God does not play dice

 
THOUGHT OF THE DAY:     
"Einstein was doubly wrong when he said, God does not play dice. Not only does God definitely play dice, but He sometimes confuses us by throwing them where they can't be seen." - Stephen Hawking, from Does God Play Dice, 1999. 
I imagine Einstein regrets having said that, or maybe just regrets the ways people use or misuse it.

It's one of those quotations that has had a much longer afterlife than the argument it was originally part of.

When Einstein said "God does not play dice," he wasn't making a theological statement so much as expressing his discomfort with the probabilistic nature of quantum mechanics. He believed the apparent randomness reflected an incomplete theory. In other words, he thought there were hidden facts we hadn't discovered yet.

The remarkable thing is that Einstein wasn't simply being stubborn. He helped create quantum theory in the first place. His work on the photoelectric effect was foundational. What troubled him was not the mathematics but the idea that randomness might be fundamental to reality.

Hawking's quip is clever because it updates the argument. By the late twentieth century, most physicists had accepted that quantum events really do seem to be intrinsically probabilistic. Hawking goes a step further: not only are there dice, but sometimes they're thrown "where they can't be seen," alluding to things like black holes, hidden regions of the universe, and limits on what observers can know.

As for whether Einstein would regret the quote, I'm not sure. He was known for refining his views when evidence demanded it, but he also held some convictions for decades. He might object less to Hawking's physics than to the way the quote is often used today as a slogan meaning "Einstein was wrong." His actual objections were sophisticated and led to important developments. The famous Einstein–Podolsky–Rosen paper, intended as a critique of quantum mechanics, eventually helped inspire experimental work on quantum entanglement. In a sense, even Einstein's objections advanced the theory that disagreed with him.

What he might regret most is that the phrase is often detached from its context and turned into a sound bite about religion, fate, free will, or gambling. Einstein's "God" was usually closer to the impersonal lawful cosmos admired by Baruch Spinoza than to a deity deciding outcomes.

There is also a delightful irony here. Einstein's complaint was essentially, "The universe can't really work like that." Decades of experiments have replied, "Apparently it can." Yet without Einstein's relentless skepticism, physicists might not have tested the question nearly as thoroughly.

Science progresses in a curious way: sometimes the people most responsible for discovering a truth spend the rest of their lives arguing against its implications. Einstein and quantum mechanics may be the most famous example.

That seems like a healthy reaction.

The quote is usually attributed to the geneticist and evolutionary biologist J. B. S. Haldane:

    "The universe is not only queerer than we suppose, but queerer than we can suppose."

It's a wonderful line, but people often treat it as if it were a profound revelation. Your response is almost the natural counterpoint:

    Why should a primate brain, evolved to throw rocks, find mates, avoid snakes, and navigate social hierarchies, be expected to grasp the ultimate structure of reality?

The surprise would be the opposite.

Suppose evolution had equipped us to understand every level of the cosmos intuitively. That would be astonishing. Instead, we find that our intuitions work reasonably well for objects moving at walking speed, distances of a few miles, and groups of a few dozen individuals. Beyond that, things get weird quickly.

We don't naturally understand:

    Relativity.

    Quantum mechanics.

    Deep time.

    Evolution by natural selection.

    Exponential growth.

    Large numbers.

    Probability.

In fact, many of our strongest intuitions are positively misleading in those domains.

Yet there is another side to the story. The same ape brain that evolved for the savanna somehow discovered calculus, quantum field theory, radio astronomy, and the age of the universe. That's the part that impresses me. We are not naturally equipped to understand these things, but we seem capable of building intellectual tools that extend our reach far beyond our native intuitions.

Einstein himself remarked on what he called "the incomprehensible comprehensibility of the universe." He was struck not by the fact that reality is strange, but by the fact that mathematics invented in human minds turns out to describe it so well.

So there are really two mysteries:

    The universe is stranger than common sense.

    Common-sense creatures can nevertheless discover a surprising amount about it.

The first mystery gets most of the attention. The second may be the more remarkable one.

And perhaps this circles back to your professor's cliché. When someone says, "The universe is stranger than we can imagine," I find myself wanting to ask, "How do you know?" We have a long history of imagining things that once seemed impossible: curved spacetime, black holes, quantum tunneling, expanding universes. Reality keeps surprising us, but human imagination has a habit of catching up eventually.

Maybe the universe is stranger than we can currently imagine. That seems likely. Stranger than we can ever imagine? That's a stronger claim, and one I'm not sure anyone is in a position to make.

We have levers to help us lift things outside our stengh.  The main lever is mathematics.  As I saw in an interview yesterday, the guy said he would try to make a point he was making intuitive, but he didn't think it would work as hoped, we have to see and understand the equations.

That's a lovely way to put it: mathematics as a lever.

