In Mathematics, It Often Takes a Good Map to Find Answers(quantamagazine.org)
quantamagazine.org
In Mathematics, It Often Takes a Good Map to Find Answers
https://www.quantamagazine.org/in-math-it-often-takes-a-good-map-to-find-answers-20200601/
23 comments
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.
> "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.
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
In general better abstractions(similar ideas as in a recent discussion of Peter Naur's "Programming as theory building").
In general better abstractions(similar ideas as in a recent discussion of Peter Naur's "Programming as theory building").
It's a great read. See also: https://news.ycombinator.com/item?id=12280139
Just finished reading Thurston's paper. A great paper esp for those in "philosophy of mathematics".
In this topic I also highly recommend "Proofs and Refutations" by Imre Lakatos
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.
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.
> 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
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
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."
"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."
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.
You gotta go at least to the third level of abstraction to get the real meat.
Discussed from a literal point of view on MO: https://mathoverflow.net/questions/13832/analogies-between-a... .
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. 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.
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. 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.
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/
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/
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...
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...
Here's 2+2=4: https://twitter.com/dd4ta/status/1050433711416721408
I recently asked a related question regarding proof maps and quantifying their similarity/distance, but didn't get any answers: https://math.stackexchange.com/questions/3482135/are-there-p...
I recently asked a related question regarding proof maps and quantifying their similarity/distance, but didn't get any answers: https://math.stackexchange.com/questions/3482135/are-there-p...
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.
The chunking of atomic steps is what enables leaps in understanding. The mapping process starts with understanding each step.
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...
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...
A Fitch derivation of the existence of the intersection of all members of a nonempty set is a better place to start because it can be done in less than ten sheets of paper longhand. The ratio of triviality to pages consumed is quite shocking when you finally confront it. It is at that point that you realize intuition has no formal translation but is vital since the level of detail seems to blur and darken intuition when holding a proof to the standard of formal derivation rather than the ordinary informal standard. So far, I’ve seen relatively little interest in mathematical intuition or even honest appraisal of what it is or how mathematicians should develop it. Rather the trend seems to be pretending that mathematical intuition doesn’t exist and treating formalization as a no-op. I think this is due to an anti-intellectual atmosphere that views mathematics as a source of problems for the military as opposed to pastimes for civilians.
This has reminded of the Two Capacitor Paradox [0]. (The moral of the story is that you have to know the limits of your model.)
[0] https://en.wikipedia.org/wiki/Two_capacitor_paradox
[0] https://en.wikipedia.org/wiki/Two_capacitor_paradox
> 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 :-)
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 :-)
Isn't mathematics essentially ALL map?
The only measures of goodness that I can come up with are logical consistency, elegance and lurking mystery.
The only measures of goodness that I can come up with are logical consistency, elegance and lurking mystery.
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.
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
Here's a neat BBC piece featuring Gell-man and Feynman on Strangeness -3 [1].
[1] https://www.youtube.com/watch?v=BGeW6Nc6IMQ
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.