Wednesday, December 21, 2011

The Mathematics Revolution

Mathematician Frank Quinn writes about the logicism that made mathematics 100% precise about a century ago:
The physical sciences all went through "revolutions": wrenching transitions in which methods changed radically and became much more powerful. It is not widely realized, but there was a similar transition in mathematics between about 1890 and 1930. The first section briefly describes the changes that took place and why they qualify as a "revolution", and the second describes turmoil and resistance to the changes at the time. The mathematical event was different from those in science, however. In science, most of the older material was wrong and discarded, while old mathematics needed precision upgrades but was mostly correct. The sciences were completely transformed while mathematics split, with the core changing profoundly but many applied areas, and mathematical science outside the core, relatively unchanged. The strangest difference is that the scientific revolutions were highly visible, while the significance of the mathematical event is essentially unrecognized. The section "Obscurity" explores factors contributing to this situation and suggests historical turning points that might have changed it. ...

To a first approximation the method of science is "find an explanation and test it thoroughly", while modern core mathematics is "find an explanation without rule violations". The criteria for validity are radically different: science depends on comparison with external reality, while mathematics is internal.

The breakthrough was development of a system of rules and procedures that really worked, in the sense that, if they are followed very carefully, then arguments without rule violations give completely reliable conclusions.
There are people who argue that Hilbert's program was a failure, but they are wrong. It successfully axiomatized mathematics. See this for details.

Quinn also explains:
Many scientists and engineers depend on math- ematics, but its reliability makes it transparent rather than appreciated, and they often dismiss core mathematics as meaningless formalism and obsessive-compulsive about details. This is a cul- tural attitude that reflects feelings of power in their domains and world views that include little else, but it is encouraged by the opposition in elementary education and philosophy. In fact, hostility to mathematics is endemic in our culture. Imagine a conversation:

A: What do you do?
B: I am a ---.
A: Oh, I hate that.

Ideally this response would be limited to such occupations as "serial killer", "child pornographer", and maybe "politician", but "mathematician" seems to work. It is common enough that many of us are reluctant to identify ourselves as mathematicians. Paul Halmos is said to have told outsiders that he was in "roofing and siding"!
Mathematicians in the movies are usually portrayed as crazy.

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