What's beautiful about mathematics?

In my first semester of my PhD program, I stayed in a graduate student apartment where my roommate was a mathematician from Thailand. On my first evening there, as we were having dinner together, he asked me what I studied. I was in a Sociology doctoral program, and told him I was interested in understanding how globalization, particularly in the form of global corporations, was shaping cultures that were undergoing rapid economic growth. He nodded in understanding.

"What about you?" I asked him.

"I cannot tell you," he replied, his words muffled by a mouthful of soup. "You would not understand."

He assumed–correctly in my case–that the puzzles that drew him to mathematics would be too complex for a layman to understand. Yet I was struck by the intense fascination and even obsession that appeared to seize him and other mathematicians I knew when confronted with what seemed to me as merely cryptic symbols.

What is it about mathematics that captures the human imagination? Why does it evoke a sense of profound beauty in the minds of those who study it? In this podcast episode, I interview mathematician Carlo Lancellotti to explore the interplay between math and physical reality, the aesthetic motivations behind mathematical pursuits, and why mathematical language is so surprisingly effective in expressing deep connections that bind seemingly unrelated phenomena.

Carlo is a Professor of Mathematics at the College of Staten Island and a faculty member in the Physics Program at the CUNY Graduate Center. His field of scholarship is mathematical physics, with a special emphasis on the kinetic theory of plasmas and gravitating systems. He has published widely in leading scientific journals, including Physical Review, Physical Review Letters, the Journal of Statistical Physics, Chaos, the Journal of Transport Theory, and Statistical Physics. He has also translated into English and published three volumes of works by the late Italian philosopher Augusto Del Noce, and written philosophical and theological essays in several outlets.

Carlo points out the often unexamined allure of beauty in mathematics. From the search for elegant formulas that simplify complex problems to the unexpected relationships that emerge, beauty in math is often synonymous with the discovery of harmony and universality. The conversation takes us on a journey through the concepts of kinetic theory, Einstein's equations, and the role of beauty in both mathematics and physics.

Join us as we discuss the hidden order and relationships that underlie the world we live in, the spiritual dimensions of mathematics, the importance of balancing concepts and skills in education, and how the beauty of math can deepen our understanding of the universe. You can watch or listen to our podcast episode below.

An unedited transcript follows.

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Brandon: Let's talk about what drew you to mathematics in the first place. What attracted you to this field?

Carlo: Well, I came from mathematics and physics. My first degree was in physics. I was always fascinated by the interplay between math and physical reality. It is truly amazing when one sees our nature, how the world is all intertwined with mathematical structures. I remember that my grandfather used to think it's a proof of the existence of God. He once told me that the fact that if you study physics, you'll find all these equations, these formulas. I don't know if that argument is philosophically rigorous, because one could say that the formulas are our own creation and that they're not exactly what nature is. They're approximate. Of course, they're approximations. But there is certainly something there. When you look at quantum electrodynamics, we have all this abstruse calculation. Then you can reproduce the experimental result to 27 digits after the dark or something enormous like that. It is very impressive.

To me, the passion for mathematics was always about the intersection of mathematics and physics. But that's not true for many mathematicians. There are many what are called pure mathematicians. They really appreciate, say, the beauty and harmony of very abstract constructions. I mean, geometrical or algebraic constructions. Since the time of Pythagoras, people have been fascinated by the pure form, the pure intellectual, the world of pure mathematical ideas. Every mathematician, or at least most of them, deep down are Platonists. They really believe in the existence of a world of ideas. One of the main characteristics of this world of ideas is harmony.

Brandon: Well, there does seem to be — at least, there's a perception. I don't if anyone has actually empirically studied this. There's a perception that mathematicians care a lot about beauty. They talk a lot about beauty. Do you think that is the case, that beauty is an important concept?

Carlo: I don't think anybody would do mathematics to get rich. There is a subsection of math, which is very applied. You can somehow gain some advantage in a technological sense. But most people do math, it's because they like it. Now, sometimes there is, let's say, not 100% good competitive aspect. Mathematics will be like a nerd Olympics, like people competing to show that they excel in solving problems, to show that they are smarter than the other guys who couldn't solve the problem. There is that component.

