7 ms·

In Mathematics, It Often Takes a Good Map to Find Answers

153 points by yarapavan 6y ago

utkarsh_apoorva 6y ago
> But imagine how poetic it would have been if the technology for constructing such a machine had been available to da Vinci all along. Very poetic indeed. Most of entrepreneurship is applying known models to new areas. Intellectually not nearly as stimulating or hard as theoretical math, but the shape and form looks similar - you do not know if a solution exists, you do not know if a problem really exists. What's funny to me is that, since it's usually applications of engineering, the technology is almost always there. It's a matter of tinkering a collection of things the right way. I ditched a career in Physics to start a company long back. This post made me think I probably haven't lost much :-)
ssivark 6y ago
I highly recommend Bill Thurston’s gem of an article On proof and progress in mathematics https://arxiv.org/abs/math/9404236 https://arxiv.org/abs/math/9404236 Talks about the human aspect of pursuing mathematical research, how they shape the attitude of the field towards a problem abs are crucial in progressing towards knowledge. Should be very readable for everyone; no formal math as such.
- sabas123 6y ago
  It was a good read, thank you for sharing. > "It was an interesting experience exchanging cultures. It became dramatically clear how much proofs depend on the audience. We prove things in a social context and address them to a certain audience. Parts of this proof I could communicate in two minutes to the topologists, but the analysts would need an hour lecture before they would begin to understand it. Similarly, there were some things that could be said in two minutes to the analysts that would take an hour before the topologists would begin to get it. And there were many other parts of the proof which should take two minutes in the abstract, but that none of the audience at the time had the mental infrastructure to get in less than an hour" I wonder if we would ever get to a point where we would find an effective and desirable mental infrastructure such that this wouldn't happen.
  - ssivark 6y ago
    Category theory is supposed to be one such tool, even though some find it very abstract. It's very much in the spirit of finding analogies among theories and analogies among analogies. (I swear I'm not trolling :P) I'm still working on my understanding of category theory, but somebody who has the mathematical fortitude might enjoy: http://groupoids.org.uk/pdffiles/Analogy-and-Comparison.pdf http://groupoids.org.uk/pdffiles/Analogy-and-Comparison.pdf In general better abstractions(similar ideas as in a recent discussion of Peter Naur's "Programming as theory building").
- raincom 6y ago
  Just finished reading Thurston's paper. A great paper esp for those in "philosophy of mathematics".
- mturmon 6y ago
  It's a great read. See also: https://news.ycombinator.com/item?id=12280139 https://news.ycombinator.com/item?id=12280139
- Errancer 6y ago
  In this topic I also highly recommend "Proofs and Refutations" by Imre Lakatos
physicsgraph 6y ago
The article is sparse on what a detailed map for mathematics would look like and merely points out that some topics have related techniques for solving them. I don't think a map for math techniques is feasible, but a map relating topics via mathematical steps is possible in Physics [1]. (Disclaimer: I'm the author of that map for Physics.) I think the reason that a map in Physics is feasible is Physicists do not use math techniques in the way mathematicians do, and the objectives are different. https://derivationmap.net/ https://derivationmap.net/
- knzhou 6y ago
  It seems to me that your map is far too detailed to use practically. You spell out every algebraic step, including stuff as simple as "divide both sides by T", so that deriving f = 1/T from T = 1/f takes about 10 nodes. This is like building a model train to a larger scale than an actual train -- what is the use? Education research tells us that what you actually want to do is the exact opposite: chunk as much as possible. You should learn algebra separately, and then use your preexisting knowledge of algebra to group f = 1/T and T = 1/f into one conceptual node. If you need 10 nodes every time something that basic is done, then your map will contain a vast amount of redundancy and be too large to use to get anywhere...
  - physicsgraph 6y ago
    I agree that navigating a map of Physics at the very lowest level would not enlighten any student or researcher. My expectation in mapping atomic steps for a wide swath of the domain might enable insights not otherwise accessible. The chunking of atomic steps is what enables leaps in understanding. The mapping process starts with understanding each step.
    - knzhou 6y ago
