Reflections on Wigner 1: Humility in Pattern Recognition - Mathematics as a Special Case

It's often asked how natural selection could have produced something like the mathematical ability of modern humans. Why can an ape, designed to mate, fight, hunt and run on a savanna, and perceive things that occur on a time scale of seconds to minutes and a size scale of a centimeter to a few hundred meters, even partly understand quarks and galaxies? Implicit in this statement is an admiration for that ability, and the power of mathematics, as well as an assumption held by physicists that should not be surprising.

The physicists' assumption is that the whole of nature, or at least the important parts of it, can be described by mathematics. In "The Unreasonable Effectiveness of Mathematics in the Natural Sciences", Wigner observes "Galileo's restriction of his observations to relatively heavy bodies was the most important step in this regard. Again, it is true that if there were no phenomena which are independent of all but a manageably small set of conditions, physics would be impossible." Another way of saying this is that those regular relationships in nature most easily recognizable by our nervous systems are those parts of nature which we are most likely to notice first; seasonal agriculture preceded gravitation for this reason. But there is a circular, self-selection issue here about the interesting correspondence between the empirical behavior of nature and the mathematical relationships humans are capable of understanding, which is that:

a) humans can understand math.

b) What we have most clearly and exactly understood of nature so far (physics) employs math

c) Therefore, math uniquely and accurate describes nature.

Point b may be true only because our limited pattern recognition ability (even including infinitely recursive propositional thinking like math within that term) only allows us to recognize a certain limited group of relationships among all possible relationships in nature. In other words, of course we've discovered physics because those relationships are the ones we can most easily recognize! It's as if someone with a ruler goes around measuring things, and at the end of the day looks at the data she's collected and is amazed that it was exactly the kind of data you can collect with a ruler.

This discussion is far from an attack on usefulness of mathematics; if you have a model that worked in the past, bet on it working in the future, and the fact that not everything in the universe is yet shown to be predictable by mathematical relationships is certainly not cause to say "We've been at it in earnest for a few centuries and haven't shown how math predicts everything; time to quit." But it also certainly isn't time to say that math can show or has shown everything important, and the rest is necessarily detail. The whole endeavor of truth-seeking I think has at least something to do with decreasing suffering and increasing happiness, both very real parts of nature, and as yet there are very few mathematical relationships concerning them. I look forward to the day that such relationships are shown, but we cannot assume that they exist, or that if they don't, suffering is unimportant.

One problem is that if indeed there are relationships in nature un-graspable by human cognition or mathematics (and note that I've made no argument as to whether those two things are the same), how could we know? It would just look like noise, and we couldn't tell if a knowable relationship was there and had yet to be pulled out, or there was nothing to know (or knowable). We might at least know whether such unknowable information, or "detail", could exist, if we had some proof within our propositional system that there are statements which are true but cannot be deduced from the system. And we have just such a proof.

If we regard mathematics as a formalist does, that math is a trick of our neurology that corresponds usefully enough to nature, the question of why math is useful at all becomes even more important. But if we inject a little humility into our celebration of our own propositional cleverness, the matter seems less pressing.

We have no reason to believe that the total space of comprehensible relationships in nature is not far, far larger than what is encompassed by "mathematics", even in math's fullest extension. If this is the case, it is easier to see how our mathematical ability is a side effect of natural selection and the nervous system it created. By giving us a larger working memory than our fellow species along with some ability to assign symbols arbitrarily, that nervous system does allows us to use propositional thinking to see nature beyond the limitations of our senses - but just barely.

In this view, we can perceive just the barest "extra" layer of nature beyond what our immediate senses do, and mathematics seems far less surprising or miraculous. There is still reason enough to investigate math's unreasonable effectiveness but we shouldn't insist on being shocked that it could have been produced by the hit-and-miss kluges of evolution. But I've made another circular assumption here, which is:

a) evolution proceeds according to natural law

b) evolution will therefore favor replicators that have some appreciatiof some of those natural laws, and modify their behavior accordingly

c) therefore, our ability to perceive the laws that have impacted our own survival, and maybe a few extra ones of the same form, should be expected


There are two mysteries then: first, that any type of regular pattern exists in nature, and second, that we are able to apprehend these patterns, particularly through mathematics. The second mystery probably disappears, seeming special only because of the likely incompleteness of math as a tool to describe nature, math's special case as a method of perception stemming from our own neurology, and the circular basis of our wonder at this as-yet early phase in our use of it. But the first question, of how or why even partly regular relationships appear to exist at all in nature, regardless of how we perceive them, remains untouched by this essay.