TITLE: Is it legitimate to specify a statement as a Theorem if it is proved using numerical methods (since it can't be proved analytically)?
QUESTION [13 upvotes]: I have some statement, which is "proved" by numerical methods (particularly, using simulations), since it isn't possible to prove analytically. Is it legitimate to articulate this statement as a Theorem, if it isn't proved analytically. If no, what should I call it? 
P.S. More details regarding the statement. 
I have some data generating process and need to prove that the generated data as a function of some variable has some properties. For example, I have $x^1,...,x^n$ generated data (n=100000) and need to show that the function $x(\gamma)$ is positive defined  over the domain. Therefore, on the generated data I apply interpolation technique in order to construct the function and I check the property.

REPLY [2 votes]: As other answers have said, a statement is either proven or unproven.  If it is proven, we call it a theorem, regardless of the methods used.  I'm answering separately to call attention to some of the pitfalls of trying to "prove" things with numerical methods.
The vast majority of real-world numerical computing (outside of the financial sector) uses IEEE 754 floating point to represent numbers which may not be integers.  In short, the floating point numbers are a finite (but large) subset of the dyadic rationals, which become sparser (at a roughly logarithmic rate) as we move away from zero.  There are also a few "special" values which are used to represent the results of invalid operations or arithmetic overflow.
Why does this matter?  It's important to understand the limitations of floating point arithmetic.  In particular:

It does not support irrational numbers whatsoever.  The Dirichlet function and its cousins cannot be usefully analyzed at all.
Numbers which are farther from zero have less precision, so functions with essential singularities will be difficult to work with in the neighborhoods of their singularities.
Limits are potentially an issue.  Normally, a limit only exists if it exists under any possible choice of epsilon, but the floating point numbers can only choose finitely many epsilons, all of which are dyadic rationals.  So if you take a limit of a function which behaves "specially" over the dyadic rationals (e.g. certain combinations of trigonometric functions), you may derive an incorrect value or incorrectly conclude that the limit exists at all.
The floating point numbers are non-associative under every operation that matters, because of intermediate rounding.  No useful algebraic manipulations will preserve equality.  So equality is basically useless and you have to approximate it by looking at whether numbers are "close enough" instead.  For some applications, this is a problem (e.g. proving that a value is never exactly zero is impossible if it comes arbitrarily close to zero).