William Saint, on Euclid's Definition of Multiplication
Quite bravely, given Voltaire’s observation, ‘He found how dangerous it is to be in the right, in matters where men in power are in the wrong’, on August the 4th 1809, William Saint wrote to the editor of A Journal of Natural Philosophy Chemistry and the Arts (1). In this letter, Saint quoted, not Sir Isaac Barrow’s English translation of Euclid’s definition of multiplication, but as we know via Sir Jonas Moore, that of ‘an unknown hand, who was ignorant of the subject’. Saint observed;
...if it were required to multiply the number 3 by the number 2, the answer would be 9 according to Euclid’s definition.
Saint also made a suggestion that would ‘fix’ Euclid’s definition of multiplication.
Perhaps this definition would be better thus: one number is said to be multiplied by another, when it is taken or repeated as many times as there are units in that other. To those, however, who may be disposed to contend, that the words "taken or repeated" do not sufficiently define the operation intended; and who may farther insist, that multiplication is only a continued addition, Euclid's definition may perhaps be preferred, if the words less one be inserted after the word multiplying.
Saint again documented the illogical nature of the English multiplication definition, in 1811, in The Monthly Magazine (2) with a review of Thomas Taylor’s book, The Elements of the True Arithmetic of Infinites (3). This was a brave act in England, where Euclid was revered. Saint would not have known Euclid’s definition arose via the writings of the London haberdasher, Henry Billingsley. Today, social media disputes typically run their course in a few days. Yet the dispute between Saint and Taylor, published in The Monthly Magazine, began in May 1811 and continued through to October that year. From this time, Saint, having been subjected to much public ridicule, appears to have no longer written about the illogical nature of the English definition of multiplication. Taylor, with his reputation at risk, kept attacking Saint until at least October 1815, insulting Saint not just in English, but in Greek, writing, ‘...since in the language of Diogenes, “ἀνθρώπους ἐκάλεσα οὐ καθάρματα”’, It was men I called for, not scoundrels (4).
John Searle, on (Euclid's) Definition of Multiplication
In 1990, Dr. John R. Searle, the Slusser Professor of Philosophy at the University of California, Berkeley, explained multiplication, based on what he had observed. In an address to the American Philosophical Association, titled, Is the brain a digital computer? (5) Dr. Searle said; [Author’s emphasis.]
Suppose that we have a computer that multiplies six times eight to get forty-eight. Now we ask "How does it do it?" Well, the answer might be that it adds six to itself seven times. But if you ask "How does it add six to itself seven times?", the answer might be that, first, it converts all of the numerals into binary notation, and second, it applies a simple algorithm for operating on binary notation until finally we reach the bottom level at which the only instructions are of the form, ‘Print a zero, erase a one.’
Dr. Searle’s footnotes to his printed speech include the following comment.
People sometimes say that it [i.e. six times eight] would have to add six to itself eight times. But that is bad arithmetic. Six added to itself eight times is fifty-four, because six added to itself zero times is still six. It is amazing how often this mistake is made.
Subsequently, Dr. Searle’s observations on multiplication, appeared in his book, The Rediscovery of the Mind, (6), and mocked, by mathematics professor, Dr. Gabriel Stolzenberg, as follows (7).
Some time ago, while thumbing through a friend’s copy of John Searle’s, The Rediscovery of the Mind, I encountered a blunder the likes of which I had never seen before. It begins:
Suppose that we have a computer that multiplies six times eight to get forty-eight. Now we ask, “How does it do it?” Well, the answer might be that it adds six to itself seven times. (The Rediscovery of the Mind: 213)
It was easy to see that if Searle had said “eight,” as the rest of humanity does, his argument would have proceeded without change. But he made it clear that he wanted his argument to be disrupted by this irrelevant deviant use of language. He did not just say “seven” instead of “eight” and go on as if nothing had happened. Instead, he tagged it with a note:
People sometimes say that it would have to add six to itself eight times. But that is bad arithmetic. Six added to itself eight times is fifty-four because six added to itself zero times is still six. It is amazing how often this mistake is made. (238)
I burst out laughing at this display of pretentious ignorance about an informal mathematical locution that is trickier than Searle supposed, capped by his conceit that he had discovered a mistake in arithmetic—or at least in the use of an arithmetical expression—that the rest of us dummies had missed. Yet, my friend, who is both a great mathematician and a great reader, had read Searle’s book without noticing this. When I asked how he could have missed it, he explained, “I assumed he was making a joke that I didn’t get.”
People blind to logic have a tendency to mock rather than think. So a question arises. What if people think they are right, only because they do not know they are wrong? As Kathryn Schulz noted, (8) being wrong feels the same as being right. It is only when you realise you are wrong, that the two feelings diverge.
1) Remarks on some of the Definitions and Axioms in Barrow’s Euclid, in A Journal of Natural Philosophy Chemistry and the Arts, Vol. 32, p. 377, William Nicholson, London, 1809. NOTE: Natural philosophy was an early term for physics.
2) The Monthly Magazine, Vol. 31, pp. 314-319, London, May 1, 1811.
3) The Elements of the True Arithmetic of Infinites, Thomas Taylor, London, 1809.
4) Colburn's New Monthly Magazine, Vol. 4. p. 4, Oct. 1, 1815. Translation from Diogenes Laertius, Lives of Eminent Philosophers, Book VI, R. D. Hicks, Loeb Classical Library, 1925.
5) Is the brain a digital computer?, Proceedings and Addresses of the American Philosophical Association, pp. 21-37, Vol. 64, No. 3, 1990.
6) See Ch. 9, The Critique of Cognitive Reason, p. 213, [footnote p. 254], in The Rediscovery of the Mind, John R. Searle, The MIT Press, Cambridge, Massachusetts, 1992.
7) Dr. Gabriel Stolzenberg is a Professor Emeritus of Mathematics. His review appears in Professor Nagel's Fashionable Nonsense, available at http://math.bu.edu/people/nk/rr/tn.html
8) Being Wrong: Adventures in the Margin of Error, Kathryn Schulz, Ecco, New York, 2010.