"Applied pure mathematics is any application of any mathematics to anything outside of the closed world of mathematics itself. You never know which weird corner of the vast libraries of “apparently useless” mathematics will help you make sense of reality."
“That is, a lot of mathematics is training people to benchmaxx”
I would go further and say a lot of education is training people to benchmaxx and this is another ill-intended effect of quantifauxcation. Liberal arts education was traded with the benchmaxx education and we have found out that machines can benchmaxx better than us.
Agreed, have always thought that practicing mathematical-problem-solving or theorem-proving would never go out of vogue since it trains the brain in a different type of thinking. Working out did not go out of vogue after most professions became sedentary. Except working out has a visible benefit on your muscles. We should make sure our mathematical thinking muscles do not atrophy.
1. Some of us in that gold rush did aim for the practical, ahem! But, yes, there was a lot of cult of personality around various problems.
2. The fetishization of puzzle-solving is brilliant. I have mumbled to myself at times about the Kagglification of research. Same idea.
3. Paul Sally was a treasure. On my first day, he walked in with a drill sergeant’s hat on, his usual eye patch, and bellowed, “This is kick ass mathematics! There are 40 of you in here. That’s 20 too many. When you fail the exam Friday of first week, you can drop the class.”
May be the classical pure vs applied is the wrong taxonomy. Hardy defended, dare I say even fetishized the search of truth regardless of whether it has an application as if that were pure. The math that is untainted by whether or not it has applications - but this changes over time. Is number theory less pure once it has application in cryptography? Is category theory less abstract now if parts of it underlies the basis of type theory (CS)?
This is why Tim Gower’s Theory Building vs Problem Solving distinction (he doesn’t take sides as he explained recently) is useful. Because this defines taste of what style appeals to you.
The natural complement to Tim Gower’s two cultures axis is the source of what drives you to build theory. Here, I see as you as unapologetically grounded in real life, much in the culture of Shannon. He founded an entire theory around a problem he was trying to solve. So, may be you are an application-inspired theory builder? And for this you need a little bit of both - persistence as in problem solving and an aesthetic sense when you are not merely satisfied when you got the trophy.
As I see it, if you are in pure "pure mathematics" the practical successes from applied pure mathematics are not going to cut it anymore once you have to argue for your existence (Tao already tried exactly that with the compressed sensing story a year ago). That of course is a problem of pure "pure mathematics" and it will be interesting to see if the field is going to fission.
As for the second point that probably requires a bit of elaboration. But in short I think, if you are in it for l'art pour l'art then the consequentialist route of justification is blocked for you. Of course it might happen, that some of your creations by chance find some application. But this is not why you did it to begin with and there are others who do exactly that. And more efficiently. Why not be honest then and join the other products of culture that are created without practical intent?
This is all true and important- but also, color me deeply skeptical that human math is toast quite yet. It's pretty clear at this point that the Usual Suspects have scooped up whole mathematics departments to train the inputs and sift the outputs of truly astonishing quantities of monkeys hammering on typewriters *to produce headlines exactly like this that land with people that equate intelligence with math*, and we are seeing the few instances where they have succeeded, and the result keeps looking like 'a person had solved most of it but left their data sitting somewhere that should have been private, but we live in hell.' They spent $22M in *electricity* to produce an unverified solution, with the help of every mathematician in the world that wasn't strapped down, and they have so much cash slopping around it seems pretty reasonable to assume there are several hundred million dollars of churning on other problems that went nowhere- perhaps because they had no chats to rip off.
There's that line in 'The Prestige' where David Bowie/Tesla notes that what he has been asked to build is not impossible, it is merely expensive, and that's sorta how I feel about this. Companies with a mythology to farm to keep the cash fires burning have latched onto a field of human endeavor that might be the most amenable to any to computer attack/assistance, and they've ecked out a couple flashy wins with dubious attribution, omitted some denominators, and did so at a cost that is simply not sustainable. Meanwhile, students still show me LLM generated worksheets of high school math where the arithmetic is wrong.
Hi professor, it's surprised to know that you view compressing sensing as some type of gold rush. Thank you for introducing the term of applied pure math and this golden post
To pile on the Wigner references, the last paragraph from "Are We Machines", 1969
> Returning to the question in the title of this section, we must conclude that, if the preceding discussion is valid, life does share with machines the property that it can be created artificially. None of this should obscure the fact, self-evident to this writer, that having emotions and desires does profoundly distinguish us from machine.
I'm sure the words to replace "emotions and desires" are in people's minds
I like this a lot and cackled at "What are millennium problems other than humanity’s final math exam?" and sobered up at "This unfortunately makes a lot of sense with the benefit of hindsight!"
