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This assemblage illustrates a philosophical point which I raise in my book Secrets of Creation concerning the relationship between addition and multiplication. The following is an adaptation of the relevant passage:

[M]ultiplication is clearly a more complicated matter than addition. Addition is well understood, but as the mysteries of the prime numbers are gradually revealed to you, it will become clear that we don’t fully understand multiplication or, more precisely, that we don’t understand the relationship between multiplica-tion and addition. Louis Kauffman, a University of Chicago mathematician whose wide-ranging interests include the philosophical foundations of mathematics, makes this point:

Multiplication is more complex [than addition]. When we multiply 2 × 3 we either take two threes and add them together, or we take 3 twos and add these together. In either case we make an operator out of one number and use this operator to reproduce copies of the other number.

The word ‘operator’ is used a lot in mathematics, incidentally—its meaning is very precise, but for our purposes it can be thought of roughly as ‘something that does something to something else.’

The point that Kauffman is making is that with multiplication, the two numbers involved are playing different roles….

Think about it this way: 2 and 3 are counting numbers. In the context of ‘3 × 2’, what are they counting?

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