[FOM] A question concerning continuous functions

[Michael Lamport Commons]

It would seem to me that one could have it either way.  One of the beauty of
Mathematics is that there is no necessity for choosing a particular axiom or
not.  One can have it both ways, but not within the same mathematical
system.

My Best,

Michael Lamport Commons, Ph.D.
Assistant Clinical Professor of Psychiatry

Program in Psychiatry and the Law
Department of Psychiatry
Harvard Medical School
Massachusetts Mental Health Center
74 Fenwood Road
Boston, MA 02115-6113

Telephone (617) 497-5270
Facsimile    (617) 491-5270

Commons at tiac.net
http://www.tiac.net/~commons/
----- Original Message -----
From: "Andrej Bauer" <Andrej.Bauer at andrej.com>
To: <fom at cs.nyu.edu>
Cc: "Arnon Avron" <aa at tau.ac.il>
Sent: Tuesday, January 28, 2003 8:59 AM
Subject: Re: [FOM] A question concerning continuous functions


>
> Arnon Avron <aa at tau.ac.il> writes:
> > 2) In my opinion, the weakest point of Intuitionism, and the reason why
> >    it will never be adopted by most mathematicians (even though many of
> >    them adhere to the importance of constructive proofs) is that it
> >    seems to totally ignore geometric intuitions and concepts. As far as
> >    I understand (and please correct me if I am wrong), intuitionists
> >    have completely abandoned the original, intuitive concept of
> >    continuity in favor of (some constructive version of) the "official"
> >    definition.  They have ended up rejecting the classical theorems
> >    mentioned above which provide (in my opinion) the main evidence for
> >    accepting this identity in the first place... Let me add that this
> >    phenomenon provides (for me) another evidence that the two notions
> >    of continuity are not identical.
>
> Permit me to correct you then :-).
>
> First, about rejection of theorems:
>
> the theorems you are referring to are not "rejected by intuitionists".
> They have just been reformulated more carefully:
>
>
> Constructive Intermediate Value Theorem:
>    Let f : [a,b] --> R be a continuous function and f(a) <= y <= f(b).
>    For every epsilon > 0 there exists x in [a,b] such that |f(x) - y| <
epsilon.
>
>
> Under additional assumptions about f it is possible to show that there
> actually exists x such that f(x) = y, e.g., if f is a polynomial or a
> sufficiently nice smooth function.
>
> To get to the classical Intermediate Value Theorem we would want a
> theorem claiming that every sequence in a bounded interval has a
> convergent subsequence, which is constructively invalid.
>
> Constructive Theorem:
>     Every _uniformly_ continuous function f : [a,b] --> R is
>     bounded. Moreover, there exists M such that f(x) <= M for
>     all x in [a,b], and for every M' < M there exists x in [a,b]
>     with M' < f(x) <= M. In other words, M is the supremum of f.
>
> Again, to conclude that there is x such that f(x) = M, we would need
> to know that every sequence on a closed interval has a convergent
> subsequence.
>
> It is a classical accident that every continuous function on a closed
> interval is also uniformly continuous.
>
>
> Second, about abonding geometric intuitions:
>
> Intuitionistic logic is _required_ for an axiomatization of the truly
> geometric intuition of "continuum" as an indivisible space containing
> infinitesimals. I am referring to synthetic differential geometry.
>
> Furthermore, when students of mathematics are presented with geometric
> constructions that shatter their geometric intuitions (a function that
> is differentiable precisely at all irrational numbers, Banach-Tarski
> paradox, non-measurable sets, etc), these are all very heavily
> classical constructions. Does this not mean that it is the fault of
> classical mathematics that human geometric intuitions are in conflict
> with the official truth?
>
> Andrej Bauer
> University of Ljubljana
> http://andrej.com/
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