the purpose of life

The Feynmannian/Sagan/Tyson "scientific" view is that the purpose of life is understanding: the world is a giant mystery, with layers and layers of intricate structure, and our purpose is to unravel a bit of that mystery.

Specific to mathematics, two ways of thinking about improving understanding, both of which are valid:

• proving novel theorems, or
• the Thurstonian view of improving the overall human understanding of mathematics, by communication and exposition.

There's another view that the purpose of life is good works; to help other people. This is the effective altruist view, basically SuccessfulFriend's view, and kind of also the view of most religions (though that's complicated). This is also the view that flows naturally from decision theory: although information and understanding do have value, ultimately that value is that they let you act more effectively. So just collecting information is pointless by itself.

• a counterpoint is that this value exists for society, but we have specialization of labor, and so there need to be some people who dedicate themselves solely to increasing and disseminating understanding. Our individual purposes can be instrumental goals relative to society's final goals.

Then there's the nihilist view, which is that there is no purpose to life. We are accidents. Reproduction and self-preservation instincts are dominant mechanisms and will take over any system that they are in, but it doesn't make them 'right'. Humanity has no special moral status. A universe full of life is not 'better' in any obvious way than a universe devoid of life. Good works are impossible because there's no plausible way to define good. This is a depressing argument but it's not obviously wrong. It's right for the same reason the atheist argument about God is right (if God is the ultimate creator, who created God? is an infinite regress). For any purported purpose X, "why should my purpose be X?" is a question that cannot be answered by appeal to X. Purposes cannot self-justify, but anything that requires justification in terms of a higher purpose cannot be the ultimate purpose.

• BUT this argument is itself kind of self-defeating. Even if we accept that there is no 'ultimate' purpose, in particular there's no reason to be depressed by this. We might as well just live joyful lives that push our elemental buttons. Even if a universe full of life is no better than a cold dead universe, it is also no worse. So this argument definitely doesn't imply you 'should' be depressed, because there is no 'should'.
• there's also the notion of 'huddling together against the dark' which came to me as a sort of motivating principle. which is kind of compatible with nilihism. But it's not obviously helpful for a career in science. Huddling against the dark basically just requires a campfire and some blankets. It's not a hopeful or expansive or progressive worldview. Which doesn't make it wrong, but it can't be complete.