[FOM] AXIOM SCHEMATA

[Donald Stahl]

There is learnability and learnability. In one sense, plenty of people have 
learned English. In another, it is plausible that no one has ever learned 
English, in that it is plausible that no one has ever learned all the words 
in the OED. When Davidson introduced the term 'learnability' into 
philosophical discussion he had it tied very closely to the finite.



Best wishes,

Donald E. Stahl 12545 Olive Boulevard
St. Louis, MO 63141-6311 USA






>From: Neil Tennant <neilt at mercutio.cohums.ohio-state.edu>
>To: Donald Stahl <philostahl at hotmail.com>
>CC: mfrank at math.uchicago.edu, corcoran at buffalo.edu, fom at cs.nyu.edu
>Subject: Re: [FOM] AXIOM SCHEMATA
>Date: Thu, 15 Jul 2004 20:43:53 -0400 (EDT)
>
>
>On Thu, 15 Jul 2004, Donald Stahl wrote:
>
> > Assuming that Tennant's questions both refer to the preference (not the
> > tendency)
>
>Yes, they do ...
>
> > I suggest (not on my own behalf) that it might have to do with the
> > idea that compositionality is supposed to be the source of novelty and
> > unlimitedness---that these are to be worked for, not assumed.  Perhaps 
>it
> > has to do with conflating theories with languages, so that the set of 
>axioms
> > corresponds to the lexicon. A non-finitely axiomatized theory would then 
>be
> > on a par with an unlearnable language.
>
>I don't see that the two would be on a par at all. On a par with respect
>to what?---learnability?
>
>But that can't be right. First-order Peano arithmetic is learnable, but
>not finitely axiomatizable. All that is important is that we should have
>an effective method for telling, of any sentence in the first-order
>language of arithmetic, whether it is an axiom. The only advantage of a
>finitely axiomatized theory is that the effective method in question will
>be more efficient than the effective method for an infinitely axiomatized
>theory. There does not seem to be any *epistemological* advantage for
>finitely axiomatized theories over the various naturally axiomatizable
>(but not finitely axiomatizable) theories in mathematics. This is because
>the latter have at most finitely many axiom schemata; and for each such
>schema, the epistemological problem of warranting its instances is uniform
>across them. (The second-order theorist would argue that this is because
>we can directly grasp the truth of the second-order quantified sentence
>for which the axiom-schema at first order is doing weaker duty.)
>
>Neil Tennant
>
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