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It's a puzzle within a puzzle as the post author describes and describes in detail in the presentation he has made. Took me back to my college days. Here's the

It's a puzzle within a puzzle as the post author describes and describes in detail in the presentation he has made. Took me back to my college days. Here's the direct link to the presentation the post author links to in the blog https://math.ucr.edu/home/baez/ramanujan/ramanujan_whittier_web.pdf https://math.ucr.edu/home/baez/ramanujan/ramanujan_whittier_...

Sniffnoy 6y ago
And here are direct links to the pair of posts where he discussed it on the n-Category Cafe: https://golem.ph.utexas.edu/category/2020/08/chasing_the_tail_of_the_gaussi.html https://golem.ph.utexas.edu/category/2020/08/chasing_the_tai... https://golem.ph.utexas.edu/category/2020/09/chasing_the_tail_of_the_gaussi_1.html https://golem.ph.utexas.edu/category/2020/09/chasing_the_tai...
ddxxdd 6y ago
I am not a fan of these types of proofs. Baez didn't explain how one would go about searching for this proof, he just happened to know that repeatedly using the differential operator on a seemingly random differential equation would create a pattern that resembles the original equation after using substitution repeatedly. In other words, this presentation doesn't seem to teach me how to think like a mathematician, it seems more like showing me how mathematicians can find solutions that nobody else could ever find in a million years. I guess that's the point when the topic is about Ramanujan, maybe?
- TheGallopedHigh 6y ago
  Unfortunately sometimes mathematics just uses tricks to do certain things. As you move along in your math learning you pick up a bag of tricks to deal with certain situations that you’ve seen before.
- jan_Inkepa 6y ago
  When you see some infinite nested pattern, you try to capture its symmetries in some way. Differential equations are one such way. Or looking for symmetries. Induction can also help, or looking for fixed points. They can all give information. It’s a bag of tools. One could go and then look at the differential equation in more detail and relate it geometrically to the series and maybe learn some cool things (but that would be a posteriori, which you don’t want!) Also, the differential equation doesn’t come from nowhere/is not random, it’s derived in the presentation from differentiating the function and seeing that it still resembles the original function in some way, allowing you to describe it with a differential equation. What he describes as a trick with solving the differential equation can be explained - if you have f’(x)=A(x)f(x)+b, that’s a strong hint there’s an exponential there somewhere; if A is x, then the chain rule hints that you have and x^2 in the exponential, etc... A lot of it (depending on what mathematician) can boil down to pattern matching and having a big-enough bag of tricks.
- dan-robertson 6y ago
  The source of the derivation of the continued fraction part is https://mobile.twitter.com/duetosymmetry/status/1302021471749971970 https://mobile.twitter.com/duetosymmetry/status/130202147174... and it was an attempt to simplify a derivation due to Jacobi. Perhaps he explains how he comes up with it, though you’ll need to be able to read Latin. Possibly this is a relatively standard technique for solving differential equations with continued fractions which people no longer use.
- MaxBarraclough 6y ago
  I'm reminded of this answer, regrettably on Quora, about how mathematicians have an unfortunate habit of tearing down the scaffolding they used to build their results. https://www.quora.com/How-did-Strassen-derive-his-matrix-multiplication-algorithm/answer/Robert-Smith-9 https://www.quora.com/How-did-Strassen-derive-his-matrix-mul...
- sn41 6y ago
  There is a lot of pattern-based theorem proving in the works of Ramanujan, Jacobi and Euler. This is beautiful mathematics, but this is not how usual mathematical research is done. Thinking like a mathematician often involves generalization, finding analogies between seemingly different theorems etc. "Good mathematicians see analogies between theorems or theories. the very best ones see analogies between analogies." Stefan Banach.
