
In Mathematics, It Often Takes a Good Map to Find Answers - yarapavan
https://www.quantamagazine.org/in-math-it-often-takes-a-good-map-to-find-answers-20200601/
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ssivark
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
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
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").

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amirhirsch
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
> 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

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mmhsieh
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
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.

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CatsAreCool
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.

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physicsgraph
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
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
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
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...

~~~
ZenOfTheArt
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.

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utkarsh_apoorva
> 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 :-)

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swayvil
Isn't mathematics essentially ALL map?

The only measures of goodness that I can come up with are logical consistency,
elegance and lurking mystery.

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amandavinci
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.

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rmrfstar
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)

