Does it make sense to divide mathematical theorems into those susceptible to experimental conjecture and those that are not?

Question

I read the following in a good book: ”let's start from zero", the authors are Vinicio Villani and Maurizio Berni (pisan mathematicians) but I don't know if the book is also marketed outside Italy.

I don't think I'm exaggerating when I say that the shocking discovery of the existence of incommensurable segments gave a decisive impulse to the development of all mathematics, understood as a theoretical, hypothetical-deductive discipline. In fact, the validity of other theorems of classical geometry (starting from those of Thales and Pythagoras) can be empirically verified on a drawing through experimental measurements, albeit approximate. Instead, the question of the commensurability or incommensurability of two segments eluded us then, as it eludes us today and will elude us forever to every attempt at experimental verification, given the inevitable imprecision of every physical measurement"

When I first read it, in my mind I enthusiastically endorsed this statement. But then I thought about it and I had some doubts. The statement seemed to imply that there are two classes of theorems, the one whose truthfulness can be suggested by so-called experimental and numerical arguments (even if obviously without a demonstration we are always at zero), and those which are totally precluded by a such approach that can suggest conjectures. But does this make sense? Really "the theorem the root of 2 cannot be expressed as a ratio between integers" is in this sense qualitatively different from the Pythagorean theorem? It is true that I can set about constructing squares on right-angled triangles and make measurements that lead me to conjecture the validity of the theorem, but I can also patiently set about searching for two integers such that the square of their ratio is two, and after long labors conjecture that such a pair of integers does not exist. All this if I didn't have an easy proof available but it's not the point here now. In short, the question is: does the division of mathematical theorems into two classes, those that can be conjectured experimentally and those that cannot, make sense?

Answer 1

Regarding the alleged "shocking discovery of the existence of incommensurable segments gave a decisive impulse, etc.": this is a bit of a historical misconception; see e.g., this answer (at history of science and math).

As far as the $\sqrt2$ example: this is similarly dubious, since there are constructive proofs of this by using lower bounds for $\sqrt2-\frac{p}{q}$ in terms of $\frac{C}{q^2}$ for a suitable universal $C$ (bigger than $\frac{1}{10}$, say). This is something testable: take your favorite rational approximation and check that it falls short of being an exact representation by a specific amount.

On the other hand, set-theoretic paradoxes such as Banach-Tarski surely cannot be predicted empirically :-) So I would conjecture that the borderline may have to do with whether a theorem can be proved in ZF or if it requires the axiom of choice.

In conclusion, when the author writes

"I don't think I'm exaggerating when I say that the shocking discovery of the existence of incommensurable segments gave a decisive impulse to the development of all mathematics, understood as a theoretical, hypothetical-deductive discipline"

he is indeed exaggerating.

Answer 2

I don't think that this is a mathematically definable division, but I would argue that almost all mathematical theorems can be conjectured experimentally.

Consider the four color theorem, which can be tested experimentally and was for over 100 years before it was proven using computers. Even now there isn't a visual "proof" of the conjecture like you might have with the pythagorean theorem, and there is no polynomial time algorithm to check if a graph is 4-colorable.

Consider Jensen's inequality, an intuitive (and visualizable) statement about convex functions (having non-negative second derivative) in probability. You can "test" this with some convex functions and certain probability distributions, but the proof is rigorous and general.

Many theorems are conjectured because someone tested some examples and saw a pattern. Surely geometry is particularly conducive to visual intuition, but I think it would be unrealistic to say that any theorem couldn't be conjectured by experimentation.

Edit. Some comments have made the point that one can answer the question "does x exist" by experimentation, and for some problems (like the existence of an odd perfect number) the proof would be extremely simple once you find the x you're looking for. Note that the negation of these statements can still be conjectured by experimentation, just not proven.