The Universe of Discourse

Tue, 23 Jan 2007

Thomas Hobbes screws up

In need of some bathroom reading last week, I grabbed my paperback copy of Thomas Hobbes' Leviathan, which is always a fun read. The thing that always strikes me about Leviathan is that almost every sentence makes me nod my head and mutter "that is so true," and then want to get in an argument with someone in which I have the opportunity to quote that sentence to refute them. That may sound like a lot to do on every sentence, but the sentences in Leviathan are really long.

Here's a random example:

And as in arithmetic unpractised men must, and professors themselves may often, err, and cast up false; so also in any other subject of reasoning, the ablest, most attentive, and most practised men may deceive themselves, and infer false conclusions; not but that reason itself is always right reason, as well as arithmetic is a certain and infallible art: but no one man's reason, nor the reason of any one number of men, makes the certainty; no more than an account is therefore well cast up because a great many men have unanimously approved it. And therefore, as when there is a controversy in an account, the parties must by their own accord set up for right reason the reason of some arbitrator, or judge, to whose sentence they will both stand, or their controversy must either come to blows, or be undecided, for want of a right reason constituted by Nature; so is it also in all debates of what kind soever: and when men that think themselves wiser than all others clamour and demand right reason for judge, yet seek no more but that things should be determined by no other men's reason but their own, it is as intolerable in the society of men, as it is in play after trump is turned to use for trump on every occasion that suit whereof they have most in their hand. For they do nothing else, that will have every of their passions, as it comes to bear sway in them, to be taken for right reason, and that in their own controversies: bewraying their want of right reason by the claim they lay to it.
Gosh, that is so true. Leviathan is of course available online at many locations; here is one such.

Anyway, somewhere in the process of all this I learned that Hobbes had some mathematical works, and spent a little time hunting them down. The Penn library has links to online versions of some, so I got to read a little with hardly any investment of effort. One that particularly grabbed my attention was "Three papers presented to the Royal Society against Dr. Wallis".

Wallis was a noted mathematician of the 17th century, a contemporary of Isaac Newton, and a contributor to the early development of the calculus. These days he is probably best known for the remarkable formula:

$${\pi\over2} = {2\over1}{2\over3}{4\over3}{4\over5}{6\over5}{6\over7}{8\over7}\cdots$$

So I was reading this Hobbes argument against Wallis, and I hardly got through the first page, because it was so astounding. I will let Hobbes speak for himself:

The Theoreme.

The four sides of a Square, being divided into any number of equal parts, for example into 10; and straight lines being drawn through opposite points, which will divide the Square into 100 lesser Squares; The received Opinion, and which Dr. Wallis commonly useth, is, that the root of those 100, namely 10, is the side of the whole Square.

The Confutation.

The Root 10 is a number of those Squares, whereof the whole containeth 100, whereof one Square is an Unitie; therefore, the Root 10, is 10 Squares: Therefore the root of 100 Squares is 10 Squares, and not the side of any Square; because the side of a Square is not a Superfices, but a Line.

Hobbes says, in short, that the square root of 100 squares is not 10 unit lengths, but 10 squares. That is his whole argument.

Hobbes, of course, is totally wrong here. He's so totally wrong that it might seem hard to believe that he even put such a totally wrong notion into print. One wants to imagine that maybe we have misunderstood Hobbes here, that he meant something other than what he said. But no, he is perfectly lucid as always. That is a drawback of being such an extremely clear writer: when you screw up, you cannot hide in obscurity.

Here is the original document, in case you cannot believe it.

I picture the members of the Royal Society squirming in their seats as Hobbes presents this "confutation" of Wallis. There is a reason why John Wallis is a noted mathematician of the 17th century, and Hobbes is not a noted mathematician at all. Oh well!

Wallis presented a rebuttal sometime later, which I was not going to mention, since I think everyone will agree that Hobbes is totally wrong. But it was such a cogent rebuttal that I wanted to quote a bit from it:

Like as 10 dozen is the root, not of 100 dozen, but of 100 dozen dozen. ... But, says he, the root of 100 soldiers, is 10 soldiers. Answer: No such matter, for 100 soldiers is not the product of 10 soldiers into 10 soldiers, but of 10 soldiers into the number 10: And therefore neither 10, nor 10 soldiers, is the root of it.
Post scriptum: The remarkable blog Giornale Nuovo recently had an article about engraved title pages of English books, and mentioned Leviathan's famous illustration specifically. Check it out.

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