Transcription
Nature is lawful, as far as we can tell. The universe and all that it is made of changes from moment to moment according to fixed, unchangeable, immutable patterns. Those patterns, as it has been tested by a couple of hundred years of observation and calculation, can be expressed in the language of mathematics. Indeed, it is this mathematics right here that describes the fundamental equations of the world. Within this mathematics, we have a description of Einstein's general theory of relativity, the standard model of particle physics, equations that describe how the force of gravity works, how the electromagnetic force works, how the strong and weak nuclear forces work, and all the particles of matter on which those forces act. This takes us tantalizingly close to the reductionist dream of a theory of everything, at least at the level of the fundamental laws and the fundamental ingredients.
Now, I can well imagine that if you're not familiar with all of this, you may still be wondering, "What does that really mean? I mean, how do these symbols describe these particles?" Well, the answer is that we can use mathematics to delineate the configuration of the particles. We can use numbers to describe where the particles are and how the particles are moving, or in the quantum mechanical language, the probabilities for those qualities. We can use mathematics to describe the intrinsic features of those particles. Right? I mean, the mass of a particle—that's a number. The amount of electric charge that a particle has—that's a number. The spin of a particle—that is a number too.
And here's the thing: we can take all of these numbers as input and use the mathematical equations to produce as output another set of numbers that delineates the configuration and the properties of those particles at any subsequent moment in time. That is how mathematics can encapsulate the patterns followed by all of nature's ingredients, the fundamental ingredients of everything, as they evolve from moment to moment through time.
Now, this is a remarkable achievement, showing how mathematics can describe the universe at a fundamental level. But even long before these modern developments, mathematics had already shown itself capable of giving deep insights into the workings of the world. I mean, even 20 centuries ago, the Greek mathematician Eratosthenes was able to use mathematics to figure out the size of the Earth. The pattern that Eratosthenes used is a very simple one. He noted that the difference in the lengths of shadows cast by two identical sticks at a given separation from one another—the difference in the lengths of those shadows—depends sensitively on the size of the Earth. Right? I mean, as the Earth gets bigger and bigger, the less and less the difference between those lengths becomes.
By actually measuring the difference in the lengths of the shadows cast by two identical sticks, actually calculating the angle of those shadows, but it's all the same in the end, Eratosthenes was able to figure out the size of planet Earth. That's a remarkable demonstration of the capacity of mathematics to reveal qualities of the world that are seemingly beyond our ability to measure.
Now, some centuries later, Isaac Newton took this idea of using math to describe reality and took it to an extraordinarily new level. Through careful observation and thought about how objects move, Newton was able to discern patterns in the motion of those objects. Perhaps most famously, he found that if you push on an object, if you exert a force on it, that object will speed up; it will accelerate, and that acceleration is proportional to the force and inversely proportional to the mass of the object—embodied in Newton's famous second law of motion.
Now, Newton also found that if you have two objects out there in space, the pull of one object on the other through gravity will be proportional to the product of their masses and inversely proportional to the square of their separation. That's Newton's famous universal law of gravity. These equations are central to what came to be called classical physics, and what a stunning picture of the world classical physics gives us! If you provide me with the positions and the velocities of all objects, all particles, we can use the mathematics of classical physics to predict the configuration, the motion, and positions of those particles tomorrow, the next day, the next year, the next millennia, or a billion millennia into the future.
Newton kind of likened the universe to a grand cosmic clockwork. You wind up the universe, and the mathematical laws then dictate how the universe ticks forward in time. Whether it's a turning cog, a spinning wheel, a tumbling raindrop, or a hurtling rock, or the orbiting moon, Newton's mathematics gave us an astoundingly precise description of the natural world.
On occasion, when Newton's mathematics seemed to be a little bit off, oftentimes it was actually just pointing toward some hidden feature of reality. Back in the mid-19th century, astronomers noticed that the orbit of Uranus had some irregularities that could not be accounted for using Newton's universal law of gravity. But they also found that those irregularities had their own inner pattern—a pattern that could be explained if one imagined that there was a hidden, hitherto unknown planet out there in space whose gravity was tugging on the planet Uranus. Indeed, the astronomers were able to calculate where that hidden planet should be, and on a September night in 1846, when those astronomers turned their telescopes to the predicted position in the night sky, they discovered the planet Neptune—a remarkable example of mathematics revealing qualities of the world.
But even Newton's ideas, Newton's mathematics, were heading toward their own reckoning. In the early years of the 20th century, scientists began to examine with precision the microscopic realm: molecules, atoms, subatomic particles. In that domain, the Newtonian ideas, however successful they had been in describing motion in the skies and motion here on Earth, were giving predictions that were completely at odds with the data. Remarkably, it just took a single generation of scientists to overhaul Newton's classical ideas, giving us the quantum mechanical description of the world—a description in which the universe evolves according to a mathematically precise game of chance, with the mathematics given by a new quantum equation: Schrödinger's equation. An equation that makes its own astounding predictions about the micro world. It tells us that particles should be able to penetrate through seemingly impermeable barriers. It tells us that two distant particles can somehow influence each other instantaneously across any distance. The math even predicts the existence of particles that had never been seen before, and all of those predictions have now been borne out by a mountain of experimental data.
The quantum mathematics is jaw-droppingly precise. We can even use it to describe certain properties of particles, such as the magnetic properties of electrons, and the mathematics makes predictions to many decimal places. When we do the experiment, the measured values agree with the theoretical prediction digit by digit by digit by digit—a clear demonstration of the power of mathematics to describe reality. Even right here, we can see the work of quantum mechanics in the fact that in this domain right now, this is what this room actually looks like. But using the power of quantum mechanics, which has given rise to the powerful computational tools of the digital revolution, we can take this room and make it look like this.
Now, all these successes of the mathematical description of the world raise a number of difficult and thorny questions. Right? I mean, is mathematics invented or is it discovered? That is, is math a product of the human imagination? Is math a language that we have developed over the course of many centuries for the express purpose of articulating the patterns that we encounter in the natural world? In a sense then, is our sense that the world is mathematical nothing but the fact that we have developed a particular tool—a kind of mathematical hammer—that makes everything look like a mathematical nail? Or, to the contrary, is math discovered? Is math out there, independent of our understanding, independent of us? Is it deeply woven into the very fibers of reality? And if that is the case, is it still possible that there's something beyond mathematics? Perhaps there's something beyond math that can illuminate the most ephemeral and precious qualities of human experience, like consciousness and the kinds of things that conscious beings contemplate, like value, ethics, morality, purpose, and meaning.
These are the kinds of questions that we're going to be talking about here today. They're difficult, they're contentious, and there is no consensus on the answers to these questions. But these questions will surely drive an energetic, vibrant conversation on the relationship between mathematics and reality.
All right, let's get to it. So with that, let me now bring in David Albert, who is the Frederick E. Woodbridge Professor of Philosophy at Columbia University and a physicist who explores the foundations of quantum mechanics, world-renowned for his insights into philosophical questions about the nature of space, time, and other problems of modern physics. Welcome, David Albert.
Next is Sheldon Goldstein, a professor of mathematics, physics, and philosophy at Rutgers University. He studies the foundations of quantum theory, probability theory, and statistical mechanics, the notion of increasing entropy, and the arrow of time. Welcome, Shelley.
