[FOM] inconsistency of P
pastudtmann at davidson.edu
Thu Oct 6 09:39:22 EDT 2011
> Could this skepticism easily creep in to throw much of empirical science into doubt?
The short answer to this question is: no. The long answer requires coming to terms with the role of mathematics in physical theory, which of course is one of the major issues in philosophy of science and so can’t be treated here in an appropriate way. But in lieu of a long answer, I will say that an empiricist of the sort I described would insist upon a distinction between using some axiom for a theoretical purpose and knowing that the axiom correctly describes mathematical reality. Does the fact that some mathematicians use the axiom of choice show that they (or we) know it to be true? Does the fact that some mathematicians use large cardinal axioms show that they know them to be true? Not obviously. At the very least, some argument would be required at this point to make a case for an affirmative answer to these questions. The same could be said about the use of axioms in physical theories. Perhaps the physical theories show that parts of the physical world are properly characterized by some axiom or other. But that does not suffice to show that anyone knows the axioms to be true of mathematical reality.
➢ Our human limitations are pretty severe. We are very small creatures with puny computational abilities and poor senses.
> In my opinion, a beautiful thing about math and science alike is how we permit ourselves to transcend our limitations through abstract reasoning and big, creative ideas.
Nothing in the empiricist position I described would suggest that anyone should (or shouldn’t) have just such an attitude.
From: fom-bounces at cs.nyu.edu [fom-bounces at cs.nyu.edu] On Behalf Of Monroe Eskew [meskew at math.uci.edu]
Sent: Wednesday, October 05, 2011 1:34 AM
To: Foundations of Mathematics
Subject: Re: [FOM] inconsistency of P
On Tue, Oct 4, 2011 at 2:13 PM, Studtmann, Paul
<pastudtmann at davidson.edu> wrote:
> And why would an empiricist draw such a line? For the simple reason that such facts are accessible through ordinary experience, while experience of the infinite is beyond the ken of finite creatures like us who are embedded in a small portion of space and time.
Could this skepticism easily creep in to throw much of empirical
science into doubt? For how could be possibly be justified in saying
that we know the chemical composition of distant stars, the mass of
Jupiter, the structure of proteins, or the radius of an electron?
These are only "known" through elaborate systems of hypotheses and
generalizations which interpret things we can see and hear. They are
far beyond our ability to experience in a direct way. Terence Tao's
lecture on the "cosmic distance ladder" seems relevant-- our knowledge
of distances (even just distances!) on a cosmic scale is obtained in a
very indirect way, and different methods apply to different orders of
Our human limitations are pretty severe. We are very small creatures
with puny computational abilities and poor senses. In my opinion, a
beautiful thing about math and science alike is how we permit
ourselves to transcend our limitations through abstract reasoning and
big, creative ideas.
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