His main conclusion seems to be (see his final paragraph in part two) that mathematics starts off with an empirical basis, but it "begins to live a peculiar life of its own" which corresponds more to a creative discipline, whose primary criteria for evaluation are aesthetic.
An excerpt from that final paragraph:
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. It would be easy to give examples, to trace specific evolutions into the baroque and the very high baroque, but this, again, would be too technical.
Seems very much along the lines of software projects: we lay down some framework that we work in which appears simple, elegant and well-suited to the conception of the product we're building. Then, as time goes one, it becomes complex as we 'patch' the imperfections of the initial system—then we refactor: a sort of paradigm shift occurs and we're working with new basic terms, hopefully restoring elegance to the system.
Many mathematicians seem to have an attitude that a well peer-reviewed proof is simply correct and etched in stone for good. This seems to me like publishing programs - some of them thousands of lines long - and expecting them to be essentially bug-free without ever running them.
Von Neumann is not exactly making the same point, but he is arguing against the sense that math is a kind of purely abstract discovery that has escaped from the messiness of day-to-day reality.
One could ask: will we ever get to a point when all mathematical proofs are rigorously checked, so that we know they're correct? I hope we do get to a point when program-checked proofs are standard, but even then there's the possibility that a proof-checker itself is buggy. So, philosophically, it seems we may never justifiably feel certain beyond all doubt that any given proof is correct.
But we can have greater confidence in a proof verified by both expert humans and by a trusted program than in a proof simply reviewed by humans. So the effort is meaningful and worthwhile - all in the context of seeing math as a field as fallible as any other.
You might be interested in this article [1]. Basically, even if many proofs are incorrect, their results are almost always still true. Also, there have been many large changes in what counts as 'rigorous' over time. I agree with you that computer-checked proofs seem like a logical next step in the rigor of mathematics.
> Many mathematicians seem to have an attitude that a well peer-reviewed proof is simply correct and etched in stone for good. This seems to me like publishing programs - some of them thousands of lines long - and expecting them to be essentially bug-free without ever running them.
I think that the important distinction is 'peer-reviewed'. A random programme published, even by a very good programmer, is indeed likely to have errors; but I think that one can have more confidence in a peer-reviewed programme (which is essentially why we trust open source over closed source, after all).
Of course there are distinctions: software is automatically tested when used, whereas mathematics is not automatically tested when read, so that one might reasonably have more confidence in software purely on the basis of its longevity than one does in mathematics; but, on the other hand, although individual pieces of reasoning can be extremely arcane, 'flow control' in mathematical reasoning tends to be much more elementary than in software, so that one is unlikely to uncover the mathematical analogue of an infrequently visited branch in a code path going untested despite heavy use of the software.
actually the amazing thing is that mathematics can be useful.
from its beginning, mathematics is not merely empirical study of everyday problems. number is believed sacred and having divine implications and some special numbers worshiped across civilizations, Greeks included. In early days, some properties of number are not taken for granted as we do today.
the duality of mathematics though, most mathematician don't have practical concerns in their minds. and this is reflected in the mathematics education. Arnold has advocated for long the intuitive teaching which basically is to introduce the ideas of mathematics using concrete tangible and often practical examples. However, in today's textbooks of all level, the Bourbaki style is still dominant.
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[ 2.6 ms ] story [ 23.1 ms ] threadAn excerpt from that final paragraph:
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. It would be easy to give examples, to trace specific evolutions into the baroque and the very high baroque, but this, again, would be too technical.
Seems very much along the lines of software projects: we lay down some framework that we work in which appears simple, elegant and well-suited to the conception of the product we're building. Then, as time goes one, it becomes complex as we 'patch' the imperfections of the initial system—then we refactor: a sort of paradigm shift occurs and we're working with new basic terms, hopefully restoring elegance to the system.
Von Neumann is not exactly making the same point, but he is arguing against the sense that math is a kind of purely abstract discovery that has escaped from the messiness of day-to-day reality.
One could ask: will we ever get to a point when all mathematical proofs are rigorously checked, so that we know they're correct? I hope we do get to a point when program-checked proofs are standard, but even then there's the possibility that a proof-checker itself is buggy. So, philosophically, it seems we may never justifiably feel certain beyond all doubt that any given proof is correct.
But we can have greater confidence in a proof verified by both expert humans and by a trusted program than in a proof simply reviewed by humans. So the effort is meaningful and worthwhile - all in the context of seeing math as a field as fallible as any other.
[1] http://www.gwern.net/The%20Existential%20Risk%20of%20Mathema...
I think that the important distinction is 'peer-reviewed'. A random programme published, even by a very good programmer, is indeed likely to have errors; but I think that one can have more confidence in a peer-reviewed programme (which is essentially why we trust open source over closed source, after all).
Of course there are distinctions: software is automatically tested when used, whereas mathematics is not automatically tested when read, so that one might reasonably have more confidence in software purely on the basis of its longevity than one does in mathematics; but, on the other hand, although individual pieces of reasoning can be extremely arcane, 'flow control' in mathematical reasoning tends to be much more elementary than in software, so that one is unlikely to uncover the mathematical analogue of an infrequently visited branch in a code path going untested despite heavy use of the software.