Next is Sylvia Jonas, who is a professor of philosophy at the Munich Center for Mathematical Philosophy, with a PhD in philosophy from Humboldt University in Berlin. She studies epistemology, the philosophy of mathematics, science, metaphysics, meta-ethics, philosophy of religion, and aesthetics. Welcome, Sylvia.
Finally, last but absolutely not least, is Max Tegmark, a physicist, cosmologist, and machine learning researcher at the Massachusetts Institute of Technology and a co-founder of the Future of Life Institute. Welcome, Max.
So thanks to all of you for joining us for this conversation about these issues that really people have thought about, struggled with, and come to a whole wide variety of perspectives on over the course of truly hundreds, if not thousands, of years in one form or another. I thought I'd begin to set the stage by giving some quotes from famous, accomplished, renowned mathematicians in the past who weighed in with their own views on some of the questions we'll be talking about, just to sort of set a bit of a backdrop of how the field has viewed these questions.
So let me start with a chronicler: "God made the integers; all the rest is the work of man." Famous mathematician G.H. Hardy said, "I believe that mathematical reality lies outside us, that our function is to discover or observe it." Andrew Wiles said, "More in our generation, to tell you the truth, I don't think I know a mathematician who doesn't think that math is discovered." And it wouldn't be right to end this list of quotes without bringing in Albert Einstein himself, who once said, "As far as the laws of mathematics refer to reality, they are not certain; and as far as they are certain, they do not refer to reality."
All right, so those are a whole variety of perspectives on the invention question, the discovery question, whether it's out there, whether it is something mathematics emerges from the machinations of the human mind. But let's begin at the beginning, if you will, by just having a little bit of the history of mathematics. I mean, oftentimes when we learn mathematics in the classroom, at least when I learned in the classroom, it was just presented as this body of knowledge. There is no sense that this knowledge even emerged from the struggles of humans thinking about things. But of course, mathematics, as we understand it, as we practice it, as we use it, is a body of insight that emerged over a long course of history.
So, Sylvia, can you just give us a thumbnail sketch of the history of mathematical development?
Sure. Yes, there are four stages in the unfolding of mathematical history. The beginning was in ancient Mesopotamia and Babylon, a tool that was used, for example, for calculating lunar calendars or measuring plots of land in order to determine who owns what. In ancient Greece, mathematics became something quite different. The ancient Greeks associated the laws of mathematics with something that they considered to be ultimate reality. Plato, for example, thought that ultimate reality is not physical reality, which is what we think today, but a sort of heaven of eternal forms—not only mathematical entities but also things like beauty, justice, truth—all of these with capital letters, as it were.
There is this famous legend that it said on top of his academy, "Let no one ignorant of geometry enter here." He believed that mathematical knowledge was the only kind of perfect knowledge, and in order to gain knowledge about any other thing, you would first have to be proficient in mathematics. So ultimate reality was seen to be in this Platonic heaven, outside and completely different from physical reality.
Jumping ahead centuries, during the time of the scientific revolution, things were completely different. All of a sudden, scientists started to discover that our physical reality follows mathematical laws and that we can actually explain empirical phenomena using mathematical laws. So in that sense, ultimate reality was now identified with physical reality, and mathematical and physical reality became two sides of the same coin. They were seen as interwoven. That picture changed again in the 19th century with the introduction of mathematical concepts that didn't have very straightforward relevance for our physical world anymore—non-Euclidean geometries, for example.
So all of a sudden, mathematicians started to feel free to explore mathematical entities that had no connection to the empirical world anymore, and that caused a complete shift in the way mathematics was seen—no longer an essential part of our physical reality, but as a tool that scientists can use, but they can also discard at their pleasure. Today, things have yet again changed. We're at a point where we can see that mathematical reality and physical reality intersect, but there are two separate realities existing independently of one another.
Great, so thank you for that history. Shelley, you know, in that history that Sylvia just recounted, and of course in your own experience as a philosopher and a mathematician who focuses on questions about the physical universe, do you see math as this tool for describing the external reality, or do you see math as something bigger than that? In the language that Sylvia was saying, touching upon a reality that perhaps goes beyond the physical reality or is parallel to the physical reality, somehow separate from the reality that's made up of tables and chairs?
Yes, I mean, I certainly see it as a tool, but I certainly see it as something beyond the tool. I guess for me, however hard it is for even me to understand, even though it's my view, I think that mathematical reality is the reality we know best—the reality that we can be most confident about. Even if we have the sense that we don't know where it is or how we could know it, given how we interact and gain knowledge, there are so many mysterious things about mathematics. But at the end of the day, I do think it's the thing we know best, and it's the reality which we can have the most confidence in.
And so mathematical reality exists in human heads for sure, right? So when we think about the ideas of mathematics, it amounts to something inside of a human brain firing one way or another—thinking about a square root, thinking about a factorial, thinking about the number 17 or whatever. So, David, are we fooling ourselves in thinking that math is somehow out there touching some deeper reality when, in essence, it's really just something that is invented and lives the only place we're certain that it lives—in the human mind?
I'm not sure that I have anything particularly deep to say about this. I should say to the audience, David always begins every answer that way, and he always has something deep. But anyway, I mean, there's a famous 20th-century philosopher named Quine. One of the things he's famous for is a criterion for what we should take to be real, for what our discourse commits us to thinking actually exists. Basically, the idea is if there's something that we find we can't do science without talking about as if it's a real thing, then we should count it as real, and otherwise we shouldn't.
Quine famously thought that certain claims about numbers are the kinds of things that you have to talk about as if the numbers are real things. Other people, people who were very interested in Quine's criteria, thought he might have made a misjudgment about this particular case. People famously like Hartry Field, for example, have tried to show that there are ways of doing science without acting as if numbers are real objects out there in the world. I think it might be useful to regard this as a scientific question and allow that to guide your beliefs about what may exist and what may not exist. The guide is essentially if you find that there's a certain kind of object whose existence you need to commit yourself to in order to get the scientific project done, then you ought to believe in that; if not, not.
So just every time you are playing with the Schrödinger equation from quantum mechanics, which is a subject that you've spent a lot of time thinking about, if you're not thinking hard about the question of invention versus discovery, do you have a sort of working methodology that's in the back of your mind that views this as a mere description of the world versus when you write down Schrödinger's equation, you're like, "This is the world"?
There are lots of physicists who like to say it's a great mystery, this question of how mathematics applies to the world, why mathematics applies to the world, why we're so lucky as for it to be the case that mathematics applies to the world. Fraga famously says, "I don't see what these people are talking about. I can't imagine a situation in which mathematics would fail to apply to the world." If somebody says to me, "I have two apples over here, and I have another two apples over here," isn't it a miracle that the fact that mathematically, abstractly, two plus two equals four applies to these apples? I'm tempted to turn the question on its head. It's very difficult for me to imagine what the world would have to be like in order for that to fail to be the case.
So, Max, jumping off from that, I know you have very strong views on the reason why mathematics does work in the manner that we've been describing. We're going to come to that later on in the discussion, but just in a more cursory way right now: math in your mind—invented or discovered?
