partial differential equations, with their much more developed theory, being used to better understand the pure mathematical object; the theory of PDEs is not getting pushed forward at all.
(4) Gowers also gives the example of the relevance of the Kakeya problem to harmonic analysis, which I agree is actually a useful result. However, the conclusion to draw from this anecdote is that a randomly chosen problem has some non-zero probability of being relevant to at least one “practical” problem. We should expect to get “lucky” like this just as well if we focus on applications.
(Edited: removed reason (5) due to insufficient justification.)
From my above comments, you have probably gathered that I have moved to more applied fields than pure math. I did this for two reasons:
(1) To be able to make any claim about the practical value of math, one needs to have an understanding of how math is actually used.
(2) I realized that without any external measures of progress, the standards for whether a problem is worthwhile come down to essentially a weighted vote.
Reason (2) is tenuous but I think clear in some fields. However, I think reason (1) is a pretty solid reason that even the purest mathematicians should have some dialogue with applied researchers (or have done applied research themselves at some point).
This requires more justification than I am currently giving it, but it is easy to find examples of math problems that don’t have even a tenuous connection to reality (I have also noticed that some pure mathematicians nevertheless argue that it does; their arguments are so weak that I can only conclude that they are trying to retroactively justify a decision they already made, and I am saying this as someone who has published papers in the field in question). The source of examples I am most familiar with is (a large portion of) enumerative combinatorics. It is possible that this is the only example, but I suspect that if I was more familiar with algebraic number theory then I could make a similar claim there.
Upon further reflection, I don’t think I can justify my claim that these sorts of problems have no connection to reality at all; perhaps a better claim is that these problems are a very inefficient way of making headway on problems that we care about, even if we extrapolate into the far future. But this would be a much subtler and difficult claim to justify, so for now I’m editing my above post to retract this statement. Since you quoted it in your response, people will still have access to it if they care.
I don’t think I can justify my claim that these sorts of problems have no connection to reality at all; perhaps a better claim is that these problems are a very inefficient way of making headway on problems that we care about, even if we extrapolate into the far future.
Upvoted for correctly understanding the issue (even while taking a position opposite to mine).
For what it’s worth, I was extremely surprised that you listed, of all things, enumerative combinatorics (i.e. counting things) as an example of a branch of mathematics with a “tenuous” connection to “reality”.
I read this essay while I was still working in pure mathematics. I’m not sure whether I agreed with it then (I think I saw it more as a cheer for the home team than something that I analyzed critically). Skimming through it now, I am very skeptical of his main claim, which is that math is so interconnected that the very applied end of math benefits from the very pure end of math. His argument breaks down for at least four reasons:
(1) He gives examples of problem P1 in field F1 being related to problem P2 in field F2, then of problem P2′ in field F2 being related to problem P3 in field F3, and concludes therefore that F1 and F3 are related. There is no reason why this should be true.
(2) Just because two problems P1 and P2 are distantly related to each other doesn’t mean that the most efficient use of resources for solving problem P2 is to work on problem P1. The obvious counterargument is that the point is not to solve a specific problem, but to advance the field of mathematics as a whole. But one always has the choice to work on problems that are more or less centrally related to mathematics as a whole, and to the parts of mathematics that have any chance of actually being applied.
(3) Gowers’ main example of a pure math problem tying in to applied mathematics is the relationship between partial differential equations and various problems in discrete mathematics. However the typical application here is that partial differential equations are used as a solution method to the discrete problem—we turn the discrete problem into its continuous analog, which is much better understood, and then bound the difference between the continuous analog and the original discrete problem. Thus in his example it is actually partial differential equations, with their much more developed theory, being used to better understand the pure mathematical object; the theory of PDEs is not getting pushed forward at all.
(4) Gowers also gives the example of the relevance of the Kakeya problem to harmonic analysis, which I agree is actually a useful result. However, the conclusion to draw from this anecdote is that a randomly chosen problem has some non-zero probability of being relevant to at least one “practical” problem. We should expect to get “lucky” like this just as well if we focus on applications.
(Edited: removed reason (5) due to insufficient justification.)
From my above comments, you have probably gathered that I have moved to more applied fields than pure math. I did this for two reasons:
(1) To be able to make any claim about the practical value of math, one needs to have an understanding of how math is actually used.
(2) I realized that without any external measures of progress, the standards for whether a problem is worthwhile come down to essentially a weighted vote.
Reason (2) is tenuous but I think clear in some fields. However, I think reason (1) is a pretty solid reason that even the purest mathematicians should have some dialogue with applied researchers (or have done applied research themselves at some point).
For what it is worth, the last time here I tried to give an example of something in algebraic number theory not mattering to reality it turned out that it actually had some sort of practical purpose.
Upon further reflection, I don’t think I can justify my claim that these sorts of problems have no connection to reality at all; perhaps a better claim is that these problems are a very inefficient way of making headway on problems that we care about, even if we extrapolate into the far future. But this would be a much subtler and difficult claim to justify, so for now I’m editing my above post to retract this statement. Since you quoted it in your response, people will still have access to it if they care.
Upvoted for correctly understanding the issue (even while taking a position opposite to mine).
For what it’s worth, I was extremely surprised that you listed, of all things, enumerative combinatorics (i.e. counting things) as an example of a branch of mathematics with a “tenuous” connection to “reality”.