If you’re only doing things that are both correct and useful, then you already knew how to solve that problem. That’s fine, but it’s not interesting.

If you’re doing things that are neither useful nor correct, then I’m not sure what you’re doing (or even why).

If you’re doing things that are correct but not useful, then you’re generating trivia. Granted, this does happen sometimes in problem solving. The solver might think, “I know this is legal, but I’m not sure whether it helps…let’s follow it for a bit and see if it turns into something useful.” I personally find this to be more of a bland problem-solving process. It often works, but is more procedural than creative.

But things that are useful but not correct–that’s where creativity lies, in my experience. “This is not true, but I wish it were; because if it were, then I could solve this problem like so… It’s not true, but I wonder if some part of it is true? Or almost true? Or if something kind of like it is almost true in a partial way?…”

The deliberate pursuit of statements that are useful but not correct can lead to beautiful, surprising solutions.

The most popular example these days is surely his computation of values of divergent infinite series, like (4)-(7) here:

http://terrytao.wordpress.com/2010/04/10/the-euler-maclaurin-formula-bernoulli-numbers-the-zeta-function-and-real-variable-analytic-continuation/

These are the “p-series” from calculus with p=0, -1, -2, … We learn in calculus class that p-series only converge with p>1. The formulas are thus incorrect.

To see the utility, we can consider a function on (1,infinity) whose input is p and whose output is the sum of the convergent p-series. This is the Riemann zeta function, and the formulas (4)-(7) suggest that it may be possible somehow to extend the definition of the function to non-positive numbers. In fact, Euler did just that and found a remarkable “functional equation” satisfied by the function that makes the assignment of values (4)-(7) much more reasonable.

It seems most or all examples of “incorrect but useful” given herein concern either approximation or intuition. A third category that might encompass or exceed both: examining incorrect deductions. These can often lead to new useful definitions or theorems. For instance, incorrect proofs of Fermat’s Last Theorem use unique factorization in number systems that resemble the integer. This leads to the idea that Unique Factorization is something that needs to be proved in such systems to be used. Or in making calculus into the more rigorous “real analysis”, certain arguments lead to the notion of a “uniform bound” being needed in some statements (the “epsilon” used needs to be independent of other choices made ), leading to definitions like uniform convergence and uniform continuity.

19 / 95, cancel out the 9s, leaving 1/5.

log(3x+2) + log(4x+1) = 2 log 11, cancel out the word “log” leaving (3x+2) + (4x+1) = 22, giving the correct answer.

In Geometry it is sometimes convenient to assume the negation of what you are trying to prove, then follow a logic chain until you reach a contradiction.

It is incorrect to assume that the square root of 2 is rational. But this leads to a logical inconsistency, enabling us to prove that the square root of 2 is irrational.