Discovered. So it's out there.
Yeah, I think when people have these arguments—"Is it invented or discovered?"—when people say it's invented, I think what they are often after is that the language of mathematics is invented. Like we invent the calligraphy by which we write the number five, because in China you can write it differently, or we invent the word for five, you know, which in Swedish is "fem." So obviously that wasn't discovered; we humans just kind of made it up. I think that's very analogous to, like, if you look in the solar system and argue about whether the planet Neptune was invented or discovered. Obviously, the name for it was invented, because in Swedish we call it "Neptunus." There's nothing fundamental about the existence of Neptune. I would very strongly argue that the existence of Neptune is not something we made up; it's just out there.
In exactly the same way, when you go and look for mathematical objects out there, like Plato, as you mentioned, Brian, was really interested in these regular shapes called Platonic solids. He could invent the name "dodecahedron" for the one that's made of 12 pentagons; he could have called it "schmodecahedron" instead if he wanted. But he couldn't invent the sixth one, because it doesn't exist, just like you can't just invent another planet in the solar system. In that sense, I feel that these mathematical structures really are discovered, even though we invent the language for talking about them.
Well, let me ask the group: do any of you allow for the possibility that one day we're having a conversation with some intelligent alien civilization, and they ask us, "Okay, what have you guys been doing to try to figure out reality?" We show them our latest, greatest mathematical description of the world, and they turn to us and they kind of roll their eyes or they sort of pat us on the head and say, "We understand that direction mathematics; we tried that for a while too, but then we found that it was a dead end, and the real way for describing reality is this," and they give us some other description of the world or way of thinking about the world that's not math, it's not even disguised math, it's just something else. Do you allow that as a possibility?
I certainly allow it as a possibility, although I wish I could imagine what that would be like. But probably it's important to distinguish between whether or not our world has to be described in mathematical terms—that mathematics is the language to express the nature of physical reality—and whether, even if physical reality could best be described in other terms, there would still be mathematical reality. The validity of mathematical reality wouldn't be diminished. You could be an imperialist about mathematics and say mathematics is everything; you could be the opposite and say mathematics is nothing. I think the reasonable view is mathematics is something but not everything.
But one of the arguments that certainly has been put forward over a long period of time for why math should be viewed as really out there is, in some sense, you know, the unreasonableness of mathematics, to quote a famous paper, in describing how things actually work in the world. So just to be a concrete example: when Isaac Newton was writing down the universal law of gravity, he had access to certain observations, certain data, to a certain degree of accuracy, a certain precision. You could imagine he wrote down this formula that was able to describe things at that level of accuracy. But then, as the years and decades go by, our ability to measure the world gets better and better, and shockingly, Newton's ideas continue to work. That leads a sense of, "Okay, this is some deep actual truth of the world, not just some human description of the world."
But then the counter-argument is, at some point, the data becomes so precise that Newton does fail, and of course we have to bring in Einstein's description of gravity—a different equation, different mathematics. So then it leads you to believe, "Well, maybe Newton's was just a provisional description made up by a brilliant human mind in the late 1600s, and it was later replaced by another description by another brilliant human mind, namely Albert Einstein." That seems to pull us back to the idea of description as opposed to the math being out there.
Sylvia, does either of those ways of looking at math pull you in one direction or another more toward it being out there versus invented, or more toward the reverse?
I certainly would also think that the success of mathematics in application, scientific applications, points towards mathematics being out there—mathematics being discovered. Sometimes, the examples you were just referring to, it looked to me like what was happening was that we were eventually revising our physical theories. But if we're talking about purely mathematical theorems, they are eternally true. Once they're approved, they don't get revised anymore. So I think it's important to draw that distinction.
And I think that applicability is really one of the most forceful arguments in favor of what philosophers call mathematical realism—the view that mathematics is out there, that mathematical objects exist independently of human minds. But it's one thing for math to exist, as both you and Shelley were referencing, as something that perhaps might be given the label "real," versus mathematics being out there as the reality that we can touch and feel and interact with, as opposed to something that we're able to cogitate about in some abstract sense.
So, I mean, I'm going to be straightforward about it: I waffle on this question. I have changed my mind over the course of my career between whether math is invented or discovered. I almost can tie it to how well my research is going. When the research is going really well, it kind of feels like all that I'm doing is discovering something that's out there. It just is falling into place, and you feel it has to be out there because it's locking together with such power and such economy, and it's so gratifying to see the equations just really doing what they're doing almost independently of the human who might be scribbling them out on a piece of paper.
At other times, I've come to the reverse, where it just feels like we're just trying to push this mathematical description on the external world, and sometimes it resists because it's not the right language or the right approach for actually describing physical arousal. So I go back and forth on this. But you personally, do you come down on that issue with one unassailable and immutable view, or do you also find it going back and forth?
I'm pretty certain that mathematics is an objective thing out there, so I've never felt very drawn to fictionalist views of mathematics. The notion of mathematical fictionalism—that really, in some sense, what we've done is we've written this wonderful rubric called mathematics, which has proven itself capable of describing the world, but we couldn't even describe the external reality without already having that rubric to begin with—how would we even interpret the motion of the moon if we didn't have the mathematical language to talk about it? How would we talk about the motion of particles if we didn't already have the vocabulary that mathematics provides us?
So is that perhaps biasing our view? It might be, though I think, to be fair, I think fictionalists about mathematics are perfectly happy to say that mathematics is perhaps an indispensable tool or a very important tool that we need to do empirical science. I just think that the fictionalist stance, the fictionalist philosophical position, arises from an unwillingness to posit the existence of abstract objects out there. Much of the fiction in this position is a reaction against mathematical realism or Platonism, even.
And so I think fictionalists' arguments have so far not succeeded in convincing me that there is something wrong with mathematical realism. David earlier mentioned Hartry Field. He is famous for having managed to nominalize Newtonian gravitational theory, which means that he managed to reformulate the theory in a way that doesn't refer at all to mathematical objects or to mathematical entities. It doesn't use any mathematical language anymore.
And that's an incredible achievement. And so that's more than just translating a mathematical sentence into an English sentence, like "force equals g times m1." I mean, all that—it's much more than that.
Yeah, there is a fairly precise and fairly compelling, once you follow the arguments, criterion that Quine wrote down to diagnose whether this or that theory that you believe in is treating this or that kind of object as existent. And Field, in what he was doing, was constrained by that and wanted to show that Newtonian mechanics could be formulated—the Newtonian mechanics or the gravitational part of Newtonian mechanics could be formulated in such a way as to satisfy the condition that, according to Quine's criterion of existence, it didn't count numbers as existing objects.
And so did that push you in a direction of thinking that math is auxiliary, that math is displaceable, to the extent that you believe something like that and to the extent that you believe that it could be extended to all of natural science?
I might say that Hartry Field himself recently is more pessimistic about the possibility of that project succeeding than he used to be. I was just bringing it up here because these questions do—do numbers really exist? So on and so forth. I think what's precious in these conversations is to find some way to get beyond people scratching their chins and looking as if they're having profound thoughts and actually find some criterion that you could get your teeth into and that you could do some work on and claim to have discovered the beginnings of answers to these kinds of questions.
