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Experts and Axioms
January/February 2010

Is the world getting hotter, and are humans responsible? Two questions there, and I don't know the answers, so I have to rely on experts. 

But can we trust the experts? I would argue that we cannot, for two reasons. One is that there is too much politics involved, but the other is that experts can be wrong, even when they all seem to agree. I'll give an example from the purest — and certainly apolitical — branch of science: mathematics.

Draw two lines in the plane, and a third line crossing them both. If it makes a right angle with the first line but not with the second, then the two lines are not parallel in the usual sense of the word, so we know they must meet if extended far enough. But how do we know this? Can we prove it?

Euclid couldn't. So when he wrote his famous Elements of geometry in Alexandria in about 300BC, he stated it as an axiom — an underlying assumption. It was his fifth in a series of five axioms. Centuries later, Arabic mathematicians tried to show this fifth axiom was a consequence of the other four. They couldn't do it. Then medieval and Renaissance mathematicians in Europe had a go, but they couldn't do it either. They were sure it was true and "proofs" were written, but on closer examination they were all false.

Then in the 1820s a young Hungarian mathematician named János Bolyai showed it could never be proved — because it wasn't true. There was a perfectly reasonable plane geometry satisfying the first four axioms, but not the fifth. In this new geometry, the angles of a triangle no longer added up to 180°, and far from being some quirky exception, this "hyperbolic plane" is extremely important in mathematics. Bolyai published his discovery as an appendix to a book by his father. It's not clear it would have been accepted on its own, yet it has been called "the most important two dozen pages in the history of thinking".

Don Quixote: Science tilts at windmills, too 

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