
On Proof and Progress in Mathematics (1994) [pdf] - maverick_iceman
http://www.ams.org/journals/bull/1994-30-02/S0273-0979-1994-00502-6/S0273-0979-1994-00502-6.pdf
======
dang
This is a beautiful piece and I could have sworn it had been discussed on HN
before, but maybe not
([https://hn.algolia.com/?query=On%20Proof%20and%20Progress%20...](https://hn.algolia.com/?query=On%20Proof%20and%20Progress%20in%20Mathematics%20%5Bpdf%5D&sort=byDate&dateRange=all&type=story&storyText=false&prefix&page=0)).

There was a good thread when Thurston died, though:
[https://news.ycombinator.com/item?id=4419566](https://news.ycombinator.com/item?id=4419566).

------
j1vms
Fascinating. Still reading through and processing this paper.

"Mathematicians are those humans who advance human understanding of
mathematics."

I would submit that there are mathematicians (great ones indeed) who did not
seek to advance human understanding of the subject, but may have regardless
done so as a side effect of the thread they tugged upon, the tapestry
unraveled. Of course, a utilitarian perspective (at least in retrospect) would
consider their ultimately accepted works to have advanced human understanding.

Yet I have observed some mathematicians regard their pursuit to be entirely
personal, something to the effect of daring to perceive the true contours of
logic and patterns in the universe, in their purest form. They revel in the
beauty of universal abstractions, the way another might admire and gaze upon a
blue lilac in one's own garden.

With this in mind, I would consider these mathematicians having only first
undertaken the more lofty goal of advancing human understanding, at the moment
they communicate their perceptions with the rest of the world (by publication
or other discourse). At that time, they have decidedly entered into the more
heavily social nature inherent in any modern intellectual discipline.

------
hyperpape
I first encountered this essay from poking around in Philosophy of Math. If
you picked up a survey of the topic, you might think it's all about formalism
vs. platonism. That's an interesting topic, but it seems to me as if there's a
contemporary trend (maybe not the dominant one) towards trying to understand
the practice of mathematics--what makes a proof good or interesting, what
mathematicians value, and how to think about mathematical understanding.

Unfortunately, the teacher I learned about this stuff from was not good at
either (a) writing down his own ideas, or (b) telling us who in the literature
was going down the same route.

However, Kenny Easwaran's paper "Probabilistic proofs and transferability"
([http://www.kennyeaswaran.org/research](http://www.kennyeaswaran.org/research))
seems somewhat in that vein. Jeremy Avigad has a paper called "Understanding
Proofs" that seems relevant, though I never finished it
([http://www.andrew.cmu.edu/user/avigad/papers.html](http://www.andrew.cmu.edu/user/avigad/papers.html)).

------
posterboy
Site's down. Google is yielding a link to the arxiv. I didn't know the site
was that old.

[https://arxiv.org/abs/math/9404236](https://arxiv.org/abs/math/9404236)

------
FabHK
Great book on that topic is "Proofs and Refutations" by Imre Lakatos. Good
read, very insightful. He shows using an extended example the iterative back
and forth between definitions, insights, theorems, proofs (which in the real
world evolves in a much more convoluted and messy way than it is ultimately
presented).

------
ZirconCode

      An interesting phenomenon in spatial thinking is that scalemakes a big
      difference. We can think about little objects in our hands, orwe can think
      of bigger human-sized structures that we scan, or we can think of spatial
      structures that encompass us and that we move around in. We tend to
      think more effectively with spatial imagery on a larger scale: it’s as if our
      brains take larger things more seriously and can devote more resources to them.
    

Really? I prefer imagining things in front of me, they become more manageable
at a scale similar to a paper or me, rather than my house. It just seems more
natural, does anyone have any insights on this?

~~~
fdej
The brain has dedicated subsystems for different aspects of spatial and visual
understanding.

Certain groups of cells (place cells, grid cells) are known to be involved
specifically in the ability to track one's own location and movement within
the external world. This ability is likely a very old and important
evolutionary invention in the development of higher animals, thus very
sophisticated and well integrated with other functions, particularly memory.
Think of the method of loci, known already in Ancient Rome and Greece, or
indeed how quickly one learns to navigate a new city (or a video game level,
say). This probably explains the phenomenon the author describes.

That said, the relative importance of the various spatial-visual subsystems of
the brain probably depends heavily on the task, and perhaps also varies
between individuals.

------
_polymer_
His thoughts on how definitions reflect different ways of thinking, as a
human, is profound to me. The importance of communication discussed in the
paper is relevant for any technical field.

