Once Before Time - Martin Bojowald [148]
Suppose we had a theory with a unique solution. Given sufficient control over the form of its solution, we can directly test the theory by comparison of its unique predictions with observations; the theory, together with its solution, is scientifically testable. The number of observations, now, is always finite even though the sheer amount of data in modern physics has grown very large. Mathematically, by contrast, we can compute arbitrarily many properties from a solution—without any upper limit. At any given time, one can always think of new observational tests of the theory that have not yet been performed, tests for which the accuracy of available measurement technologies may not yet suffice. One can never completely verify the theory, but at most prove it false should any test not be passed2—a well-known fact in science that now, in the context of the uniqueness of theories or solutions, acquires new meaning.
Since we can never completely test a unique theory, we always have the option of slightly changing it, for instance by choosing different parameter values or delicately switching its underlying principles. If this is done with sufficient care, the uniqueness of its solution can be maintained, and one can remain in agreement with all experimental tests performed. With further physical progress, of course, an observation no longer compatible with every solution of all formerly possible theories would be made at some time in the future; some solutions will be ruled out. And if every one of the considered theories has a unique solution, whole theories would be falsified when their solutions failed. The wiggle room for differences in successful theories decreases as science goes on, but will never be restricted to just one possible theory. If one has a theory with a mathematically unique solution, the theory, understood in the physical sense, cannot be unique.
From the logical inversion of this statement it follows that a unique theory cannot have a unique solution, though we certainly have to keep in mind the fine differences of mathematical and physical uniqueness. Interestingly, recent developments seem to confirm this inference partially and in an astounding manner for a theory—string theory—for which mathematical uniqueness has been claimed. In the search for those solutions of this theory that potentially describe at least the simplest properties of experimentally known elementary particles, a whole landscape of slightly different solutions has opened up—a field of solutions of unimaginable size, whose number would dwarf that of all the protons in the universe. And every one of the solutions could be compatible with all experiments done so far. (Although this number is enormous, and damning to the uniqueness of the theory’s predictions, just being able to estimate the number of solutions is an impressive feat.)
One is reminded of a precedent from philosophy, with morals taking the role of quantum gravity. Over the centuries, many philosophers had attempted to construct a unique theory of morals founded on clear principles. Best known among the examples is perhaps Kant’s categorical imperative as a general principle whence individual rules of conduct should be derivable. Here we have the same problem as in string theory: The noble principles of theory allow innumerable down-to-earth solutions without being of real help in selecting a manageable set. Nietzsche recognized this most clearly:
… a vast new panorama opens up for him, a possibility makes him giddy, mistrust, suspicion and fear of every kind spring