From Trespassing on Einstein’s Lawn: A Father, a Daughter, the Meaning of Nothing and the Beginning of Everything, by Amanda Gefter. Copyright © 2014, by Amanda Gefter. Excerpted by permission of Bantam Books, a division of Random House, Inc. All rights reserved. No part of this excerpt may be reproduced or reprinted without permission in writing from the publisher.
If I had come to London to ponder the nature of reality, I had clearly come to the right place. In my philosophy of science class, we discussed it endlessly. Is there a reality? Is it sitting “out there,” independent of us? If so, what is it made of ? How can we distinguish it from mere appearances? Is there any hope we’ll ever know it at all?
In class, we debated the merits of realism and antirealism. Realism is the commonsense belief that scientific theories describe true things about the world—a real world that exists whether or not we’re looking—and that electrons, quarks, dark matter, and whatever other objects appear in our best theories, whether or not they can be observed directly, are real objects, the true ontological furniture of a singular, mind-independent world.
Antirealism is an umbrella category for all sorts of ideas that reject realism in one way or another. There’s Kantean antirealism, which says that while there is a real world out there independent of us, there’s no way for us to know it. There’s Berkeley’s esse est percipi, the more radical claim that behind appearances lurk more appearances, that objects are made not of atoms but of thoughts. There’s social constructionism, which says that reality and truth are whatever we agree to call reality and truth, a theory that reminded me of something my New School postmodernist friends would say and then argue that it had to be true because they believed it, even after I had pointed out that by not agreeing with them I had, by their own definition, proven them wrong. On the saner side there’s instrumentalism, which simply states that science is a tool for predicting the outcomes of experiments; whether or not there is a reality, and whether or not we can access it, is entirely beside the point.
I had already discovered that instrumentalism was a common position among physicists, who always seemed to squirm at any mention of the R-word. It’s the philosophers’ job to worry about reality, they’d say. We just calculate and predict and test.
No matter how many times I heard that, it always struck me as total bullshit. Okay, maybe if you were an electrical engineer or a surgeon or a meteorologist you’d just be concerned with predictions and the outcomes of experiments, but the people I was hearing this from were physicists. Theoretical physicists. People who were dealing with black holes and multiple universes and glitches in the simulation. Maybe when you work in theoretical physics, you feel the need to overcompensate by pretending to be as no-nonsense as a refrigerator repairman, but at the end of the day, who are you kidding? You stay awake nights worrying about how matter behaves at length scales a millionth of a billionth of a billionth of a billionth of a centimeter in six extra dimensions undetectable by any foreseeable experiment, but you don’t care at all what reality is? Please.
Given my propensity for worrying about simulations and shadows and butterfly dreams, I wouldn’t have guessed that I would find myself advocating a strict realist view. Then again, I was a self-proclaimed reality hunter, so entertaining any antirealist ideas would be like shooting myself in the foot. Besides, at times the arguments for antirealism struck me as utterly absurd. The pinnacle of absurdity came one afternoon when a girl in my class argued her antirealist position from a feminist standpoint.
“Wait, did she just say ‘feminist’?” I asked the guy next to me. “Feminist physics?” I couldn’t imagine where this was headed.
“Not only is science a socially constructed enterprise, it is also explicitly male-centric,” she explained to the class. “Think about the terminology. Particles are represented as balls, and they interact with each other through forces.”
Seriously? Balls? I coughed to cover my snickering. Judging by her expression, this was a very serious matter.
“So if physics is socially constructed,” one guy began, “regardless of whether it’s constructed by men or women, you don’t think it corresponds to reality at all?”
“No, I don’t,” she replied.
I couldn’t stop myself from joining in. “So how exactly do, say, airplanes fly?”
“Because we all agree that they do,” she responded. I blinked. “Are you serious?”
Somehow, seemingly instantaneously, the classroom had divided itself into teams—realists versus antirealists. We even shuffled our desks around to make it known exactly which side of this fight we were on.
