## Tuesday, October 16, 2007

### Erasmus' slippages in math and logic

In Erasmus' satirical essay The Praise of Folly, Goddess Folly exposes in a jokingly manner how the entire humankind is subject to her dominion. Near the end of the book (chapter 63 according to the Spanish translation that I have) we read:
Ecclesiastes says in his first chapter, "The number of fools is infinite;" and when he calls it infinite, does he not seem to comprehend all men, unless it be some few whom yet 'tis a question whether any man ever saw?
Here Erasmus seems to imply that detracting an infinite quantity (the fool) from a set (all men) can only leave finitely many remaining elements, which of course is not true. About a hundred years later, Galileo shows a clearer understanding of a related problem in his celebrated paradox, which observes that natural numbers can be put in a one-to-one relation with square numbers, even though the latter form a strict subset of the former. Galileo prudently chooses to restrict terms like "greater" or "equal" to finite quantities alone so as to escape the riddle. We will have to wait another 250 years for Cantor to show us the way into the paradise of infinity; so, we can't really blame Erasmus too hard on this one.

A few sentences later we find the following apparent glitch:
For by the moon interpreters understand human nature, and by the sun, God, the only fountain of light; with which agrees that which Christ himself in the Gospel denies, that anyone is to be called good but one, and that is God. And then if he is a fool that is not wise, and every good man according to the Stoics is a wise man, it is no wonder if all mankind be concluded under folly.
The flawed reasoning can be rendered in logical notation like this (leaving out the bit on the goodness of God):

xGood(x)
xWise(x) → Fool(x)
x Good(x) → Wise(x)
implies
x Fool(x)

which is a non sequitur: there could be evil wise. The argument would be correct, however, if we had the following instead of the third premise:

x Wise(x) → Good(x)

So, before declaring Erasmus guilty of logical mistake, let's examine the original latin source (bolds mine):
Siquidem Lunam humanam naturam interpretantur, Solem omnis luminis fontem, Deum. Huic adstipulatur quod ipse Christus in Euangelio negat, quemquam appellandum bonum, nisi Deum unum. Porro si stultus est, quisquis sapiens non est, et quisquis bonus, idem sapiens, auctoribus Stoicis, nimirum mortales omneis Stultitia complectatur necessum est.

As far as I can tell, the correct interpretation of the marked part is x Good(x) → Wise(x), so Erasmus got it wrong. Unlike the case with infinity, here we can't concede the author the excuse of having been born too early: he should have known better.