Modeling aesthetics in mathematics

What exactly is beautiful math?

[A]bove all, adepts [of mathematics] find therein delights analogous to those given by painting and music. They admire the delicate harmony of numbers and forms; they marvel when a new discovery opens to them an unexpected perspective; and has not the joy they thus feel the esthetic character, even though the senses take no part therein? Only a privileged few are called to enjoy it fully, it is true, but is not this the case for all the noblest arts?

-Henri Poincaré, The Value of Science

One expects a mathematical theorem or a mathematical theory not only to describe and to classify in a simple and elegant way numerous and a priori disparate special cases. One also expects “elegance” in its “architectural”, structural makeup. Ease in stating the problem, great difficulty in getting hold of it and in all attempts at approaching it, then again some very surprising twist by which the approach, or some part of the approach, becomes easy, etc. Also, if the deductions are lengthy or complicated, there should be some simple general principle involved, which “explains” the complications and detours, reduces the apparent arbitrariness to a few simple guiding motivations, etc. These criteria are clearly those of any creative art.

-John von Neumann, The Mathematician

The moral: a good proof is one that makes us wiser.

-Yuri Manin, A Course in Mathematical Logic for Mathematicians

My hypothesis is that generally when people talk about beauty in mathematics they’re talking about things that teach us something useful for proving new facts. For example, proving a difficult but simple theorem is useful because its difficulty means it may imply other previously difficult theorems, and its simplicity means it may show up and be used often. A theorem that establishes a connection between two previously disparate areas of mathematics is considered beautiful, and such a connection allows knowledge from one are to be applied to the other, potentially cracking new problems. An unexpected proof – “an unexpected perspective” or “surprising twist” – offers something new to be learned, something that can then be used for other problems.

Quote of the day: Yuri Gurevich

I remember, in a geometry class, my teacher wanted to prove the congruence of two triangles. Let’s take a third triangle, she said, and I asked where do triangles come from. I worried that there may be no more triangles there. Those were hard times in Russia and we were accustomed to shortages. She looked at me for a while and then said: ‘Shut up’.

-Platonism, Constructivism, and Computer Proofs vs. Proofs by Hand

Link of the day: Learn the Greek alphabet

A handy flashcards web app for memorizing all the Greek letters

Link of the day: Jon Skeet speaks

Jon Skeet on the tricky edge cases that can show up with basic data types and how they model reality. Back to basics: the mess we’ve made of our fundamental data types

Command line Grooveshark post-Grooveshark

In this post I’ll share a way to get a Grooveshark-like experience with a Linux command-line application.

Step 1: Build a YouTube playlist with some music you like.

Step 2: Go to your channel, select Playlists, find the one you just made, and click View full playlist. Make sure the privacy setting is either public or unlisted. Copy the playlist ID in the URL (after list=). For example, the playlist ID of one of my playlists is PLmaRvdyzIrIGkRbl7jJdzEYrRcXe4od9G.

Step 3: Install mpv and mps-youtube.

Step 4: Run

mpsyt pl <playlist ID>, dump, \*

You’re streaming the playlist! Press <space> to pause/play, < to play previous track, > to play next track.

To play the playlist shuffled, run this instead:

mpsyt pl <playlist ID>, dump, shuffle \*

I have this alias in my .zshrc file:

alias playlistName="mpsyt pl PLmaRvdyzIrIGkRbl7jJdzEYrRcXe4od9G, dump, shuffle \*"

so I can get music playing with just one command.

If, as in Grooveshark, your YouTube playlists are public, you can open mpsyt and run userpl <YouTube username> to see your YouTube playlists and select one to play.

The good thing is that the whole videos aren’t streamed, just high-quality audio.

Among other features, mpsyt allows you to search for YouTube videos and create local playlists (not connected to a YouTube account), which you can do if you want to avoid the YouTube web interface completely.

Status of the day

Wojciech Szpankowski:

From Wojciech Szpankowski’s book, drawn by Philippe Jacquet:

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