Choose Your Superpower

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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

Comment with your reasoning below.

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14 thoughts on “Choose Your Superpower

  1. 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.

  2. 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?

  3. 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.

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

  4. 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!

  5. 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.

  6. 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?

  7. 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.

  8. 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.

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

  10. 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.

  11. 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.

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