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The following quote is attributed to G.H. Hardy, a British mathematician:

The mathematician’s patterns, like the painter’s or the poet’s must be beautiful; the ideas like the colours or the words, must fit together in a harmonious way. Beauty is the first test: there is no permanent place in the world for ugly mathematics.

What exactly did Hardy mean by "ugly mathematics"? For example, would Hardy find the number π ugly if he found out about the tau manifesto?

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  • The tau manifesto is art that uses mathematics as its subject matter - and as art it works well. As mathematics it can't be taken too seriously, sociologically pi is too well entrenched for tau to take over.
    – Mozibur Ullah
    Commented May 26, 2014 at 3:08
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    However, the point made by the Tau Manifesto does point to an example pertinent to the OP. Expressions involving Tau would give rise to 'prettier' mathematics (though calling the Pi variants 'ugly' is a step too far: "marginally inelegant" would be more fair).
    – Niel de Beaudrap
    Commented May 27, 2014 at 8:27
  • @de Beaudrap:sure; but prettier isn't beautiful in the sense of being deep; the discovery of i and thus the complex plane & analysis is a beautiful extension of the mathematics of the real; that said if one could wave a magic wand and turn all instances of pi to value of tau it would only be a good thing; changing it to tau is mnemonically bad though as pi sounds like pie which are generally made as round things.
    – Mozibur Ullah
    Commented Jun 30, 2014 at 1:33

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I think by 'ugly' he meant two things: 'relying on hidden or unwarranted assumptions' and 'unnecessarily complicated'. The irony IMO is that he was an advocate of ugly mathematics. In The Man Who Knew Infinity, about Ramanujan and his partnership with Hardy, it explains how Hardy supported the new discipline of 'real' analysis which was meant to replace the old ways of explaining calculus. However, those old ways which were unnecessarily complicated were only in use because mathematicians were afraid of using infinitesimals. So Hardy replaced one ugly mathematics with another equally ugly version. Once infinitesimals were accepted with the emergence of smooth infinitesimal analysis and non-standard analysis the whole dispute was cleared up.

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  • Thanks for an excellent answer and welcome to the community! :)
    – user132181
    Commented Jun 29, 2014 at 8:16
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    I agree with your first sentence, but the rest I think can't be quite correct; Hardy wasn't, as a mathematician, interested in foundations of calculus; but what could be done with calculus in number theory - this is basis of his collaboration with Ramanujan; further the foundational questions that were raised by calculus led quite directly to topology - which is a major discipline in its own right.
    – Mozibur Ullah
    Commented Jun 30, 2014 at 1:38
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    Hardy does in the end defines elegance as simplicity, power, and depth, and this relates to philosophical discussion of simplicity and elegance in, for instance, plato.stanford.edu/entries/simplicity (or wikipedia).
    – Blaisorblade
    Commented Mar 20, 2015 at 9:27

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