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There are many mathematicians who talk about the particular beauty of a subject. They may say a particular result is pretty. It may be beautiful.

It seems to me play a fundamental role in the composition of new mathematics. I know that there are theories of aesthetics, I imagine they refer either to music or to art, but do they apply to mathematics?

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    Maybe check this out:en.wikipedia.org/wiki/Mathematical_beauty. Euclid's Elements, for example, is not an analysis of aesthetics in mathematics, though it is quite aesthetically pleasing: there are numerous ways of doing the propositions; yet, Euclid seems to prefer aesthetically pleasing demonstrations. To a modern mind, it might also be apparent that, say, his presentation of Postulate Five is a backhanded way of preserving the beauty of of system. But what "new mathematics" are you thinking of? Simplicity and beauty have always pervaded mathematical thought. – Jon Sep 29 '12 at 20:09
  • And often, they go hand in hand: the simpler, the more aesthetically pleasing. Complexity is ugly: the simpler a formulation, the greater chance of generalization. – Jon Sep 29 '12 at 20:11
  • @Jon: I think thats a common enough view. But I've also Voevodsky (Fields Meddalist) say that the frontiers of mathematics are so abstract and takes so much training that he fears for their continued existence. But maybe he's also commenting on the support for pure research in mathematics in the anglo-american world which I think it pretty poor. – Mozibur Ullah Oct 2 '12 at 11:27
  • i think i saw russell claim that aesthetics define our selection of a problem ("initial hypotheses"). see page 2 of the analysis of matter – user6917 Aug 28 '14 at 19:18
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You may be aware of Schmidhuber's Beauty Postulate:

Among several patterns classified as "comparable" by some subjective observer, the subjectively most beautiful is the one with the simplest (shortest) description, given the observer's particular method for encoding and memorizing it.

His home page has links to papers written about this, with reference to science and mathematical beauty.

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The answer lies in value.

We use the term in both mathematics as well as in meta-ethics. Truth and goodness are analogous terms. Now leap to aesthetic value, goodness in ethics and beauty in aesthetics are analogous.

It is not controversial to say that ones perception of beauty is subjective. It is, however controversial to say that a value in mathematics is subjective. The mathematician seems to be expressing a subjective opinion about the beauty of an objectively true mathematical value.

Now, to say a work of art is beautiful, over time, seems to become an expression of an objective value, or truth. It is uncontroversial to feature, say, a work of Van Gogh in a museum gallery. Intersubjectivity, a collectively agreed upon objectivity, is a discredited idea, (see Mackie, inventing Right and Wrong, 1976,) so what is happening?

Searle, in an essay in Philosophy in a new century(2008), chapter 9, explains a distinction to be made when assigning the terms objective and subjective.

The notion is to distinguish, for both terms, their ontological and epistemological status.

The short answer is that value is ontologically subjective, that is, it exists within our minds. It can be experienced as having an epistemically objective status, that is, truth, whether its object is ontologically objective or subjective. That is, whether the object exists within or without our minds. Gravity, for instance, is without, while the value of a dollar is within, and both are epistemically true, or objective.

The controversy lies in whether one holds that mathematics is ontologically objective or subjective. To go into that would be the long answer. (I posted something along those lines in answer to the question whether mathematics is invented or discovered.)

I go with value, mathematical, ethical and aesthetic all being ontologically subjective and therefore analogous and interchangeable. You can't argue with Bach, for example.

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