Berlinski on the Idea of "Proof"
Mr. Berlinski defines for us what a proof should do: It "compels belief." But Mr. Berlinski seems to have a controversial understanding of this compulsion. For him it is more psychological than logical, more rhetorical than mathematical, more feeling than fact. Take the author's discussion of Euclid's first proof, a demonstration of the proposition that on a given straight line segment it is always possible to construct an equilateral triangle. After agreeing with modern mathematicians that there are numerous logical lapses in Euclid's argument, Mr. Berlinski nevertheless concludes that his proof succeeds -- because, again, its combination of logic and illustration "compels belief."This -- the idea that "proofs" are ultimately about compelling the emotions -- rings true to me. Of course that doesn't invalidate logical demonstrations, or arguments and evidence, but it somewhat downgrades the awe in which we may be obliged to hold them. It should also put in their place those bullies and manipulators, like Richard Dawkins, who claim that anyone who doesn't meet their standards of orthodox belief has effectively damned himself as "ignorant, stupid or insane (or wicked, but I'd rather not consider that)."
Ms. Snyder, who wrote The Philosophical Breakfast Club: Four Remarkable Friends Who Transformed Science and Changed the World, also reflects interestingly on Lincoln's preoccupation with Euclid:
One of the more curious historical revelations of Steven Spielberg's "Lincoln" is that America's 16th president was obsessed by a Greek mathematician from the fourth century B.C. While traveling from town to town as a young lawyer riding the Eighth Circuit in Illinois, Abraham Lincoln kept a copy of Euclid's geometrical treatise, "The Elements," in his saddlebag; his law partner Billy Herndon related that at night Lincoln would lie on the floor reading it by lamplight. Lincoln said he was moved to study Euclid by his desire to understand what a "demonstration" was, and how it differed from any other kind of argument.Snyder reminds us that the 19th century mind was troubled by the realization that non-Euclidean geometries were not unthinkable:
In an age of political upheaval, this mathematical revolution provoked a great deal of anxiety. If the axioms of Euclidean geometry were not necessarily true but only true of Euclidean space -- which was only one type of possible space among many -- what could be necessarily true? As the philosopher Bertrand Russell succinctly put it: "If mathematics was doubtful, how much more doubtful ethics must be."Hence the appeal to a man like Lincoln, and a lot of us today:
It may be that what Lincoln found in Euclid was not only the notion of a logical demonstration but also the comfort of certainty, necessity and stability in a transformative and tumultuous time.