Antirealism had seemed a rather insane position until I felt the sting of its best right hook: every previous scientific theory ever devised in the history of science has, until now, turned out to be wrong. So what kind of morons would we have to be to believe that our current theories are the exception, the one time mankind—or womankind— has ever gotten it right? And if theories are always turning out to be wrong, how can they possibly be telling us anything about the true nature of reality? I learned that this rather fatal blow is known in philosopher-speak as the “pessimistic meta-induction,” which just means that with some solid inductive reasoning it becomes obvious that science is a hopeless enterprise.
It was a depressing thought, but luckily realism had its own uppercut ready, the argument that I had unknowingly made against the girl who was mad about balls: if scientific theories don’t describe at least part of the true reality out there, then the success of technology— not to mention the success of a theory’s bold, novel predictions that go way beyond whatever observations were fed into it in the first place— has to be chalked up to a miracle.
Okay, so all theories turn out to be wrong, but the technologies we build based on those theories miraculously work. The pessimistic meta-induction and the no-miracles argument formed a kind of stalemate, and philosophers had been bickering about it ever since. One philosopher, however, had found a middle ground. He happened to be sitting in an office down the hall.
I had barely unpacked my things when I started hearing the noises. Scurrying noises. A rustling. On a few occasions I swore I saw a blur of motion out of the corner of my eye. Then one night, lying in the loft, half asleep, I heard a guttural sound, the kind of sound a cat makes before it jumps, a sort of revving of the motor. It startled me and I sat up without thinking, bashing my head into the ceiling. By the time I managed to turn on the light, whatever had made the sound was gone.
It wasn’t hard to guess what was happening. This was London, after all. I had read somewhere that wherever you stand in the entire city, you’re never more than twenty yards from a rat. There were 50 million of them. That was like seven rats per human. Could seven rats even fit in my flat? Not if this one was big enough to make guttural sounds, I thought. I tried to go back to sleep, assuring myself, unconvincingly, that rats can’t climb ladders.
In the morning I went to the hardware store, where I found a disconcertingly large selection of rodent control devices, a whole wall of them. I was gazing at it in awe and confusion when the sales guy asked if he could help.
“I don’t want to be cruel,” I said. “I mean, I want them out. If I could reason with them, I would. I just want something that doesn’t make me a terrible person.”
He nodded. “Then I’d avoid the glue traps.”
He showed me a trap that consisted of a box that you rig with bait and when the rat goes in to get it, it triggers this sort of garage door that falls shut, locking the rat inside, where it waits for you to take it outside and set it free. Not into the wild per se, but at least headed for someone else’s flat. I bought two.
I heard them rustling around down there as I drifted off to sleep that night. Esse est percipi. Esse est percipi. I chanted the phrase like a spell, hoping it might transform any ontologically valid rats into vaporous thoughts I could sleep off by morning. Perhaps the real estate agent had meant to say that this flat was modern and mind-dependent. Reassuringly, I had yet to actually perceive any living creatures; their existence was nothing more than a pessimistic induction. Cogito ergo rats. Maybe the programmers were messing with me. Maybe the strange sounds were just glitches in the simulation. Or maybe my dad had been right and this place was subject to quantum fluctuations, the sudden but fleeting appearance of rodents from an ever-churning vacuum. Maybe as long as I didn’t observe them, they’d be stuck suspended in a kind of quantum mousetrap, half real, half illusion. Schrödinger’s rats.
But in the morning, when I observed them, the traps were empty.
John Worrall had a sweet look about him, like the kind of guy who could broker peace among feuding academics, or the kind who would one day become the leader of a philosopher-of-science-made rock band called Critique of Pure Rhythm. He had started out in statistics but been lured to philosophy by Karl Popper, who had founded the philosophy of science department here. In 1989 Worrall published an article in the journal Dialectica arguing for a middle ground between realism and antirealism. He called his view structural realism, and claimed that it held the best of both worlds: it could explain science’s success without invoking miracles and account for its pessimistic progression from one wrong theory to the next.
The problem, Worrall explained, was that realists were being realists about the wrong things. In fact, “things” were precisely the problem. Realists talked about a real mind-independent world, out there, composed of real things such as atoms and tables and rats. But when you look closely, scientific theories aren’t about “things” at all. They’re about mathematical structure.