So much math is undefined symbol-shuffling obscurantist pseudo-rigor, that most people in math-heavy fields don't seem to have any notion of what it means to understand something. If you illustrate something in a clear, intuitive way without all the symbology, they'll say it isn't math. E.g. three orthogonal views of a helix, sine, cosine, circle, has most of the meaning in trigonometry, as well as complex numbers and the root of Grassmann, Clifford, and Geometric Algebras.
Or the story of the mathematician hired by Edison, given his first problem: find the volume of a light bulb to figure how big the factory vacuum pumps will need to be. Hours later, he has a pile of caliper readings and is numerically integrating by hand, when Edison stops by to find out what happened to him. Edison weighs the bulb, fills the lightbulb with water, weighs it again, makes a note and walks out without a word.
Or my demo for elementary-age electronics students of a bandpass filter made from a $0.60 quad op-amp chip and some audio pots (volume knobs, stereo, log-taper variable resistors) with an old HP FFT signal analyzer. The knobs controlled the amplitude, frequency and bandwith. Sweeping the input frequency gives very slow readings of the filter spectrum envelope, using random noise input gives the results instantly, twiddle rhe knobs and you can see the envelope shift or change shape. Turn the bandwith all the way down, the filter turns noise into a pure sine wave of any frequency you like; turn the bandwith all the way up and it turns into a cos^2 comb filter, which you can shift and scale, that looks like pure noise in the time domain 'scope trace, but has hidden structure. Something sinething time-energy duality quantum Heisenburg.
This is such a profound reflection on the difference between math as a competitive quiz show and mathematics as a genuine cultural and technological language for understanding. The trap of 'open problem culture' and trophy hunting often overshadows the true beauty of applied pure mathematics—which is building mental models to make sense of reality rather than just chasing abstract trophies. For anyone looking to dive deeper into this mindset and learn how to approach applied mathematics through self-study, I’ve shared a practical guide here: https://freemathematicsbooks.substack.com/p/how-to-self-study-applied-mathematics
"This is why I like (and have been using throughout) Jordan Ellenberg’s term applied pure mathematics."
STOP TRYING TO MAKE FETCH HAPPEN
Never! I will Stiglers Law this into existence.
"Applied pure mathematics is any application of any mathematics to anything outside of the closed world of mathematics itself. You never know which weird corner of the vast libraries of “apparently useless” mathematics will help you make sense of reality."
MAH GAWD THAT'S OPERATIONAL RESEARCH'S MUSIC!!
I really enjoyed this and overall agree. You say
“That is, a lot of mathematics is training people to benchmaxx”
I would go further and say a lot of education is training people to benchmaxx and this is another ill-intended effect of quantifauxcation. Liberal arts education was traded with the benchmaxx education and we have found out that machines can benchmaxx better than us.
Agreed, have always thought that practicing mathematical-problem-solving or theorem-proving would never go out of vogue since it trains the brain in a different type of thinking. Working out did not go out of vogue after most professions became sedentary. Except working out has a visible benefit on your muscles. We should make sure our mathematical thinking muscles do not atrophy.
1. Some of us in that gold rush did aim for the practical, ahem! But, yes, there was a lot of cult of personality around various problems.
2. The fetishization of puzzle-solving is brilliant. I have mumbled to myself at times about the Kagglification of research. Same idea.
3. Paul Sally was a treasure. On my first day, he walked in with a drill sergeant’s hat on, his usual eye patch, and bellowed, “This is kick ass mathematics! There are 40 of you in here. That’s 20 too many. When you fail the exam Friday of first week, you can drop the class.”
About 10-15
Wait, now I must know: How many made it to week 2?
May be the classical pure vs applied is the wrong taxonomy. Hardy defended, dare I say even fetishized the search of truth regardless of whether it has an application as if that were pure. The math that is untainted by whether or not it has applications - but this changes over time. Is number theory less pure once it has application in cryptography? Is category theory less abstract now if parts of it underlies the basis of type theory (CS)?
This is why Tim Gower’s Theory Building vs Problem Solving distinction (he doesn’t take sides as he explained recently) is useful. Because this defines taste of what style appeals to you.
The natural complement to Tim Gower’s two cultures axis is the source of what drives you to build theory. Here, I see as you as unapologetically grounded in real life, much in the culture of Shannon. He founded an entire theory around a problem he was trying to solve. So, may be you are an application-inspired theory builder? And for this you need a little bit of both - persistence as in problem solving and an aesthetic sense when you are not merely satisfied when you got the trophy.
As I see it, if you are in pure "pure mathematics" the practical successes from applied pure mathematics are not going to cut it anymore once you have to argue for your existence (Tao already tried exactly that with the compressed sensing story a year ago). That of course is a problem of pure "pure mathematics" and it will be interesting to see if the field is going to fission.
Can you point me to Tao's compressed sensing reflection?
