Davies On the Unreasonable Effectiveness of Mathematics

Davies On the Unreasonable Effectiveness of Mathematics February 18, 2018

In Mind of God, agnostic physicist Paul Davies explores a number of deep questions, one of which takes aim at Eugene’s Wigner’s question on the “unreasonable effectiveness of mathematics” in describing the natural world. Restated by Einstein, “how is it possible that mathematics, a product of human thought that is independent of experience fits so excellently the objects of physical reality?”

He presents various perspectives on the “is mathematical invented or discovered?” question. For many with Platonic leanings, mathematics is obviously discovered. Here Davies lays out the thought of Roger Pemrose, with things like complex numbers and “the Mandelbrot set”, along with his mathematical breakthroughs on black holes and space-time singularities, that mathematics is certainly discovered.

He also provides the counter argument that mathematics is invented, just the way our cognitive faculties express scientific truths abstractly. This may sound initially attractive, that mathematics is the language we’ve invented for this abstraction, so no wonder why it’s incredibly effective. But something seems awry there. In Davies’ words:

“So is the success of mathematics in science just a cultural quirk, an accident of our evolutionary and social history? Some scientists and philosophers have claimed that it is, but I confess I find this claim altogether too glib, for a number of reasons. First, much of mathematics that is so spectacularly effective in physical theory was worked out as an abstract exercise by pure mathematicians long before it was applied to the real world. The original investigations were entirely unconnected with their eventual application… And yet we discover, often years afterward, that nature is playing by the very same mathematical rules that these pure mathematicians have already formulated. (151)

He continues:

The mystery becomes even deeper when we take account of the existence of mathematical geniuses… The astonishing insight of mathematicians such as Gauss and Riemann… Probably the most famous case is that of the Indian mathematician S. Ramanujan… Born in India in the lat nineteenth century, Ramanujan came from a poor family and had only a limited education. He more or less taught himself mathematics and, being isolated from mainstream academic life, he approached the subject in a very unconventional manner… To this day nobody really knows how he achieved his extraordinary feats… “(153-154)

The challenge for mathematics being an invented language to describe empirical facts is that mathematics is time and time again in front of empirical discovery and/or intimately connected or adding to empirical discovery. This not only returns us to Einstein’s question (“how is it possible that mathematics, a product of human thought that is independent of experience fits so excellently the objects of physical reality?”), but seems to further distance us from the mathematics is just invented solution. Restated by Mario Livio (for a deeper dive, check out my review of Is God a Mathematician):

“It is difficult to see how this ability could have led to abstract mathematical theories of the subatomic world, such as quantum mechanics, that display stupendous accuracy… Researchers use mathematical models to product new phenomena, new particles, and the results of never-before-performed experiments and observations. What puzzled Wigner and Einstein was the incredible success of the last two processes. How it is possible that time after time physicists are able to find mathematical tools that not only explain the existing experimental and observational results, but which also lead to entirely new discernment and new predictions?”

Personally, alongside Davies (and Livio), I find the mathematics is just invented solution extremely wanting. At the same time, I agree that a Platonic realism of mathematics is also problematic (mathematical truths in a separate immaterial realm). Holding an Aristotelian realism (metaphysically, morally, and mathematically), which I do, I find an extremely balanced alternative of both Platonic realism and irrealism. There is mathematical ontology in matter (again, not in some Platonic realm, nor just invented by our minds). Like with morality, we don’t invent it. There is intrinsic value in nature. We have the ability to abstract this knowledge epistemologically from nature because it’s in nature ontologically.

Image credit:

Author: ASU / Tom Story

Source: Wikimedia Commons

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