Certainly, the aesthetic motivation is one of the top three, at least. People do like math, again, because they like to see how things are related in the structures that are very harmonious and very surprising. The great mathematicians are people who discover connection between fields that are apparently unrelated, and you'll see that there is this thing that pops up where you don't expect it. You can study algebra doing geometry. You can study geometry doing algebra. A method that was devised to study certain curves comes up in number theory. That's not my field. I do mathematical physics, but I know that people get very excited about these things. Certainly, there is an aesthetic component to that.

Brandon: What are the kinds of aesthetic criteria that mathematicians use? I mean, are there sort of a common set of things that mathematicians find beautiful in their work? You mentioned harmony. But what does it mean to a mathematician to find something beautiful?

Carlo: Well, there are different things. Like, what is universality? You discover, for example, this certain property that you discovered in one example is amazingly general. I don't know. Like the great theories of probability. You get the central limit theorem or laws of large numbers to an astounding generality. I don't know. People in number theory, they discovered this very surprising relationship between sequences of numbers that you would really try something that seems very complicated. The distribution of prime numbers are based on very simple statistical laws. I don't know. People have tried to prove the Riemann hypothesis, which is a conjecture in number theory linked to complex analysis using statistics or statistical mechanics. So, there is the thrill of universality and the thrill of unexpected connections. I think that's what really gets people going. Again, everybody is on feel.

I studied a lot of statistical mechanics, which is a branch of physics, which is very mathematical. It's remarkable what you can say, for example, about how things happen. In delimiting, we should take the number of particles going to infinity. Because in reality, you never have an infinite particle. You have 10 to 24 particles. But mathematically, you can take the limit as N goes to infinity of your equations. That converges to a very simple limit. This simple limit manifests itself in the physics. That's very fascinating. It's very beautiful. I don't know how else to put it.

Brandon: Is there anything more you could say about your own experience of the work you've done? Where is it that you've encountered in your own work what you would consider beautiful?

Carlo: Well, I started, as I said, from physics. I did my first thesis in the kinetic theory of gases. I have a clear memory. I was amazed like wow. It was the first time I saw the Boltzmann equation. The Boltzmann equation is this famous nonlinear equation due to Ludwig Boltzmann. He was a great Austrian physicist. It describes the evolution of the distribution function for a set of particles in phase space. So, you're taking a set of particles. These particles are positioned in momentum. You set the limit of the number of particles to infinity. You want to study the density of the particle in six-dimensional space. That's a complete mess. I mean, why would you expect that there was any way to do that? Then Boltzmann came out with this super elegant equation, which is a complicated equation. But the fact that it can be done, that the human mind can write down a law to describe something like that is — when I saw that equation, I was surprised. Wow. I would never imagine that one could come up with something like that. It is a physical achievement. So, that has been kind of my interest.

Also, the results of fascination in the fact that you can come up with equations, but then you cannot solve them. You move one inch, and you discover new vistas of unsolved problems. Most of them are very hard. And so, there is both the surprise of discovery and the realization of your ignorance. There is something there. I don't know. Again, when I was younger, I tried to study very simple solutions to describe a shockwave in a rarefied gas, describe what it was in an equation. That's the simplest possible problem and so complex, that really you can study this with a numeric receipt. There is a very simple profile, but you cannot come up with a formula. And so, there is, I don't know. It is the beauty that the model captures. But at the same time, it escapes you. So, it's a very interesting experience.

Brandon: Talk about the relationship between beauty and truth in mathematics. Because a number of people have argued that beautiful equations are a guide to truth, or at least a reliable heuristic. Others think that this is a source of bias, and we can't really—

Carlo: Like Dirac. Well, if there was no hope of finding a beautiful formula, people would not do this job. Because, at the end of the day, the operation of mathematical physics or theoretical physics is to replace something very complicated with something very simple. That's another aspect of the beauty of it. You have a huge, complicated physical problem, and then you discovered that there is one formula that, at least in principle, contains it. If there's one formula, it was 100 pages of formula, nobody would care to discover it. You know what I mean? The simplification is part of the beauty of the process. It's the reason to do it. If you couldn't find something like that, there will be no point. It would just try to do a numerical simulation, particle by particle. But that would not be as interesting.

