"Look at it this way. When I read a math paper it's no different than a musician reading a score. In each case the pleasure comes from the play of patterns, the harmonics and contrasts... The essential thing about mathematics is that it gives esthetic pleasure without coming through the senses." (Rudy Rucker, A New Golden Age)
"'I find herein a wonderful beauty,' he told Pandelume. 'This is no science, this is art, where equations fall away to elements like resolving chords, and where always prevails a symmetry either explicit or multiplex, but always of a crystalline serenity.'" (Jack Vance, The Dying Earth)
The preceding dialogues are both from works of fiction. That being said, they may in fact truly represent how the majority of mathematicians experience their work. For example, Rudy Rucker is himself a retired professor of mathematics and computers (as well as a science fiction author). My own instructor of advanced statistics would end every proof with the heartfelt words, "And that's the beauty."
I've heard that kind of sentiment a lot. But I never experienced mathematics that way. I now have a graduate degree in mathematics and statistics, and currently teach full-time as a lecturer of college mathematics, and these kinds of declarations still mystify me. Math has never felt "beautiful" or "poetic". I would never in a million years think to describe math as "pleasurable" or "serene".
Math drives me mad.
My experience of mathematics is this: Math is a battle. It may be necessary, it may be demanding, it may even be heroic. But the existential reality is that if you're doing math, you've got a problem. You very literally have a problem, something that is bringing your useful work to a halt, a problem that needs solving. And personally, I don't like problems; I am not fond of them; I wish they were not there. I want them to be gone, eradicated, and out of my way. I don't like puzzles; I want solutions. And once you have a solution, then you're not doing math anymore. So the process of mathematics is an experience in infuriation.
So, again: Math is a battle. It is a battle that feels like it must be fought. It can feel like a violent addiction; hours and days and nights disappearing into a mental blackness, unable to track the time or bodily needs. Becoming aware again at the very edge of exhaustion, hunger, filth, and collapse.
At worst, math can feel like a horrible life-or-death struggle, clawing messily in the midst of muddy, bloody, poisonous trenches. At best, it may feel like an elegant martial-arts move, managing to use the enemy's weight against itself, to its destruction.
I love seeing a powerful new mathematical theorem. But not because it "gives esthetic pleasure"; I have yet to see that. Rather, because a powerful theorem is the mathematical equivalent to "Nuke it from orbit – It's the only way to be sure". A compelling philosophy.
On the day that you really need math it will be a high-explosive, demolishing the barrier between you and where you want to go. Is there a pleasure in that? Perhaps, but not from the "play of patterns, the harmonics and contrasts". Rather, it's because blowing up things is cool. Like at a monster-truck rally, crushing cars is cool. Math may not be beautiful or fun for us, but it is powerful, and that's what we need from it.
Of course, I also don't know how to a read a music score, so I'm similarly mystified if that's the operating analogy for most mathematicians. Perhaps I'm missing something essential, but I have to stay true to my own experience. If math is going to be useful or worthwhile then it must literally rock you in some way, relieve an unbearable tension, and change your perception of what is possible.
And so, the battle continues.