PLATONISM AND NATURE
Some Platonists argue that mathematics is eternal, exact and perfect — that its truths are discovered by humans rather than invented. A corollary of the Platonist paradigm is that nature is a somewhat imperfect approximation to the eternal mathematical truths. Although many are swayed by these arguments, it is possible that the Platonist philosophy has things backwards.
An alternative paradigm asserts that it is nature that is an eternal magnificent perfection, and that pure mathematics is an inherently abstract and imperfect enterprise. According to this alternative philosophy, when mathematics is applied to modeling nature, the analytical models are:
(1) Artificial (in the non-pejorative sense of the word, and more in the sense that the models are invented rather than discovered), and
(2) Approximate (in the sense that they cannot in principle provide a complete representation of nature’s infinite complexity).
The Platonist paradigm seems to be motivated by a fervent hope that mathematics offers exact answers and absolute truths. Unfortunately, it seems more likely that such things as exact answers and absolute truths will always remain beyond human reach. Perfect circles and absolute certainty probably exist only in the “world” of the imagination. We would do well to be mindful of the distinction between what is real and what is an abstraction.
Once it was thought that the Solar System is like a giant “clockwork” that was stable and completely predictable in its motions. Then the great mathematician Poincare showed that even a 3-body system was not integrable, that the Solar System was certainly not a stable clockwork. His work led us from a Platonic view of nature towards a more realistic view that recognizes the importance and universality of nonlinear dynamical systems and deterministic chaos. That revolution is still ongoing in some areas of science, especially in the physics of the microcosm.
Mathematics is a truly sublime subject of study and it plays an extremely important role in modeling nature. Yet perhaps applied mathematics is significantly more limited than the Platonists and the Bourbaki physicists are willing to acknowledge.
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