[FOM] Remedial mathematics?

[Jeremy Shipley]

Thanks Tim,

I see your broader point, and I agree that if ordinary mathematical
standards accepts primitive knowledge of abstract particulars (Z, N,
the iterative hierarchy) then there is an abundance of trivial
consistency proofs.  I suppose I am "revolutionary" enough a
philosopher to be willing to explore alternatives to ordinary
standards. I do not want to take up too much of your time responding
to a post referencing epistemological views that may be less familiar
to fom folks, but I would like to elaborate on the relationship
between finitism, fallibilism, and essentialism as I see it.

Regarding how things stand without an absolute refutation of the
skeptic: First, I would say that I don't think this is the only
philosophical motivation for essentialism; Second, I would argue that
both essentialism and existentialism are adaptable to fallibilist
epistemological views. For example, I do not see any barrier to
holding that knowledge of consistency is prior to knowledge of
existence (or possible existence) while also acknowledging that
knowledge of consistency lacks the infallible justification envisioned
by Hilbert's acceptance of finitist restrictions. One might, seeing
the power of ZFC for formalization projects and its role as a standard
of relative consistency in fom research, draw on Crispin Wright's
notion of epistemological entitlement (relative to an inquiry) to
warrant belief that con(ZFC). In another thread some one suggested
that contextualist and subject-sensitive invariantist epistemologies
may be relevant. I agree. Of course, fallibilist epistemological views
could just as well be pressed for accepting existence claims (prior to
axiomatization) as suggested by your example of Z, but because I think
that there are motivations for Hilbert's views (or views in the
neighborhood) that are not epistemological I don't see why the failure
of finitism should lead us to abandon essentialism.

But maybe what I'm saying is very close in many respects to your point
of view and I am just taking the wrong lesson from your example using
Z.

-Jeremy

On Tue, May 24, 2011 at 9:29 AM, Timothy Y. Chow <tchow at alum.mit.edu> wrote:
> On Mon, 23 May 2011, Jeremy Shipley wrote:
>> Existentialism (Frege): Existence (ie, intuition of logical and
>> geometric objects) precedes essence (ie, consistent axiomatic
>> systemization).
>>
>> Essentialism (Hilbert): Essence precedes existence.
> [...]
>> it strikes me as odd that fom practitioners should appeal to
>> existentialism, even if it is accepted in ordinary mathematical
>> reasoning. I would have thought that even if ordinary mathematics is
>> existentialist, most fom practitioners would be essentialist. Otherwise
>> why care about consistency proofs in the first place, which find their
>> relevance, or at least their historical motivation, within an
>> essentialist program?
>
> Part of the problem here is that you're anachronistically lifting an old
> debate into the modern world.  In the old days, it wasn't so clear that
> every mathematical argument could be easily formalized in a variety of
> ways, nor was it clear that we couldn't hope for a finitary proof of the
> consistency of infinitary set theory.  F.o.m. practitioners may in the
> past have looked to consistency proofs as providing a road from deep
> skepticism about infinitary reasoning to absolute certainty in its
> legitimacy, but thanks to Goedel they now recognize that as a chimera.
> They are therefore frustrated when mathematicians who ought to know better
> continue to cling to the old way of thinking, insisting that the only way
> to "know for sure" that some system is consistent is to give a finitary
> proof of it, while simultaneously failing to practice what they preach
> (since mathematicians continue to claim to "know for sure" lots of things
> without any finitary proof).
>
> Getting back to your question, I do not think that f.o.m. practitioners
> are "appealing to existentialism."  They simply recognize that the
> "trivial proof" of the consistency of PA that I sketched can be formalized
> in ZF (for example), and don't worry about it further.  ZF is stronger
> than PA, but so what?  Again, that should bother you only if you think
> there's something illegitimate about infinitary reasoning and retain a
> craving for a finitary proof of the consistency of PA.  Neither the
> existence of a finitary proof of the consistency of PA, nor the
> consistency of PA itself, is considered an interesting mathematical
> problem any more, because the issues are well understood.
>
> *Relative* consistency of various formal systems is still interesting from
> a f.o.m. perspective because they help us paint a picture of the landscape
> of formal systems and where they stand relative to each other.  But f.o.m.
> practitioners no longer regard consistency proofs as functioning as
> definitive refutations of skeptical doubts.
>
> Tim
>



-- 
Jeremy Shipley
Ballard and Seashore Doctoral Research Fellow
Department of Philosophy
The University of Iowa

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