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The debate about flight vs. invisibility is as old as time (and is the subject of a great This American Life segment, hosted by John Hodgman). Today, I have a different question: which MATHEMATICAL superpower would you prefer?

Super Approximation: the ability to immediately answer any numerical question to within 20% accuracy

Super Visualization: the ability to picture extra spatial dimensions in your mind

Super Counterexamples: the ability to immediately furnish the counterexample to any statement where one exists

Super Counterexamples is far and away the best choice — for one, you instantly know the answer to many (all?) open problems, or at least have significant insight. RH? You can either furnish a nontrivial zero off the critical line, or you know it’s true. P=NP? Either find something in NP that’s not in P, or you know for sure none exists.

More mundanely, if you’re a student, you get an immense advantage on every problem of the type “prove or find a counterexample…”. Either you magically know a counterexample, or you turn it into a (psychologically much easier) “prove this statement” problem.

I suppose a case could be made for approximation (could you accurately approximate the number of planets in our galaxy containing life? the number of gods? Become a reasonably good chief commissar of a Soviet-style command economy?). But the pure mathematician in me is overruling the utilitarian.

Okay. Like the above commenter my immediate thought was that super-counterexample would be clearly the most powerful. But then I thought – powerful for what? It is the most powerful for /finding answers to questions/. You could gain hella factual knowledge in epsilon time. But it wouldnt make me know how things work. The learning isn’t in /knowing/ the counterexample. It’s in /looking for/ the counterexample. Neither is the fun in knowing the answers to things – it’s in finding out new shapes things can be, new ways things can behave. So for developing my soul and finding eudaimonia as a human and a mathematician, super visualization is CLEARLY the best choice. I don’t want to be Proofwiki, I want to be Desmos.

Super visualisation. The first one a computer can do and the last one is just to be awkward. But this one would really blow my mind. Almost metaphysical!

Definitely, want the visualization power. My skills there aren’t great and I feel like it would be extremely satisfying to have enhanced abilities in that regard.

I waffled between Super Visualization and Super Counterexamples. On the one hand, if a counterexample existed for a conjecture I made, I would know it instantaneously and if one didn’t, I would only have to search for the proof.

On the other hand, I’m a complex analyst who focuses on functions of several complex variables. That means that the spaces I study have at least 4 real dimensions, so super visualization would really help my intuition out.

As an analyst, “super approximation” would be my preferred super power, but I would want the form of super approximation which immediately finds the right upper and/or lower bounds so that I can take a limit as $\varepsilon$ goes to zero. That’s how approximation works, right?… right?

Super visualisation although a neat hobby isn’t really useful in everyday life and is socially useless as no one else would get your vision. Super counterexamples, unless used very sparingly, is socially obnoxious – it gives you the ability to nit pick any argument to death. Super approximations on the other hand is useful in everyday life and unlikely to lead to becoming a social outcast and allows me to say with complete certainty that 80% (+/-20%) of the people who have voted on this poll already are wrong.

Feynman claimed to have something similar to super approximation, which worked well until someone asked him not for a regular solution, but for, IIRC, ten digits of the solution, starting 20 digits after the decimal point.

I selected Super-visualization, but like the arguments for the counter example super power. The super-visualization would be mind-blowingly interesting, but the counter example super power is just so downright pragmatic and highly useful.

Without a doubt, any of these three would be awesome to have.

For me, it really comes down to either visualization or counterexamples; and I would probably choose counterexamples, for more mundane but perhaps more practical reasons than previously mentioned.

I tend to work with discrete mathematics, which often defies visualization. And, I often work with computer software and especially concurrency and distributed systems, which is a rather difficult topic with huge practical consequences. The counterexample superpower would let me immediately spot flaws in the concurrent algorithm I’m attempting to implement.

Super Counterexamples is far and away the best choice — for one, you instantly know the answer to many (all?) open problems, or at least have significant insight. RH? You can either furnish a nontrivial zero off the critical line, or you know it’s true. P=NP? Either find something in NP that’s not in P, or you know for sure none exists.

