[FOM] On Existence of Mathematical Objects
Dmytro Taranovsky
dmytro at mit.edu
Fri Oct 3 15:57:30 EDT 2003
In my last posting, I demonstrated that it is generally agreed that we
can refer to mathematical objects, and I noted that this implies that
mathematical objects metaphysically exist. Vladimir Sazonov objected,
claiming that mathematical theories are like works of fiction, where we
can talk about the characters without claiming that they exist.
However, "object exists" means "object is" which means "object is
something", so "object does not exist" means "object is not" which means
"object is not something" which means "object is nothing". If you refer
to an object, you do not refer to nothing, so you claim or assume that
the object exists. On the other hand, the fact that the set of all real
numbers exists does not mean that it has corporeal existence--it
certainly does not; and although, technically speaking, the fact that
we can meaningfully refer to a character in a work of fiction implies
that the character exists, a fictitious character cannot cause anything,
so, as a practical matter, we can assume its nonexistence.
An objection to Platonism is the claim that there are multiple universes
of sets and that, for example, the Continuum Hypothesis is meaningless
because it is true in some universes and false in others, and in talking
about the Continuum Hypothesis, we fail to specify which universe we are
talking about. The objection appears reasonable--after all, sentences
about fictional characters are often meaningless because the work of
fiction examined only partially specifies the fictional universe that
the work is about and various universes consistent with the work of
fiction may contradict each other--except for the fact that in talking
about sets, we refer to the unique maximal universe that consists of all
sets that exist. Unlike a work of fiction, where parts of the plot may
be unspecified, the invocation of maximality and totality causes the
universe of sets to be unique. For example, suppose that the notion of
the set of all real numbers is vague. In that case, there are two
different sets, R1 and R2, each of which is the set of all real
numbers. By the axiom of extensionality, there is a real number r that
belongs to only one of the two sets. However, the set that does not
contain r cannot be the set of all real numbers, contradicting the
assumption.
The issue of multiple universes of sets does explain the fact that so
many important questions are undecidable in ZFC. Figuratively speaking,
the deductive apparatus of ZFC does not know that ZFC is meant to be the
theory of all sets and considers a statement to be a theorem only if it
is true in every universe that satisfies the basic closure conditions
enumerated in ZFC. The solution is to find new set existence axioms and
closure properties: Since the universe consists of all sets, it should
satisfy strong closure properties, and since every set that exists in
some universe actually exists, it should also satisfy strong existence
axioms. The universe, being unique rather than random, should also be
canonical. The difficulty is that we can know which statements are
subtle approximations to the claim that every set exists or that the
structure of the universe is canonical only after we carefully research
their consequences and their subtle relationship with other
propositions. Only recently did it become clear that projective
determinacy is the correct existence axiom for second order arithmetic.
The vast realms of the set theoretical universe are yet to be explored.
Dmytro Taranovsky
http://web.mit.edu/dmytro/www/main.htm
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