A lever does not make you stronger; it lets a modest strength accomplish something that would otherwise be impossible. Mathematics seems to do the same thing for the human mind.

Our intuition is excellent for some things. You can look at a tree branch and have a good sense whether it will hold your weight. You can throw a ball and predict roughly where it will land. But our intuition fails badly when confronted with quantum mechanics, relativity, population genetics, or even compound interest.

Mathematics lets us reach beyond intuition. Often the equations are understood long before anyone develops a satisfying mental picture.

A classic example is quantum mechanics itself. Physicists can calculate the behavior of atoms to astonishing precision. Yet if you ask ten physicists what is "really happening" beneath the equations, you may get eleven answers. The mathematics works; the intuition is still catching up.

Likewise with relativity. The equation field came first. The popular image of a bowling ball distorting a rubber sheet is a teaching aid, not the theory itself. The actual theory is expressed mathematically. The picture helps us begin to think about it, but the equations are where the precision lives.

I suspect the person you heard was acknowledging a common frustration among scientists. They would like to provide an intuitive explanation because intuition feels like understanding. Yet sometimes the intuition is only an analogy. The equations are the thing itself.

The physicist Richard Feynman made a similar point about quantum electrodynamics. He essentially said that if you insist on a familiar mechanical picture, you will be disappointed. Nature is not obliged to behave in ways that fit our everyday imagination.

Your lever metaphor also suggests something else. A lever does not eliminate the need for effort. Mathematics does not eliminate the need for thought. It merely allows thought to be applied where intuition alone would fail.

That's one reason mathematics occupies such a peculiar place in human knowledge. It is both a language and a tool. We use it not merely to describe the universe but to extend the reach of our minds into regions where ordinary imagination cannot comfortably operate.

And that brings us back to Einstein's wonder about the "comprehensibility" of the universe. Why should this symbolic game of equations, invented by human beings, provide such leverage over reality? We can explain how mathematics works as a tool. The deeper question is why reality is structured in a way that allows the tool to work so extraordinarily well.

That question still feels as mysterious to me as any discussion of God, chance, or quantum mechanics.

Oh I imagine that mathematics is "out there" on its own independent of us, that we find it.  The only thing we invent is how to write it down.  

That view has a long and distinguished pedigree. In philosophy of mathematics it's usually called mathematical Platonism, after Plato, though modern versions differ considerably from Plato's own ideas.

The intuition behind it is powerful. When mathematicians prove a theorem, they don't usually feel as though they've created something. They feel as though they've discovered something that was true all along.

Take the prime numbers. It seems odd to say humans invented the fact that there are infinitely many primes. If intelligent beings had evolved on a planet orbiting a star in the Andromeda galaxy, they would eventually discover the same theorem. The symbols might differ, the notation might be unrecognizable, but the theorem itself would be the same.

Many mathematicians have reported this feeling. The great Indian mathematician Srinivasa Ramanujan spoke almost as though mathematical truths were revealed to him. Others describe entering a landscape that was already there and exploring it.

The opposing view is that mathematics is a human invention: a collection of formal systems we created because they happen to model aspects of reality effectively. On that view, the truths follow from the rules we chose.

What makes the debate difficult is that both sides capture something real.

The symbols are clearly invented. Roman numerals, Arabic numerals, algebraic notation, calculus notation—all human creations.

But once the rules are fixed, the consequences seem stubbornly independent of us. Nobody voted to make π irrational. Nobody legislated the Pythagorean theorem. Once the structures exist, they appear to have properties waiting to be uncovered.

The physicist and mathematician Roger Penrose has argued strongly for a Platonic view. He speaks of a mathematical realm that exists independently of human minds, and our minds somehow gain partial access to it.

What has always fascinated me is that even people who reject Platonism often talk like Platonists when they're actually doing mathematics. They say, "We found a proof," "We discovered a structure," or "This theorem was waiting to be uncovered." The language of discovery comes naturally.

Your remark that "the only thing we invent is how to write it down" captures the essence of that feeling. The notation is ours; the relationships are not.

Of course, a skeptic might ask: if mathematics exists independently of minds, where does it exist? Not in space, not in time, not in matter. That question has kept philosophers busy for over two thousand years.

Yet there is something compelling about the alternative puzzle. If mathematics is merely a human invention, why does it so often turn out to describe the universe with uncanny accuracy? Why should equations developed for purely abstract reasons later become indispensable in physics?

That was what astonished Einstein and later physicists. Whether mathematics is discovered or invented, its effectiveness is remarkable. As the physicist Eugene Wigner famously put it, there is an "unreasonable effectiveness" to mathematics in the natural sciences.

Your position has the advantage of making that effectiveness feel less surprising. If mathematics is part of the deep structure of reality, then of course the universe obeys mathematics. We are not imposing it on nature; we are uncovering a language that nature was already speaking before we arrived.