Maybe at the end of the day this isn't going to pan out; this isn't going to be the right way to decide which things exist and which don't, and there are going to be these intuitions of the kind that Shelley evoked—that this is something mysteriously that we feel we know with a level of certainty that we don't experience in any other way. And at the same time, it's not something that we can tell a story about, as we can with Neptune.
And with reference to what Max was saying, I think there's a big disanalogy between Neptune and claims about the existence of the number five or something like that. We know how to tell a story about how we know about the existence of Neptune. It's a causal story. It has to do with Neptune's having direct physical influences on a chain of things that eventually ends up inside our heads. In the case of mathematics, we don't have a story like that—a causal story that links the existence of the number five to a certain brain state of mind.
So there's a real dilemma there. There are real powerful intuitive forces pulling you in two different directions at the same time, and that's a tension in which it's imaginable just to remain forever. The thing that's refreshing about an observation like Quine's is that it might give you another direction, another slightly different way of raising these questions.
So, Max, the number five presumably has never done anything to you. You've never felt the number five; you've never reacted the way you would if a black hole were next to you.
What's that?
Well, there you go. So it has affected you in some metaphysical way. But does that description that David just gave make you rethink it all—the analogy with Neptune that you gave?
I would push back, actually, on this a bit. I think the reason that the business with causality comes in and so much in your arguments that you so eloquently made there, David, is because causality only makes sense even if once you have the concept of time. Right? First things are like this, and then that somehow causes things to be like that at a later time. The mathematics in our universe is at a much deeper level that even transcends time.
If you come back and ask questions about the things that mathematicians ultimately really study in mathematical structures, there isn't necessarily even any time there. Like, look at the cube. You know, if you study the cube, what properties does it have? It has six faces, it has 12 edges, it has eight corners. Was the cube created causally? Did the cube once have only five faces? No, the cube exists completely outside of time and space as well, for that matter.
And when we look at how our physical space is thought about in science, right? Einstein came along and said, "You know, time we should think of just as the fourth dimension of this timeless thing called space-time," which is—and if the reality they were having this discussion in right now were a movie, then space-time would be the entire DVD.
So what I'm saying here, Brian, in answer to your question, is these mathematical structures that mathematicians study—they don't exist in time or in space. But space and time exist inside of some of them, like inside of the four-dimensional space-time of Einstein, which mathematicians refer to by the nerdy name of Minkowski space.
And five is just like that. Mathematicians don't just study five; they study this one mathematical object called the integers. And five of them—five is part of that. Where the integers ever created? There's no reference to time in there at all, and therefore there's no element of causality in it. It's in that sense that I feel these mathematical structures just exist, and we're not obliged to explain what caused them to pop into existence because they never did.
If I can jump in, the causal story wasn't about questions about how they got there. It came up in connection with questions about how we know about them. We have a how information about them—if we're thinking of them as things outside of our heads, no less than tables and chairs are outside of our heads. Okay? There's a question that arises about how they got from their positions—how information about them got from outside of our heads to inside of our heads.
In the case of tables and chairs, we have a story about that—how that happens. That story involves causation. I completely agree with you that causation is already a higher-level notion; it's based on mathematical ideas, so on and so forth. Nonetheless, once we have this causal language, it gives us a satisfying story about how we found out about these tables and chairs.
There may also be a causal story about how the tables and chairs got there, but that's not what we're concerned with at the moment. We're concerned with how we found out about them. I think the cube would still have six faces even if it never got into our head. You know, if you rewind five billion years ago and there were no humans, if there were some aliens who built this supercomputer that started classifying geometric objects, it would still get the cube on the list, and the computer with the calculator has got six faces, and it has eight corners and twelve edges, even if there were no heads at all in existence in the universe at that time.
And it's in that sense that I feel that mathematical structure has those properties, and data came along, and we found out about them. But we didn't make it pop into existence; we just discovered its existence.
Something that I think—and of course, Sylvia knows much, much more about this than I do—but something that I think has often puzzled people about mathematical knowledge is that that kind of an account of how this stuff got into our heads doesn't seem to be available in that case. We cannot tell a story that is convincing to us by our today's standards of how we have epistemic access to these mathematical objects that are purportedly out there.
I just wanted to point out that Plato, of course, had a story about it. He thought that our souls, before we came into existence, before we were born, our souls were communing with the eternal forms. During life, our mathematical knowledge is a recollection of that time. But also, mathematicians like Gödel, for example, thought that we have our five senses with which we perceive physical objects, but we also have mathematical intuition. He believed that there is this mathematical landscape out there, and we perceive it with an additional sense, perhaps we could call it our mathematical sense.
He thought there was nothing particularly perplexing about telling such a story. I think it's just that nowadays we're used to asking for an empirical story, an empirical explanation for kinds of things that we can or cannot know, and that seems very difficult in the case of mathematics.
The reason we have trouble being satisfied with how we come to know mathematics is precisely because we know it so intimately and so well. This would require a lot of elaboration, but I really do think that's what's going on. It's a bit like consciousness. If we know something directly and intimately, maybe we shouldn't expect to be able to give an account of how we know it—an account of the kind which we feel so satisfied about for example knowing that Neptune was there. We do the analysis, and we're so satisfied it comes out right, but that's because we knew it indirectly, not intimately.
And that's true of so much of physics. You know, we've had some conversation about math describing the external world that we can touch and feel and observe and measure. And you just brought up consciousness, Shelley, which is the next section, the next chapter that I wanted to move into, which is the paradigm of equality of the world that we can't grasp in the way that we can more familiar physical objects. It's something that, as you say, we have deep knowledge of, we have intimate connection with, and yet it's something that most people, I think intuitively, would think stands outside of mathematics—not something that can be described in that particular language.
So is that enough to really understand consciousness, or is consciousness something that is much more perplexing than any of the qualities of the world that we've been talking about so far?
Here's what I tell my students: I basically, with regard to physics, I'm basically a pretty strong reductionist. So at the end of the day, our simple hope will be at the end a simple t-shirt equation which captures all the physics. We don't have anything like that yet, presumably, but the equations we do have are beautiful, and they explain most of what we say around it. Basically, I say that, you know, the physics—fundamental physics—should explain everything. But then I tell my students, "Once and then, leave that out and forget that for the rest of my course." I tell them, "Yes, but I lied to you: consciousness is left out. Physics does not explain everything."
So you really think that consciousness is left out?
Absolutely. I'm absolutely convinced of that.
Why?
Let me just say this is not a challenge. If you could convince me of that, I would be eternally grateful, because my view is a much more the first view—the reductionist view—that at the end of the day, we're bags of particles governed by physical law, and that's all there is to it. It's a kind of bleak, but I think beautiful way of looking at the world. But tell me why I'm wrong.
Look, I couldn't convince you. Look, I had exactly that view for many, many years. I think sometime after I finished graduate school, I suddenly—for me, it was just an eye-opening experience saying, "Oh, I was wrong all these years." I was so strongly arguing against all these people who said, "No, no, consciousness can't be accounted for by physics or mathematics." I couldn't understand why people were saying that, and suddenly, you know, the phase transition occurred in my thinking.
Do you remember what induced that phase transition?
