[FOM] philosophical literature on intuitionism
Frank Waaldijk
frank.waaldijk at hetnet.nl
Tue Oct 14 14:08:29 EDT 2008
dear Thomas,
you wrote about intuitionism:
> I'm curious to know what the people who dreamt this stuff up actually
> thought they were doing.
>
> Where is the best place to start? Is it Dummett's book? Did Brouwer write
> anything one might want to read? I seem to remember there is an essay in
> one of the collections (Benacerraf and Putnam?). One of my colleagues
> here says that Intuitionism is really a form of solipsism, and that for an
> intuitionist to countenance any kind of interpretation into classical
> logic (or vice versa) is to undermine the solipsism and would not be
> welcomed by the true believers. I do remember reading that Brouwer was
> hostile to attempts to axiomatise constructive logic..
Perhaps I could pose a counterquestion about classical mathematics?
You see, I'm curious to know what the people who dreamt up classical math,
especially unrestricted ZFC, actually thought they were doing.
Where would you point me for the best place to start? Is it the collected
works of Cauchy? Did Hilbert write anything one might like to read? I seem
to remember there is an essay by Poincaré...One of my colleagues used to say
that classical math is really a form of Platonism, and that for a classical
mathematician to countenance any kind of interpretation in the real physical
world (or vice versa) is to undermine the Platonism and would not be
welcomed by true believers. I do remember that Hilbert was hostile to
attempts to constructivize logic.
OK. Just to show you that your question is phrased with the common bias held
by classical mathematicians. Mostly this bias is held fiercely, but without
having given Brouwer's mathematics any real consideration.
What were these people who `dreamt this stuff up' actually thinking? Put
simply, very simply, the matter boils down to this. Brouwer showed very
convincingly that most of classical mathematics is a hopelessly Platonistic
mind game. The large majority of classical results cannot be applied to
measurable, real world physics. They cannot be turned into algorithms to be
implemented on a computer. They hold no meaning outside a completely
formalistic setting. Still, if one would insist that such a setting is the
only way to do mathematics, then it is perhaps instructive to know that
intuitionism can be brought into just such a formalistic setting, as
consistently as classical mathematics. So why choose classical mathematics
over intuitionism? As a formal game, what would be the higher worth of one
over the other?
However, Brouwer insisted that mathematics should be constructive. In this
way, results from mathematics would correspond to our intuition (hence the
name I suppose) of the real world, especially our concept of time. (Notice
that the concept of time is still incredibly vague in modern physics-as far
as I'm aware).
The result is that practically all of intuitionism is applicable to real
world physics, and that practically all intuitionistic results can be
implemented on a computer. OK, Brouwer is not the easiest reading. But later
developments have shown that a lot of Brouwer's more mystical formulations
can be replaced by simple axiomatics, which have been shown to be as
consistent as classical mathematics (to put it simply). And for instance
Heyting has written a very good introduction to intuitionism.
All in all, there is now a very respectable body of constructive mathematics
(which grew out of intuitionism), and its uses in both physics and computer
science are ever increasing. It's really but a matter of time before the
classical bubble bursts, if you ask my honest opinion. Might still take
another century, but burst it will.
Hope this was a useful steer!
Kind regards,
Frank Waaldijk
http://home.hetnet.nl/~sufra/mathematics.html
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