Now, the thing is that's absolutely inevitable. The point is that, of course, our mind is not the mind of God, so to speak. Sometimes there is the possibility of falling in love with a beautiful formula and expecting that that should be it. Because the formula is very mathematically elegant in its simplicity in its richness of its structure, then that's the way the world should be. Again, a famous example in my field is the Maxwell particles in the 19th century. After Boltzmann wrote his famous equation, the great English British but he was Scottish. Anyway, British physicist, Maxwell, discovered that the equations simplify enormously. You'll find some super elegant solutions.

If the particles in question are attracted with a force that decays with the fifth power, the radius, so these are called the Maxwellian module. So, every kinetic theorist thinks that If I was God, particles in the world would attract and repel each other with a force that decays with the fifth power of the radius, which is not the way it is. Because typically, particle like electrons or ions, they use the second power, the Coulomb force between charged particles. It was one of our square, not one of our five. That's a pity, because kinetic theory would work much better.

There is, generally speaking, the situation in which, for some strange reason, a lot of physical problem sit at the most hard to capture choice of parameters. A favorite example is, again, the 1/r2 interaction, which is typical of gravity and electrostatic interaction. I know at least three different theorems, that everything works for r say less than two or bigger than two. But when you get to r=2, everything, all of your math starts falling apart. And so, there is a way in which reality sometimes does not correspond to the simplest what we would correspond beautiful because we understand it, but sits at the boundary of it. This drives mathematicians crazy sometimes.

The typical example of the beauty is the way people have been trying to — like, again, theoretical physics, the standard model and stuff. People use that as a criterion. Sometimes it works. Always impressive the way the number three, for example, pops up in very strange ways in physics. There's three dimensions. That's very, if you wish, Trinitarian because there is one space in three dimensions. It is inseparably one and three. It is clearly one space, but it's clearly triune. It clearly goes in three directions. Or you can find like today, the example of the quarks. The quarks are organized in three triplets, three squares. There are all these beautiful numbers. Of course, the people invented the — people discovered it, of course. They wouldn't care about the number three in particular. But they certainly were much more attracted to a theory with three triplets than if it had been six times, seven or something like that. There is a certain elegance, a certain beauty too, for example, simple numerical groups.

Brandon: Yeah. How do people perceive this? I suppose in theoretical physics, one of the challenges is this sort of drive towards simplicity, towards unification, even trying to develop a unified theory? Physics is something, again, that's very contested. Is that worth investing in? Is there a reason to believe that reality at its core must be so simple as to be contained in a single equation or something like that, or is that a waste of — how do you think of that?

Carlo: Well, I remember you were telling me about vagueness sentence about the miraculous effect of mathematics. Reality hints. There is, again, this experience. Reality hints to very simple, elegant general principles which, at the same time, always escape us. And so, whether it's worth it, I think it is worth it. It was worth building great cathedrals. If it is in a strict practical, technological sense, some of these things at some point are not worth it. But it is one of the few areas in which our time does something gratuitous, in which we do something just for the sake of the beauty of it.

I don't begrudge the people who want to do that. I mean, the other sciences could be useful. Of course, within reason is not correct. It is a spiritual activity in some sense. It is also like poetry or art. I think it should be pursued within reason. Whether reality, ultimately, we are going to find the formula for everything. I don't think so. I mean, there is simply the problem of scale. There was an interesting book by — I don't know if you know David Bohm. You know David Bohm, the American physicist? His case was that physics will always be incomplete because there is always the next scale. You can always look at something bigger, something smaller, something on shorter time, something on a longer time. All your models at some point break. So, if you go down, closer and closer to the Big Bang, closer and closer to the microscopic Planck scale, at some point, you need to change the model because it will start working. And so, there is, in that sense, a certain infinity to physics. But what is remarkable is that within certain scales, you can describe the war with really elegant, simple mathematical models. That's an amazing thing.

Brandon: Yeah, how do you make sense of that? Because from, say, a strictly evolutionary standpoint, there's no sort of reason to expect a priori that reality, at any level, could be described with what human beings, this sort of insignificant creature on a planet that's on an insignificant galaxy somewhere and in space can actually grasp the workings of what seemed to be laws that cover the expanse of the universe. How do you, as a mathematician, make sense? How do the other mathematicians make sense of this unreasonable effectiveness of mathematics?