Yeah, but I don't remember. I know it was reading Wigner's remarks on the mind-body problem, but when I went back to look at it later to see what was it precisely in there that did the job for me, I couldn't see what it was. So, no, to me, it's not clear what it was.
But certainly, my thinking just changed dramatically. I mean, I think David Chalmers does a great job in three or four hundred pages of explaining why it is that he thinks that consciousness transcends physics and mathematics. But yeah, you would like me to convince you of it. I mean, I find this is the hardest thing in the world to convince anybody.
Yeah, yeah, yeah.
You can't remember an hour or two hours, three hours, weeks, maybe months can do it of discussion. It's maybe useful just to list a couple of reasons that people do give, whether any of these reasons capture what's going on with Shelley. There are several different kinds of reasons that people cite for being deeply puzzled about how a reduction of consciousness to physics would work.
Somebody says to me, "A bunch of particles arrayed like this and moving in this way—that's a table." Okay? And if I go back to them and I say, "Why is that a table?" There's a lot they can say. Well, look, solve the equations of motion; you'll find that particles assembled in this way will hold up plates and glasses, and you can be convinced that the way we ourselves recognize what a table is is by means of its causal interactions with us and with the rest of the world. You can give a very convincing story about why an assembly of particles arranged in that way, moving in that way, is a table.
If you try to play the same game with, say, the sensation of green or the sensation of red, okay, there's a feeling that somebody tells me the sensation of green is a bunch of ions in your head going this way, and the sensation of red is a bunch of ions in your head going this way. It's just hard to see why there's something greener about this kind of motion and redder about this kind of motion. It feels like a case of comparing apples and oranges. You just don't know what to say; you don't see what they have to do with one another. You don't see what the connection is.
And this is what Shelley talked about in terms of the intimacy of our acquaintance with these things. Unlike identifying something as a table or a chair or an orangutan or a computer or anything like this, plausibly we pick out all of those things by means of their causal effects on us. The sensation of redness doesn't seem like that. We don't distinguish red from green in terms of their different causal effects on us. We distinguish them from one another somehow just by the state of being in them.
Conscious phenomenal states seem like the only things in the world that have that kind of relationship to us. So this is one reason why people are puzzled. I don't know if it's Shelley's reason; it's certainly there are other ones people cite.
Suppose I say of myself, "I'm just a machine. I'm just a collection of billiard balls bouncing around like this." And somebody says, "Why do you believe that?" You say, "I think I have good reasons for believing that our scientific investigations of the world suggest that something like that is true." And the guy says, "Gee, I'm puzzled. You just claimed that you're the kind of physical system whose properties, including your beliefs, are just determined by the physical laws."
Okay? But now you're acting as if there's something else that goes into determining your beliefs—some kind of sensitivity to what's reasonable. But if your model of yourself is the correct one, that model denies that there is any kind of additional sensitivity to reasons.
If somebody asked me why I believe that two plus two is four, I have the feeling that I can see how it couldn't not be the case that two plus two is four. This is the kind of certainty that Shelley refers to. But if I really take it seriously that everything about me is just covered by the initial conditions and the laws of physics, then there's only one explanation available of why I believe that two plus two is four—that's because of the initial conditions of the world and the laws of physics.
Okay? And there's nothing about the truth that enters into the explanation of my belief that two plus two is four.
So I would like to inject some optimism here, both to David, who was worried that his bouncing particles couldn't really be trusted in their views about the outside world, and to you, Shelley, who are worried that consciousness can forever be on physics and reductionism, and also to you, Brian, who was using the word bleak to describe reductionism.
So if we start with you, David, I think that the reason you should actually put some trust into what your bouncing quarks and electrons are having you tell us about the state of the world is because, as Brian said, there is this additional story you can add on top of it—the super beans on just particle physics. It just follows from it, and it's the story of Darwinian evolution.
We obviously—there were probably a lot of organisms who were really, really lousy at taking information from the environment and coming up with correct predictions, and they got eaten or starved to death. And those who were still around have evolved to actually tend to have more reasonable beliefs, mostly about what's going on.
Just to—I just want to get to Shelley first. Also, for this notion that we're forever doomed to never be able to explain consciousness, I think history actually gives a lot of hope here. So Brian mentioned Galileo before. If Galileo threw a grape and a hazelnut, right, he could predict wonderfully how they would both move in a parabola and when they would hit the ground, but he had no clue why the grape was green and soft and the hazelnut was brown and hard.
And that really seemed to most scientists at the time to be just beyond physics. How could he ever predict that it was going to be green? But then Maxwell's equations came along and showed that colors and light can be described by mathematics. The Schrödinger equation of quantum mechanics was discovered and helped us calculate why the hazelnut is hard and the grape is soft.
And gradually, physics went from this situation where it could describe almost nothing except motion of the phenomena in the world to where we are today, where it can describe almost everything—from the subatomic world to our expanding universe and the formation of black holes.
Interestingly, what's left—the final biggest bastion of ignorance that we have to try to conquer—are intelligence and consciousness, which is exactly where we're going in now. And we will come back and talk about this later, but my guess is that both intelligence and consciousness are particular kinds of information processing that we can ultimately also describe with math.
In fact, that's the reason why AI has been so successful recently in making some progress on the intelligence side. And there are mathematical theories by Giulio Tononi and others also that try to do with consciousness. But I agree we should be humble; we're absolutely not there yet. But we should be optimistic that there's hope.
And last, Brian, you said it felt bleak to acknowledge that you're just a blob of quarks and electrons evolving according to the standard model of particle physics. I object to you.
So I shouldn't say "bleak." Many people see it as bleak. My own view is it's spectacular that particles and bags and collections of products do the things that we do. So my view is actually quite the opposite. But I don't know where you come down on it.
So Richard Feynman, who's inspired me enormously, has this very beautiful counter-argument to this. You know, he was arguing with an artist who said that physics just ruined everything by trying to describe it reductionistically. You hold up a flower and say, "Look how beautiful it is!" But you, as a scientist, take this all apart, and it becomes dull and thin.
First of all, I can't appreciate the beauty of a flower at the same time I see much more about the flower than he sees. I could imagine the cells in there, the complicated actions inside, which also have a beauty. I mean, it's not just beauty at this dimension of one centimeter; there's also beauty in the smaller dimensions, the inner structure, also the processes—the fact that the colors in the flower evolved in order to attract insects to pollinate it—all kinds of interesting questions, which the science knowledge only adds to the excitement, the mystery, and the awe of a flower.
It only adds. I don't understand how it subtracts. The Feynman rose story is a beautiful one that I've recounted many times too, so I totally agree with that perspective.
Sylvia, I just want to turn to you on this question of the self-reflective inner awareness that we human beings have within our heads. Do you see that as part of a mathematical description of the world, or for instance, would you imagine that other languages or other perspectives need to be brought in to have a full understanding of that quality of the world?
I sometimes feel that the puzzlement about consciousness and what we can achieve with mathematical descriptions of consciousness arises partly from different meanings that the word "description" has. So we can think of all kinds of descriptions—descriptions in different languages. So if we think of conscious experience, we could think of trying to find a mathematical description of that, and that would help us understand, for example, what distinguishes conscious beings from unconscious objects in the world.
