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| In a message dated 10/31/2002 11:32:13 AM Central Standard Time, cowan@hidden.email writes: << {{}, {{}}, {{{}, {{}}}} >> This is natural number 3, not 2. << I recognize only one universal: numbers >> That is, I suppose, the imaginaries or whatever lies beyond them. But what then are the others: not a selection from these, since we do not have the means to make the selection even in theory. << . The distinctions between different kinds of numbers are purely functional: I no more think that integers are not reals than I think that primes are not integers. >> But the primes are a definable subset of the naturals (not integers, by the way, or the definition fails), the integers are not such of the reals. And what, exactly, are these functions that distinguish them? Successor, addition, subtraction, multiplication, division, exponentiation and radicination? Counting, accounting, herding, cutting, ...? << > That is the sort of thing that > can happen only in the confused world of shadows, not in The Real World. Au contraire. To identify numbers with sets may be useful, bu to say that numbers *are* certain sets is rank {consequenti,operation}alism. >> I don't think I identified numbers with sets, unless by careless phrasing. However, the fact that they map naturally onto grossly different kinds of sets is indicative of their rather different natures. If you want to get seriously operational, then you are back to Kronecker, "God gave us the natural numbers, all else is the work of man," which is also a tradition to which Plato would subscribe (on other days -- when he thinks that 2 is the first number, for example). In any case, the issue (can we get back to that for a moment) here is linguistic, namely, how to indicate briefly what sort of numerical expressions we expect in a given context. Diffferent indicators suggest different expectations, and violating those expectations can interfere with communication (not to mention -- or because -- being totally inappropriate). |