A mathematical structure is a set of isomorphic elements, each of which can be perfectly mapped onto the next. The notations 25, 52, and (27 − 2) all share the same mathematical structure. The structure isn’t any particular number—it’s the whole set of equivalent representations of a number, the steady, singular truth behind a multitude of mere appearances. Sets are more fundamental than the numbers themselves.
All of mathematics—all of structure—comes down to sets? I wrote in my notebook. I remembered reading somewhere that the entire number line could be built from the empty set: the set containing nothing. Inside the empty set is nothing. Zero. But the set that contains the empty set is not empty. It contains one element: the empty set. It’s the number 1. Not merely equal to 1, but the very definition of the number 1. The set that contains both the empty set and the set that contains the empty set is 2. Ad infinitum. Or ad nothing.
The number line was nothing more than a series of nested sets, and in its hidden center was nothing. Worrall said that physics was about mathematical structure. Set theory said that mathematical structure was about nothing.
The idea that you could build the number line from the empty set—was it a clever trick or was it telling us something profound about the universe? Was it telling us how to turn nothing into something? Put brackets around it. A boundary. Somethingness emerges from a change in point of view. Inside to outside.
I wasn’t sure how you’d apply that lesson to something like a universe, something that doesn’t have an outside. One-sided coin, the side of things. How do you make something out of that? Even if you could, you’d still be stuck with Russell’s paradox. The barber shaves every man who doesn’t shave himself—so who shaves the barber? If he shaves himself, he doesn’t shave himself, and if he doesn’t, he does. The issue wasn’t about facial hair. It was about the paradoxes that arise if sets can contain themselves. When you take the view from outside the brackets and try to shove it back inside.
Worrall attributed structural realism to Henri Poincaré, who in 1905 wrote, “Equations express relations, and if the equations remain true, it is because the relations preserve their reality. . . . The true relations between these real objects are the only reality we can attain.” Theories are just sets of mathematical relations—equations related by isomorphisms. By equals signs. Quantum field theory doesn’t talk about hard little (cough) balls called particles; it talks about “irreducible representations of the Poincaré symmetry group.” If it makes it easier to picture those irreducible representations as little spheres, that’s your right. But if that picture doesn’t hold up in light of new evidence, don’t get mad at the theory. Quantum field theory is a group of mathematical structures. Electrons are little stories we tell ourselves. Of course, we need stories. There’s a reason “42” is not a satisfying answer to life, the universe, and everything. Structure alone doesn’t quench our existential thirst. We want meaning. And for our brains, meaning comes in the form of stories.
Still, it’s important to separate what theories mean to us from what they actually say. That was Worrall’s point. Theories never talk about objects—only our interpretations of theories do. Theories themselves only talk about mathematical structure. And if we’re realists about structure, the pessimistic meta-induction no longer applies.
When theories turn out to be wrong, Worrall said, it’s usually our interpretative story that’s wrong—not the structure. Take gravity. According to Newton, gravity is a force that masses exert on one another from a distance. According to Einstein, it’s the local curvature of spacetime. The two ideas are contradictory. Both couldn’t be right, so clearly, the antirealists said, Newton’s theory wasn’t describing reality at all, a fact that made it pretty hard to explain how he was able to predict the motions of the planets. Worrall disagreed. If you take away the interpretations and just look at the math, it’s a whole different game. When gravity is weak and velocities are low, Einstein’s equations give way to Newton’s. Newtonian gravity is the low-energy limit of general relativity. Newton had the wrong story but the right structure—only it turned out to be a tiny corner of something much, much bigger. We don’t need miracles to understand why Newtonian gravity worked; it was successful because it had homed in on a small piece of reality’s structure. Einstein discovered a bigger piece, and there’s still more to be found.
The same went for quantum mechanics. Although its description of the world is drastically different from that of classical mechanics— where particles have simultaneously defined positions and momenta, cat obituaries are far more straightforward, and demons can predict the future to infinite accuracy—its mathematical structure reduces to that of classical mechanics when physical systems are large compared to the size of Planck’s constant. As one theory surrenders to the next, physical interpretations are left behind in ruin, but mathematical structure persists. Scientific progress isn’t a parade of miraculously wrong theories—it’s an optimistic snowball, gathering the structure of reality as it rolls. Several more rustling nights were followed by several more ratless mornings.
