FOM: December 1 - December 22, 1998

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FOM: "Consistent":Three Questions.

        [Thanks to Davis, Tait, Poston,  et al.;  here I think is a 
slightly better framing of my concerns.]

        Three questions,  one technical,  two historical-philosophical, 
about "consistency".

        Is there an example of,  or can one prove the existence of,  a 
theory framed in a higher order classical logic [or at least a logic so 
understood that the class of semantical logical consequences and the 
class of syntactic consequences do not coincide;  but hopefully the 
logic and the theory would be nontrivial wrt my question] that is
        (a) inconsistent,
        (b) a contradiction is not derivable.

        In German one has, _konsistent_ and _widerspruchslos_.  I'm 
curious about the history of these terms, in the context of logic and/or
mathematical demonstration.  In GA III, Hilbert uses the second.  (Well,
 I find _Widerspruchsfreiheit_ and _Widerspruchslosigkeit_;  is there a 
difference in connotation?)  I don't find 'konsistent'.
        Cantor speaks of "(in)konsistente Vielheit".  In, e.g., _Cantor 
an Dedekind 28 Juli 1899_,  Cantor has Omega' as "keine konsistente 
Vielheit" because it supports "ein Widerspruch".  Is Cantor's use of 
'konsistent' rather than 'widerspruchslos/frei' stylistic only,  a 
matter of diction?
        In English one says "consistent" rather than 
"contradiction-free" because of the awkwardness of the latter;  but one 
wonders if this doesn't sometimes conceal some important distinctions.

        This goes to the heart of my concern:  a _reductio ad absurdu_m 
argument in the primary sense shows a proposition to be false by showing
that it implies something "absurd".  The latter meaning not only "false"
but also "not true to" or even "dissonant with" (etc.).  It does seem 
that "negation" in intuitionistic reasoning is most resonantly 
understood in these terms (reducible to something intuitionistically 
absurd; "false" for "absurd" won't do,  of course).  Here one then has a
use for "consistent"--true to intuitionistic reasoning -- which only by 
slight of hand is explicable in terms of "contradiction-free".
        With classical logic,  one is perhaps even required to restrict 
"absurdities" to contradictions;  then of course one can use RAA 
arguments as proofs rather than as disproofs only (it is only by a kind 
of ruse that in intuitionistic reasoning what is essentially a disproof 
of p is made to appear to be a proof of "-p".).
        I am still wondering if for example supplying the upper 
half-plane model for Bolyai-Lobach. geometry does not _fundamentally_ 
(from the point of view of the mathematical achievement or effect) show 
B-L geometry to be "consistent" more in the sense of "consistent with 
mathematics",  that is,  it does bring B-L geometry fully into the fold 
of mathematics.  Hanging out here (if I may speak colloquially) is the 
thought that in a fundamental sense,  consistency proofs ought to lend 
mathematical signficance to a subject whose mathematical significance is
in doubt.  (Whereas one can worry over the significance of just 
supplying indirectly -- as in Go"del's completeness proof -- a model in 
the natural numbers that has no striking mathematical significance.)    
       Of course,  what I driving me is the great discomfort I've always
felt with mathematics that is too logic driven,  perhaps because I place
a great value on understanding in some full or perhaps melodramatic 
sense.  (The line about depriving a mathematician of the law of the 
excluded middle is like depriving a boxer of their fists has struck me 
as issuing from the sort of person who goes to boxing matches in the 
hopes of seeing a smashing knock-out punch,  or listens to symphonies 
impatient for "the good parts".)       

Robert Tragesser       


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