[FOM] Fwd: invitation to comment
silver_1 at mindspring.com
Sun Oct 9 23:39:11 EDT 2011
I don't recall exactly what Arnon Avron wrote, but I remember agreeing with him completely. I'm not claiming that what I'm about to write is exactly what he thinks. To me, the question of whether numbers exist is really a question about whether abstract entities exist (unless you can arrive at a principled way to separate some abstract entities from others). If one were to deny their existence, one would have to look for existence problems in the formation of a symbolic language, the description of an axiom, a proof, the description of theory P (PA), etc. first. I'm not claiming that this cannot be done and that abstract entities are forced upon us, I'm saying only that *everything* in PA earlier than the postulation of numbers must be examined. If it's determined that abstractions are already accepted as (in some sense) existing, then the problem of the existence of numbers in PA is not the numbers themselves, but all the abstract elements before numbers are even reached.
As a hint toward the direction I'm heading: even persons are abstract entities in the sense of their being assumptions about "sense data" (as Russell called them). We do not encounter *people* in immediate experience, we encounter their noises, looks, smells, etc. From this evidence, we postulate people (Russell again). So, if one were trying to be strict about this, almost nothing would count as existing. If numbers *do not* exist, while all other abstract entities in PA *do* exist, I would like to see how.
Otherwise I think we're stuck with accepting numbers or we must repudiate not only numbers, but every other item that's abstract in any theory of them. Quine developed a quite clumsy set theory (_Set Theory and Its Logic_), trying his best to be nominalistic as long as he could. Of course, he ultimately failed and was required to admit abstract entities at a certain point.
So, either numbers exist or no abstract entities exist (and there are lots of other abstract entities in mathematical theories), unless, as I indicated, some *principled* way were developed which would to separate numbers from other abstract entities--which I think is impossible.
I believe Avron scoffed at the "story view" of numbers. That is, in the Sherlock Holmes *story* Watson was his companion, but neither Sherlock nor Watson truly exist. The "story view," as expressed in FOM was itself not correct and would have to be cleaned up, even if someone were to accept it. As Carnap explained (e.g., in "Empiricism, Semantics, and Ontology"), one can temporarily accept a given framework and thus declare that every object in that framework exists. However, we normally do not accept his strict view, thus numbers ride along with humans as existents. If not, what is the principled reason for accepting one yet not the other?
More information about the FOM