I hunted around the flat, looking for any rat-sized entryways. I taped up the tiniest cracks in the walls and stuffed the openings around pipes with steel wool. To be extra-cautious, I stacked books all along the perimeter. Just in case they could jump the books, I created various obstacles for them to encounter on the other side. The whole setup got quite elaborate, with makeshift forts and moats and the garage-door traps in the center. The rats might be clever and resilient, I thought, but I had physics books and duct tape and opposable thumbs.
Still, the rustling continued, and one night I was awoken by the thump of a book falling from its fortress. In the morning I saw that it was Julian Barbour’s The End of Time. I wondered if the rats were trying to tell me something.
According to Worrall, I didn’t have to lend any ontological credence to the individual rats—all I had to worry about were the structural relations among them. That made me feel slightly better, but I still wished I had it in me to be a social constructionist. Then I could get rid of the damn things just by refusing to believe they were there— philosophical extermination. Unfortunately, I believed that physics was what made planes fly and rats scurry. Given the data points of shuffling noises, motion in my peripheral vision, falling books, apocalyptic messages, and London, I had to face facts: the existence of rats, quantum or otherwise, was the simplest explanation.
If I couldn’t excise them with Occam’s razor, I was going to have to resort to more conventional methods. “Okay,” I told the hardware guy, “give me the traps that will kill them. But kill them quick, so they don’t suffer.”
He helped me load my basket with rat traps—standard springloaded mousetraps, only bigger. I bought seven.
I went home and rigged the traps. It wasn’t as easy as it looked. They’re supposed to be this perfect, impossible-to-improve-upon invention, but I nearly lost a finger. Eventually I got them all set up and baited them with peanut butter. I had read somewhere that rats love peanut butter. Then I grabbed my suitcase and got the hell out of there.
I was sitting in a Japanese restaurant in Holborn, in central London, waiting for Michael Brooks.
After laying the traps, I had checked in to a hotel a few blocks down the road on Notting Hill Gate. I didn’t want to be around when the rats discovered the peanut butter and I figured I’d enjoy a few extra square feet. After settling in, I had emailed Brooks about a New Scientist article and mentioned that I was now living on his side of the pond. “Since you are here in London,” he had replied, “why don’t we meet for lunch?”
Brooks arrived at the restaurant along with Valerie Jamieson, another physics editor from New Scientist, who introduced herself with a melodic Scottish brogue. We ordered drinks and sushi, which arrived at our table on a large wooden boat. As we plucked fish from the deck with our chopsticks, we chatted about life in London and in the universe at large.
“What’s your view of inflation?” Brooks asked me.
Having just shoved a piece of salmon into my mouth, I had a moment to think. Inflation. On one hand, I understood the appeal. As Guth liked to say, it was the ultimate free lunch: a universe that blossoms from some primordial seed and just keeps on growing, gravity’s negative energy offsetting the limitless creation of infinite space, which stretches quantum ripples into astronomical veins, the gravitational lifeblood of stars and galaxies.
On the other, inflation couldn’t explain why the universe exists at all. From whence the primordial seed? It assumed the existence of the inflation from the start, not to mention the very laws of physics, and in its heart it wasn’t quantum. It didn’t account for what internal observers could see or explain why nothing looks like something. Its logic was Boolean, its view God’s-eye, its approach bottom-up; it was helpless in the face of quantum dragons. Besides, there was that disturbing low quadrupole. WMAP hadn’t found any large-scale temperature fluctuations—not what you’d expect from an inflated universe.
I swallowed the salmon. “I think it has more problems than people admit.”
I felt weird offering my opinion, as if maybe I wasn’t supposed to have one, and as the conversation continued, I couldn’t help feeling a little guilty. Brooks and Jamieson had PhDs in physics, and they were real journalists to boot. I was nothing but a poseur trying my best to fit in. But the strange part was, I felt like I did fit in. As we compared views of inflation and its discontents and swapped stories about our run-ins with quirky cosmologists, it dawned on me that there was a whole community of people out there—writers—who actually wanted to talk about physics over sushi. Science journalism was supposed to be my disguise, but the mask fit a little too perfectly today.
