Is God a Mathematician_ - Mario Livio [117]
The lesson here is clear. The mathematical tools were not chosen arbitrarily, but rather precisely on the basis of their ability to correctly predict the results of the relevant experiments or observations. So at least for this very simple case, their effectiveness was essentially guaranteed. Humans did not have to guess in advance what the correct mathematics would be. Nature afforded them the luxury of trial and error to determine what worked. They also did not have to stick with the same tools for all circumstances. Sometimes the appropriate mathematical formalism for a given problem did not exist, and someone had to invent it (as in the case of Newton inventing calculus, or modern mathematicians inventing various topological/geometric ideas in the context of the current efforts in string theory). In other cases, the formalism had already existed, but someone had to discover that this was a solution awaiting the right problem (as in the case of Einstein using Riemannian geometry, or particle physicists using group theory). The point is that through a burning curiosity, stubborn persistence, creative imagination, and fierce determination, humans were able to find the relevant mathematical formalisms for modeling a large number of physical phenomena.
One characteristic of mathematics that was absolutely crucial for what I dubbed the “passive” effectiveness was its essentially eternal validity. Euclidean geometry remains as correct today as it was in 300 BC. We understand now that its axioms are not inevitable, and rather than representing absolute truths about space, they represent truths within a particular, human-perceived universe and its associated human-invented formalism. Nevertheless, once we comprehend the more limited context, all the theorems hold true. In other words, branches of mathematics get to be incorporated into larger, more comprehensive branches (e.g., Euclidean geometry is only one possible version of geometry), but the correctness within each branch persists. It is this indefinite longevity that has allowed scientists at any given time to search for adequate mathematical tools in the entire arsenal of developed formalisms.
The simple example of the pebbles in the vase still does not address two elements of Wigner’s enigma. First, there is the question why in some cases do we seem to get more accuracy out of the theory than we have put into it? In the experiment with the pebbles, the accuracy of the “predicted” results (the aggregation of other numbers of pebbles) is not any better than the accuracy of the experiments that had led to the formulation of the “theory” (arithmetic addition) in the first place. On the other hand, in Newton’s theory of gravity, for instance, the accuracy of its predictions proved to far exceed that of the observational results that motivated the theory. Why? A brief re-examination of the history of Newton’s theory may provide some insight.
Ptolemy’s geocentric model reigned supreme for about fifteen centuries. While the model did not claim any universality—the motion of each planet was treated individually—and there was no mention of physical causes (e.g., forces; acceleration), the agreement with observations was reasonable. Nicolaus Copernicus (1473–1543) published his heliocentric model in 1543, and Galileo put it on solid ground, so to speak. Galileo also established the foundations for the laws