You can entirely duplicate (and surpass) Super-Approximation using Super-Counter-Examples:
"The human population of no English county called starting with "K" is greater than 1,000,000"
"The population of Kent is 1,515,328"
Even if your counterexample is the most generic ever ("Kent") you can iterate till you get an exact number.
The ability to find the counterexample would be great. My former manager believed that change is always good and the possible payoff was always worth the havoc in constantly changing our procedures and methods. He would not allow productive employees to keep using the tools that made us productive.
So much of modern approaches to semantics and automated reasoning are couched as properties of continuous high-dimensional spaces, that being able to visualize them might actually help distinguish artifacts of the particular examples chosen from real properties of the spaces.
My wife already has the counterexample to any arguement i have ever posed. Combined with my power of visualization, we would be unstoppable.
>I feel like the counterexample superpower is at the mercy of Gödel, though. What if that statement is false, >but doesn't have a counterexample?
If there is no counterexample... then it is true!
You are confusing true for provable.
As a high school math teacher, counterexample for sure. For all those times the student wants to know if their method will always work, and I "know" it won't, but can't find a convenient way to explain why not.
I'm a physicist. Counterexamples and numerical approximation would be neat enough, but I'd really prefer the super visualization.
> I think counterexamples are even stronger, since you can find counterexamples to statements like "no number is within epsilon of [quantity you want to approximate]"
I feel like the counterexample superpower is at the mercy of Gödel, though. What if that statement is false, but doesn't have a counterexample?
Like any good mathematician, counterexample - a constructive disproof of the Riemann Hypothesis or a non constructive proof!
Approximation and counterexamples seem way overpowered compared to visualization. In particular, approximation lets you know the correct answer to any conjecture, since any decision problem can be phrased as a question whose answer is either 0 or 1 (which are separated by more than 20%). I think counterexamples are even stronger, since you can find counterexamples to statements like "no number is within epsilon of [quantity you want to approximate]"
Approximation, because recursion makes it arbitrarily accurate, and physics is a thing