Essentially all of my organized exercise comes from lap-swimming. I like to keep track of how many laps I have swum at one go, and to make it more interesting I have taken to focusing on a particular canonical way of referencing each positive integer, based on multiplicative generation.

‘One’, of course, is unique: the identity element for the multiplicative group over the positive integers. Prime numbers I visualize by way of an assertion of the form “Prime number 1 is 2”. (In principle, at least—when the numbers get large this tends to turn into sort of a phone number: 511, 613, 717, 819, 923 and so on. Eventually these start to sound like numbers of years, especially 1861 and 1967.)

Composite numbers which are powers are canonicalized to “3 cubed”, “2 to the 5th power” and the like, always using the smallest possible base. Other composite numbers just get realized as “*i* times *j*”, where *i* is less than *j* and both are maximally close to the square root of their product (so “8 times 9” trumps “6 times 12”).

There are various patterns to be encountered in this counting sequence, such as the “four 13s” (3 times 13, 5 times 8 (the factors sum to 13), prime #13 is 41, 6 times 7 (the factors sum to 13)). I learned the term ‘semiprime’ for a number which is the product of two primes: the rare occurrences of three semiprimes in a row are notable (33, 34, 35; 85, 86, 87; 93, 94, 95; the next one is more laps than I ever swim).

I imagine Pythagoras and his crowd as tremendously impressed with the magic and complexity of the multiplicative group. What if we couldn’t just summon up integers for use by means of positional notation but had to construct them multiplicatively, visiting a notional numerical supermarket that only stocks primes?

I would suppose that a defender of classical theism like Ed Feser would have to say that the primeness of all primes, no matter how large, is continuously and simultaneously ‘known’ to God; to my mind this kind of ‘knowledge’ has nothing to do with knowledge as accessible to the human intellect.

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