More mundanely, if you’re a student, you get an immense advantage on every problem of the type “prove or find a counterexample…”. Either you magically know a counterexample, or you turn it into a (psychologically much easier) “prove this statement” problem.

I suppose a case could be made for approximation (could you accurately approximate the number of planets in our galaxy containing life? the number of gods? Become a reasonably good chief commissar of a Soviet-style command economy?). But the pure mathematician in me is overruling the utilitarian.

Bill Thurston had super-visualization.

Super-logic — the ability so spot the holes in my proofs.

Super intuition — The ability to sense that a proposition is true or false before attempting to prove or disprove.

Super-elegance — The ability to succinctly and creatively state my arguments. (vs. the super-elegance that super-models have.)

Super-knowing where to start.

Super-getting it — every open cover has a finite sub-cover? What the heck does that even mean?

Okay. Like the above commenter my immediate thought was that super-counterexample would be clearly the most powerful. But then I thought – powerful for what? It is the most powerful for /finding answers to questions/. You could gain hella factual knowledge in epsilon time. But it wouldnt make me know how things work. The learning isn’t in /knowing/ the counterexample. It’s in /looking for/ the counterexample. Neither is the fun in knowing the answers to things – it’s in finding out new shapes things can be, new ways things can behave. So for developing my soul and finding eudaimonia as a human and a mathematician, super visualization is CLEARLY the best choice. I don’t want to be Proofwiki, I want to be Desmos.

Well said! I think you might have changed my opinion. I voted counterexample, but I think you provided me the counterexample to my statement!

Super visualisation. The first one a computer can do and the last one is just to be awkward. But this one would really blow my mind. Almost metaphysical!

Definitely, want the visualization power. My skills there aren’t great and I feel like it would be extremely satisfying to have enhanced abilities in that regard.

I waffled between Super Visualization and Super Counterexamples. On the one hand, if a counterexample existed for a conjecture I made, I would know it instantaneously and if one didn’t, I would only have to search for the proof.

On the other hand, I’m a complex analyst who focuses on functions of several complex variables. That means that the spaces I study have at least 4 real dimensions, so super visualization would really help my intuition out.

As an analyst, “super approximation” would be my preferred super power, but I would want the form of super approximation which immediately finds the right upper and/or lower bounds so that I can take a limit as $\varepsilon$ goes to zero. That’s how approximation works, right?… right?

Super visualisation although a neat hobby isn’t really useful in everyday life and is socially useless as no one else would get your vision. Super counterexamples, unless used very sparingly, is socially obnoxious – it gives you the ability to nit pick any argument to death. Super approximations on the other hand is useful in everyday life and unlikely to lead to becoming a social outcast and allows me to say with complete certainty that 80% (+/-20%) of the people who have voted on this poll already are wrong.

OK, I voted for one of the 3, but screw that, I wanna FLY (defy gravity)!

Feynman claimed to have something similar to super approximation, which worked well until someone asked him not for a regular solution, but for, IIRC, ten digits of the solution, starting 20 digits after the decimal point.

Super approximation allows for iteration, achieving arbitrary precision (find x to 20%, say x1: then find x-x1 to 20%, etc…)

I selected Super-visualization, but like the arguments for the counter example super power. The super-visualization would be mind-blowingly interesting, but the counter example super power is just so downright pragmatic and highly useful.

Without a doubt, any of these three would be awesome to have.

For me, it really comes down to either visualization or counterexamples; and I would probably choose counterexamples, for more mundane but perhaps more practical reasons than previously mentioned.

I tend to work with discrete mathematics, which often defies visualization. And, I often work with computer software and especially concurrency and distributed systems, which is a rather difficult topic with huge practical consequences. The counterexample superpower would let me immediately spot flaws in the concurrent algorithm I’m attempting to implement.