      Well, I recommend doing a concrete, nontrivial derivation from start to finish just to see how this approach scales. As a basic example that is typically covered in about half a page in books, try doing a full derivation of the wave equation for a wave on a string. I would bet that once you set up the 1000 nodes required to do this, you'll be completely exhausted, and moreover will have gotten no new insight! If you're not tired yet, try deriving the equation describing waves on a stiff rod -- it'll take at least 1500 nodes, most of which will be exactly the same as the ones for the wave equation. Furthermore, this excessive mathematical structure hides the physical assumptions that really drive the validity of these equations. A real string doesn't actually obey the wave equation perfectly. The reason has to do with physical aspects of the string itself, not minutiae in the mathematical derivation of the wave equation. I can't think of an example where progress in physics was stalled because somebody tried to divide both sides of an equation by T and failed...
mmhsieh 6y ago
The difficulty in coming up with a good map of mathematics is summarized by this quote by Banach: "A mathematician is a person who can find analogies between theorems; a better mathematician is one who can see analogies between proofs and the best mathematician can notice analogies between theories. One can imagine that the ultimate mathematician is one who can see analogies between analogies."
- jordigh 6y ago
  Haha, it's like Maclane said: "I did not invent category theory to talk about functors. I invented it to talk about natural transformations." You gotta go at least to the third level of abstraction to get the real meat.
- JadeNB 6y ago
  Discussed from a literal point of view on MO: https://mathoverflow.net/questions/13832/analogies-between-analogies https://mathoverflow.net/questions/13832/analogies-between-a... .
amirhirsch 6y ago
I liked this article. One notable point that felt like it was missing in the article is that the Prime Number Theorem, that the count of primes grow like (n / ln n) was provided such a map by Riemann in the letter in which he put forward his infamous eponymous hypothesis. That letter introduced the idea of using analysis to the Prime Number Theorem, extending the groundbreaking work of Riemann's friend Dirichlet who introduced the world to analytic number theory in Dirichlet's Theorem on the infinitude of primes in arithmetic progressions. It would take nearly half a century for mathematicians to digest the application of Fourier Analysis put forward by Riemann, and the proof of the Prime Number Theorem came only in the early 1900's. By then the analytic machinery would have been more commonly taught -- probably largely due to the advent of electrical engineering. Erdos and Selberg eventually put out fully arithmetic proofs of the Prime Number Theorem. And generally the helicopter analogy from the article probably doesn't apply so well to mathematics because you can probably always reduce theories and encapsulate all the dependent proofs to arithmetic first principles, but of course you already have the map. Recently the proofs of the Sensitivity Conjecture by Hao Huang and of the Bounded Gaps Between Primes by Yitang Zhang surprised mathematicians in how little new machinery these seemingly intractable problems required -- in the case of Zhang application of "hard work" on top of GPY and Hao Huang, a single clever insight.
- tzs 6y ago
  > and the proof of the Prime Number Theorem came only in the early 1900's That's just a little too late. It was proved in 1896 independently by Hadamard and de la Vallée Poussin. Hadamard, J. "Sur la distribution des zéros de la fonction zeta(s) et ses conséquences arithmétiques (')." Bull. Soc. math. France 24, 199-220, 1896 de la Vallée Poussin, C.-J. "Recherches analytiques la théorie des nombres premiers." Ann. Soc. scient. Bruxelles 20, 183-256, 1896
CatsAreCool 6y ago
I liked this article since it points out a problem in math where it can be hard to know what is currently known. Perhaps a result can be proven using a little known proposition in a completely different area of math, but it is hard to find that result in the literature. That is one reason I came up with https://mathlore.org https://mathlore.org. It is a place to collect mathematical info (with links to articles for a deeper look) so you or others can find it later when you need it. It supports of public collection of math info as well as allowing you to build your own private collection so you can keep track of what you have learned. The hope is it will be useful to others to help learn math and prove new theorems.
swayvil 6y ago
Isn't mathematics essentially ALL map? The only measures of goodness that I can come up with are logical consistency, elegance and lurking mystery.
rmrfstar 6y ago
Maps turn out to be useful in experimental physics too. Here's a neat BBC piece featuring Gell-man and Feynman on Strangeness -3 [1]. [1] https://www.youtube.com/watch?v=BGeW6Nc6IMQ https://www.youtube.com/watch?v=BGeW6Nc6IMQ
amandavinci 6y ago
The analogy of maps and boats is just an instance of the exploration-exploitation idea. I have seen instances of this pop up every time we discuss problem solving in some form. It falls in the perennial variety of ideas that can't be revisited enough.