Carlo: Yeah, I don't even know whether it's really that reasonable in the sense that the world presents itself to us with a remarkable consistency. I don't know. It's not like a dream. Listen to me. I have a dream in which, in a dream, things keep changing randomly. One second, you're close. The next second, you're naked. The next second, you're back in Europe. Then next second, it feels like the world is like a random association of image. Even without mathematics, the world does present itself to us as very stable, in some sense. Most geologically stable. It doesn't act at random. There are obvious regularities. When there’s regularities, there is math. I mean, it is not surprising in that sense. What could be a world, like a random sequence of dreams? I don't know. It's a difficult question, I guess. But what is striking is how, again, it can be boiled down to certain very simple general features. Because you could have irregularity that, again, fits in. In order to describe the world, you need to have a huge number of equations, for example. Not just a few. That seems to me, again, you could describe — I don't know.

Another very striking, for example, set of equation is general relativity, Einstein's equation of general relativity. You can write them with one, two. I cannot count for you. But six or seven letters. All of the physics of very complicated systems are in those six or seven Greek letters or whatever they are if you interpreted them correctly. It's unreasonably if you wish simple model. I think that the simplicity is more striking than the existence of a model when it is simple. But then again, also, the simplicity is again is the ceiling. Because then when you try to solve the equations, they are unbelievably difficult. What I'm saying is that there is a simplicity of concept that contains an extreme complexity of behavior. I guess that's what's interesting to me.

Brandon: What are the limits then to what mathematics can tell us about reality?

Carlo: Well, it's abstract. I mean, this is true of physics in general. It abstracts one aspect of life. I always give the example of a beautiful statue. You go to the Louvre, and there is the Venus de Milo. She's sexy. She touches about Greek history. There is history. There is the human aspect. You can measure the electrical conductivity of the Venus de Milo. But you are abstracting. Physics does not ultimately know reality. The scholastics would use the word "qua." For Thomas Aquinas and company, qua means with respect to, in the sense of. Physics only knows reality in as much as it can be measured in certain ways. You can measure length, weight, conductivity, luminosity. You can measure things.

But, of course, our experience of reality is much more than the experience of reality as measured your best fantasy ideas. Beautiful experience. It's useful experience of reality. In terms of personality, we experience reality, even physical reality. We just don't experience it as measurable the way physics knows it. I mean, physics cannot tell you again if a body that falls down a window is a true gem that's worth a million dollar or a stone. Do you know what I mean? It can tell you the chemistry but not the value, for example. Questions of value. Your economic dimension is not really — I'm talking too much about the very simple part, that physics abstracts one aspect of reality. But in order to do that abstraction, to extract this aspect of measurability and reproducibility, you have first to take in the world as a whole. And so, the mistake is to reduce that experience to the ensuing process of quantitative abstraction. Does it make sense?

Brandon: Yeah, so, those abstractions are helpful to provide certain kinds of descriptions, but they don't really exhaust other kinds of categories. Because you're seeing value.

Carlo: Yeah, exactly. Provide the kind of descriptions according to some aspect of the experience. If I jump off the window, physics can tell me how long it will take me to hit the ground. But that's not a lot about me.

Brandon: Right. We typically associate beauty with art, with visual art or music or architecture, or fashion and so on. But if we took, say, mathematics or even physics as a starting point for understanding beauty, what could that give us? Could we learn something about beauty that we typically don't think about?

Carlo: Well, again, I think that there is — an aspect of beauty is universality again. I mean, the fact that there is something that in particular you find something which is universal. The other aspect is that, again, the effort of finding unsuspected relationships, the way things are connected. I remember it was I think the great Italian philosopher, Rosmini, who said that the definition of beauty is unity in diversity. That's an interesting concept because beauty is never uniform, right? Your beauty can never be just a blank slate, or it cannot just be a complete jumble of differences.

Beauty always involves relationship. Beauty always involves a relationship between parts. You think of a beautiful Greek temple. It's all about the proportion of the different parts of the columns and the top. They are organized according to a certain proportion. Beauty really is capturing this unity and this relationship between a diverse experience of reality and bringing it to unit. If that's the definition of beauty, that's certainly found in mathematics and physics, because it is precisely that. It takes in a limited real, the real quantity. It does establish connections between different statements, different observations, and brings them together in a unified picture. I think that's what it is.