As I snagged a piece of port-side tuna, I couldn’t help but wonder what my father was doing now, on the other side of the ocean. It was morning there. He was probably getting ready for work.
One . . . two . . . three. Turn the key. Take a deep breath. Open the door.
After a week living it up in a hotel, it was time to return to my miniature flat and dive back into ultimate reality. But as I stood frozen outside my door it occurred to me that when I had set up the traps, I hadn’t fully considered the end result. I had wanted the rats gone, but they weren’t gone. They were right there on the other side of the door, possibly seven of them, with snapped necks and shocked faces, the traps sprung and sated like bottomed-out guillotines, the morbid remains of a rodent revolution, a noble troop brought down by Sainsbury‘s peanut butter. And what exactly was I supposed to do with them? Hold a mass funeral? Fire twenty-one shots from a tiny, tiny cannon? Run?
One . . . two . . . three . . . Fuck. Is there anything in there I can’t live without?
After several more failed attempts, I finally turned the key, cringing as I pushed open the door. Inside, I surveyed the gruesome scene. It was even more horrifying than I had imagined. Every last lick of peanut butter was gone, and the traps, still rigged, were empty.
Worrall’s structural realist philosophy had struck a chord in me. If I wanted to find the truth about ultimate reality and the nature of the something that allegedly came from nothing, it was going to be crucial to separate our descriptions of the world from the world itself, what physics really says from the meanings we ascribe to it. But I was confused. Worrall had said that theories talk about mathematical structure, and not about objects. Did that mean that objects don’t exist at all, or merely that our scientific theories can never tell us which objects are the real ones? Was it a claim about what we can know or about what actually exists? Was it epistemological or ontological?
“Epistemological,” Worrall answered definitively when I asked him. “I have a lot of trouble with the idea of relations without relata. And anyway, I feel that we should generally be silent about metaphysical issues. We think about what reality is probably made of via physics. All structural realism does is insist that we should not think that we have any grasp of reality over and above what our current theories tell us.”
At first, Worrall’s objections to an ontological structural realism seemed fair enough. After all, what could it mean to talk about relations without relata? If the world is made of mathematical relationships, mathematical relationships among what?
Maybe they’re not among anything. Maybe the relationships are all that exist. Maybe the world is made of math. At first that sounded nuts, but when I thought about it I had to wonder, what exactly is the other option? That the world is made of “things”? What the hell is a “thing”? It was one of those concepts that fold under the slightest interrogation. Look closely at any object and you find it’s an amalgamation of particles. But look closely at the particles and you find that they are irreducible representations of the Poincaré symmetry group—whatever that meant. The point is, particles, at bottom, look a lot like math.
If structure is all that our theories can ever tell us about the world, forever veiling some unknowable ontology beneath, then our pursuit of ultimate reality was completely hopeless. Accepting Worrall’s epistemological structural realism was like retreating right back into Bostrom’s computer waving a simulated white flag.
On the other hand, if structure is all that exists—if the world really is made of math rather than things—then physics can tell us everything there is to know about ultimate reality. Ontic structural realism was our only hope. My father’s and my mission hung in the balance.
“Does anyone think that structural realism is ontological?” I asked my philosophy professor after class one day.
He thought a moment, then nodded. “You ought to talk to James Ladyman.”
The fact that all the peanut butter was gone was pretty damning evidence that the rats were ontologically valid, but I knew that I couldn’t logically defend my inference to the best explanation. Sure, it seemed the most likely conclusion, but blunting Occam’s razor was the undeniable fact that there were an infinite number of possible unobservables that could explain the peanut butter’s disappearance—though I was having trouble imagining what the hell they might be. Was British peanut butter especially prone to rapid evaporation? Had seven dollops of anti–peanut butter spontaneously sprung from the vacuum, annihilating the store-bought stuff in a sudden burst of light? This underdetermination of theory by the data was bolstered by the null results of the traps, which just sat there, empty, rigged, full of potential energy, itching to go kinetic. I had learned in philosophy class that inductive reasoning was totally indefensible; all the clues in the world just wouldn’t amount to much. The only way to claim that the rats were categorically real was to logically deduce their existence from some set of self-evident axioms, to render them necessary and not merely contingent. Of course, by those standards, a rat could be sitting in front of me waving its paw and my existence claims still wouldn’t hold water. I could hear the contingent bastards scratching at the walls, scurrying in the ceiling two feet above my head.
