[lit-ideas] Re: Imagination and maths [and maybe more.....]

  • From: Robert.Paul@xxxxxxxxxxxxxxxxxx (Robert Paul)
  • To: lit-ideas@xxxxxxxxxxxxx
  • Date: 19 Dec 2004 16:16:47 PST

M. Chase writes:

I'm sure Robert is aware of Hilary Putnam's work on quantum 
logic, which dates from the 1960's. In "Is logic empirical?" in R. Cohen and 
M. P. Wartofski (eds.), Boston Studies in the Philosophy of Science 
(Dordrecht, Holland: D. Reidel, 1968). Reprinted as "The logic of 
quantum mechanics" in H. Putnam, Mathematics, Matter and Method, 
Cambridge University Press (1976) he writes inter alia Logic is  
as empirical as geometry. We live in a world with a non-classical 
M. Chase is too kind. However, paraphrasing authority--in this case Putnam (who
was re-working Quine)--is no substitute for actual argument and examples. It is
often said that quantum physics and its logic have somehow shown that the law of
the excluded middle [not-(p and not-p)] must be abandoned; that it is 'false';
or that (as one of Andreas' sources suggests) it no longer (!) applies to
'reality' or to 'the world we live in.' Putnam wants to claim that the
propositions of logic are every bit as empirical as 'Mike has two euros in his
pocket.' His reason for claiming this would seem to be that one can imagine
examples in which (some forms of) _applied_ mathematics do not, in the case of
certain quantum phenomena, 'work' when the traditional law of the excluded
middle is invoked. (I say 'invoked' because the claim is that it cannot, in the
end, be _used_.) And this may well be true: let's just stipulate that it is
true, and that the logic and mathematics of quantum physics must resort to other

However, whether certain forms of _applied_ mathematics 'work' in a quantum
universe, is not itself a mathematical or logical claim. It is an empirical
claim, on all fours with 'That elephant won't fit into the trunk of that
Volkswagen.' It tells us nothing about which models or modes of applied
mathematics do 'work' in everyday life (counting change, charging someone with
bigamy, dividing a plate of cookies 'evenly'); and it needs to be argued that
anything that is true of applied mathematics is true of pure mathematics 'in

As long as I'm making bold claims about areas in which I scarcely dare to tread
I'll suggest something more: the results of Putnam (Quine; Brouwer; the 'later'
Wittgenstein) depend upon our being able to make assertions in a natural
language and to deduce consequences from them. If those who are ready to ascend
to the New World of logic believe that once there they can throw away their
ladders, so be it. In the meantime, for Putnam, as for anyone, setting out valid
arguments is impossible without a belief in the law of non-contradiction. As
Aristotle said, 'You might reject this; but you can't argue with me about it.'
[Metaphysics, somewhere]

Robert Paul
The Reed Institute
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