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A quantitative representation, inherited from our evolutionary past, underlies our intuitive understanding of numbers. If we did not already posses some internal non-verbal representation of the quantity "eight", we would probably be unable to attribute a meaning to the digit 8. We would then be reduced to purely formal manipulations of digital systems, in exactly the same way that a computer follows an algorithm without ever understanding its meaning.

I would like to suggest that these mathematical entities are so difficult for us to accept and so defy intuition because they do not correspond to any preexisting category in our brain. Positive integers naturally find an echo in the innate representation of numerosity; hence a four-year-old can understand them. Other sorts of numbers, howev...

Place-value coding is a must if one wants to perform calculations using simple algorithms. Just try to compute XIV * VII using Roman numerals! Calculations are also inconvenient in the Greek alphabetical notation, because nothing betrays that number N (50) is ten times greater than number E (5). This is the main reason the Greeks and the Romans never performed computations without the help of an abacus. By contrast, our Arabic numerals, based on the place-value principle, make the magnitude relations between 5, 50, 500, and 5,000 completely transparent. Place-value notations are the only ones that reduce the complexity of multiplication to the mere memorization of a table of products from 2 * 2 up to 9 * 9. Their invention revolutionized the art of numerical computation.

We haven't quite answered our question, though: Why is this type of list so difficult to learn? Any electronic agenda with a minuscule memory of less than a kilobyte has no trouble storing them all. In fact, this computer metaphor almost begs the answer. If our brain fails to retain arithmetic facts, that is because the organization of human memory, unlike that of a computer, is associative: It weaves multiple links among disparate data. Associative links permit the reconstruction of memories on the basis of fragmented information. We invoke this reconstruction process, consciously or not, whenever we try to retrieve a past fact. Step by step, the perfume of Proust's madeleine evokes a universe of memories rich in sounds, visions, words, and past feelings.

Associative memory is a strength...

Do you see the problem? This child is not responding at random. Every single answer obeys the strictest logic. The classical subtraction algorithm is rigorously applied, digit after digit, from right to left. The child, however, reaches an impasse whenever the top digit is smaller than the bottom. This situation calls for carrying over, but for some reason the child prefers to invert the operation and subtract the top digit from the bottom one. Little does it matter that this operation is meaningless. Indeed, the result often exceeds the starting number, without disturbing the pupil in the least. Calculation appears to him as a pure manipulation of symbols, a surrealist game largely devoid of meaning.

Where do these bugs come from? Strange as it might seem, no textbook ever describes the c...

In all truth, matters are trifle more complex because only a certain version of Peano's axioms that mathematicians call "first-order Peano arithmetic" suffers from this infinite expansion of nonstandard models. Yet this version is generally thought to be the best axiomatization of number theroy that we have.

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