153 points by yarapavan 6y ago

> But imagine how poetic it would have been if the technology for constructing such a machine had been available to da Vinci all along. Very poetic indeed. Most of entrepreneurship is applying known models to new areas. Intellectually not nearly as stimulating or hard as theoretical math, but the shape and form looks similar - you do not know if a solution exists, you do not know if a problem really exists. What's funny to me is that, since it's usually applications of engineering, the technology is almost always there. It's a matter of tinkering a collection of things the right way. I ditched a career in Physics to start a company long back. This post made me think I probably haven't lost much :-)

ssivark 6y ago

I highly recommend Bill Thurston’s gem of an article On proof and progress in mathematics https://arxiv.org/abs/math/9404236 https://arxiv.org/abs/math/9404236 Talks about the human aspect of pursuing mathematical research, how they shape the attitude of the field towards a problem abs are crucial in progressing towards knowledge. Should be very readable for everyone; no formal math as such.

sabas123 6y ago
It was a good read, thank you for sharing. > "It was an interesting experience exchanging cultures. It became dramatically clear how much proofs depend on the audience. We prove things in a social context and address them to a certain audience. Parts of this proof I could communicate in two minutes to the topologists, but the analysts would need an hour lecture before they would begin to understand it. Similarly, there were some things that could be said in two minutes to the analysts that would take an hour before the topologists would begin to get it. And there were many other parts of the proof which should take two minutes in the abstract, but that none of the audience at the time had the mental infrastructure to get in less than an hour" I wonder if we would ever get to a point where we would find an effective and desirable mental infrastructure such that this wouldn't happen.
- ssivark 6y ago
  Category theory is supposed to be one such tool, even though some find it very abstract. It's very much in the spirit of finding analogies among theories and analogies among analogies. (I swear I'm not trolling :P) I'm still working on my understanding of category theory, but somebody who has the mathematical fortitude might enjoy: http://groupoids.org.uk/pdffiles/Analogy-and-Comparison.pdf http://groupoids.org.uk/pdffiles/Analogy-and-Comparison.pdf In general better abstractions(similar ideas as in a recent discussion of Peter Naur's "Programming as theory building").
raincom 6y ago
Just finished reading Thurston's paper. A great paper esp for those in "philosophy of mathematics".
mturmon 6y ago
It's a great read. See also: https://news.ycombinator.com/item?id=12280139 https://news.ycombinator.com/item?id=12280139
Errancer 6y ago
In this topic I also highly recommend "Proofs and Refutations" by Imre Lakatos

physicsgraph 6y ago

The article is sparse on what a detailed map for mathematics would look like and merely points out that some topics have related techniques for solving them. I don't think a map for math techniques is feasible, but a map relating topics via mathematical steps is possible in Physics [1]. (Disclaimer: I'm the author of that map for Physics.) I think the reason that a map in Physics is feasible is Physicists do not use math techniques in the way mathematicians do, and the objectives are different. https://derivationmap.net/ https://derivationmap.net/

knzhou 6y ago
It seems to me that your map is far too detailed to use practically. You spell out every algebraic step, including stuff as simple as "divide both sides by T", so that deriving f = 1/T from T = 1/f takes about 10 nodes. This is like building a model train to a larger scale than an actual train -- what is the use? Education research tells us that what you actually want to do is the exact opposite: chunk as much as possible. You should learn algebra separately, and then use your preexisting knowledge of algebra to group f = 1/T and T = 1/f into one conceptual node. If you need 10 nodes every time something that basic is done, then your map will contain a vast amount of redundancy and be too large to use to get anywhere...
- physicsgraph 6y ago
  I agree that navigating a map of Physics at the very lowest level would not enlighten any student or researcher. My expectation in mapping atomic steps for a wide swath of the domain might enable insights not otherwise accessible. The chunking of atomic steps is what enables leaps in understanding. The mapping process starts with understanding each step.
  - knzhou 6y ago
    Well, I recommend doing a concrete, nontrivial derivation from start to finish just to see how this approach scales. As a basic example that is typically covered in about half a page in books, try doing a full derivation of the wave equation for a wave on a string. I would bet that once you set up the 1000 nodes required to do this, you'll be completely exhausted, and moreover will have gotten no new insight! If you're not tired yet, try deriving the equation describing waves on a stiff rod -- it'll take at least 1500 nodes, most of which will be exactly the same as the ones for the wave equation. Furthermore, this excessive mathematical structure hides the physical assumptions that really drive the validity of these equations. A real string doesn't actually obey the wave equation perfectly. The reason has to do with physical aspects of the string itself, not minutiae in the mathematical derivation of the wave equation. I can't think of an example where progress in physics was stalled because somebody tried to divide both sides of an equation by T and failed...