Brandon: Yeah, that's really helpful. One of the things that we have been studying scientists for a few years and when they talk about beauty, whether biologists or physicists, the common experience of beauty where they really find profound beauty in their work is in understanding. This is the ability to grasp the hidden order or the inner logic of systems, and to say, "Oh, that's how things are." There's a coming together of things but also an experience of something being revealed to you. Reality somehow is revealing itself or disclosing itself, and grasping a sense of how things come together in that revelation. That seems to be the phrasing of this is how things are. I found it echoed by poets I've interviewed, for instance, as well as scientists. So, there seems to be something in common across fields.

I want to ask about whether there's any sort of spiritual dimension to mathematics. For one reason, I think of people like Frank Wilczek, a Nobel Prize winner, who said that nature embodies beautiful ideas and those ideas are expressed in mathematics. Though Galileo thinks that mathematics is the language with which God is writing the universe. But I also think about Islamic art over the centuries, particularly in their golden era when so much of Islamic art was essentially like complex mathematics. It seems embodying itself into various patterns. I wonder whether you see any connection between mathematics and then the religious or spiritual sense in human beings?

Carlo: Well, as I said before, mathematicians tend to be very platonic. I mean, they tend to believe that there is a world of ideas. I always say that one of the trends in modern physics has been towards increasing mathematization of physics. Meaning, that in the physics of the 18th century, in a sense, the bodies were extraneous. It was the physics of artillery. You shoot cannon balls; you study the trajectories. You use the math to do that. But at the end of the day, the body, the solid rigid body, the cannonball is a given matter. It's a primitive function.

Now, the trend of physics has been towards breaking down matter into, again, mathematical structures. If you look now at modern physics, it's not clear that there is matter. There’re all these representations of the Poincare group they're called, or whatever they are when you study electron. At the end of the day, to understand our model, you have this very sophisticated mathematical structures. Then somehow, the way I think it was structured manifests itself as a tennis ball. But in reality, it has been disembodied by modern physics. Because it's become so clearly identified with this surface together mathematical things. I think that that's the reason why. Mathematicians and physicists, I read some statistics that–you are the expert–they tends to be more religious than biologists. Is that right?

Brandon: It seems to be to some degree, yeah.

Carlo: I think that if it is the case, that's because ultimately all their work is about assuming that the mathematical organization is primitive. It's the root of reality. While to some degree, biology is more like 18th century science. It's like a machine. You’re like a mechanician that dismantles a watch, a clock, and see all the different parts move together. Then you bring them back together, but it's not as mathematized as physics, for example, or even chemistry. I think, in that sense, mathematicians in physics would have a very hard time to say that the mathematical realities don't exist. Because it looks like, in fact, mathematical realities exist and then give us the impression of matter. But matter has become a meaningless concept. What does the word matter add to the standard model? You could argue that once you have all this group theory and the Lagrangians and all these things, then there is the world. But why is it material? This is also a problem.

At the end of the day, I think in the modern world, material just means predictability. Because we can mimic determinism, maybe perhaps even that after quantum mechanics is problematic. But, again, two phases. Matter has been boiled down to math. Then, can you really say that math does not exist? Well, we live immersed in it in some sense, at least from a perspective of a theoretical physicist.

Brandon: Right. Well, there are a lot of people who would say that they're not math persons. They would say that they don't quite get math, or it's not for them, et cetera. Are there ways in which beauty could draw people to mathematics or to science in ways that currently we don't quite see in our education system?

Carlo: Yeah, that's what we hope. I'm a math professor. We always try to share our intuitions for applications. Math shows up here. It shows up there. Math comes up everywhere. It's beautiful. Some people see that, but other people don't. I mean, everybody's different. And so, I work every day with some families, the students who just doesn't care about math at all. There is a bit of a push in our culture to try to explain that sociologically because nobody ever showed you. We never had good teachers. Or you're a woman, they don't take you seriously. Some of that could be true. But it's also true that math is not to everybody's taste. Because some people enjoy abstractionist, enjoy patterns more than others do. Some people are more pattern-oriented. At the end of the day, mathematics has a lot to do with the recognition of this universal structures that are abstract. Again, some people have a greater affinity for that. Did I answer the question?