And so the goal of such a description would be to enlighten us on the physics underlying conscious experience. But we could have a totally different goal for our description. For example, to replicate or to induce a particular phenomenal experience in another being that we're trying to describe our conscious experience to. And in order to achieve that goal, a different kind of description might be much more suitable.
So let's say if you had never in your life tasted vanilla ice cream, and I was going to try to describe to you what it tastes like, then perhaps a mathematical description would not be the best one. Or let's think about a different example—ice cream is a little bit too trivial. Think about the emotion of fear.
I could give a mathematical explanation of what happens in a subject's brain when that subject is in a state of fear. I could describe, you know, which neurons are firing and what exactly has happened on the particle level in that person's organism. But in order to make you understand, assuming that you're a being who's never had—who's never felt fear—in order to explain to you what fear is like, it might be more useful to use a completely different kind of description using a different kind of language.
Let's say the language of music. It's famously—and this also relates to what we just heard about, you know, disputes between physicists and artists. Famously, art can reveal and communicate, let's say, knowledge, can communicate things, can reveal things to us that cannot be revealed in a different way using a different language.
So I think we always have to ask what exactly is it that we want to achieve with a particular description. And if what we want to achieve is to get a better sense in terms of the underlying physics of consciousness, then my intuition is more on Max's side. I think we can be hopeful that empirical science will at some point be able to figure out a mathematical model for consciousness.
If it's about making another sentient being understand what my inner life is like, I'm much more skeptical.
What about morality? Just turning the gears a little bit further, human minds, human brains are able to have a moral sensibility of right and wrong and good and evil and all those qualities that are so vital to a rich life here on planet Earth. Do morality and those notions stand, in your view, outside of the mathematical description or the mathematical universe, or are they subsumed within it in some deep way that might not be the most efficient or effective way of describing them, but nevertheless they are accounted for in a mathematical way of thinking about the world?
I think mathematics gives us an interesting paradigm or an interesting model that can help us to try and understand how morality fits into our empirical world. So we've discussed this earlier in the program that, you know, some people are drawn to the idea that there exist mathematical objects. Other people object to that idea and think that's completely crazy because those objects would be so vastly different from the objects we're familiar with—physical objects.
And pretty much the same thing can be said about moral entities. You know, like moral values, for example, would be what I call a moral object. And so there are some people out there—let's call them moral realists—who believe that morality is something much like mathematics. It's eternal; there are eternal truths out there. And what we're trying to do when we think about morality is to sort of dock onto those truths—like to capture those truths and represent them in our beliefs.
But that, of course, entails that moral objects exist, and it's very confusing to some people to imagine what that would look like. I think that thinking about mathematics and seeing that we have good reason to believe in mathematical realism and we shouldn't be afraid of positing those entities can actually help us overcome our scruples also in the moral arena.
So one of the main arguments that is usually cited in favor of moral anti-realism—in favor of the view that morality is something invented, something that we constructed for some evolutionary purpose—one of the main arguments is always disagreement. We vastly, or human beings, vastly disagree on what are the moral truths; therefore, there cannot be any substantial truths out there.
Now, I think there are a number of reasons why that argument is flawed, but the comparison to mathematics can lend credibility to the view of moral realism. And here is the comparison: if the reason to reject realism about morality is fundamental disagreement, we have the same also in the mathematical arena. It's true that mathematics is often viewed as this sort of disagreement-free cosmos where all that mathematicians are doing is discovering new mathematical theorems or proving new mathematical theorems.
But in fact, when we look at the foundations of mathematics, of set theory, for example, we can see that there is pretty vast disagreement on pretty fundamental questions also in mathematics. For example, on the question, "How many mathematical universes are there?" And so I think that this main argument against moral realism can be discarded once we turn to mathematics.
So that's just one way in which the analogy between mathematics and morality can be enlightening.
Well, I'm not all that comfortable with the plurality of mathematical truths in the sense that genuinely conflicting mathematical truths—that's something I have trouble understanding. I do think the analogy between mathematics and morality is a useful one, just as Sylvia said. And of course, the big difference is that I, at least for most people—and that's, I think, well, I don't know about most people, for me in any case—mathematical truths force themselves on me with a compulsion which is at least somewhat lacking when it comes to moral truths.
I'm not a moral realist, but nonetheless, the mathematical realism that just seemed to—that I thought I feel much more strongly. Maybe a distinction, however, is relevant—not sure it's a sensible one or not—with a distinction which would apply both to mathematical reality and moral reality.
And the distinction is this: what is mathematical realism? What is moral realism? In both cases, you might say the mathematical realism you believe in—the objectivity of mathematical propositions or moral truths—you believe, for example, somehow that there really is—it's not that there are an infinite number of primes, and there always were, and there didn't have to be people around for that to be true. There simply are an infinite number of primes.
That's certainly a view that a mathematical realist would have. Now, for many mathematical realists, if not most, and many would—David might say, "How will you—how would you—maybe you're forced to have this view." But I know many people would say, "But I do not want to insist that there are mathematical objects somehow. I don't believe there's a separate realm of mathematical objects. There's a realm of mathematical truth."
I can certainly say that somehow or other, there's definitely an infinite number of primes, but I don't believe prime numbers exist. The mathematical objects themselves don't exist. I think that's a view some people have.
David, any thoughts on the morality-math analogy?
There's an argument that mathematics is indispensable. Some kind of mathematical realism is indispensable to the scientific project, and I take it there's no analogous argument in the moral case. We can teach people a whole course of theoretical physics without mentioning the words "good" or "bad."
But we can't do it without mentioning the word "five," right?
Right. Can I just—
Yeah, yes.
Judge our ontological commitments to mathematical entities with reference to the scientific project, and that's because the scientific project seems like an intrinsically indispensable project—that science is our paradigm for knowledge at the moment. But here is an alternative indispensability argument for moral entities, or a criterion, as David called it.
If we cannot not quantify over moral entities or normative entities in a project of our lives that is intrinsically indispensable—perhaps not science, but something else—then that might also give us a reason to believe in the existence of moral entities. And so it has been argued—this is not my argument; this is David Enoch's argument—it has been argued that quantifying or referring to normative entities like reasons or values is indispensable to the intrinsically indispensable project of deliberation.
Human beings cannot fail to deliberate. We deliberate all the time in order to come up with—well, in order to decide how to behave and how to interact with the world. And in this indispensable project of deliberation, we cannot fail to refer to normative entities. And so that's a sort of analogous indispensability argument that I think has definitely some force.
This question is actually incredibly timely because we're now putting rather mathematical-based objects—artificial intelligence—right in charge of ever more decisions. They're out there now deciding who gets a loan, who does not get a loan. They're having influence on who gets probation, who stays in jail. Very controversially, some people are even wanting to build weapons that themselves decide who lives and who dies.
Right? And then this question of sort of splits into two parts. One is how can you actually mathematically codify an ethical moral system so that you can explain it to a machine—a self-driving car or whatever? And then the separate question is, well, okay, if you can quantify mathematically all these different systems of morality, well, which one is the best? This one—clearly there's no consensus on.
But I want to inject just a little bit of optimism here too, because frankly, I don't think the reason our world is in such dire straits right now is because we can't find—we haven't managed to answer exactly that question. Almost everybody on this planet agrees that, hey, it's better if humanity does not go extinct than if we do go extinct. It's better to have less poverty than to have more poverty.
