Posted by: Alexandre Borovik | November 20, 2008

Niceness theorems

A remarkable paper by Hazewinkel. From introduction:

I aim to raise a new kind of question. It appears that many important mathematical objects (including counterexamples) are unreasonably nice, beautiful and elegant. They tend to have (many) more (nice) properties and extra bits of structure than one would a priori expect.

The question is why this happens and whether this can be understood .



  1. Maybe it’s because most people don’t advertise the ugly objects, and even if they do, these objects don’t get much publicity.

  2. If you compare mathematics with, say, visual artefacts, you discover that mathematics has no analogues of “kitsch”. Indeed, ugly objects in mathematics are not advertised, but this only replaces the one question with another: why does mathematics have a shared an more or less universally accepted set of aesthetic criteria?

  3. I don’t know.. I haven’t seen a really nice theoretical background to Ramsey theory. As far as I can see, it’s a bunch of isolated little “kitschy” results connected by twiddling a small number of parameters.

  4. i am not a mathematician, but have an interest in applied math (and did a little).

    i actually don’t believe this, any more than if one says ‘musicians agree on a particular criteria of musical aesthetic’. (or views about natural or human beauty, or art.)

    for example, i liked the old fashioned math logic and set theory, and developments, but category theory leaves me cold (the diagrams are nice, but thats all).

    the fundamental theorem of arithmatic to me is the most ‘nice’. and, infinity of primes, etc.

    once you hit analyses its like construction—it looks nice at the end, but its often a mess along the way, along with some hand waving (or ‘prayer’ /miracles uction/engineering).

    stochastic processes,random graphs, math physics, discrete math, and Group theory have nice parts. things like knot diagrams and number theory look real nice, but since i don’t understand them well they look more like a thicket—if you are a rabbit and know a trail, ok, but i don’t and am too slow, lazy, etc to find one.

  5. Mathematical beauty is of course in the mind of the beholder, that is to say the understander. Rota comments that the beauty of a mathematical fact is strongly related to its power to enlighten, but quite typically the moment of satori demands extensive prior preparation and cultivation on the part of the witness.

    Perhaps some areas of mathematics are just not (or are not yet) spectator sports: you have to “live the life” to appreciate some forms of beauty. For example, someone who struggles to prove some hard result in Ramsey theorem, only to have an Erdos or Ron Graham come along and show how to do it, and the guy thinks, “Oh my god! That’s ingenious!” I’d think it would be quite difficult to explicate that sort of aesthetic appreciation in terms of universally accepted criteria. (But nor could I bring myself to apply the word ‘kitsch’ to an arena of such apparent intellectual struggle — I’d sooner apply it to situations involving rather baser and more “universal” impulses — like wonderment at those “crystal balls” which ultimately devolve on the fact that 9 is 1 less than 10.)

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