Trespassing on Einstein's Lawn (20 page)

The more I thought about it, I realized that such underdetermination in ontology runs rampant in physics. It reminded me of Dirac's holes. In the early days of quantum mechanics, Paul Dirac had come up with an equation that made the Schrödinger equation compatible with special relativity. The only problem was that the equation allowed particles such as electrons to have negative energy, something that clearly didn't happen in the real world. To save his equation, Dirac imagined that the quantum vacuum was a sea in which every possible negative energy state was already filled, leaving only positive energy states accessible to electrons. But a new problem arose when Dirac realized that, if excited, the negative energy states could transform into positive energy states, leaving an empty hole in the negative energy sea. The hole would have all the properties of an electron, but with a positive charge.

With his holes, Dirac had predicted the existence of antiparticles. What Dirac had considered a positively charged hole physicists nowadays think of as a positron—an object in its own right, not merely a hole. But the point is, the math never changed. Only the interpretation did. Physicists could just as well stick with the hole picture and they'd still come up with all the predictions for anything they might test in a lab. You can think of a positron as a thing or as an absence, two ontologies about as opposite as you can find, but from the point of view of mathematical structure, they're exactly the same. I wanted to run to philosophy class to tell my classmate the good news:
You don't have to talk about particles as little balls! You can talk about them as
holes
!

“How do you define structure?” I asked Ladyman.

“I'd say it's a system of relations. But then people say, ‘Well, a system of relations is among objects so related,' ” he said, echoing Worrall's critique. “But quantum mechanics and general relativity don't
seem to be based on an ontology of objects first and then relations between them sort of sitting on top. It's really more the other way round. The objects are just nodes in the relational structure or something.”

Balls and holes are merely
descriptions;
they're instantiations of structure, not the structure itself. The real thing is a mathematical relationship. If you're a realist about structure, the underdetermination crisis is averted.

“Does that mean the physical world is made of math?”

“It might be that at a certain level of description it becomes impossible to adequately represent the world other than mathematically. If you read popularizations of, say, quantum field theory, at a certain point the writer has to say, ‘We can't explain this but it turns out that such and such …' The resources they've got to communicate are not adequate because they make people think that we're talking about little particles, and we're not. So the more fundamental a description of reality becomes, the more mathematical it becomes, and the distinction between the abstract and the concrete becomes sort of unstable. On the other hand, I don't want to say that the concrete universe is made of maths. But its nature might be so far removed from our commonsense notion of a concrete physical object that maybe it is less misleading to say it's made of maths than to say it's made of matter. These are very difficult issues. I really don't know.”

“The way I picture it is like reality is the bottom layer, and then you have a layer of mathematics on top, and there's a one-to-one mapping between the two,” I said. “And on top of that you have language, but there's not a one-to-one mapping between the mathematics and language, so something gets lost in translation, like you said. But then my question is, if there's really a one-to-one mapping between math and reality, doesn't that by definition mean that they are the same thing?”

“I suppose the problem at the moment is that we don't have a one-to-one mapping, because even our best theories aren't completely accurate,” Ladyman said. “So yeah, you might think, if we eventually did have a one-to-one mapping, what would be the grounds for denying that reality was mathematical? I'm not really sure. I suppose I'm very skeptical of anything in philosophy that purports to explain the difference
between abstract maths and maths that's substantiated. Because in the end, what could we possibly explain that difference in terms of? Like, I reject the question, ‘What breathes fire into the equations?' Because anything you say is just gonna be figurative, right? Because you'd say, ‘Well, there's the abstract maths and then the actual universe is a sort of substructure of all the possible structure there could be. So what's the difference between the uninstantiated structure and the instantiated structure?' Well, the philosopher will say there's a primitive instantiation relation or something—you could invent some metaphysical language to talk about it, but to me that's no different from saying that some of the maths has pixie dust in it. It's not going to do any work. Because what could it possibly connect to that would have any meaning? If you ask questions in science like ‘What causes an earthquake?' you appeal to conceptual resources and those are nonempty because they're tied to observation. But maths—pure maths isn't tied to observation. If the theory of everything is a mathematical theory, how would you test it? It would have to have some content that has to do with something other than mathematics.”

“I've heard some people say that if you really had a theory of everything, it wouldn't be testable,” I offered up.

“Right, hmmm,” Ladyman said, thoughtful. “That's interesting.”

I could hardly believe I was defending the notion that the world was made of math, given my teenage years as a strict nonbeliever. I was glad my mother wasn't there to get the satisfaction.

But like Ladyman, I didn't see what the other option could be, not if we followed Worrall's advice and listened to “what our current theories tell us.” As far as I could see, our current theories really were telling us that reality is made of math. That objects give way to equations, that thingness melts to abstraction. Given the drastic underdetermination of ontology in general relativity and quantum mechanics, Ladyman's version of structural realism seemed to be the only lifeboat capable of keeping us afloat in a sea of existential crisis and contradiction. As I thought about it, I realized how surprising that was. I mean, you'd think it would be the other way around—that as our theories of physics got better, snowballing ever closer to ultimate reality, they'd offer us increasingly clearer pictures of the objects that ultimately constitute
reality. Instead it seemed that the only clear-cut message they offered was that “objects” aren't the right ontology at all. Not only was physics undermining every intuition we have about the world, it was also weeding out philosophies. From where I was sitting in a nondescript room in a nondescript hotel, ontic structural realism seemed to be the only one left standing.