mmhsieh 6y ago

The difficulty in coming up with a good map of mathematics is summarized by this quote by Banach: "A mathematician is a person who can find analogies between theorems; a better mathematician is one who can see analogies between proofs and the best mathematician can notice analogies between theories. One can imagine that the ultimate mathematician is one who can see analogies between analogies."

jordigh 6y ago
Haha, it's like Maclane said: "I did not invent category theory to talk about functors. I invented it to talk about natural transformations." You gotta go at least to the third level of abstraction to get the real meat.
JadeNB 6y ago
Discussed from a literal point of view on MO: https://mathoverflow.net/questions/13832/analogies-between-analogies https://mathoverflow.net/questions/13832/analogies-between-a... .

amirhirsch 6y ago

I liked this article. One notable point that felt like it was missing in the article is that the Prime Number Theorem, that the count of primes grow like (n / ln n) was provided such a map by Riemann in the letter in which he put forward his infamous eponymous hypothesis. That letter introduced the idea of using analysis to the Prime Number Theorem, extending the groundbreaking work of Riemann's friend Dirichlet who introduced the world to analytic number theory in Dirichlet's Theorem on the infinitude of primes in arithmetic progressions. It would take nearly half a century for mathematicians to digest the application of Fourier Analysis put forward by Riemann, and the proof of the Prime Number Theorem came only in the early 1900's. By then the analytic machinery would have been more commonly taught -- probably largely due to the advent of electrical engineering. Erdos and Selberg eventually put out fully arithmetic proofs of the Prime Number Theorem. And generally the helicopter analogy from the article probably doesn't apply so well to mathematics because you can probably always reduce theories and encapsulate all the dependent proofs to arithmetic first principles, but of course you already have the map. Recently the proofs of the Sensitivity Conjecture by Hao Huang and of the Bounded Gaps Between Primes by Yitang Zhang surprised mathematicians in how little new machinery these seemingly intractable problems required -- in the case of Zhang application of "hard work" on top of GPY and Hao Huang, a single clever insight.

tzs 6y ago
> and the proof of the Prime Number Theorem came only in the early 1900's That's just a little too late. It was proved in 1896 independently by Hadamard and de la Vallée Poussin. Hadamard, J. "Sur la distribution des zéros de la fonction zeta(s) et ses conséquences arithmétiques (')." Bull. Soc. math. France 24, 199-220, 1896 de la Vallée Poussin, C.-J. "Recherches analytiques la théorie des nombres premiers." Ann. Soc. scient. Bruxelles 20, 183-256, 1896

CatsAreCool 6y ago

I liked this article since it points out a problem in math where it can be hard to know what is currently known. Perhaps a result can be proven using a little known proposition in a completely different area of math, but it is hard to find that result in the literature. That is one reason I came up with https://mathlore.org https://mathlore.org. It is a place to collect mathematical info (with links to articles for a deeper look) so you or others can find it later when you need it. It supports of public collection of math info as well as allowing you to build your own private collection so you can keep track of what you have learned. The hope is it will be useful to others to help learn math and prove new theorems.

swayvil 6y ago

Isn't mathematics essentially ALL map? The only measures of goodness that I can come up with are logical consistency, elegance and lurking mystery.

rmrfstar 6y ago

Maps turn out to be useful in experimental physics too. Here's a neat BBC piece featuring Gell-man and Feynman on Strangeness -3 [1]. [1] https://www.youtube.com/watch?v=BGeW6Nc6IMQ https://www.youtube.com/watch?v=BGeW6Nc6IMQ

amandavinci 6y ago

The analogy of maps and boats is just an instance of the exploration-exploitation idea. I have seen instances of this pop up every time we discuss problem solving in some form. It falls in the perennial variety of ideas that can't be revisited enough.