[FOM] intuitions of logic in Helsinki and Cambridge
Gabriel Stolzenberg
gstolzen at math.bu.edu
Mon Feb 27 23:47:18 EST 2006
This is in reply to Panu Raatikainen's comments of February 27
about my posting, "intuitions of logic in Chicago and Cambridge."
Panu raises excellent points. I see now that I failed to make
clear that, unlike the members of this list, the folks who I am
talking about don't spend their time thinking about foundations of
mathematics.
Panu begins by quoting me and then comments on what I say.
> > In fact, classical mathematicians sometimes use their logical
> > intuitions to "prove" the law of excluded middle. Although they
> > don't realize it, they use excluded middle reasoning to prove the
> > statement of the law of excluded middle.
>
> Isn't this a little bit uncharitable. Even if some have proceeded like
> this, an adherent of classical logic certainly need not to do that.
As I indicated above, I'm not talking about people who think about
the law of excluded middle. I'm talking about folks who reason
according to it without thinking about it, without even being aware
that they are reasoning according to it. So I don't think they are
what you mean by an "adherent."
Psychologically, such reasoning seems to be an involuntary and
unreflective response to a certain kind of challenge, a response that
usually begins with "Suppose not." My point was that, because this
kind of excluded middle reasoning is involuntary and unreflective,
it sometimes is evoked inappropriately, e.g., by a challenge to prove
the law of excluded middle.
> Rather, one can derive LEM from the Principle of Bivalence, which
> in turn seems to be analytically built in to the classical, realist
> conception of truth.
I didn't mean to sugggest that I was seriously challenging classical
mathematicians to prove the law of excluded middle. (If I was, then
Kreisel might have been right when he told me that I was mentally ill.
This was on the basis of my review of Bishop's book in the Bulletin
of the AMS. I found this fascinating, so I asked him what his method
of diagnosis was. He said, "Statistical." At this point, I realized
that I shouldn't be having this conversation, so I kicked him out of
the room. Verbally, not physically.)
I just wanted to see how, in certain situations (chatting in a common
room, over dinner in a restaurant, etc.), classical mathematicians would
respond if they thought that this was what I was doing. And, in my very
small sample, I found that it was the involuntary, unreflective response
that I described above.
> However, I think that it is very difficult to argue against these
> ideas without already presupposing the intuitionistic interpretation
> of logical constants.
If, by "to argue against these ideas," you mean arguing in favor
of rejecting the law of excluded law, recall that, in constructive
mathematics, the law of excluded middle is happily neither accepted
nor rejected. If you start out as a classical mathematician, as I
did, you don't acquire a constructive mindset by rejecting the law
of excluded middle. It doesn't work that way! (What intuitionists
do is another matter.)
Gabriel Stolzenberg
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