Nicholas Nassem Taleb's latest book (Antifragile) continues to provide fascinating unreferenced claims.

In the middle of an argument that much of the important knowledge in the world is practical, not theoretical, in which he argues that the mediaeval architects responsible for some of Europe's great cathedrals did not gain their knowledge from theory, he makes the following assertion:

...according to the medieval historian Guy Beaujouan, before the thirteenth century no more than five persons in the whole of Europe knew how to perform a division.

His general argument, that theory frequently postdates successful practice, probably deserves a separate question (which I will submit when I can formulate it carefully). But that specific assertion, imperfectly referenced, that only a handful of people in Europe, had mastered the mathematics of division by 1200 AD seems to be surprising. Is it true?

  • Build a time machine and make a census?! ;) – 0xC0000022L Feb 24 '13 at 23:45
  • Yes matt, it could be true because only after 1150 Arabic numerals was introduced into Europe with Gherard of Cremona's translation of Ptolemy's Almagest. Before that time in Europe we had the Roman numbers that, as is well known, do not allow to perform division: "cogito ergo sum". – Carlo Alterego Feb 25 '13 at 0:23
  • Yes, but that would exclude Spain which was under Al-Andalus rule from 711 AD. – denten Feb 25 '13 at 0:56
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    @Carlo: Cogito ergo sum? That is a non-sequitur. – Oddthinking Feb 25 '13 at 1:59
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    @Oddthinking Did you intentionally use a Latin term to say that Carlo seems to have problems understanding Latin? Slow clap! – Konrad Rudolph Feb 25 '13 at 10:38
up vote 27 down vote accepted

This is not my field, but I have taken graduate-level courses on Andalusian history and the history of mathematics.

From what I can tell, the claim is at best imprecise and at worst an outright misrepresentation of history. For example, in an article published by the British Society for the History of Science, "From Abacus to Algorism: Theory and Practice in Medieval Arithmetic," Gillian Evans writes that:

The treatises on the abacus that seem to have proliferated in the late eleventh and early twelfth centuries comprise more or less extended preliminaries on the nature of number and the usefulness of knowing how to calculate upon the abacus. They usually give the Arabic numerals and their names, along with the names and symbols for the Roman duodecimal fractions, but they prefer, for the most part, to use Roman figures for the integers in their worked examples and in their multiplication tables. They confine themselves, typically, to multiplication, division and fractions, and, strictly, to the mechanics of the abacus.

In the "Arithmetic of the Medieval Universities," an article published by the National Council of Teachers of Mathematics, Dorothy V. Schrader writes:

The outstanding mathematical genius of the period was Gerbert, who taught the quadrivium with marked success in the cathedral school at Rheims from 972 to 982. Gerbert improved the abacus by placing symbols at the top of each column, and so extended its use. He developed a method of division, making possible all four fundamental operations on the abacus.

The sources I quote here suggest that even before the widespread introduction of Arabic numerals in Christian Europe, division was widely known, routinely performed on the abacus (and using hands), attested to in the literature of the period, and taught in the universities--all well before the 13th century.

Additionally, for even more examples of the above consult the freely-available Rara Arithmetica.

  • + OK. I'm impressed. – Mike Dunlavey Feb 25 '13 at 2:23
  • +1 and I wish I could upvote more: it is always surprising to get a good quality answer this quickly on an arcane subject. – matt_black Feb 25 '13 at 16:21
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    @DVK: but it does show that the concept and the need of "division" was known at that time and well understood by many more than just five people, which is what I believe Taleb is claiming. If you meant whether a specific division algorithm, namely the long division, was known, then yes it is insufficient; but long division is just one algorithm among many albeit the most popular one in current tunes. And, just like any good mathematical algorithm, long division is just as mechanical as using abacus. – Lie Ryan Feb 26 '13 at 0:15
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    @DVK Division on the abacus is quite difficult, requiring a solid understanding of theory. I do not know what Taleb may or may not have meant. The answer is confined to the historical record. The claim as it is stated is patently misleading. – denten Feb 26 '13 at 2:22
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    @DVK - Your complaint is like complaining that British people know how to drive in the left lane so that does not mean that they know how to drive since we drive in the right lane. It is still division. It was not spelled out as long division but rather that they knew how to divide one number by another. – Chad Feb 26 '13 at 18:00

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