

The Mathematician (1947) - danso
http://www-history.mcs.st-and.ac.uk/Extras/Von_Neumann_Part_1.html

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dalke
Those interested in von Neumann as a person may be interested in this 1966
film about him, at
[https://www.youtube.com/watch?v=VTS9O0CoVng](https://www.youtube.com/watch?v=VTS9O0CoVng)
.

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marvel_boy
Thanks for the link. I enjoyed every second of this awesome film.

~~~
dalke
You're welcome!

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westoncb
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.

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tylerneylon
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.

~~~
JadeNB
> 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.

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hahan
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.

