On Tue, Jul 7, 2009 at 7:45 PM, Dan Asimov <dasimov@earthlink.net> wrote:
Forgive me if I'm repeating myself, but I feel inclined to reply to the one part of what Jim quoted that I understand -- a quote from an earlier post by Andy:
<< Back on March 25, Andy Latto wrote:
My view is that philosphers often worry too much about ontology (which things "exist" and which don't), when it really isn't a very interesting or important question, and it's just the nature of our language that misleads us into thinking it is.
Personally, I find existence (of certain things discussed in philosophy, anyway) to be of the utmost interest to me, much as is the case in mathematics.
In mathematics, there are questions of existence that are interesting, and others that are uninteresting. Questions of what exists within a particular axiomatized mathematical system are often interesting: Does there exist an algorithm A that decides the traveling salesman problem in polynomial time? Does there exist an even number that is not the sum of two primes? Do there exist a, b, c, N, integers > 2, such that a^n + b^n = c^n are examples of the sort of existence question that mathematicians find interesting. When mathematicians first started using complex numbers, and found them useful for many purposes, but there was question as to whether they really existed. At some point, people realized that any statement about complex numbers is equivalent to a related statement about ordered pairs of reals. Assuming you're already convinced that the theory of the real numbers is without contradiction, this answers the interesting existence question, "Does there exist a proof of a contradiction using complex numbers?". But the question "Do complex numbers really exist, or is it only the ordered pairs of real numbers that exist" is an example of another sort of existence question, one that may be of interest to some philosophers of mathematics, but not usually to working mathematicians. The entities that exist in models of the ZF axiomatization of set theory are the sets. The entities that exist in models of the GB axiomatization include both the sets and the classes, where the classes are collections of sets that are "too large" to be sets, such as the class of all sets, and the class of all sets that are not members of themselves. Do classes "really exist"? Once it's been proved that the same statements about sets are stateable and provable in ZF and GB, and that the two theories of set theory are equiconsistent, I don't think that most mathematicians think the question "Yes, but do proper classes (that is, classes that are not sets) *really* exist to be a interesting one; many would not even consider it a meaningful one. My opinion is that philosophers worry to much about existence questions that are like the second sort of mathematical existence problem, and the appropriate answer is "who cares? there are equally good descriptions of the universe, with the same predictive power, that include and that fail to include those entities." Andy Latto andy.latto@pobox.com