Friday, November 02, 2007

I've got an axiom to grind

“Mathematical correctness gone mad."

It's a phrase almost all of us have used at one time or another. In fact, it's a cliché.

But like all clichés, there is more than an element of truth behind it.

In Britain today, mathematical correctness has gone mad. And it is driving people crazy.

There are so many examples of this obsession with mathematical correctness; it's hard to know where to start.

But here are just a few, to get us going.

Axioms in traditional thought are "self-evident truths", but that can be a problem because at a formal level, an axiom is just a string of symbols, which has an intrinsic meaning only in the context of all derivable formulas of an axiomatic system.

Hilbert's program tried to put all of mathematics on a firm axiomatic basis, but according to Gödel's incompleteness theorem every (sufficiently powerful) axiomatic system has undecidable formulas; and so a final axiomatization of mathematics is impossible.

It would be easy to go on and on, because the examples are countless.

In some cases, the exercise of mathematical correctness is relatively harmless. It provides good knocking copy for fulminating commentators, or for late-night radio chat shows.

But in many other cases it is not. We cannot simply dismiss every example as an isolated case of stupidity or zeal or plain barminess.

The systematic spread of mathematical correctness has a corrosive effect on our society.

It makes people imagine that mathematics, as far as its formal content is concerned, can be reduced to set theory in the sense that every mathematical statement or proof could be cast into formulas within set theory.

It provides officials with an excuse to meddle and interfere in people's lives, where they have no business to.

It leads to expensive, time-consuming and pointless litigation.

It plays into the hands of extremists.

And it undermines people's respect for the institutions of our country.

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