FOM: human well-being; constructivism; anti-foundation
amah8857 at fau.edu
Thu Dec 28 10:27:48 EST 2000
Apologies for such late comments, I just saw it in the achieve of FOM.
> > This is a reply to Randall Holmes' message of Tue, 30 May 2000
> > 13:05:25 -0600. It also comments on messages of Frank, Sazonov,
> > Hazen, and Mahalanobis.
> > ----------
> > 2. CONSTRUCTIVIST PHILOSOPHY
> > I said that constructivistic mathematics is based on a subjectivistic
> > philosophy, according to which mathematics consists of mental
> > constructions in the mind of the mathematician.
> > In support of my view, Ketland quoted Heyting to show that Brouwer's
> > intuitionism is based on exactly this kind of subjectivism. And a
> > couple of people pointed out that Dummett is also somewhere close to
> > the subjectivist camp.
> > Against my view, several people noted that one can ``be interested
> > in'' or ``work on'' intuitionistic systems, without actually
> > ``believing in'' the underlying philosophical ideas.
What do you mean by intuitionistic system? Is it working with
Intuitionistic logic? If so, then what do you call subjectivism there?
Yes, historically it is right that Brouwer first started Intuitionistic
Mathematics, and Heyting abstracted it to get Intuitionistic Logic
(remember Brouwer never believed in logic). I am not very familiar with
Brouwer's work, but I see no signs of subjectivism in Intuitonistic Logic.
Can you please explain.
If a question is.
What is the underlying philosophical idea behind any constructivism?
Then my answer will be:
when do you say a object to exist, or may be when do you believe that all
the terms of a binary sequence is zero, this is my take from
any philosophy of constructivism.
So if you are saying that underlying philosophical idea behind
intuitionistic logic is subjectivism, then I differ respectfully. I think,
I believe in underlying philosophical idea behind Intuitionism, as what is
meant by proving existence.
> > I concede this point, but I say that it has nothing to do with the
> > philosophical/foundational issue. Intuitionistic systems of
> > mathematics were originally introduced in service of a Kantian or
> > subjectivist philosophy. If these formal systems take on a life of
> > their own, that does not erase the philosophical issues that gave rise
> > to them. In particular, if intuitionistic logic and type theory are
> > convenient for computer-aided algebra or computer-aided proof systems
> > such as Nuprl, that has no necessary connection to the philosophical
> > issue, which remains vital for f.o.m.
Is your point, if I may ask! That subjectivism is interesting and merits a
study. If so then I can't agree more, but I will also point that
it probably has very little or nothing to do with modern day constructivism
or Intuitionistic logic as it stands today.
> > Also opposing my view, Mahanalobis cited Bridges/Richman to the effect
> > that there are alternative ``varieties of constructivism'' (e.g., that
> > of Bishop) which may not be based on Brouwer's subjectivist
> > philosophy.
> > I would point out that the Bridges/Richman book explicitly eschews
> > philosophical concerns. ``We are writing for mathematicians rather
> > than for philosophers or logicians.'' (Page 2). It is true that
> > Bridges/Richman distinguish three schools which they call INT, BISH,
> > RUSS (Brouwer's intuitionism, Bishop-style constructivism, Russian
> > constructivism). However, they do so at a purely mathematical,
> > non-philosophical level:
If there is a distinction in the mathematical level then that implies a
distinction in the philosophical level.
> > BISH = classical mathematics with intuitionistic logic
> > INT = BISH + fan theorem + continuity principle
> > RUSS = BISH + Church's thesis + Markov's principle
As you put it, is wrong! Classical mathematics is inconsistent with
continuity principle which makes INT inconsistent. I don't think BISH can
be explained as classical mathematics with Intuitionistic logic, because
there are concepts in BISH which are absent in classical mathematics.
Moreover BISH balances between three (?) contradictory mathematics, RUSS,
> > In particular, Bridges/Richman do not comment on *why* one might
> > choose one system over another, except to say that BISH may be better,
> > because it assumes less.
Probably it is worth noting that these are contradictory systems, so it is
not one over another but one against another! BISH not only assumes less
but also it is the core of these three systems, any theorem if proved in
BISH and expressible in INT, RUSS and CLASSICAL, then the same proof goes
on in these three cases.
> > Is the philosophy that underlies Bishop-style mathematics really so
> > very different from Brouwerian subjectivism?
YES it is, BISH is consistent with RUSS, nobody till date has objected to
RUSS as subjective.
>If it is, then I think
> > there is need for a much fuller explanation. Bishop's discussion at
> > the beginning of his book on constructive analysis seems woefully
> > inadequate. In particular, it does not answer the obvious objections
> > that can be made by ultrafinitists, formalists, recursive function
> > theorists, et al.
I can think of ultrafinitists objection to BISH, What are the objections
of formalists and recursive function theorists to BISH?
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