Reduction and Elimination in Philosophy and the Sciences
Reduction and Elimination in Philosophy and the Sciences
Reduction and Elimination in Philosophy and the Sciences
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The Key Problems of KC<br />
Matteo Plebani, Venice, Italy<br />
1. The key problem of KC<br />
Accord<strong>in</strong>g to Floyd <strong>and</strong> Putnam, we can extrapolate from<br />
Wittgenste<strong>in</strong>’s ‘notorius’ remarks on Gödel’s <strong>the</strong>orem some<br />
philosophically <strong>in</strong>sightful remarks. Let “P” be <strong>the</strong> Godelian<br />
sentence for <strong>the</strong> logical system of Pr<strong>in</strong>cipia Ma<strong>the</strong>matica<br />
(PM). Wittgenste<strong>in</strong>’s key claim (KC) runs as follows:<br />
KC: If one assumes that ¬P is provable <strong>in</strong> PM, <strong>the</strong>n<br />
one should give up <strong>the</strong> “translation” of P by <strong>the</strong> English<br />
sentence “P is not provable”<br />
The key problem of KC is <strong>the</strong> follow<strong>in</strong>g simple remark (call<br />
it KP): KC is compatible with realism. In this context, “realism”<br />
is <strong>the</strong> claim that ma<strong>the</strong>matics is <strong>the</strong> study of a well–<br />
def<strong>in</strong>ed doma<strong>in</strong> of abstract objects that exist <strong>in</strong>dependently<br />
of our thought, language or experience; it implies <strong>the</strong> view<br />
that arithmetic is <strong>the</strong> study of <strong>the</strong> st<strong>and</strong>ard model N of<br />
natural numbers.<br />
However, this problem is considered <strong>in</strong> strict relation<br />
to <strong>the</strong> goal of provid<strong>in</strong>g arguments to clarify <strong>and</strong> support<br />
Wittgenste<strong>in</strong>’s stance concern<strong>in</strong>g Gödel’s <strong>the</strong>orem.<br />
O<strong>the</strong>rwise KC would be an <strong>in</strong>terest<strong>in</strong>g remark. It is true<br />
that it provides genu<strong>in</strong>e <strong>in</strong>sight <strong>in</strong>to <strong>the</strong> philosophical<br />
mean<strong>in</strong>g of Gödel’s <strong>the</strong>orem, but it certa<strong>in</strong>ly throws little<br />
light on Wittgenste<strong>in</strong>’s thought. In order to underst<strong>and</strong> why<br />
this is <strong>the</strong> case, it is important to follow Timothy Bays’s<br />
(Bays 2006, p.6) recollection of <strong>the</strong> three uncontroversial<br />
ma<strong>the</strong>matical results upon which KC is based:<br />
1. If PM � ¬P, <strong>the</strong>n PM is ω-<strong>in</strong>consistent.<br />
2. If PM is consistent but ω-<strong>in</strong>consistent, <strong>the</strong>n all of<br />
<strong>the</strong> models of PM conta<strong>in</strong> non-st<strong>and</strong>ard natural<br />
numbers—i.e., elements which <strong>the</strong> model treats as<br />
natural numbers but which do not correspond to any<br />
of <strong>the</strong> ord<strong>in</strong>ary natural numbers.<br />
3. The translation of P as “P is not provable” depends<br />
on <strong>in</strong>terpret<strong>in</strong>g P at <strong>the</strong> “natural numbers<br />
alone.” If we <strong>in</strong>terpret P at a non-st<strong>and</strong>ard model—<br />
i.e., at one of <strong>the</strong> models described <strong>in</strong> 2—<strong>the</strong>n <strong>the</strong>re<br />
is no reason to th<strong>in</strong>k that this will lead to a translation<br />
of P as “P is not provable.”<br />
Bays goes on to criticize <strong>the</strong> passage between 1-3 <strong>and</strong><br />
KC. He ma<strong>in</strong>ta<strong>in</strong>s that we shouldn’t give up our translation<br />
of P as “P is not provable”, <strong>in</strong> <strong>the</strong> case PM turns out to be<br />
ω-<strong>in</strong>consistent, because “<strong>the</strong>re’s no reason to constra<strong>in</strong><br />
our translation of P to <strong>the</strong> class of models which happen to<br />
satisfy PM” (Bays 2006, p.6). I agree with him on this latter<br />
po<strong>in</strong>t, but still th<strong>in</strong>k that <strong>the</strong> merit of KC is to underl<strong>in</strong>e that<br />
<strong>the</strong> equivalence between P <strong>and</strong> “P is not provable” holds<br />
only <strong>in</strong> <strong>the</strong> st<strong>and</strong>ard model. And we can th<strong>in</strong>k of cases <strong>in</strong><br />
which this does matter 1 . But <strong>the</strong> issue is that <strong>the</strong> existence<br />
1 We won’t discuss this po<strong>in</strong>t here, but we can give a sketch of our argument.<br />
Follow<strong>in</strong>g an h<strong>in</strong>t from Mart<strong>in</strong>o 2006, we th<strong>in</strong>k this could play an important role<br />
<strong>in</strong> <strong>the</strong> formulation of Godel’s second <strong>in</strong>completeness <strong>the</strong>orem. The problem,<br />
roughly stated, is this: an idealised ma<strong>the</strong>matician without any spatial or<br />
temporal limitation, could acknowledge <strong>the</strong> consistency of a system s as<br />
logical consequence of its axioms (that means: <strong>in</strong> every model – if <strong>the</strong>re are –<br />
of <strong>the</strong> axioms of S, it is true that S is consistent). But not <strong>in</strong> every model for S<br />
<strong>the</strong> arithmetic sentence that should express <strong>the</strong> consistency of S (call it Con)<br />
