[FOM] "Mathematician in the street" on AC

steve newberry stevnewb at att.net
Mon Aug 17 12:45:45 EDT 2009


I send you this directly because Martin has, for some time now, bounced everything I have attempted to say on this topic.


With respect to the word 'true' (or 'valid') it may be remarked that there are at least two major categories of truth:  Deductive truth, which can be Deductively proven (and is hence "Deductively-valid") from within Russell's Ramified Theory of Types (without the Axiom of Reducibility),  (or from within Gentzen's Calculus of Sequents), and such truths are inherently tautological.  Deductively-valid premises can only have Deductively-valid (tautological) consequences.  [Here, following the usage of Rasiowa and Sikorski, I extend the usage of 'tautological' to include predicate logic.]

The second major category of truth differs from the first in that it is inherently CONTINGENT upon one or more empirical facts that effectively entail the existence of at least one infinite counter-model which has no finite submodels.  Such a truth is valid-on-every-finite domain, (and is hence "Inductively-valid') and may only be Inductively proven by formalisms which (unlike RRTT and GCS) admit impredicative constructions (perhaps in the guise of the Axiom of Reducibility), and/or The Axiom of Choice in any of its myriad forms. It is trivial, but nonetheless worthy of note that contingent consequnces can only be materially implied by contingent premisses.

Using a different language and symbolism, L"owenheim (On Possibilities ..." remarked of the existence of Deductive (non-contingent) propositions of Logic, which propositions he described as "...Identical...", and non-Deductive propositions, which he divided into two partitions, as having (in his notation) either "...halting subscripts..." or "...fleeing subscripts..."  This was a remarkable insight, which (arguably?) presaged G"odel's Incompleteness Theorems.

(The above paper may be found in van Heijenoort's compendium survey "From Frege to G"odel".)

It may also be noted that Deductive-validity falls under the rubric of Classical Logic, whereas Intuitionistic Logic falls under the rubric of Inductive-validity.

(I use 'valid', 'validity' to connote the absence of counter-models w.r.t. a domain.)

Many of the questions which are raised on the FOM list can be profitably examined from the perspective of these two categories of truth.


Steve Newberry

 On Sat, 8/15/09, Timothy Y. Chow <tchow at alum.mit.edu> wrote:

> From: Timothy Y. Chow <tchow at alum.mit.edu>
> Subject: Re: [FOM] "Mathematician in the street" on AC
> To: "Foundations of Mathematics" <fom at cs.nyu.edu>
> Date: Saturday, August 15, 2009, 9:18 AM
> On Sat, 15 Aug 2009, A J Franco de
> Oliveira wrote:
> > 1) is there a common understanding amongst FOMs of
> what "true" means
> > when applied to AC, say?
> > 2) If so, what exactly is it?
> This seems to be a special case of the question schema, "Do
> all 
> philosophers agree about X"?  Generically, the answer
> is no to all 
> instances of this schema.
> > 3-4) Same question for MISs, respectively.
> When doing math, as opposed to when they have their guard
> up in a 
> conversation that is perceived to be philosophical, I claim
> that 
> mathematicians' use of the word "true" is roughly
> Tarskian.  That is, they 
> will assert that "X" is true under the same conditions that
> they will 
> assert X.
> Thus we're reduced to asking when mathematicians will
> assert X.  The most 
> common case is that they'll assert X when they know how to
> prove X, or are 
> confident that someone competent has proved X.  But
> they'll also sometimes 
> assert X if X is conjectural but strongly
> supported---though only if it's 
> clear from context that they're not claiming to know a
> proof of X.
> Tim
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