[HN Gopher] In Mathematics, It Often Takes a Good Map to Find An...
___________________________________________________________________
In Mathematics, It Often Takes a Good Map to Find Answers
Author : yarapavan
Score : 87 points
Date : 2020-06-03 18:34 UTC (4 hours ago)
(HTM) web link (www.quantamagazine.org)
(TXT) w3m dump (www.quantamagazine.org)
| physicsgraph wrote:
| 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/
| knzhou wrote:
| 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...
| 7373737373 wrote:
| 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...
| physicsgraph wrote:
| 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 wrote:
| 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...
| Koshkin wrote:
| 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
| ZenOfTheArt wrote:
| 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.
| CatsAreCool wrote:
| 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.
| amirhirsch wrote:
| 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 wrote:
| > 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 Vallee Poussin.
|
| Hadamard, J. "Sur la distribution des zeros de la fonction
| zeta(s) et ses consequences arithmetiques (')." Bull. Soc.
| math. France 24, 199-220, 1896
|
| de la Vallee Poussin, C.-J. "Recherches analytiques la theorie
| des nombres premiers." Ann. Soc. scient. Bruxelles 20, 183-256,
| 1896
| mmhsieh wrote:
| 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 wrote:
| 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.
| utkarsh_apoorva wrote:
| > 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 wrote:
| I highly recommend Bill Thurston's gem of an article _On proof
| and progress in mathematics_ 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.
| mturmon wrote:
| It's a great read. See also:
| https://news.ycombinator.com/item?id=12280139
| raincom wrote:
| Just finished reading Thurston's paper. A great paper esp for
| those in "philosophy of mathematics".
| sabas123 wrote:
| 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 wrote:
| 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").
___________________________________________________________________
(page generated 2020-06-03 23:00 UTC)