“I think that it is a relatively good approximation to truth—which is much too complicated to allow anything but approximations-that mathematical ideas originate in empirics, although the genealogy is sometimes long and obscure. But, once they are so conceived, the subject begins to live a peculiar life of its own and is better compared to a creative one, governed by almost entirely aesthetical motivations, than to anything else and, in particular, to an empirical science. There is, however, a further point which, I believe, needs stressing. As a mathematical discipline travels far from its empirical source, or still more, if it is a second and third generation only indirectly inspired by ideas coming from “reality” it is beset with very grave dangers. It becomes more and more purely aestheticizing, more and more purely I’art pour I’art. This need not be bad, if the field is surrounded by correlated subjects, which still have closer empirical connections, or if the discipline is under the influence of men with an exceptionally well-developed taste. But there is a grave danger that the subject will develop along the line of least resistance, that the stream, so far from its source, will separate into a multitude of insignificant branches, and that the discipline will become a disorganized mass of details and complexities. In other words, at a great distance from its empirical source, or after much “abstract” inbreeding, a mathematical subject is in danger of degeneration. At the inception the style is usually classical; when it shows signs of becoming baroque, then the danger signal is up.”
John Von Neumann, written in his article The Mathematician
I’m inclined to disagree. Deep abstraction gives us powerful tools for solving less abstract problems, including those that come out of the empirical sciences. Even fields developed with a deliberate eye to avoiding practical applications have sometimes turned out to make significant contributions to the sciences (I understand knot theory, for example, began this way, but has since turned out to have important applications in biochemistry).
You make a strong point, however; the question as to whether we can or cannot improve the efficiency of mathematical research appears to be an open one. I think that perhaps the real issue is that we don’t have a correct reductionist account of mathematics, and thus are not able to see clearly what we are doing when we build our theories. If we had a better road-map, I think that at the very least we could tie mathematics down to level 1/level 2 space so that we could have a better idea as to how we can measure the profitability of various possible lines of inquiry.
Well, my idea would be along the lines of thinking of mathematics as a combination of certain types of cognition combined with some sort of social feedback loop. We are phenomena and so are our actions. We do mathematics, therefore we should be able to make an empirical study of it.
I suppose that I would like an empirical dissection of mathematics as it is practiced by humans, something that would allow us to measure the statistical usefulness of various areas of mathematical thought. Do you think that this can’t be done? Do you think that it has already been done? If so I would be interested to hear your views, but I was under the impression that numerical cognition was still an open field. Or do you not think that this is related to numerical cognition?
I’m really not sure what to make of your reply. Even if you do
define mathematics as the study of systems with a complete reductionist account
shouldn’t we be able to give a reductionist account of the methods we use to give a reductionist account? This is one of those things that I do not feel I have a clear method for thinking around, yet it still seems like a problem, and a not-quite-mysterious one to boot. Intuitively it seems that we should be able to give a neurological account of mathematical thought and gain an empirical understanding of why some mathematics appears/is more applicable than others and what sorts of applications we might expect out of certain types of mathematical thought.′
Any insight you feel like sharing on this topic would be greatly appreciated. Am I just confused by some embedded mysterious question?
John Von Neumann, written in his article The Mathematician
I’m inclined to disagree. Deep abstraction gives us powerful tools for solving less abstract problems, including those that come out of the empirical sciences. Even fields developed with a deliberate eye to avoiding practical applications have sometimes turned out to make significant contributions to the sciences (I understand knot theory, for example, began this way, but has since turned out to have important applications in biochemistry).
You make a strong point, however; the question as to whether we can or cannot improve the efficiency of mathematical research appears to be an open one. I think that perhaps the real issue is that we don’t have a correct reductionist account of mathematics, and thus are not able to see clearly what we are doing when we build our theories. If we had a better road-map, I think that at the very least we could tie mathematics down to level 1/level 2 space so that we could have a better idea as to how we can measure the profitability of various possible lines of inquiry.
What do you mean, we don’t have a correct reductionist account of mathematics?
I could define mathematics as the study of systems with a complete reductionist account.
Well, my idea would be along the lines of thinking of mathematics as a combination of certain types of cognition combined with some sort of social feedback loop. We are phenomena and so are our actions. We do mathematics, therefore we should be able to make an empirical study of it.
I suppose that I would like an empirical dissection of mathematics as it is practiced by humans, something that would allow us to measure the statistical usefulness of various areas of mathematical thought. Do you think that this can’t be done? Do you think that it has already been done? If so I would be interested to hear your views, but I was under the impression that numerical cognition was still an open field. Or do you not think that this is related to numerical cognition?
I’m really not sure what to make of your reply. Even if you do
shouldn’t we be able to give a reductionist account of the methods we use to give a reductionist account? This is one of those things that I do not feel I have a clear method for thinking around, yet it still seems like a problem, and a not-quite-mysterious one to boot. Intuitively it seems that we should be able to give a neurological account of mathematical thought and gain an empirical understanding of why some mathematics appears/is more applicable than others and what sorts of applications we might expect out of certain types of mathematical thought.′
Any insight you feel like sharing on this topic would be greatly appreciated. Am I just confused by some embedded mysterious question?