Exploring the Meaning of Mathematical Existence

[zpconn]

This is a bit philosophical.

What does it mean to say that a mathematical object exists?

To add some concrete thoughts, I recently read the following:

"The empty set has the property that for all objects x, the statement 'x is in the empty set' is false."

But this statement reeks of all sorts of logical complexities to me. For one, x is left completely undefined and ambiguous. Typically, the phrase "for all..." in mathematics is used to say that every member of a certain set has a certain property. But in that case, which set does x belong to? It's well-known that there is no "universal set" of all mathematical objects in standard ZF set theory. So if x is not specified as belonging to a specific set, and yet it cannot belong to some "universal set" becase such a thing is contradictory under the standard ZF axioms, in what way does x meaningfully exist? And furthermore, how is the quoted statement above meaningful at all when the primary object of interest, x, cannot be established to even meaningfully exist?

Now, as far as I can tell, the quoted statement could be reworded to produce something that's not so logically pernicious. For example, this seems an improvement: "Suppose A is any non-empty set. Then for each member x of A, the statement 'x is in the empty set' is false." But still, the quoted statement raises some concerns about the exact meaning of mathematical existence.

 

[exk]

The empty set can be thought of in terms of vacuous truths. More info here http://en.wikipedia.org/wiki/Vacuous_truth (look at the math section especially).

From the wiki:
An even simpler example concerns the theorem that says that for any set X, the empty set is a subset of X. This is equivalent to asserting that every element of the empty set is an element of X, which is vacuously true since there are no elements of the empty set.

 

[Hurkyl]

This sounds more like a matter of grammar than of philosophy. Bounded quantifiers are simply not a requirement of logic. In fact, some formal languages don't even contain a membership operator -- in them, you can't even express the notion of a bounded quantifier! Even if you do have bounded quantifiers, there's no reason the range of the variable should be an object of the theory you're studying.

 

[zpconn]

I've never heard of bounded quantifiers. Thanks for the information.

You mention that some formal languages are possible that do not contain bounded quantifiers. But is mathematics, if we treat it as a "formal language," such a language? It's not often in mathematics that one sees a statement referring to an object x without providing at least some sort of description of x--in effect stating that x is a member of some set (such as the set having only members that satisfy the given criteria).

The problem, I suppose, ultimately falls down to the vagueness of the term "mathematical object." The term seems loaded with problems. The difficulty is this. I don't know of any completely rigorous and general definition of the term. Without a general definition, the term is left to be defined basically by all the notions that we have deemed "mathematical objects"--that is to say, defined by example. But in 100 years, there will certainly be more notions that are considered as mathematical objects than there are today--and hence the term "mathematical object" has no purely static, absolute meaning.

Is there some formal and rigorous way for dealing with such broad notions in mathematics?

I'm probably being far too pedantic here, but it's an interesting thought either way. I see no problem with innocent speculation.