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Experiment and `Theory'

We have now examined two views of experimental mathematics but we appear to be no closer to a definition than when we began. However, we are now ready to begin in full our exploration of experiment. In Advice to a Young Scientist, P.B. Medawar defines four different kinds of experiment: the Kantian, Baconian, Aristotelian, and the Galilean. Mathematics has always participated deeply in the first three categories but has somehow managed to avoid employing the Galilean model. In developing our notion of experimental mathematics we will try to adhere to this Galilean mode as much as possible. We will begin with the Kantian experiments. Medawar gives as his example:

generating `the classical non-Euclidean geometries (hyperbolic, elliptic) by replacing Euclid's axiom of parallels (or something equivalent to it) with alternative forms.' ([13] pp. 73-74)

It seems clear that mathematicians will have difficulty escaping from the Kantian fold. Even a Platonist must concede that mathematics is only accessible through the human mind and thus at a basic level all mathematics might be considered a Kantian experiment. We can debate whether Euclidean geometry is but an idealization of the geometry of nature (where a point has no length or breadth and a line has length but no breadth?) or nature an imperfect reflection of `pure' geometrical objects, but in either case the objects of interest lie within the minds eye.

Similarly, we cannot escape the Baconian experiment. In Medawar's words this

is a contrived as opposed to a natural happening, it ``is the consequence of `trying things out' or even of merely messing about.'' ([13] p. 69)

Most of the research described as experimental is Baconian in nature and in fact one can argue that all of mathematics proceeds out of Baconian experiments. One tries out a transformation here, an identity there, examines what happens when one weakens this condition or strengthens that one. Even the application of probabilistic arguments in number theory can be seen as a Baconian experiment. The experiments may be well thought out and very likely to succeed but the ultimate criteria of inclusion of the result in the literature is success or failure. If the `messing about' works (e.g., the theorem is proved, the counterexample found) the material is kept; otherwise, it is relegated to the scrap heap.

The Aristotelian experiments are described as demonstrations:

apply electrodes to a frog's sciatic nerve, and lo, the leg kicks; always precede the presentation of the dog's dinner with the ringing of a bell, and lo, the bell alone will soon make the dog dribble. ([13] p. 71)

The results are tailored to demonstrate the theorems, as opposed to the experiments being used to devise and revise the theorems. This may seem to have little to do with mathematics but it has everything to do with pedagogy. The Aristotelian experiment is equivalent to the concrete examples we employ to help explain our definitions, theorems, or the problems assigned to students so they can see how their newly learned tools will work.

The last and most important is the Galilean experiment:

(the) Galilean Experiment is a critical experiment -- one that discriminates between possibilities and, in doing so, either gives us confidence in the view we are taking or makes us think it in need of correction. ([13] p. )

Ideally one devises an experiment to distinguish between two or more competing hypotheses. In subjects like medicine the questions are in principal more clear cut (the Will Roger's phenomenon or Simpson's paradox complicates matters

Does this medicine work (longevity, quality of life, cost effectiveness,etc.)? Is this treatment better than that one? Unfortunately, these questions are extremely difficult to answer and the model Medawar presents here does not correspond with the current view of experimentation. Since the spectacular `failure' (i.e., it worked beautifully but ultimately was supplanted see [12]) of Newtonian physics it has been widely held that no amount of experimental evidence can prove or disprove a theorem about the world around us and it is widely known that in the real world the models one tests are not true. Medawar acknowledges the difficulty of proving a result but has more confidence than modern philosophers in disproving hypotheses. If experiment cannot distinguish between hypotheses or prove theorems, what can it do? What advantages does it have? Is it necessary?

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