HOBOKEN, AUGUST 10, 2024.  JP Andrew, a philosopher with the tag @2Philosophical_, asserts on Twitter/X that the “unreasonable effectiveness” of math is “evidence for Theism”--that is, God. Let me push back against that argument.

Andrew is alluding to “The Unreasonable Effectiveness of Mathematics in the Natural Sciences.” In that 1959 lecture, physicist Eugene Wigner calls our ability to “divine” the mathematical laws of nature a “miracle.”

Ironically, Wigner spends much of his talk qualifying that statement. He warns that just because a mathematical theory works—that is, matches observations--doesn’t mean it’s true. Ptolemy’s geocentric picture of the solar system worked but turned out to be false. Same with Nils Bohr’s early model of the atom.

One effective theory might also be incompatible with another, implying that one or both are wrong. That’s the case with general relativity and quantum theory. “All physicists believe that a union of the two theories is inherently possible and that we shall find it,” Wigner says. “Nevertheless, it is possible also to imagine that no union of the two theories can be found.”

Sixty-five years after Wigner’s lecture, physicists still haven’t unified relativity and quantum theory, and some wannabe unifiers have wandered off into untestable speculation. Plus, no one agrees on what quantum theory says about the world.

Physicist Nigel Goldenfeld elaborates on Wigner’s caveats in a recent conversation with mathematician Steven Strogatz. Those who tout math’s “unreasonable effectiveness,” Goldenfeld says, focus on successes and overlook failures.

“We’re talking about problems where we’ve been lucky to make an impact. And so our sample is skewed,” Goldenfeld says. “You could also ask what is the reason for the unreasonable ineffectiveness of mathematics in biology,” he remarks.

Physics formulas like the Dirac equation, which combines quantum theory and special relativity, reveal little about biological phenomena like the human mind. “Just knowing the basic forces between atoms,” Goldenfeld says, doesn’t explain “why you can think.”

Even modeling something as strictly physical as a phase transition is hard. Models of a metal’s loss of magnetic properties as it heats up consist of approximations of approximations of approximations of what supposedly goes on at the level of atoms. The models work but no one knows why, Goldenfeld says, or whether they mirror reality.

Another fascinating recent exploration of math’s effectiveness is Why Does Math Work If It’s Not Real? by mathematician Dragan Radulovic. Most mathematicians, he says, pursue math for its own sake. We just “doodle with our equations and play with our theorems, never intending them to be applicable.” And yet “math is not only applicable, but it seems to govern the very laws of our universe,” he writes.

Distinguishing math from science, Radulovic proposes an analogy between nature and games like chess. Scientists are trying to discover the basic rules of the game within which we find ourselves.

Mathematicians, in contrast, are trying to discover all logically possible games, whether or not they correspond to our reality. The odds are against any given mathematical invention proving useful to scientists, and yet time and again they do. Radulovic excels at showing how seemingly esoteric, impractical inventions, such as imaginary numbers and non-Euclidian geometry, end up solving problems in physics and other fields.

“It almost seems as if some magic hand guided the ancient mathematicians” toward formulas that would help future scientists. Unlike burning bushes and parting seas, Radulovic says, mathematics is a “real miracle,” and “the book of mathematics is written by the very creator; no matter who or what that is.”

And yet Radulovic, like Wigner and Goldenfeld, implicitly undercuts the theism theory. He notes that mathematics is riddled with pitfalls and paradoxes, like Gödel’s proof about the limits of proof. Is God messing with us?

And most of us can’t grok math without long, grueling training. If God truly cares about us, and wants us to understand Her/Him/They/It via mathematics, why make math so hard for most of us to grasp? That doesn’t seem right. [See Radulovic’s response to this column below.]

That brings me to the problem posed by all theisms, not just the mathematical version. If God created us, and loves us, why is life so painful and unfair? This is the problem of evil. One popular answer is that God gave us free will, so we are free to be mean to each other. Hence, evil.

Wait, so Ukrainians and Palestinians are dying because God gave Putin and Netanyahu free will? That doesn’t seem right. Nor does the free-will hypothesis explain all the suffering inflicted by “acts of God” like earthquakes, volcanoes and tsunamis.

