Stephen G Simpson
simpson at math.psu.edu
Tue Apr 20 15:37:16 EDT 1999
Reply to Volker Halbach 19 Apr 1999 22:53:28.
Thanks very much for your sketch of the considerations that you say
lead many contemporary academic philosophers to anti-foundationalism.
I want to understand this phenomenon. Your remarks will certainly
help to advance the discussion.
I'd like to get clear on exactly what academic philosophers mean by
``foundationalism''. It may or may not be worth while to distinguish
various senses of ``foundationalism''. Shapiro defines it one way,
Ferguson defines it another way, and you define it a third way.
Shapiro says it's foolish to look for certainty in mathematics,
Ferguson says that philosophical considerations regarding mathematics
are automatically worthless, you say it's fruitless to look for
fundamental beliefs in terms of which other beliefs can be justified.
Am I wrong in thinking there's a common denominator here? From my
perspective, it's getting hard to distinguish anti-foundationalism
> I will try to stick to a definition of foundationalism (in
> epistemology) as it can be found in textbooks.
For those of us who are not academic philosophers, what textbook are
you referring to here? A textbook-style statement of the
anti-foundationalist position would certainly be helpful.
> Do you believe in the axioms of ZFC and claim that they are
> self-justifying or self-evident and not in need of any
> justification by appeal to some other principles?
These are good f.o.m. questions. By asking these questions you are
acting like one of the (dreaded?) foundationalists. In other words,
you are engaged in f.o.m. inquiry. A typical f.o.m. activity is to
study these questions as well as similar questions regarding
alternative foundational schemes other than ZFC.
My question is, why are academic philosophers such as Shapiro opposed
to this kind of activity?
To answer your question:
The idea of relying on ZFC as *the* foundation of mathematics is the
outcome of one particular foundationalist (f.o.m.) program which holds
a dominant position at this particular point in time. Call it the
program of set-theoretic foundations. Historically, I view it as a
descendent of the logicist program. Others may view it differently.
In any case, the undeniable point in the late 20th century is that the
ZFC foundational doctrine has achieved and holds an extremely large
measure of acceptance in the mathematical community, to the point
where one could say that it is ``the official'' or ``the approved'' or
``the accepted'' f.o.m. doctrine at this point in time. This is what
I think Friedman was getting at when he pointed out that ZFC axioms do
not need to be explicitly cited, but any additional or alternative
foundational assumptions do need to be explicitly cited, in papers
submitted for publication in, say, Annals of Mathematics. In this way
the mathematical community has achieved a certain consensus about
standards of evidence, specifically about criteria for deciding when a
theorem may be regarded as having been proved.
>From the viewpoint of mathematics, the ZFC consensus may be seen as a
good thing, because it protects mathematics from the possible bad
consequences of foundational warfare. Within the framework of this
consensus, it's usual to view ZFC as a good system in that it is
relatively clear, simple, flexible, nice, mathematically powerful,
etc. Under this view, a ``working mathematician'' may reasonably take
the axioms of ZFC as basic or self-evident or self-justifying. By
doing so, he will be working in a good framework for mathematics, and
he will avoid difficult philosophical issues. This is an acceptable
stance for a ``working mathematician'' to take. Such a ``working
mathematician'' is not a foundationalist.
At the same time, other people are free to oppose the consensus and be
upset about the status quo. For instance, the intuitionists will
oppose the ZFC consensus. The subject which adjudicates between these
varying positions regarding the ultimate basis for mathematics is
called f.o.m. The view that there is such a subject as f.o.m.,
i.e. that it is desirable to study and adjudicate these issues on the
basis of more fundamental principles, is known as foundationalism.
> papers that purport to provide evidence for the ZFC axioms. Thus
> the ZFC axioms are not basic beliefs: they are justified by appeal
> to other beliefs.
Yes. The point of these papers is to justify ZFC in terms of more
basic beliefs. That is f.o.m. activity.
> I am not anti-foundationalist, but I also find it very
> unsatisfying to accept axioms without having any reason to do so.
Good. That means you have the f.o.m. spirit. You are not willing to
blindly accept a foundational doctrine, for instance the ZFC
consensus, without further examination. That's a good thing. That's
> I also do not believe that anti-foundationalism and second-order
> logic are tied together in some way.
Shapiro tries to marry them in his book. But I really think that's a
side issue. The really important issue raised by Shapiro is, is
foundationalism (i.e. f.o.m.) a good thing or not? I say it is. What
do you say?
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