Tuesday, September 07, 2010

Euler’s ‘2’ continued

For a more eccentric analogy (see previous post for previous analogy), let a family of cooks in some dull country be introduced to the meaning of ‘orange’ by means of some imported carrots. Since our cooks want to refer only to ordinary objects, not to such things as properties, which seem to them hardly things at all, they reduce talk of orange things to talk of their carrots. But they would clearly be wrong to take us to be referring to their carrots with our uses of ‘orange’. Indeed, we are not even referring to the colour of their carrots, which might turn yellow.
......In view of the way things are—e.g. the cells of the human retina—a more realistic reduction would reduce orange to the two primary colours red and yellow. And to do something similar for the referent of ‘2’ would take us, not to standard set theory but to psychology. The letters ‘M’ and ‘N’ are angular and black and are, collectively, 2 letters, and it is by means of such examples that we came to know what ‘2’ means. Much as shapes and colours are predicated of ordinary objects, the natural numbers are predicated of finite sets.
......Could such properties be collections? Well, there is a philosophical tradition of reducing properties (e.g. orange) to extensions of properties (the class of all orange things), but there is a well known problem with reducing 2 to the class of all pairs. Set-theoretic paradoxes show that such classes are, if absolutely general (not just the class of pairs of ordinary objects), indefinitely extensible. There is no pre-existing class of all pairs to reduce 2 to. Our reductionists therefore reduce 2 to a particular pair-set. Not being eccentric, they don’t reduce it to something concrete, like a pair of carrots (although that has obvious reductionist benefits), but to something as abstract as numbers are thought to be.
......However, it is not so much a discovery as a technicality to use {Ø, {Ø}} rather than {{Ø}}, or at least, such is the choice not to use some other set, class or category theory, or indeed, constructive mathematics. The set-theoretic axiom of infinity, in particular, is true by definition of all standard sets, but is not so much a discovery as a guess about the natural numbers. Now, that assertion only makes sense insofar as numbers are not sets, but that just means that our reductionists risk losing the ability to assert that the axiom of infinity is only a guess; about what would it be a guess? Such reductionists therefore put themselves in the position of those nineteenth century scientists who, for good reasons, took ‘space’ to mean Euclidean space. Those reasons were just not good enough; and note that various supertasks and other paradoxes currently give us cause to question the truth of the axiom of infinity, construed as a property of the non-set-theoretic natural numbers.

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