[FOM] Mathematical Knowledge

praatika@mappi.helsinki.fi praatika at mappi.helsinki.fi
Sat Mar 24 04:18:22 EDT 2007

Paul Studtmann wrote:

> The question is this: what is the strongest mathematical system that
> mathematicians regularly employ that we know to be consistent.

It all depends on how exactly one understands "know" here. If one requires 
a proof for the knowledge of consistency, I don't think we can get much 
beyond PA. However, all ordinary mathematics can be developed in ACA_0, 
which is a conservative extension of PA.  So one possible reply would be: 
ACA_0 is the strongest mathematical system that mathematicians regularly 
employ (that is, outside set theory), and we know (I think) it is 

However, we have lots of scientific knowledge which is more empirical in 
character, and we do not have proof for these results. So I think it is 
artificial to restrict "knowledge" to mathematically provable. We can 
think that a stronger theory, e.g. ZFC, formalizes an intuitive notion 
(here: the iterative concept of set); moreover, we have not been able to 
reproduce any known paradoxes or to derive a contradiction in it; And we 
may take thisto be sufficient for "the knowledge of consistency". 
Analogously, in empirical science, we don't verify theories but only test 
them and get inductive support for them (so far a theory has passed all 
tests); usually, one talks about knowledge here.     
In fact, Martin Davis wrote here some time ago (Fri May 30 2003) :

"If the underlying system is PA (first order number theory), I believe 
that one can claim that the evidence for its consistency is simply 
overwhelming. Even the trivial consistency proof based on the standard 
model uses far less than much well-accepted ordinary mathematics. And the 
various proof-theoretic epsilon-zero consistency proofs by Gentzen, 
Ackermann, and Gödel are entirely compelling.
    For higher order systems like type theory or ZFC, I know no reason for 
believing in their consistency other than the fact that the axioms are 
satisfied by our intuitive Cantorian picture of sets of sets of sets of 
.... To someone who has no doubt that the properties of this construct are 
objective (even if only partially determinable by us) the matter is 
unproblematic. Others have to live with the uncertainty that is with us in 
most aspects of the human condition."

So I whole-heartedly agree with him here.

> (I am also interested in the opinions of those who think that the
> question is somehow misguided, perhaps because the terms employed are
> insufficiently precise, or some other such reason.)  

Yes, that's what I think. 

> So, if someone thinks that we should not talk about knowing whether some
> theory is consistent but rather in terms of degrees of confidence that
> some theory is consistent, I am happy to hear what that person has to 
> say.

Yes, I think it is a matter of degree.

Best, Panu

Panu Raatikainen

Academy Research Fellow, The Academy of Finland
Docent in Theoretical Philosophy, University of Helsinki

Department of Philosophy
P.O.Box 9
FIN-00014 University of Helsinki

e-mail: panu.raatikainen at helsinki.fi

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