I think the fundamental flaw in this line of reasoning is that you are forgetting just how recently calculus was invented. It was not at all that either Leibniz or Newton sat down at the desk and decided to develop the calculus. I can hardly think of a more apt moment to quote Newton himself: “What Descartes did was a good step. You have added much several ways, and especially in taking the colours of thin plates into philosophical consideration. If I have seen a little further it is by standing on the shoulders of Giants.”
I would argue that the deterministic developments of what we currently consider to be “the great mathematicians” were, more or less, low-hanging fruit. That, in fact, turning a corner with Godel and Turing, mathematics research, especially in probabilistic modeling, is actually accelerating.
As to whether the pursuit of pure research is the right career choice for a given person, or whether that person can adequately assess how much of a contribution he or she can make in a research field, I feel that there are many additional factors to be weighed. For instance, many people who pursue research also pursue creative endeavors of other types and would prefer to have an unstructured career that affords them time. Being a research professor tends to concentrate stress and working hours into clumps throughout the year, leaving other spans of time with significantly fewer responsibilities. If a person wants to make a small mark in a research domain and also wishes to seriously pursue an art, such as writing or photography, this is a perfectly sensible career choice. If, however, they are strictly trying to maximize the social benefit of their having existed on the Earth, then research may not be the right pursuit. But, in that case, the computational effort expended to try to model one’s own eventual contribution could be highly variable and depends on certain types of honesty which people, especially at the ages when long-term career choices are made, rarely employ.
I would also be interested in thinking more about a type of Moore’s Law related to the dramatic effect of mathematical progress. If you normalize the “amount of improvement” due to a “tiny mathematical victory” in modern research by the amount of improvement of say calculus of variations, you certainly won’t feel that you’ve accomplished a lot. The tiny blood vessels in the ends of your fingertips might not be seen to be as important as your femoral artery to a layman, but a medical scientist knows that, to some degree, “it don’t work that way.”
As an applied mathematician, I disagree very strongly with these claims. See the paper “The Dawning of the Age of Stochasticity” by David Mumford:
http://www.dam.brown.edu/people/mumford/Papers/OverviewPapers/DawningAgeStoch.pdf.
I think the fundamental flaw in this line of reasoning is that you are forgetting just how recently calculus was invented. It was not at all that either Leibniz or Newton sat down at the desk and decided to develop the calculus. I can hardly think of a more apt moment to quote Newton himself: “What Descartes did was a good step. You have added much several ways, and especially in taking the colours of thin plates into philosophical consideration. If I have seen a little further it is by standing on the shoulders of Giants.”
I would argue that the deterministic developments of what we currently consider to be “the great mathematicians” were, more or less, low-hanging fruit. That, in fact, turning a corner with Godel and Turing, mathematics research, especially in probabilistic modeling, is actually accelerating.
As to whether the pursuit of pure research is the right career choice for a given person, or whether that person can adequately assess how much of a contribution he or she can make in a research field, I feel that there are many additional factors to be weighed. For instance, many people who pursue research also pursue creative endeavors of other types and would prefer to have an unstructured career that affords them time. Being a research professor tends to concentrate stress and working hours into clumps throughout the year, leaving other spans of time with significantly fewer responsibilities. If a person wants to make a small mark in a research domain and also wishes to seriously pursue an art, such as writing or photography, this is a perfectly sensible career choice. If, however, they are strictly trying to maximize the social benefit of their having existed on the Earth, then research may not be the right pursuit. But, in that case, the computational effort expended to try to model one’s own eventual contribution could be highly variable and depends on certain types of honesty which people, especially at the ages when long-term career choices are made, rarely employ.
I would also be interested in thinking more about a type of Moore’s Law related to the dramatic effect of mathematical progress. If you normalize the “amount of improvement” due to a “tiny mathematical victory” in modern research by the amount of improvement of say calculus of variations, you certainly won’t feel that you’ve accomplished a lot. The tiny blood vessels in the ends of your fingertips might not be seen to be as important as your femoral artery to a layman, but a medical scientist knows that, to some degree, “it don’t work that way.”