So the reason we, as a species, are still epically failing at avoiding, you know, the risk of nuclear war and avoiding people starving to death all the time isn't because of this moral disagreement. It's more about the—for some reason, we just really haven't gotten our act together in having—and in the way we manage our planet.
And people often push in a certain direction because they think it's going to be in a way they think is morally good, and it isn't. So I think the optimistic part is I don't think we're limited by our understanding of math and morality and our opportunity to make a better future.
So with that optimistic note, I want to turn to the final section. Max, one of the ideas that you are a proponent of in terms of discussion—we've had a little bit around a corner where we've been discussing sort of the universe and math, either as a description of it or somehow being part of it or somehow being, you know, deeply connected with physical reality.
You go further and suggest that perhaps we should turn that question around and view math as the real reality, which every so often perhaps looks like the physical reality that we experience, but it's the real uber, you know, bottom line, most basic reality that exists. I don't know if I'm describing that accurately, but if you could give us your sense of this view of mathematics, and then we can have some of the folks weigh in as to whether that's a view that resonates with them or not.
That's right. So if you look at the spectrum of views you can have from "It's all in our heads," you know, all just invented, to somehow it's in some sense out there, I'm as far as you can go, I think, on the opposite extreme, saying not only does mathematics describe our universe, but our universe is actually mathematical in the sense that we live in a gigantic mathematical object.
And when you first hear that, frankly, it sounds ridiculous. It sounds absurd, right? If you all look around in the room you're sitting right now, if this is all supposed to be mathematical, well, I mean, where is the math? You know, I'm looking around in my room; outside my window now I see a tree here. You know, if something is entirely mathematical, it means that all its properties are mathematical properties. It has only mathematical properties.
But that tree there, it's green and brown, and it's leafy. You know, that doesn't sound like mathematical properties at all. So what am I even talking about? Well, when I look at the tree again, though, through my eyes as a physicist, I see a blob of quarks and electrons. And what are the properties of an electron? Minus one, one half, one.
And of course, we physicists have made up nerdy names for those properties, like electric charge, spin, and lepton numbers. But that was just the language we made up to refer to those numbers. The properties are just numbers. And you, Brian, know as well as anybody that the only difference between an electron and an up quark is what numbers its properties are.
And then what about—so if all the stuff in our space has only mathematical properties, then what about the space itself? What properties does space have? Well, if it has the property three, for example, that's the largest number of fingers I can put that are perpendicular to each other. Again, we made up a fancy name for that; we call it the dimensionality of the space. But that word is just the word we made up, like the word Neptune. The property of space is three.
And then we heard earlier, you—the non-Euclidean geometry came up. We now know that space also has properties of curvature and topology, which you study in graduate math classes. They're entirely mathematical properties. So if you accept the idea that all the stuff in space, as well as space itself, actually seems, as far as we can tell as physicists, to have no properties at all except mathematical properties, then it starts to sound a little bit less insane that maybe our entire physical reality is, in fact, a mathematical object.
Max, is your view compatible with the view that—which you don't presumably don't have—but is it compatible with the view that consciousness transcends mathematics and physics?
I would guess that consciousness and intelligence are certain kinds of information processing—not all kinds of information processing, but the one day someone will discover an equation, and maybe Giulio Tononi already has it, such that when the information processing obeys that, there is a subjective experience there.
Like David Albert so eloquently described, information itself, of course, as we learned from Claude Shannon, shouldn't be described entirely mathematically as well.
So, by Max, along the lines of the, you know, the minus one and the half, you know, in terms of electric charge and spin and so forth, of course, I totally understand where you're coming from. And yes, if you were to ask me, "What is an electron like? What really is it?" I would ultimately be forced to give the description that you are rehearsing. You know, I'd give you its charge as a number; I'd give you a spin, and I wouldn't be able to talk about electron-ness in any other more fundamental, less mathematical way.
But that still could be a human description of the electron as opposed to the real intrinsic electron-ness of that particle, right? It could still be us humans imposing a description. We developed this tool, and it does a good job—a great job—but it could still be that.
No, it's really fascinating. I think historically—and I'm so grateful, Sylvia, for you giving us this historical perspective—how again and again, if you took just traditional materialism, saying, "Oh, there's this stuff," as if we knew what that really was, right? How all the non-mathematical properties we used to talk about, one by one, have just sort of melted away, and we just seem to be left with only the mathematical ones.
Even the fact that there is a particle there, an electron, we know now, of course, is also an oversimplification. There is this thing called a quantum field, and how many electrons are in there? It just tells you something about that state in the field. It's like how much the guitar string is vibrating—how excited is some mode of this entirely mathematical thing?
And then came quantum mechanics and said, "Well, actually, you describe the whole thing with a wave function that lives in this place called the Hilbert space—completely mathematical things."
And I think I find it very intriguing that throughout the history of physics, whenever we have found that some mathematical description of the world was inadequate, it wasn't quite right, and we had a revolution and replaced it by something else, that something else was also mathematical.
And the most ambitious attempt so far to find a single set of equations you can put on a t-shirt—maybe in the future, string theory, as you know better than anyone else on this panel, Brian—is entirely mathematical too. I really don't see any evidence so far from nature that there's anything non-mathematical about the world.
Now, if I understand you, you go much further than what you've so far recounted. You would actually go so far as to say that all of mathematics is—as real, or I should say, any piece of mathematics is as real as any other piece of math. You don't distinguish between math that's relevant to reality and math that isn't relevant. Is that an accurate description?
That's right. They sometimes call that the level four multiverse. So when we talk about something existing, if we say that pink elephants don't exist, what we secretly tend to mean by that is, "Why, they don't exist here on Earth or anywhere where we've looked."
But maybe there's another planet really, really far away where you actually have pink elephants somewhere else, right? And we've gone through this progression in history where, again and again and again, we've come to realize that the totality of physical reality that exists is just way bigger than we thought.
Then everything we knew about is just a small part of something grander—a planet, you know, solar system, a galaxy, a galaxy cluster, our observable universe. It's probably much bigger than the part that we can see that light has reached us from, etc.
And if what I'm saying—what I'm guessing is true, then it's bigger still. So you have all of these different mathematical structures that mathematicians, like Shelley, can study—the cube, a Columbia manifold, you know, Minkowski space, three plus one-dimensional manifold, the Hilbert space, etc.
If they're, you know, Hilbert, the mathematician who was quoted earlier, famously said that mathematical existence is just a freedom from contradiction. There are a lot of different objects that exist mathematically, and my guess is that we inhabit one of them.
And if there are other ones which are also complicated enough that some of their inhabitants are conscious and have experiences, they're going to feel that their existence is physical, and just the same as we feel that ours exists physically.
And in just a sense, the same sense that if you, Brian, were actually right now in some future, very high-tech computer game, you know, this governed by mathematical laws, you know, you wouldn't feel any different. You would feel, "Of course, my little game world here exists, and I can punch the table; it'll punch me back."
So yes, if this is true, I think this Copernican revolution needs to go one step farther, and there is—the ultimate reality is still much bigger than we thought.
