FOM: mathematical induction
Stephen G Simpson
simpson at math.psu.edu
Mon Mar 1 18:06:54 EST 1999
Michael Detlefsen 1 Mar 1999 12:03:54:
> Steve wants to stir the pot in a way that I don't want to and never
> intended to stir it. So, he presents my view as one of antipathy to
> logic ...
C'mon, Mic. You are openly trying to rehabilitate Poincare's
outdated, virulently bigoted, anti-mathematical-logic arguments. (I
think one could make a case that a lot of the anti-f.o.m. bigotry that
exists today among hard-core mathematicians, especially differential
geometers, is traceable to Poincare.) Why then do you take umbrage at
being accused of antipathy to logic?
> "There is something carried by reasoning by mathematical induction
> that does not seem to be sheerly a product of its 'logical
> content'. ...
Yes, mathematical induction isn't a purely logical principle. It's a
*mathematical* principle, a basic property of the natural number
system. Nevertheless, proofs involving it take place within a logical
framework. In particular, they can be formalized in (well-known
formal systems based on) the predicate calculus. So? What's the big
problem? Why do you try to insist that logic (i.e. the predicate
calculus) is somehow inadequate to this situation? It's totally
unjustified.
> Steve says that I give no 'real' example that illustrates my
> point.
You didn't give any example of a mathematical proof of a mathematical
proposition using mathematical induction. I gave such an example and
asked you to explain your point in terms of it, but you didn't do so.
The `examples' that you present are only imaginary thought experiments
-- hypothetical philosophical descriptions of `knowers' with non-human
capabilities. I don't see how they apply to mathematics.
> Similar arguments can be given for knowers not having infinitary
> powers, but I'll not go into that now.
Maybe you had better go into that, in order to bring this discussion
down to earth. At the same time, could you please illustrate the
arguments with at least one real example?
-- Steve
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