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If an argument looks promising but needs a hypothesis, assume the hypothesis for now but try to remove it later

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Quick description

When trying to solve a problem in mathematics, one often encounters the difficulty that a tool or argument that looks promising requires some additional hypothesis which is not actually present in the problem at hand. For instance:

  • A tool may only be rigorously justified if a certain function f is continuous, but in the given problem the function f is only known to be measurable (say).

  • The argument one has in mind may work well if f is somehow "dispersed", "uniformly distributed", or "pseudorandom" in the sense that a certain norm \|f\| that measures this dispersion or uniformity is small; however, with the hypotheses at hand it is possible that this norm is large instead.

  • One has an argument which would work well if the objects at hand were of a concrete nature (e.g. finite-dimensional matrices), but they are only given in a more abstract form (e.g. abstract group elements, or operators on an infinite-dimensional space).

  • An argument would work well if all the objects in a certain class commuted with each other, but unfortunately in general one expects the relevant objects to be non-commutative.

  • An argument which would work well if a certain error term was in fact zero, but in practice it will be non-zero.

In such cases, one should not necessarily abandon all hope of the argument actually solving the problem. Sometimes what one can do is temporarily assume whatever hypothesis is necessary to make the desired argument work, solve the problem conditionally on that hypothesis, and then go back and think about how to modify the argument so that the additional hypothesis is not needed, or to find ways to amplify the conditional result to an unconditional result. (And even if one cannot solve the full problem this way, one may be able to get an interesting partial result that someone else may be able to use or build on.) For instance

  • If one can figure out how to solve a problem for continuous f, and now wants to handle measurable f, one might try creating an epsilon of room and approximating the measurable function by a continuous one, and using some sort of limiting argument.

  • If one has an argument which works when a certain object f is "dispersed" or "uniform", try thinking about what would happen in the opposite case when f is "concentrated" or "non-uniform". One might be able to deduce some usable structural information on f in this case, which may in turn suggest a new and different argument to handle this other case.

  • If one has an argument that works well in a concrete setting, one can try abstracting that argument by replacing concrete objects with abstract ones wherever possible, and in particular trying to isolate the particular axioms of the concrete framework that were actually used in your argument.

  • If you have an argument that works for commutative settings, but you need it for a non-commutative one, you might try rearranging the argument to use as little commutativity as possible (e.g. by carefully distinguishing between left translations and right translations, etc.). Sometimes this means that you have to unpack or "deconstruct" the proof of some standard result in the commutative setting in order to understand what portion of it can be salvaged for the non-commutative one.

  • If you have an argument that works when a certain error term vanishes, reintroduce the error term into the argument, give that term a name that suggests smallness (e.g. \varepsilon), and keep track of how the lower order terms involving \varepsilon propagate through the argument.

Prerequisites

Example 1

(Roth's theorem is, once again, a poster child for this sort of strategy. Other suggestions welcome)

General discussion

Of course, sometimes the hypothesis one is imposing is so strong that it is unlikely that it will shed much light on what to do once the hypothesis is removed. For instance, if an argument needs some sort of "miraculous cancellation" in which two extremely dangerous terms in an expression are assumed to "magically" cancel each other out, but one has no heuristic justification as to why such a cancellation should occur (e.g. an analogy with a simpler situation in which such a cancellation is known to happen), then it may be that this argument is of no use in the general case. (And it could also be, of course, that the general case is simply false.) So one should maintain a realistic sense as to how mild or how strong a hypothesis is before one invests too much time into the special subcase of the problem that assumes that hypothesis.

See also If you don't know how to make a decision, then don't make it, Temporarily suspend rigor and First pretend that a normal subgroup is trivial.