So am I an atheist? No, because like JP Andrew I find this world suspiciously good. I’m impressed by how the Hudson River looks from the tip of Anthony’s Nose; by how my son and daughter tease me and each other when they visit me in Hoboken; by how it feels playing hockey with my old buddies on Lake Alice.

In short, I’m impressed by all the things that make life worth living, including science, about which I’ve been writing for more than 40 years. But a God who deserves credit for the good stuff deserves blame for the bad stuff.

I’ve pitched my own psychedelic solution to the problem of evil. But no theology really makes sense to me, not even my own. The older I get, the more the world baffles me. That’s why I call myself an agnostic.

JP Andrews acknowledges the problem of evil, sort of. Leaf blowers and most music after 1970, he says, provide “evidence for Atheism.” I agree on leaf blowers. I would add nuclear weapons, the invention of which was enabled by physicists’ equations. Is the hydrogen bomb a miracle?

Postscript: Thanks to my pal Richard Gaylord for alerting me to the Goldenfeld podcast and Radulovic book, both of which I highly recommend.

Radulovic responds: After I sent this column to Dragan Radulovic, author of Why Does Math Work If It’s Not Real?, he replied:

Thanks. The book is written so it has many layers, at the surface there is a list of fun math anecdotes, next layer contains some deeper math/science related subjects, and then, there is a layer with some theological connotations (there is a fourth layer, yet to be discovered by my audience). Interestingly, on Amazon I got some very nice reviews, but none of them picked on theological aspect of things. So, kudos to you.

I asked Dragan to tell me about the “fourth layer,” and he replied:

Ok here is a short version. I “sprinkled” several (3-5 depending on how one counts) open questions throughout the book. I believe that some of them might unlock a door for potentially serious work. One never knows. Maybe a young and eager scientist/mathematician out there might get inspired - and frankly I would not mind collaborating on some of those projects.

One has to do with the inherent incompatibility between discrete math and continuous math. That alone is not new and not a surprise. However, continuous math is used in essentially all Physics calculations, but at the same time quantum mechanics insists on a discrete math. But these two are incompatible!?  Obviously, scientists are aware of this, but they brush it off since Planck’s constant is so small. But try simulating this discrete universe on computer and you quickly learn how these “minute” errors add up. Besides, even if it is a “small” issue, it is an issue. How does the nature solve it?

Another one has to do with the inclusion of Brownian motion into the three Newton’s laws. Maybe this is just mathematical peculiarity, but through the years we have learned that often a mathematical pedantry opens a door to some new concept in physics – black hole, antimatter, neutrino… This Brownian motion and Newton laws topic I explored a bit with my colleague (Goran Peskir – Univ Manchester) and Goran even publish a paper on this.

Also, an open project could be building the mathematical tree of life. I hope that a math historian with a much wider background than mine can build such a hierarchical display. This visualization can be very valuable. Because very often, a well displayed concept allows us to see better the big picture – and then some new ideas and concepts come about.

Physicist Frank Tipler also emailed me this response:

I have a chapter in my book The Physics of Christianity on the Problem of Evil, but really the Problem was solved by Plato more than two thousand years ago: God, being all Good, wants to maximize the amount of good in reality.  This means that He has to actualize all universes that have more good than evil in them.  The Best of All Possible Worlds has been actualized; it's just not ours, which still has more good than evil in it, as you point out. Plato really went further.  He said God actualized ALL possible worlds, period.  The reason?  Existence is a good in itself, and even a universe with nothing but evil in it should still be actualized in order to maximize the good in reality. I point out in my own chapter on the Problem of Evil that this resolution addresses the problem of natural evil, like the Great Lisbon Earthquake that so bothered Voltaire.

Further Reading:

Is Ultimate Truth an Equation? Nah

Is the Schrödinger Equation True?

The “Horgan Surface” and “The Death of Proof”

Should Machines Replace Mathematicians?

Pluralism: Beyond the One and Only Truth

Theories of Consciousness, Gaza and My Cognitive Dissonance

Can Beauty Redeem the World?

On God, Quantum Mechanics and My Agnostic Schtick

My free, online book My Quantum Experiment also touches on the mysteries of mathematics.