That was another question in the back of my mind -- famously, the four color theorem was proved by computers through exhaustive analysis (checking every possibility). At the time, it was controversial as a "proof" since it didn't really take the usual form of a proof.

I've often wondered "Why can't we do something like that, but for all instances of things like the Collatz conjecture?" Of course, it's computationally infeasible. But that raises the question: Suppose the four-color theorem was only able to prove 95% of cases rather than 100%. Isn't it at least sort of valuable to do so? Or is that last 5% all the difference?

I suppose what bugs me about math is, physicists are proved wrong all the time. Mathematicians are rarely proved mistaken, because they construct assumptions that you can't disagree with. There's no chance for empirical data to falsify one's assertions.

But that's an odd kind of debate, and not too productive. I just can't help but wonder about stuff like this.

The four-color theorem had the preliminary challenge of creating a method to identify all the relevant cases. That also required a mathematical theory. (I don't actually understand how it was done!)

If you didn't have that, you would have an infinite search for possible maps that violate the conjecture, because you wouldn't be able to divide them into a finite number of equivalence classes, or enumerate a finite number of possible counterexamples, or whatever.

For Collatz, I don't think we have any lemma that gives a path to saying "a counterexample must be one of these 2¹⁰⁰ integers" or the like. So without such a thing, checking every possibility would mean checking every integer. (Well, there are definitely lemmas that make it unnecessary to check large numbers of integers, so I should say instead that checking every possibility would still mean checking an infinite number of cases.)

As an interesting example where we did have a finite bound that was still out of reach to computationally exhaust, see the Weak Goldbach Conjecture:

https://en.m.wikipedia.org/wiki/Goldbach%27s_weak_conjecture

It was known to have at most finitely many counter-examples by the 1930’s, and by the 1950’s it was known that the largest counter-example had to be < 3^3^15. In 2002 that was lowered to about 10^1346. Still well out of reach of any computer!

Only in 2012 was an unconditional proof given.

It can be valuable, but it's not as valuable as covering all cases.

If I could prove collatz for 99% of numbers it would be great.

If i could additionally prove which 99% converge, let's say all numbers not divisible by 100, it would be enormous because even showing this much is likely to be a stepping stone to a full proof (what property is it about the number 100 that excludes all non-divisors from diverging?)

> I suppose what bugs me about math is, physicists are proved wrong all the time. Mathematicians are rarely proved mistaken, because they construct assumptions that you can't disagree with. There's no chance for empirical data to falsify one's assertions.

Right, and that's the whole point and beauty of it.

Math is simply not a science, not in the sense of "natural science". Math research does not follow the scientific method, it is not an iterative effort to construct (wrong, but useful) models of some external universe like science does. Math is a thing of its own, and it's best just to not compare it to physics or biology or gender studies or whatever.

"famously, the four color theorem was proved by computers through exhaustive analysis (checking every possibility)"

This is impossible. The plane can be arbitrarily large with an arbitrary number of regions. It can be exhaustively checked for n number of regions up to a certain n. But not for arbitrary n.

Not every possible graph, just every "interesting" class of subgraph that can be extended or combined with others to obtain a coloring of any planar graph. These are a large but finite number; if a graph is planar it cannot have too many edges.

You're adhering to certain misconceptions for no good reason. An exhaustive check of all the cases in the model that supports the four-color conjecture is the "empirical" (here, meaning experimental) data required to falsify conjectures. And even physicists don't deny the value of falsification in narrowing down models to truth. But I think they are much more loose in what they are willing to stipulate as true if it means advancing other conclusions.

Furthermore mathematicians do accept certain kinds of probabilistic arguments, but they need to take a form of proving that an object will be guaranteed to have a certain property as a parameter on the object approaches a statistical limit. So partial evidence can sometimes point to the possible truth of a mathematical conjecture, but you are still burdened with the requirement of deductive demonstration if you desire logical certainty.

The tautological notion you take of math is a half-truth. On the one hand, models in math seek to be sound over the objects they specify. This is opposed to physics where deliberate simplifications must be made to make your idealizations of a system manageable or relevant. At the same time, no mathematical system is complete. So there will be truths that require other axiomatizations to access via proof. This is one source of mathematical creativity requiring judgment beyond following tautologies.