is true. This expla<strong>in</strong>s why <strong>the</strong> ma<strong>the</strong>matician doesn’t draw it as a conclusion<br />
from <strong>the</strong> axioms of S. This could give an idea of a contest <strong>in</strong> which <strong>the</strong> transla-<br />
of non-st<strong>and</strong>ard model could perhaps pose a problem on<br />
choos<strong>in</strong>g of how to translate P, but it is perfectly compatible<br />
with <strong>the</strong> fact that P is true <strong>in</strong> N iff P is not provable <strong>in</strong><br />
PM <strong>and</strong> that if PM is consistent, than P is true <strong>in</strong> N (<strong>and</strong><br />
not provable <strong>in</strong> PM).<br />
In o<strong>the</strong>r words, not one of <strong>the</strong>se results provides any<br />
ground for scepticism concern<strong>in</strong>g <strong>the</strong> existence of <strong>the</strong><br />
st<strong>and</strong>ard model N, because all <strong>the</strong> results are obta<strong>in</strong>ed<br />
us<strong>in</strong>g model-<strong>the</strong>oretic mach<strong>in</strong>ery, <strong>and</strong>, as has been argued<br />
by many, model <strong>the</strong>ory is <strong>the</strong> realist framework par<br />
exellence.<br />
There is only a way <strong>in</strong> which Floyd <strong>and</strong> Putnam’s<br />
suggestion might be saved: it is possible (although<br />
questionable, see Rodych 2003) that Wittgenste<strong>in</strong> claimed<br />
someth<strong>in</strong>g like KC; if so, he certa<strong>in</strong>ly had a great <strong>in</strong>sight<br />
<strong>in</strong>to Gödel’s results. But he might have made such<br />
remarks only <strong>in</strong> order to highlight an important fact that<br />
could be acknowledged also from a realistic viewpo<strong>in</strong>t.<br />
This is tantamount to claim<strong>in</strong>g that KC does not help us<br />
underst<strong>and</strong> what Wittgenste<strong>in</strong>’s stance about Gödel’s<br />
<strong>the</strong>orem was.<br />
2. The myth of prose<br />
Why do Foyd <strong>and</strong> Putnam th<strong>in</strong>k that KC is a philosophical<br />
claim of “great <strong>in</strong>terest” (p. 624)? Because <strong>the</strong>y believe it<br />
helps to avoid a mis<strong>in</strong>terpretation of Gödel’s result:<br />
That <strong>the</strong> Gödel <strong>the</strong>orem shows that (1) <strong>the</strong>re is a<br />
well def<strong>in</strong>ed notion of “ma<strong>the</strong>matical truth” applicable<br />
to every formula of PM; <strong>and</strong> (2) that if PM is<br />
consistent, <strong>the</strong>n some “ma<strong>the</strong>matical truths” <strong>in</strong> that<br />
sense are undecidable <strong>in</strong> PM, is not a ma<strong>the</strong>matical<br />
result but a metaphysical claim. But that if P is provable<br />
<strong>in</strong> PM <strong>the</strong>n PM is <strong>in</strong>consistent <strong>and</strong> if ¬P is<br />
provable <strong>in</strong> PM <strong>the</strong>n PM is ω-<strong>in</strong>consistent is precisely<br />
<strong>the</strong> ma<strong>the</strong>matical claim that Gödel proved.<br />
What Wittgenste<strong>in</strong> is criticiz<strong>in</strong>g is <strong>the</strong> philosophical<br />
naiveté <strong>in</strong>volved <strong>in</strong> confus<strong>in</strong>g <strong>the</strong> two, or th<strong>in</strong>k<strong>in</strong>g<br />
that <strong>the</strong> former follows from <strong>the</strong> latter. But not because<br />
Wittgenste<strong>in</strong> want to simply deny <strong>the</strong> metaphysical<br />
claim; ra<strong>the</strong>r he wants us to see how little<br />
sense we have succeeded <strong>in</strong> giv<strong>in</strong>g it.<br />
That’s an application to <strong>the</strong> case of Gödel’s <strong>the</strong>orem of a<br />
general way of read<strong>in</strong>g Wittgenste<strong>in</strong>’s remarks on <strong>the</strong><br />
Foundations of Ma<strong>the</strong>matics: I will call it “<strong>the</strong> myth of<br />
prose”. Accord<strong>in</strong>g to <strong>the</strong> myth of prose, <strong>the</strong> task of philosophical<br />
<strong>in</strong>vestigation of Ma<strong>the</strong>matics is to dist<strong>in</strong>guish between<br />
<strong>the</strong> real ma<strong>the</strong>matical content of a <strong>the</strong>orem <strong>and</strong><br />
some philosophical <strong>the</strong>sis often associated to it from<br />
ma<strong>the</strong>maticians when <strong>the</strong>y expose it <strong>in</strong>formally. This apparently<br />
sensible approach leads to an implausible result.<br />
As has been argued by many, <strong>the</strong>re is a perfectly legitimate<br />
ma<strong>the</strong>matical sense <strong>in</strong> stat<strong>in</strong>g that Gödel’s <strong>the</strong>orem<br />
shows that if PM is consistent, than <strong>the</strong>re are sentence<br />
that are both true <strong>and</strong> undecidable <strong>in</strong> PM. Certa<strong>in</strong>ly, <strong>in</strong><br />
Gödel’s orig<strong>in</strong>al paper (Gödel 1931) <strong>the</strong> <strong>the</strong>orem is formulated<br />
<strong>in</strong> syntactical terms, us<strong>in</strong>g <strong>the</strong> notions of consistency<br />
tion of <strong>the</strong> predicate “Proof”, <strong>in</strong>volved <strong>in</strong> <strong>the</strong> construction of P <strong>and</strong> Con, as<br />
“provable” should be given up.<br />
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