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Mathematical proof is foolproof, it seems, only in the absence of fools

Brian Hayes

Reading Euclid Backward

Here is how proof is supposed to work, as illustrated by an anecdote in John Aubrey's Brief Lives about the 17th century philosopher Thomas Hobbes:

He was 40 yeares old before he looked on geometry; which happened accidentally. Being in a gentleman's library in..., Euclid's Elements lay open, and 'twas the 47 El. libri I. He read the proposition. "By G—," sayd he (he would now and then sweare, by way of emphasis), "this is impossible!" So he reads the demonstration of it, which referred him back to such a proposition; which proposition he read. That referred him back to another, which he also read. Et sic deinceps, that at last he was demonstratively convinced of that trueth. This made him in love with geometry.

What's most remarkable about this tale—whether or not there's any trueth in it—is the way Hobbes is persuaded against his own will. He starts out incredulous, but he can't resist the force of deductive logic. From proposition 47 (which happens to be the Pythagorean theorem), he is swept backward through the book, from conclusions to their premises and eventually to axioms. Though he searches for a flaw, each step of the argument compels assent. This is the power of pure reason.

For many of us, the first exposure to mathematical proof—typically in a geometry class—is rather different from Hobbes's middle-age epiphany. A nearer model comes from another well-worn story, found in Plato's dialogue Meno. Socrates, drawing figures in the sand, undertakes to coach an untutored slave boy, helping him to prove a special case of the Pythagorean theorem. I paraphrase very loosely:

Socrates:Here is a square with sides of length 2 and area equal to 4. If we double the area, to 8 units, what will the length of a side be?

Boy: Umm, 4?

Socrates: Does 4´4=8?

Boy: Okay, maybe it's 3.

Socrates: Does 3´3=8?

Boy: I give up.

Socrates:Observe this line from corner to corner, which the erudite among us call a diagonal. If we erect a new square on the diagonal, note that one-half of the original square makes up one-fourth of the new square, and so the total area of the new square must be double that of the original square. Therefore the length of the diagonal is the length we were seeking, is it not?

Boy: Whatever.

At this point I trust we are all rooting for the kid. I would like to be able to report that the dialogue continues with the boy taking the initiative, saying something like, "Okay, dude, so what's the length of your erudite diagonal? It's not 4 and it's not 3, so what is it, exactly?" Alas, Plato reports no such challenge from the slave boy.

The problem with the Meno proof is exactly the opposite of the one I faced when I was an untutored wage slave. Whereas I was too inept and intellectually ill-equipped to craft a proof that would persuade my colleague (or even myself, for that matter), Socrates is a figure of such potent authority that the poor kid would surely go along with anything the master said. He would put up no resistance even if Socrates were proving that 1=2. It's hard to believe that the boy will go on to prove theorems of his own.

Sadly, Hobbes didn't get much more benefit from his own geometry lesson. He became a notorious mathematical crank, claiming to have solved all the most famous problems of classical geometry, including the trisection of the angle, the squaring of the circle and the doubling of the cube. These claims were a little less foolish in the 17th century than they would be now, since the impossibility of the tasks had not yet been firmly established. Nevertheless, Hobbes's contemporaries had no trouble spotting the gaffes in his proofs.

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