[FOM] intuitions of logic in Helsinki and Cambridge

praatika@mappi.helsinki.fi praatika at mappi.helsinki.fi
Tue Feb 28 06:13:25 EST 2006

Gabriel Stolzenberg <gstolzen at math.bu.edu>:

> 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.
>    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.
> 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.

OK, I then misundertood your point. Obviously you're right here. 
> 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.)

So how do you think it works? (honestly, I am iterested.)  

I had myself in mind people such as Dummett, Prawitz and Martin-Lof who 
argue against LEM by questioning the Principle of Bivalence. The attempts 
to make this more precise easily leads to all sorts of circularities (as 
I've tried to argue).

Best, Panu

Panu Raatikainen
Ph.D., Academy Research Fellow,
Docent in Theoretical Philosophy
Department of Philosophy, 
University of Helsinki

Visiting Fellow, 
Institute of Philosophy,
School of Advanced Studies, 
University of London

E-mail: panu.raatikainen at helsinki.fi

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