And could you say a bit more about why building potential intellectual scaffolding won't cut it for the pure mathematicians?
Sure, it is a series of posts (I guess they are called toots) on his mastodon beginning with https://mathstodon.xyz/@tao/114967650999562435
As for the second point that probably requires a bit of elaboration. But in short I think, if you are in it for l'art pour l'art then the consequentialist route of justification is blocked for you. Of course it might happen, that some of your creations by chance find some application. But this is not why you did it to begin with and there are others who do exactly that. And more efficiently. Why not be honest then and join the other products of culture that are created without practical intent?
Wonderful post - thx
This is all true and important- but also, color me deeply skeptical that human math is toast quite yet. It's pretty clear at this point that the Usual Suspects have scooped up whole mathematics departments to train the inputs and sift the outputs of truly astonishing quantities of monkeys hammering on typewriters *to produce headlines exactly like this that land with people that equate intelligence with math*, and we are seeing the few instances where they have succeeded, and the result keeps looking like 'a person had solved most of it but left their data sitting somewhere that should have been private, but we live in hell.' They spent $22M in *electricity* to produce an unverified solution, with the help of every mathematician in the world that wasn't strapped down, and they have so much cash slopping around it seems pretty reasonable to assume there are several hundred million dollars of churning on other problems that went nowhere- perhaps because they had no chats to rip off.
There's that line in 'The Prestige' where David Bowie/Tesla notes that what he has been asked to build is not impossible, it is merely expensive, and that's sorta how I feel about this. Companies with a mythology to farm to keep the cash fires burning have latched onto a field of human endeavor that might be the most amenable to any to computer attack/assistance, and they've ecked out a couple flashy wins with dubious attribution, omitted some denominators, and did so at a cost that is simply not sustainable. Meanwhile, students still show me LLM generated worksheets of high school math where the arithmetic is wrong.
Hi professor, it's surprised to know that you view compressing sensing as some type of gold rush. Thank you for introducing the term of applied pure math and this golden post
To pile on the Wigner references, the last paragraph from "Are We Machines", 1969
> Returning to the question in the title of this section, we must conclude that, if the preceding discussion is valid, life does share with machines the property that it can be created artificially. None of this should obscure the fact, self-evident to this writer, that having emotions and desires does profoundly distinguish us from machine.
I'm sure the words to replace "emotions and desires" are in people's minds
I like this a lot and cackled at "What are millennium problems other than humanity’s final math exam?" and sobered up at "This unfortunately makes a lot of sense with the benefit of hindsight!"
I did not however care for the recent Fields Medal Winner's statement (the authors could benefit from reading this post) because I found it mostly to be preaching to the choir. https://computingandsociety.substack.com/p/the-debate-about-ai-companies-proving.
At least, something that doesn't feel me with despair.
*fill
whoa! honors analysis with the math pirate himself?
So much math is undefined symbol-shuffling obscurantist pseudo-rigor, that most people in math-heavy fields don't seem to have any notion of what it means to understand something. If you illustrate something in a clear, intuitive way without all the symbology, they'll say it isn't math. E.g. three orthogonal views of a helix, sine, cosine, circle, has most of the meaning in trigonometry, as well as complex numbers and the root of Grassmann, Clifford, and Geometric Algebras.
Or the story of the mathematician hired by Edison, given his first problem: find the volume of a light bulb to figure how big the factory vacuum pumps will need to be. Hours later, he has a pile of caliper readings and is numerically integrating by hand, when Edison stops by to find out what happened to him. Edison weighs the bulb, fills the lightbulb with water, weighs it again, makes a note and walks out without a word.
Or my demo for elementary-age electronics students of a bandpass filter made from a $0.60 quad op-amp chip and some audio pots (volume knobs, stereo, log-taper variable resistors) with an old HP FFT signal analyzer. The knobs controlled the amplitude, frequency and bandwith. Sweeping the input frequency gives very slow readings of the filter spectrum envelope, using random noise input gives the results instantly, twiddle rhe knobs and you can see the envelope shift or change shape. Turn the bandwith all the way down, the filter turns noise into a pure sine wave of any frequency you like; turn the bandwith all the way up and it turns into a cos^2 comb filter, which you can shift and scale, that looks like pure noise in the time domain 'scope trace, but has hidden structure. Something sinething time-energy duality quantum Heisenburg.
This is such a profound reflection on the difference between math as a competitive quiz show and mathematics as a genuine cultural and technological language for understanding. The trap of 'open problem culture' and trophy hunting often overshadows the true beauty of applied pure mathematics—which is building mental models to make sense of reality rather than just chasing abstract trophies. For anyone looking to dive deeper into this mindset and learn how to approach applied mathematics through self-study, I’ve shared a practical guide here: https://freemathematicsbooks.substack.com/p/how-to-self-study-applied-mathematics