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Experts and axioms By Mark Ronan
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”.
After 2,000 years, Bolyai had clearly demonstrated that Euclid’s fifth axiom was not a consequence of the other four, but he was staggered, and somewhat upset, to find that the greatest living mathematician—Carl
Friedrich Gauss—already knew it. Gauss was a man with a huge reputation who didn’t want to get into arguments with his contemporaries. The consensus among experts was that the fifth axiom was provable—this was an item of faith—so he had decided to leave his work to be published after his death. Only when a man less than half his age had shown the consensus was wrong did Gauss admit he already knew that it was wrong.
Now we have a consensus that CO ² emissions are the cause of global warming, and contrarian voices are shouted down. Politics has entered the scientific debate, and knights in shining armour have risen to protect us, even if like dear old Don Quixote they are merely tilting at windmills.
Civil society By Barton Swaim
Samuel Johnson’s Dictionary contains some lovely lines. One of them, as I learned recently from Jeffrey Meyers’s excellent biography, is Johnson’s entry for the word “stammel”: “Of this word I know not the meaning.” For some time I’ve felt that way about the word “diversity”, at least as it’s commonly used. All I know is what it doesn’t mean. It hasn’t got much to do with difference or variety. In American universities, most humanities departments have imposed upon themselves the rule that no person shall join the faculty whose opinions stray significantly from the rest of the department. This is done in order to achieve diversity.
I suspect something similar is about to happen to the word “civility”. It’s on the lips of all the smart people just now. Jim Leach, the chairman of the National Endowment for the Humanities, has just kicked off a “50-state civility tour”. The president of the university where I live recently published a column headed “Toward a more civil discourse.” He has “created a group of university and community representatives,” he informs us, “to recommend how reasoned and civil debate can become the norm for resolving some of society’s most polarising issues.”
All the talk about civility first arose last summer in the US, when a number of local “town hall meetings” on the subject of health-care reform were “disrupted” by people who had got the idea that they were going to be forced to pay other people’s hospital bills. Then, during a presidential speech to a joint session of Congress, Representative Joe Wilson shouted “You lie!” when President Obama stated that illegal immigrants would not be covered under the Bill then being considered.
Before all this, nobody among the Left-leaning opinion-makers seemed to care much about civility. Governor Howard Dean, a former presidential candidate and now chairman of the Democratic Party, once suggested that President George W. Bush might have known about the September 11 attacks before they happened—and almost nobody among his ideological sympathisers could bring themselves unequivocally to deplore the remark. That is just one among many instances of what could accurately be called uncivil discourse during the Bush years.
But what’s troubling about all this isn’t the inconsistency or the partisanship. Those vices are inevitable in politics. The frightening thing about this new interest in civility is the word’s nebulousness in the mouths of those who use it. Civility, in the political sense, is the recognition that those with whom we disagree belong to our polity just as much as we do, and that their views, however misguided, are expressed in good faith. It isn’t a
Don Quixote: Science tilts at windmills, too