World famous Christian apologist William Lane Craig has a fun new argument: the universe is describable by math, and this cries out for designer. He’s impressed by “the uncanny effectiveness of mathematics.” He said:

It was very evident to me that [naturalists are not] able to provide any sort of an explanation of mathematics’ applicability to the physical world. . . .

The theist has explanatory resources that are not available to the rationalist.

So mathematics does impressive things; therefore, God? And if the theist has useful “explanatory resources,” I wonder if they’re built on evidence.

I’ll resist the temptation to respond for the moment. Let’s first fill out the argument.

The Unreasonable Effectiveness of Mathematics

Uncharacteristically, Craig brought expert backup this time. He points to a 1960 paper by Nobel Prize-winning physicist and mathematician Eugene Wigner, “The Unreasonable Effectiveness of Mathematics in the Natural Sciences.” Wigner says, “Mathematical concepts turn up in entirely unexpected connections.” More to Craig’s point, he says:

The enormous usefulness of mathematics in the natural science is something bordering on the mysterious and there is no rational explanation for it.

Some examples of the applicability of math to the physical world include the ideal gas law, PV = nRT; the inverse-square law; Ohm’s law, V = IR; Newton’s law of gravity, F =Gm₁m₂/r²; and Maxwell’s equations:

These and myriad other examples illustrate math’s power in describing nature. Wigner concludes:

The miracle of the appropriateness of the language of mathematics for the formulation of the laws of physics is a wonderful gift which we neither understand nor deserve.

Wigner points to the mystery but only hints in the religious direction. It’s Craig who is determined to resolve the mystery with God.

Meta observations

This Argument from Math is just a variant of the Transcendental Argument (discussed at length here), which in turn is just one of many arguments where Christian apologists can only hope to provoke the reaction, “Say, that is curious!”

These are caltrop arguments—they succeed not because they’re correct but because they’re confusing. A successful argument instead follows Hoare’s Dictum: you can make your argument so simple that there are obviously no errors, or you can make it so complicated that there are no obvious errors.

Note too that this Argument from Math is just a deist argument. If you found it convincing, you could only justify becoming a deist. At that point, you’re no closer to Christianity than to Pastafarianism.

The puddle problem

We may find ourselves in the situation of Douglas Adams’ puddle that thought that its hole was made to fit it perfectly, rather than the other way around.

Reality is the hole, and math is the puddle—reality is what it is, and the math adapts as necessary. If one formulation of a law does a poor or incomplete job of explaining the physics (say, when Newton’s law of gravity didn’t work perfectly in environments with extreme gravity), the math can be changed (in this example, by adding corrections to account for General Relativity).

We don’t start with math and then marvel that the universe comports to it; instead, we see what the universe does and then invent stuff (tensors, quaternions, differential equations) that economically describes what we see. Math is a description of reality.

Also note that math has been tuned by reality to be simple. Mathematicians came across matrix operations so often that they developed shorthand versions—the del operator (∇), for example. Expand that out into a more elementary formulation, and it’s not so simple anymore. Or, replace an advanced mathematical idea with its explanation (“What does ‘integral’ mean?”), and you’ve got a textbook chapter—again, not so simple. It’s simple only in its terse form, unhindered by explanations that we laymen would need, but that hides the complex mountain on which it’s built.

Wigner said, “The only physical theories which we are willing to accept are the beautiful ones.” Here again, this may not be nature giving us miraculous math but scientists being trained by reality to see what works (and is therefore beautiful) and what doesn’t.

Physicist Max Tegmark responded to Wigner’s idea. He said that a question like “Why is math so good at describing reality?” is like “Why is language so good for conveying ideas?” Language was tuned and adapted to be good for what we need it to do, and the same is true for math.

What is surprising?

Wigner said that Newton’s law of gravity “has proved accurate beyond all reasonable expectations,” but what are these reasonable expectations? That the universe is mathematically describable is surprising only if we expect it to be otherwise (I’ve discussed a related topic here). What then should we expect? Should we expect the same laws of nature but different fundamental constants? Different constants in different parts of the universe? Different laws? Or maybe a structure so chaotic that no equation would be accurate for more than an instant?

Why are any of these possibilities more expected than what we actually have? What’s unreasonable about how math works in our world? Once we study hundreds of other universes, we’ll get a sense of what they look like to compare with our own, but without this data, we have nothing to go on, and we have no grounds on which to formulate “reasonable expectations.”

That’s a big burden on Craig’s shoulders, which he doesn’t even acknowledge. I doubt he has even thought of the problem, and he certainly doesn’t respond to it.

“God did it” is simply a synonym for “we don’t know.” That explains nothing.

To be concluded in part 2.

.

(This is an update of a post that originally appeared 02/07/15.)

Image from stuartpilbrow, CC license

.

Consider this parable:
A certain mathematician, in a philosophical mood one day, wondered what grounded his mathematics. The math works, of course, but he wonders if he’s missing something foundational.
He consults a friend of his, a theologian. The theologian knows almost nothing about mathematics, but he knows his Christianity.
The mathematician says, “Mathematics is like an inverted triangle with the most advanced math along the wide top edge. The top layer is grounded on the math below it, which is grounded on what is below, and so on through the layers, down to arithmetic and logic at the point at the bottom. And that’s where it stops.”
The theologian nodded his head wisely. “I see the problem—what does the bottom rest on?”
The mathematician was silent.
“In your view, it rests on nothing,” said the theologian. “It just sits there in midair. But the problem is easily resolved—mathematics and logic comes from God. There’s your grounding.”
“Are you saying that I need to convert to Christianity to be a mathematician?”
“No, just realize that you are borrowing from the Christian worldview every time you make a computation or write an equation.”
Satisfied that this nagging problem has been resolved, the mathematician returns to his work and thinks no more of it.
The End.
So, is the mathematician any better off? Is he faster or more accurate or more creative? Do his proofs work now where they hadn’t before? In short, did he get anything of value from the whole episode? Not at all.
And note, of course, that the axioms at the bottom of the triangle aren’t taken on faith, they’re tested. “1 + 1 = 2” has worked on everything so far, but we’ll take notice if we find a situation where it doesn’t. Some mathematical claims are proven and some are tested, but each is reliable.
I’ve heard this “grounding” or “atheists borrow from the Christian worldview” idea many times, but I’ve yet to discover what this missing thing is that is being borrowed.
If we imagine that 1 + 1 equals 2 only because God says so, that means that a universe is possible where 1 + 1 doesn’t equal 2. That’s a remarkable claim, and I’d like to see it supported by the theologian rather than simply asserted without evidence.
“God did it” is nothing more than a restatement of the problem. “God did it” is precisely as useful as “logic and arithmetic are simply properties of our reality” or “that’s just the way it is” or even “I don’t know.” An interesting question has been suppressed, not resolved. In fact, by the theologian’s own logic, his answer rests in midair because he provides no reason to conclude that God exists. His claim is no more believable than that of any other religion—that is, not at all.
The person who stops at “God did it” has stated an opinion only—an opinion with no evidence to back it up. It doesn’t advance the cause of truth one bit.
Mathematics is tested, and it works. Scratch your head about what grounds it if you want, but God is an unnecessary and unedifying addition to the mix.

God is an ever-receding pocket
of scientific ignorance.
— Neil DeGrasse-Tyson

(This is a modified version of a post originally published 11/25/11.)