Creating Hypotheses

Problem solving is a creative enterprise

Most treatments of the philosophy and practice of science deal with the deductive part. It's fine to be admonished to use multiple working hypotheses, but where do they come from? While my difficulty on other pages has been to restrain myself from listing too many references, my problem here is the opposite. Witness the following titles: How to Solve It (Pólya 1945, 1973), The Search for Solutions (Judson 1980), How to Model It (Starfield et al. 1994) and Strategies for Creative Problem Solving (Fogler 1995). Where good problems come from would appear to be a closely guarded secret. Curiously, but clearly related to the acceptability of induction as a means of proof, virtually all the systematic guidance that I can find comes directly or indirectly from mathematics. The fundamental techniques are laid out by Pólya (1945; 1988). They are dissected and somewhat enhanced in subsequent treatments (e.g., Schoenfeld 1985), usefully generalized beyond mathematical applications by Adams (1974) and applied to ecological modeling by Starfield et al. (1994). Creativity in problem solving can be and is taught. Nearly all of us have taken creative writing courses, for example. Why shouldn't one be able to foster scientific creativity in the same way, i.e., by studying good and bad examples and practicing?

Things to try, once you have formulated an interesting question, are many (Schoenfeld 1985). The ones that work most often for me in biological oceanographic applications are given in boldface type. Draw a diagram if at all possible. Write an explicit equation. Reformulate the problem by change of perspective (e.g., prey's vs. predator's) or notation. Examine special cases; choose special values as examples and to get a feel for the problem, and examine limiting cases to bound the possibilities. Try to simplify the problem, including application of dimensional analysis and scaling arguments. Whether simplification works or not (but especially if not) try to generalize the problem. Try analogies (but be careful to distinguish analogies from homologies). In particular, construct an analogous problem with fewer variables. Hold all but one variable constant and determine that variable's impact. Replace conditions by equivalent ones. Relax a condition, and then try to re-impose it to see what happens. Exploit related problems (not just in oceanography). For biological oceanographic problems go in particular to terrestrial ecology and earth sciences and limnology and to engineering. Look for problems that have similar forms, givens or conclusions. Reduce the problem to absurdity. Work backward from each of the likely solutions to see if you arrive at the original problem. Introduce auxiliary arguments into the problem. Argue by contradiction (If this proposition is false, then...). Restate the problem. Change the problem and (or) the data to get better convergence [not to be construed as a suggestion to alter data in your notebook but rather as a suggestion to alter the problem phrasing or data to be collected before they get to your notebook or the normalization or standardization of the variables after they are recorded].

Advanced problem solving is problem creation

When you have some tentative solutions (alternative hypotheses), do they withstand comparison with extant data? Do they pass dimensional analysis and simple scaling arguments? Now, for the really fun part, do they predict anything that has not been observed but that could be?

What is most interesting about these various approaches is that their exercise gives the ability not only to answer posed problems but to pose answerable problems, i.e., to devise testable, alternative hypotheses. That is, the learning, doing and teaching of creative problem solving is probably the best means toward learning, doing and teaching hypothesis generation.

Not everyone is cut out to be a theoretician (thank goodness), and some of the most prolific theoreticians are incapable of distinguishing good ideas from bad. Nor is theory always the limiting ingredient in determining the rate of scientific advance. An exercise that I truly enjoy and encourage you to try is to choose any scientific subspecialty (small set of problems) with which you are familiar and ask what is currently limiting the rate of advance in understanding. For example, in the study of deposit feeding explicit, predictive theory was absent until the late `70s. In seabed acoustics (with real, messy media like muds inhabited by animals), by contrast, what today limits the rate of progress is the shortage of laboratory facilities where simple manipulative experiments can be carried out.

As an exercise, pick an area of science for which you can develop some passion. Then ask whether progress currently is limited by lack of theory, lack of laboratory facilities (or observations) or lack of particular field observations (or tools). If you have trouble coming up with ideas independently, mine the discussion and introduction sections of relevant papers. Can you extend their ideas or modify them through one of the other convolutions (above) upon which mathematicians rely? In real life, people talk to colleagues. That's an essential part of what makes some institutions great. Talk to other students and faculty about your ideas. Articulation and writing each cause subtle or dramatic transformations of ideas. Some vanish; others crystallize. You want to try ideas out on colleagues before you try them out on formal reviewers or program managers.

References

Adams, J.L. 1974. Conceptual Blockbusting. W.H. Freeman & Co., San Francisco. 138 pp.

Fogler, H.S., and S.E. LeBlanc. 1995. Strategies for Creative Problem Solving. Prentice-Hall, Englewood Cliffs, NJ. 203 pp.

Judson, H.F. 1980. The Search for Solutions. Holt, Rinehart and Winston, New York. 211 pp.

Pólya, G. 1945. How to Solve It. Princeton Univ. Press, Princeton. 204 pp.

Pólya, G. 1988. How to Solve It, 2nd Ed. Princeton Univ. Press, Princeton. 253 pp.

Schoenfeld, A.H. 1985. Mathematical Problem Solving. Academic Press, NY. 409 pp.

Starfield, A.M., K.A. Smith and A.L. Bleloch. 1994. How to Model It: Problem Solving for the Computer Age. Burgess International Group, Inc., Edina, MN. 206. pp. Reprint of 1990 edition from McGraw Hill, NY.