FOM: constructivist philosophy

Stephen G Simpson simpson at
Thu Jun 1 19:59:37 EDT 2000

A response to Fred Richman's posting of today.

 > It seems to me that the philosophical discussion of intuitionistic
 > systems should focus on how they are currently viewed and used, not
 > on some "original intent".

Why not focus on both?  We can discuss Brouwer's subjectivist
philosophy and its relationship to his original intuitionistic
mathematics, and to Heyting's original formalization of it, in terms
of intuitionistic logic.  We can also discuss how latter-day
constructivists view their own work, and how this fits into the wider
context of mathematics, science, and intellectual life.

 > I'm not quite sure what philosophical issues are.  Many years ago,
 > when I worked one summer in a lab, I was told that there were two
 > philosophies about the machine shop: [...]

Colloquially, ``a philosophy'' means any relatively broad principle or
broad way of looking at things.  In the current discussion here on
FOM, when I spoke of philosophy, I was referring to the broadest
principles of all: a comprehensive, integrated world-view of man,
nature, existence, the mind, ethics, politics, art, etc.  (Of course I
am aware that most present-day academic philosophers are averse to
``system building'', i.e., any comprehensive world-view.)

It seems to me that there are stong connections between philosophy and
f.o.m., and we can explore these connections here on the FOM list.

For example, we know that Bishop has stated (rather vehemently) that
nonconstructive existence proofs constitute fraud.  Why?  If we prove
the absurdity of (forall x) not Phi(x), why does Bishop think this is
not a proof (or at least very strong evidence) that (exists x) Phi(x)?
Is there a metaphysical/epistemological assumption that leads Bishop
to his position?  What would the assumption mean for general science?
The moon exists, yet we did not construct it, and no fraud is
involved.  Why and how does Bishop think mathematics is different from
the rest of science?  What kind of constructivist principle is at work
here?  And, how are the constructions to be carried out?  If they are
purely mental, isn't that pure subjectivism, at least in the realm of

 > If intuitionistic logic is undergoing a renaissance because of
 > computer algebra, then that is what needs to be clarified, [...]

This is also a perfectly good topic for discussion.  There are
software packages that use Heyting's formalism of intuitionistic
logic, and it is reasonable to ask why.  The reasons could be
philosophical, non-philosophical, or even anti-philosophical.

 > Is the dichotomy between hard and soft analysis a philosophical
 > issue? [...]  The underlying metaphysics (?) of category theory
 > might be summarized by the statement that whenever you study
 > mathematical objects, it is essential that you also study the maps
 > between them. [...]

I don't see these issues (hard vs soft analysis, objects vs maps, etc)
as truly philosophical.  They do not seem to flow from any
comprehensive world-view or conflict of world-views.  Of course we can
still discuss them on the FOM list, but let's try to relate them to
issues and programs in f.o.m.  As they stand, I would identify them as
relatively narrow, methodological issues.  They involve limited claims
to the effect that certain methods will more quickly lead to better
and more comprehensive results in specific branches of mathematics.
Such claims do not deny the validity of other methods, even in those
specific branches, and they do not say much about the logical
structure of mathematics as a whole.

-- Steve

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