I use beauty as "Pleasing to the perception" which doesn't limit it to sight, sound, touch, taste, or any other physical sense. To take your example, Cantor's

I use beauty as "Pleasing to the perception" which doesn't limit it to sight, sound, touch, taste, or any other physical sense. To take your example, Cantor's diagonalisation - after a time - takes on a life of its own. Treating math as simply pushing symbols around makes it devoid of meaning, and it's meaning that makes it possible to do. Creating a proof requires more than exploring all the ways of pushing symbols around - you need a sense of "rightness" to guide you. Similarly, writing a program is more than banging out something that's syntactically correct. You form it from your internal sense of what's right. For reference, I did a PhD in 1983-88 specialising in Graph Theory and Combinatorics. Since then, in addition to my day job, I've given around 100 presentations a year on what math is really about. Patterns, predictions, testing, proofs, structure, and why things work. Whay are all primes of the form 4k+1 equal to the sum of two squares? You can puch symbols around all you like, but finding a proof requires seeing structure, form, pattern and "truths" underneath the simple statements. Following someone else's proof is usually unenlightening. There really is beauty and elegance there.