As I walked the London streets, the sky a dull gray overhead, the pavement slick with rainwater, I looked around at the so-called world. It was crazy to think that everything—the majestic townhouses and double-decker buses, the sprawling green of Hyde Park and the white stone at Marble Arch—was made not of physical things, but of math. Then again, wasn't that exactly what Wheeler had been saying all along?

It from bit:
the world is made of information. Not
described
by information, but
made
of information.
A house is made of bricks but the bricks are made of information.
And what was information if not mathematical structure?

Being a realist about objects was kind of like believing that
love
and
amor
are two totally different things just because they look and sound different. You have to know the rules of translation between English and Spanish to discover that the two words are equivalent—there's a one-to-one isomorphic mapping from one word to the other, a mapping that preserves some underlying structure, not
love
or
amor
but the concept to which they both refer.
Love
and
amor
are words. Descriptions. What's real is what survives the translation, the structural relationship between them. We can't give it a name. Giving it a name would trade structure back for description. Giving it a name would require choosing a single language, a preferred coordinate system, violating general covariance, breaking the symmetry of a linguistic spacetime.

Science is about structure. The stories we tell and the images we create to describe the structure are up to us. The key is to not mistake description for reality. But how do we sort them out? We have to look at all the varied descriptions and find their common denominators, the structure they share, the thing that remains unchanged when you go from one description to the next. And then it hit me.

* * *

I nearly ran from the cab to the door, hurriedly dragging my suitcase behind me, and rang the bell.

On the other side of the door, Cassidy launched into her best rendition of a ferocious bark. “You're a good girl,” I heard my mother reassuring her as she made her way toward the door.

“Oh my God!” my mother shouted when she discovered me standing on the other side, suitcase in hand. “What are you doing here?”

She tried to hug me, but Cassidy pushed past her, hopping and whimpering, her butt wiggling so fast that for a second she lost her footing. She jumped up, put her paws on my chest, and licked my chin. “Cassideeeeeeeee!” I squealed, grabbing her floppy ears and planting a kiss firmly on her snout. She wiggled with delight, then bolted into the yard to pee.

As I gave my mother a big hug, I saw my father emerge from the doorway behind her, trying to figure out what the commotion was about.

“Surprise!” I said.

He hugged me, looking happy and shocked. “What are you doing here?”

I grinned. “I know what we're looking for.”

“Are you hungry?” my mother asked as my father grabbed my suitcase from me and carried it into the house.

I followed them inside. Cassidy trotted alongside me, her tail happily whacking my legs as I walked.

“You must be hungry after your flight,” my mother continued. “I can't believe you flew all the way here without telling us.”

I could see the anger dawning on her face.

“In this family,” she said with a stern voice, staring me down, “we do not fly across the ocean without telling each other.”

“Sorry,” I said. “It was a last-minute decision.”

“Too last-minute to make a phone call?”

“I wanted to surprise Dad. I had an epiphany.”

“Epiphanies can be shared over the phone.”

“I guess,” I said, pouting. “It would have been so much less dramatic.”

I followed her into the kitchen, where my father joined me at the table. Cassidy flopped down on the floor at my feet.

“So are you hungry?”

“I've been in England,” I said. “I'm
starving.
”

“What's the epiphany?” my father asked.

“I can make chicken,” my mother said, peering into the refrigerator. “And I have those spicy noodles you like. Let's see what else. There's fruit salad. There's peanut butter.… ”

Cassidy's ears perked up, but I shuddered at the thought. “No peanut butter. Never peanut butter.”

“What's the epiphany?” repeated my father.

“I can make a salad with feta and walnuts.”

“That sounds great.”

“What about dressing? I have a raspberry vinaigrette—”

“For the love of God,
what
is the epiphany?”

“Okay,” I said, turning to my father. “Are you ready for this?”

He offered up a look of cartoon-like suspense.

“Something is only real if it's invariant,” I said.

He stared off into space, mumbling the words back to himself. “Something is only real if it's invariant.… ”

“Think about it. Invariant means it's the same in every reference frame. It's a feature of the world that all observers would agree on. It's how we intuitively define what's ‘objective.' Here's the reality test. If you can find one frame of reference in which the thing disappears, then it's not invariant, it's observer-dependent. It's not real.”

He sat quietly for a moment, thinking. “So if something is invariant, it's real. And if it's observer-dependent, it's what? An illusion?”

“Yeah. I mean, it's not like a hallucination, it's not subjective. But it's not
ultimately
real.”

“Like a rainbow.”

“Exactly! It's caused by physics, it's not subjective, but it's not real, either. Right? Wait. How do rainbows work?”

“The Sun shines from behind you and the light gets refracted by water in the atmosphere.”

“Right, okay. So you need the Sun and you need the water, so it's objective, but it's dependent on your reference frame. If you move to another spot you might not see it anymore. It's a legit physical phenomenon, but it's a product of your viewpoint. There's no tangible, physical rainbow-colored object hanging in the sky. You can't grab hold of it. It's like a mirage. It's not real.”

“It's like the color of a galaxy,” my father said. “Color isn't a real
feature of the galaxy; it's a feature of how the galaxy is moving relative to an observer. The relative motion changes the wave's frequency, and frequency is what we see as color. So if a galaxy is redshifted, we know it's moving away from us. If it's blueshifted, it's moving toward us. It's a Doppler effect. It's observer-dependent.”

I nodded. “If we want to find ultimate reality, we have to eliminate all the features of the universe that are observer-dependent until we're left with the ones that are truly invariant.”