“Okay,” I told the hardware guy, “I’ll take the glue traps.”
“I’ll tell you what reality is not. It’s not made of little things.”
James Ladyman was sitting on the floor of his hotel room. “We can’t help but think that way, but it’s not what reality is like.” I was swaying in a creaky swivel chair. We had met in the bar of the Holiday Inn, where Ladyman was staying while he was in town for a conference on metaphysics. Despite Worrall’s warning that we should be silent on metaphysical issues, it seemed a whole slew of philosophers weren’t ready to keep their mouths shut. The hotel bar had proved too noisy for a discussion of the nature of reality, so we had retreated to his room, where he was now sitting on the floor, stretching his legs. With a headful of dreadlocks spilling halfway down his back, it would be easy to mistake Ladyman for the bongo drummer in a reggae band, though his British accent carried the distinct melody of academia. “But how do you go from saying ‘structure is all we can know’ to ‘structure is all that exists’?” I asked.
“The motivation for me was looking at contemporary physics and realizing that it doesn’t support an intuitive picture of unobservable objects. You could say particle physics is about mesons and quarks and baryons and electrons and neutrinos and so on, but when you get beyond the pictures they draw and just look at the theories, it’s very difficult to interpret those theories as being about particles, right?” Ladyman said. “So the thing about particles is that they’re not particles. . . . If you want to know what the ontology is, look at what the theory is saying. Don’t try to overlay the mathematical structure with some kind of folky, homely imagery.”
“So physics itself steered you to interpret structural realism ontologically?” I said, grinning.
Worrall had developed structural realism to respond to a philosopher’s squabble. If Ladyman’s version was driven by physics rather than pure philosophy, it stood a better chance of being true.
“Both quantum mechanics and relativity profoundly challenge our intuitive idea that the world is made of objects,” he said. “Quantum particles have all sorts of problems about their individuality: entangled states, quantum statistics. Then in general relativity, spacetime points don’t seem to be the ultimate reality; the reality is something more like a metric field. In both cases we are pushed away from an ontology according to which you drill down and find little things that everything is built of.”
It was a good point. Not only were quantum statistics weird, but they made it pretty much impossible to think of particles as “things.” If you have two electrons, there’s no way to distinguish between them. Electrons have no known substructure; they’re defined solely by their rest mass, spin, and charge, which are the same for every electron. Electrons, by definition, are identical. Of course, you’d think you could distinguish them just by their locations in space and time—an electron here is not the same particle as an electron there, by virtue of their being in different places. That trick might have worked in classical physics, but not quantum. Quantum particles don’t have well-defined positions in spacetime, only probabilities for appearing in various locations, the locations themselves smeared out by uncertainty. The result is that quantum physics renders elementary particles literally indistinguishable, a fact that becomes pretty important when you’re calculating probabilities. If each of the seven rats in my flat inevitably ended up stuck to a glue trap, then I’d say there was a one in seven chance of finding a given rat on a given trap. But if the rats really were quantum, there would be a 100 percent chance of finding any rat on any given trap. If you’re placing bets, knowing whether you’re dealing with classical or quantum statistics makes a pretty big difference. And what would it even mean to call a rat a “thing” if it has no individuality on which to pin its “thingness”?
General relativity only exacerbated the situation. My father had taught me that to keep accelerated and inertial reference frames on equal footing—to turn a curve into a line—you have to bend the paper. The problem is that you can bend it in endlessly different ways and produce the same results, a fact made possible by Einstein’s central principle, general covariance. Different configurations of the paper can all correspond to the exact same physics, a kind of underdetermination that led not only Ladyman but also Einstein himself to believe that the paper itself—the “thingness” of spacetime—wasn’t ultimately real. The only reality lay in the spatiotemporal relationships traced by the paper’s curves. The metric. The structure.
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 common-sense 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.