David, any thoughts on whether we might be living in sort of the mathematical reality of all equations being instantiated in some way, shape, or form?
I mean, first of all, they don't—I all I'm going to be able to express about this is a certain kind of bafflement about what's going on and how I'm supposed to understand it. I mean, when, Max, when you say our description of the world has gotten more and more mathematical, more and more of the non-mathematical things have been eliminated, I'm not sure I understand what that means.
You talk about, you know, Plato and Aristotle sitting around or cavemen sitting around. Nobody would have doubted that the tree that they're looking at has some volume, okay? And you can assign a number to that, okay? What number you assign is going to depend, of course, on your choice of units, and that's no less true for the electron spin than it is for the people sitting around looking at a tree.
But do we think they would have denied that the tree has a volume? Do we think they would have denied that the tree has a mathematical description? It's just that you seem to want to collapse any distinction between the mathematical description of these things and the things being mathematically described.
And there I just really get confused. It's as if, you know, if somebody comes up to me and tells me all of these perplexities that you're having are going to evaporate if you can just get it into your head that a cucumber is really a Ferrari. Okay? And if you can just get your head around that, everything's going to be fine.
And I sit there saying, "Cucumber, Ferrari, cucumber." All I have to do is figure out how to see that a cucumber is really a Ferrari. I can't do it. I don't know what I'm being asked. I don't know what I'm being asked to imagine.
And I don't know what I'm being asked to imagine when somebody tells me, "No, it's not just that the tree has a volume that can be expressed by a number; it's that the tree is the number." Like I say, I'm just being thick-headed here. I'm just being stupid. I don't understand what it is that I'm asked to imagine.
So it's a fair question, but I think it's one that has a good answer. There is something that's really fundamentally changed in our understanding of trees in the past thousands of years. In the beginning, when Archimedes was studying trees and Galileo was studying trees, there were some very limited aspects of the properties of the tree that they could describe mathematically.
They could describe how long it would take for a branch to fall down if it fell off and stuff like that, but they had no idea why the wood was so rigid and stiff and why the bark looked brown and so on. There were measurements you could make of the tree that you just had no mathematical way of calculating the answer to.
That's just really changed. Today, really, there are still—there are—there's no evidence, I would say, that there is—that you couldn't, if you were to sufficiently—if you were to—that you couldn't, at least in principle, derive all properties of a tree from the standard model of particle physics.
We know in math just because you have the fundamental equations does not mean you can always calculate things accurately. You know, we can't even calculate the mass of the proton relative to the neutron accurately, even though we really think we know QCD, right?
We're in computing exactly how the tree is going to grow up just from knowing its DNA. You know, right now it's beyond our ability to compute, but I would say there's the evidence that there's something more to the tree than a quark blob has sort of melted away.
That's not true of all of physics. Consciousness and intelligence, especially consciousness, I think we have to be very open-minded that maybe we're missing something big. But certainly with trees, I think there's every reason to believe that we've got the fundamental description of it right, and it's mathematical.
And now we're struggling with how we can link that to the sort of higher-level processes that are going on of a growing living thing.
Max, if I can just push back a little bit. Look, everybody would have agreed thousands of years ago that trees have tiny little pieces, okay? And that trees are, in some sense, the sum of their tiny little pieces, okay?
Probably they would have agreed that the pieces are, in some sense, simpler than the whole tree itself. Still, I don't understand how that takes us one inch in the direction of saying something like the tree just is numbers.
We're getting better and better at not only describing the tree using numbers but predicting how the tree is going to behave in a way that ineluctably makes use of numbers, okay? But if somebody says, "Here, you got two apples here," so I don't know what the next move is supposed to be.
Is it supposed to be? Is it supposed to be?
So really, all you have here is the number two if you really think of it. I really—it really feels like cucumbers and Ferraris.
I don't want to talk smack about cavemen or physicists from 2000 years ago, but they did not understand that trees are made of air primarily—that they actually take carbon dioxide out of the air and sunlight. They take energy to bind it in and so on.
These are new things. They were just missing. They didn't have a complete understanding of the makeup of trees that we now do, right? And whereas now I think we're at the point where there really is no evidence that trees are made of a new kind of quark that hasn't yet been discovered or that you have to overthrow string theory to describe trees.
The second point that you raise, which is also very important, is I think the distinction between saying that something is described by math and saying that it is mathematical. And that might be related to your cucumber-Ferrari issue.
So right, in mathematics, there's a very beautiful body at work on equivalence. There are many different ways in which you can mathematically describe, for example, the integers. Peano wrote down five axioms, and there are certain descriptions. You can start in many different places; you can use different symbols, different notation.
And when we talk about a mathematical structure, what we actually mean isn't any one description of it, but we mean the equivalence class of that thing which is described by all the equivalent descriptions. That's the thing that exists.
So there isn't, in Plato's mathematical realm, two different cubes. There is just one mathematical structure of the cube, right? And in the same way, if it turns out our physical world—if you think of a mathematical structure as just a bunch of abstract elements with relations between them, if that corresponds one-to-one with the abstract elements with elements in that we call physical things, like superstrings or quarks or whatever, and relations between them, then they are one and the same thing.
And it's in that sense that I say that our physical—I think our physical world really is a mathematical structure. You know, you could even write a computer program that starts to just classify one by one different structures that exist in mathematics. My guess is that there is a mathematical structure which is, in fact, the one that we live in.
And since we live in it, we have come up with also the human words to refer to the entities we talk about. We call them electrons and quarks or whatever, but they are just the names that we humans have come up with for referring to abstract mathematical entities.
Well, I've certainly enjoyed our conversation with all of you—mathematical objects, if you allow me to use that language. Clearly, this is an argument that could keep on going for some time, and it's a sharper version of the very question we began with in terms of, "Is math a description? Is math a human invention? Is math actually out there?" Or, in Max's vision, "Is math the be-all and end-all of reality?"
Wherever we come down on questions like that, I think all of us here—and hopefully everyone who's watching this—can get a sense of how beautiful mathematics is, how powerful it is at describing the world. Perhaps it's even more than that.
I began with some quotes at the beginning of our conversation. I want to end with a few quotes that capture that spirit. First, from Bertrand Russell: "Mathematics rightly viewed possesses not only truth but supreme beauty—a beauty cold and austere like that of a sculpture, without appeal to any part of our weaker nature, without the gorgeous trappings of painting or music, yet sublimely pure and capable of a stern perfection such as only the greatest art can show."
Mariam Murzakani said, "You have to spend some energy and effort, but if you do, you can see the beauty of mathematics." From G.H. Hardy, "The mathematician's patterns, like the painter's or the poet's, must be beautiful. The ideas, like the colors or the words, must fit together in a harmonious way. Beauty is the first test; there is no permanent place in the world for ugly mathematics."
And finally, Paul Erdős: "Why are numbers beautiful? It's like asking why is Beethoven's Ninth Symphony beautiful? If you don't see why, someone can't tell you. I know numbers are beautiful. If they aren't beautiful, nothing is."
So with that, let me just thank all of you for this wide-ranging conversation on math and reality. Thanks so much for joining us. Enjoy the conversation, and maybe we'll have the pleasure of picking up these controversies sometime in the future. Thank you.