A good number of very high profile philosophers and mathematicians have drawn attention to what they see as the intrinsic beauty in mathematical solutions.

For example :

"It seems to me now that mathematics is capable of an artistic excellence as great as that of any music, perhaps greater; not because the pleasure it gives (although very pure) is comparable, either in intensity or in the number of people who feel it, to that of music, but because it gives in absolute perfection that combination, characteristic of great art, of godlike freedom, with the sense of inevitable destiny; because, in fact, it constructs an ideal world where everything is perfect but true."

Bertrand Russell (1872-1970), *Autobiography*, George Allen and Unwin Ltd, 1967, v1, p158

Not so often mentioned, however, is that where there is beauty there can also be ugliness – or worse. For instance, take as an example the paper : ‘A GHASTLY GENERALIZED N-MANIFOLD’- by professor Robert J. Daverman and Dr. John J. Walsh (published in the ILLINOIS JOURNAL OF MATHEMATICS, Volume 25, Number 4, Winter 1981)

*[Note:* The full paper can be accessed by clicking 'Full-text: Open access PDF file' via the link above.]

Less mathematically gifted readers may not find the ghastliness immediately apparent though, indeed the word ‘ghastly’ appears only in the title of the paper, and thus at the risk of irritating those who are familiar with 2-ghastly spaces in acyclic manifold cell-like decompositions, and who will no doubt find the inherent ghastliness to be self-evident, reprinted below is a concise explanation that Professor Daverman has kindly supplied.

“It is ghastly because it contains no cuber of dimension 2, 3 …, or n-1, where N is the dimension of the ghastly object.”

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