- seppel 6y ago
  > I am not a fan of these types of proofs. Baez didn't explain how one would go about searching for this proof, he just happened to know that repeatedly using the differential operator on a seemingly random differential equation would create a pattern that resembles the original equation after using substitution repeatedly. Indeed. But without knowing what really happend here, I suspect the following: Somebody (I guess Ramanujan), came up with the random differential equation, then found two equal solutions that looked structurally different. So he asked how you can prove that they are equal. So, I'm not sure here, but in general, this is how many mathematical puzzles are created: You come up with a random proof by starting somewhere in the middle that then transforming your equation in two totally different directions. If you now throw away the middle part, you have a hard to prove fact.
- gus_massa 6y ago
  > he just happened to know that repeatedly using the differential operator on a seemingly random differential equation would create a pattern that resembles the original equation after using substitution repeatedly Yep. One of the main differences of advanced Math is that the problem gives no clues about the solution. You look at the problem, you pick one of the standard tricks from your backpack of trick, and try to hit the problem as hard as you can. Sometimes the trick solves the problem. Sometimes the trick simplifies the problem. (Sometimes it is not obvious that it is a simplification). Sometimes the trick does nothing, so you just pick another trick from your backpack of tricks... If that doesn't work, you call a friend that has another backpack full of tricks ... The idea is that in a Math BS or PhD you see a lot of tricks, and get some advice about where each one can be useful. Transforming an infinite sum to a function is an standard trick. Transforming that to a differential equation is not so standard, but I've seen it before. And sometimes no trick solves the problem, so you must invent a new trick. After a few year, if the trick is useful in other problems it will become popular and it will be added to the standard curriculum of a major in Math, or to the advanced classes for PhD, or just be a standard trick in a small niche.
  - kkwteh 6y ago
    There are two ways to solve a math problem and you described one of them: hit the nut with a hammer as hard as you can until it breaks. Grothendieck preferred the second method of slowly raising the ocean and soaking the nut for a few years, until one day it opens all by itself.
    - jan_Inkepa 6y ago
      Homeopathic approaches can feel just as arbitrary, though. “How did the mathematician know to take a bazillion tiny steps to get here?”, “Which steps in this process are the ones that do anything?”, and “Why can’t they just get to the point?!” are all questions one might ask oneself while in the midst of reading such material.
  - mrdmnd 6y ago
    This is the best comment I've ever seen describing the actual practice of mathematics.
- scythe 6y ago
  >Baez didn't explain how one would go about searching for this proof, he just happened to know that repeatedly using the differential operator on a seemingly random differential equation would create a pattern that resembles the original equation after using substitution repeatedly. If you're not used to working with continued fractions, the techniques are very counterintuitive. None of the usual calculus tricks for infinite sequences work normally. It's probably not best to read that part as an introduction to the theory of continued fractions. As Baez himself admitted, even he had a hard time understanding Laplace's proof. The rest of the development, I assume, was not so confounding. It's a little of both: here's something you might understand, there's some black magic.

