Lev Landau once defined science as “the means to satisfy one’s personal curiosity at the state’s expense”. In my student days, we were made to understand this was a joke, but I think that Landau was dead serious. In Soviet Union, the state was the only source of money. Luckily, in Western society, scientists can satisfy their curiosity at the expense of state, academia, charities, private companies or mad financiers. (People skilled in grant writing are doing a good job of exploiting as many of these resources as possible, although the ‘curiosity satisfaction’ is not, as a rule, listed among the goals of the proposed research.)
Sure, there could be other reasons to go to science, but only one decent reason to do it is to satisfy one’s curiosity. Here’s the proof. When the Nobel laureates or some other prestigious award winners — in other words, the folk who is said to achieve something in science — give their Nobel Lectures or interviews for media, they always mention their curiousity (as a driving force of scientific endeavour). They never say “I always wanted to get a Nobel Prize” or “I like to be invited to give talks in nice locations” or “That will show them” or “I was dodging the army draft” or “I had nothing better to do”. No: “Since I was a child, I was curious about this and that”, etc.
Landau’s definition also can help to figure out what is science and what is not. Take, for example, bioinformatics and computational biology — some people think these are synonyms. No they are not. Computational biology is science. Bioinformatics is technology. Try to formulate a question which betrays a convincing degree of personal curiosity and you’ll see what I mean.
Curiosity, then, is the governing motive of mathematics, the one that shapes it into a pursuit and a passion. It is not, of course, the only motive; nor is it in every particular case the strongest motive. There is, for instance, vanity, the desire to make a name for oneself. And we should not pretend that academics are above such sordid desires; anyone who has ever had dealings with academics knows that they are often almost obsessed with the possibility of being well respected, and that this motivation in at least many cases overtops even curiosity as a driving force in their work. Academia is filled to the brim with vanity; on a Humean view, this is one of the reasons it works in the first place.