With all due respect, I find declarations like

> Math is an end in itself; if you want to help the world that should be on your own time.

the marks of an idealist who doesn't do math "for a living." I am a mathematician. I've been doing it for years. You argue for a purity that is naive and counterproductive.

Mathematicians have all sorts of motivations to do mathematics. Intrinsic beauty is certainly primary, but in order to continue in this job that doesn't pay all that well and requires sacrifices our families don't understand, we've needed to come to some terms with our roles in the world. We've needed to justify our apparent uselessness, because some of us in conscience can't be useless people and can't morally continue to do pure math if it is indeed contributing nothing of value to the world. Doing math (or writing music, or making art) for the sole purity of thought, the simple beauty of it, is allowed only to people with a certain sort of psychological and financial privilege. I was not raised with that privilege.

The intrinsic beauty of math and the fact that it's a contribution to the world are not in contradiction. Bach wrote beautiful music that has changed the way we hear and the way we think, changed the path of human civilization. He did it for a paycheck. He did it for the audiences who would hear it then. He did it for the beauty. People who write programming languages because they want more beauty in programming do it for themselves and others. If Bach's music wasn't shared, if Ruby just sat hidden on a hard drive, neither of them would have made a lick of difference in the world and I would argue they'd have no value. Mathematics exists without and beyond us. Our discoveries, and the way they're shared, are what make them valuable to human life.

I do math because I desire to better the world: not by ending child abuse, but by discovering and then sharing the beauty of new mathematics. That's why we write papers, you know -- not just for jobs and tenure. Sharing has its own benefits, as in encountering the ideas of others we are sparked into new inspiration.

Move beyond the political and charitable in thinking about how one might better the world. Many software developers are interested in bettering the world and are doing it through their work, even if it's not an app for water in Africa. Can you so readily dismiss all of them? or is it ok because software development is a dirty business that contrasts with that pure garden of mathematics?

It seems we completely agree, but you have a peculiar way of putting things. I'm not the one accusing you of writing papers "just for jobs and tenure"... you are! In the same post! ;-)

I think it's good that you're "sharing the beauty of new mathematics" with your peers -- that's what I've been talking about all along.

But I also think it's presumptuous to want to have a job that "betters the world"; most jobs don't make any difference in the state of the world; many worsen it; and of course a lot of people don't even have a job in the first place.

What's more, history shows that most or all of math will be useful, eventually; the way it's put in the OP, it sounds like math should turn into some kind of vocational school producing teachings that should be immediately applicable; don't be in such a hurry.

You don't know what the future will need anyway; you only know the needs of the present, which are a very bad predictor of the future. It can be argued that by thinking about the present less, one helps the world more.