Sniffnoy 6y ago

And here are direct links to the pair of posts where he discussed it on the n-Category Cafe: https://golem.ph.utexas.edu/category/2020/08/chasing_the_tail_of_the_gaussi.html https://golem.ph.utexas.edu/category/2020/08/chasing_the_tai... https://golem.ph.utexas.edu/category/2020/09/chasing_the_tail_of_the_gaussi_1.html https://golem.ph.utexas.edu/category/2020/09/chasing_the_tai...

ddxxdd 6y ago

I am not a fan of these types of proofs. Baez didn't explain how one would go about searching for this proof, he just happened to know that repeatedly using the differential operator on a seemingly random differential equation would create a pattern that resembles the original equation after using substitution repeatedly. In other words, this presentation doesn't seem to teach me how to think like a mathematician, it seems more like showing me how mathematicians can find solutions that nobody else could ever find in a million years. I guess that's the point when the topic is about Ramanujan, maybe?

TheGallopedHigh 6y ago
Unfortunately sometimes mathematics just uses tricks to do certain things. As you move along in your math learning you pick up a bag of tricks to deal with certain situations that you’ve seen before.
jan_Inkepa 6y ago
When you see some infinite nested pattern, you try to capture its symmetries in some way. Differential equations are one such way. Or looking for symmetries. Induction can also help, or looking for fixed points. They can all give information. It’s a bag of tools. One could go and then look at the differential equation in more detail and relate it geometrically to the series and maybe learn some cool things (but that would be a posteriori, which you don’t want!) Also, the differential equation doesn’t come from nowhere/is not random, it’s derived in the presentation from differentiating the function and seeing that it still resembles the original function in some way, allowing you to describe it with a differential equation. What he describes as a trick with solving the differential equation can be explained - if you have f’(x)=A(x)f(x)+b, that’s a strong hint there’s an exponential there somewhere; if A is x, then the chain rule hints that you have and x^2 in the exponential, etc... A lot of it (depending on what mathematician) can boil down to pattern matching and having a big-enough bag of tricks.
dan-robertson 6y ago
The source of the derivation of the continued fraction part is https://mobile.twitter.com/duetosymmetry/status/1302021471749971970 https://mobile.twitter.com/duetosymmetry/status/130202147174... and it was an attempt to simplify a derivation due to Jacobi. Perhaps he explains how he comes up with it, though you’ll need to be able to read Latin. Possibly this is a relatively standard technique for solving differential equations with continued fractions which people no longer use.
MaxBarraclough 6y ago
I'm reminded of this answer, regrettably on Quora, about how mathematicians have an unfortunate habit of tearing down the scaffolding they used to build their results. https://www.quora.com/How-did-Strassen-derive-his-matrix-multiplication-algorithm/answer/Robert-Smith-9 https://www.quora.com/How-did-Strassen-derive-his-matrix-mul...
sn41 6y ago
There is a lot of pattern-based theorem proving in the works of Ramanujan, Jacobi and Euler. This is beautiful mathematics, but this is not how usual mathematical research is done. Thinking like a mathematician often involves generalization, finding analogies between seemingly different theorems etc. "Good mathematicians see analogies between theorems or theories. the very best ones see analogies between analogies." Stefan Banach.
seppel 6y ago
> I am not a fan of these types of proofs. Baez didn't explain how one would go about searching for this proof, he just happened to know that repeatedly using the differential operator on a seemingly random differential equation would create a pattern that resembles the original equation after using substitution repeatedly. Indeed. But without knowing what really happend here, I suspect the following: Somebody (I guess Ramanujan), came up with the random differential equation, then found two equal solutions that looked structurally different. So he asked how you can prove that they are equal. So, I'm not sure here, but in general, this is how many mathematical puzzles are created: You come up with a random proof by starting somewhere in the middle that then transforming your equation in two totally different directions. If you now throw away the middle part, you have a hard to prove fact.
gus_massa 6y ago
> he just happened to know that repeatedly using the differential operator on a seemingly random differential equation would create a pattern that resembles the original equation after using substitution repeatedly Yep. One of the main differences of advanced Math is that the problem gives no clues about the solution. You look at the problem, you pick one of the standard tricks from your backpack of trick, and try to hit the problem as hard as you can. Sometimes the trick solves the problem. Sometimes the trick simplifies the problem. (Sometimes it is not obvious that it is a simplification). Sometimes the trick does nothing, so you just pick another trick from your backpack of tricks... If that doesn't work, you call a friend that has another backpack full of tricks ... The idea is that in a Math BS or PhD you see a lot of tricks, and get some advice about where each one can be useful. Transforming an infinite sum to a function is an standard trick. Transforming that to a differential equation is not so standard, but I've seen it before. And sometimes no trick solves the problem, so you must invent a new trick. After a few year, if the trick is useful in other problems it will become popular and it will be added to the standard curriculum of a major in Math, or to the advanced classes for PhD, or just be a standard trick in a small niche.
- kkwteh 6y ago
  There are two ways to solve a math problem and you described one of them: hit the nut with a hammer as hard as you can until it breaks. Grothendieck preferred the second method of slowly raising the ocean and soaking the nut for a few years, until one day it opens all by itself.
  - jan_Inkepa 6y ago
    Homeopathic approaches can feel just as arbitrary, though. “How did the mathematician know to take a bazillion tiny steps to get here?”, “Which steps in this process are the ones that do anything?”, and “Why can’t they just get to the point?!” are all questions one might ask oneself while in the midst of reading such material.
- mrdmnd 6y ago
  This is the best comment I've ever seen describing the actual practice of mathematics.
scythe 6y ago
>Baez didn't explain how one would go about searching for this proof, he just happened to know that repeatedly using the differential operator on a seemingly random differential equation would create a pattern that resembles the original equation after using substitution repeatedly. If you're not used to working with continued fractions, the techniques are very counterintuitive. None of the usual calculus tricks for infinite sequences work normally. It's probably not best to read that part as an introduction to the theory of continued fractions. As Baez himself admitted, even he had a hard time understanding Laplace's proof. The rest of the development, I assume, was not so confounding. It's a little of both: here's something you might understand, there's some black magic.