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On Proof and Progress in Mathematics (1994) [pdf] (ams.org)
63 points by maverick_iceman on Aug 13, 2016 | hide | past | web | favorite | 8 comments



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...).

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


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.


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) 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).


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


  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?


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.


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).


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.




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