Brandon: Well, you sort of. Do you have maybe any tips? Anything that's worked in your experience with your students perhaps in terms of better appreciating the beauty of mathematics?

Carlo: Yeah, I think that certainly will be good that mathematics be presented in close connection, for example, to physics in high school. That will be a good thing. Also, I think the most important thing is that mathematics should be presented conceptually. Because the big weak spot, at least living in the US, that I see in K-12 math education is that there is a very mechanical presentation of math. Math is about manipulating symbols and solving equations by doing step A, step B, step C. While mathematics is primarily about concept, the ideas. It should be presented as such and the question of numbers. That's easier said than done. Because then, there is the opposite extreme which has been, in math education, which multiplication suffers from fads to an unbelievable extent.

Some of the fads have been to totally fuzzy it up and focus on concept without delivering skills. You have to find the balance. People have been working on that. Because, of course, math is a big, strategic interest with respect to the economy, technology. Everybody thinks that you have to teach math. But teaching of math is still, in some ways, problematic. I think that part of it is precisely this kind of utilitarian approach that math is for technology. And so, you have to teach people certain skills. You have to teach people to understand why people came up with math. So, you need to have a conceptual or, let's say, even philosophical approach to teaching. But this seems to be very hard to do. I don't know why. It's a good question.

Brandon: Think of even better appreciating art or even becoming a more competent artist, it seems we are living in an age that's much more specialized than when you think of periods like the Renaissance, where all the great artists were also scientists. There wasn't this sort of clear separation. It seems that knowledge of mathematics would be helpful if you wanted to be a musician or an artist.

Carlo: So, you're out there with advocating a connection with the teaching of mathematics and the teaching of art or the teaching of mathematics and the teaching of science or both.

Brandon: Well, I think with the science now, I don't know if that separation is there. Maybe with biology, there are people I know who move into one field or the other because they don't like the memorization of biology. They like the equation solving. And so, they move into something like mathematics or physics. But in terms of art, the separation seems very clear for a lot of people that I'm not a math person. I'd rather go and be a musician. Then, well, to really to become a first, for instance, a good composer, you need to understand some of the mathematics. So, I wonder whether you see ways in which it's not simply about technology but that we could also mathematics can improve our appreciation for art and even other kinds of fields?

Carlo: Right. Yeah, I think you should be positive. If you look at proportion, for example, if you look at a great art or perspective, the projective geometry is everywhere in art. But more than anything, what's the hardest thing is to help people see the beauty of math as math. Why? It's because it is abstract. So, it takes a certain mind to do that. To tell you the truth, I don't think that everybody needs to be a great mathematician, of course.

Brandon: Sure.

Carlo: But it's important when somebody is taught something, they understand. If they understand something very simple, it's better than doing more without understanding. So, I think there is a real race sometimes to push people as far as they can, without maybe just doing less better. I think that that will probably be helpful. Because then, when people understand something, then they become more self-confident. They can move on and do more. But it is easy just to push people alarm class after class, and it builds on itself. It's very unforgiving. At some point, you stop understanding. Then you proceed by inertia and all these former manipulations, but it doesn't go very far.

I mean it is a general problem in education. I think that education being kind of utilitarian and technical, you measure the results without really appreciating where people are. That's, for example, I think it's sucks in the US. All mathematics testing is written. Well, in my time, I'm from Italy. I think the same was true of most of Europe. In order to get the math education, you have to take it since a very early age. You have to explain what you're doing orally, oral examination. The way you're tested affects the way you understand. If you have to explain why you're doing something, what's the definition behind it, what's the theorem that you are going to use, immediately, you are forced to think in a way different than if you have to sit down and just solve equations. I think that, ideally, one should have also that kind of conceptual oral testing, for example. But at this point, it's impossible. When people get to college, it's too late. It's a nightmare. It's not impossible. It's hard. Sorry. I went off on a tangent.

Brandon: No, that's it. I think, yeah, being able to sort of explain and have a narrative around it seems certain an alternative to how we're doing things now. Well, Carlo, thank you for your time. This has been really fascinating. Great to learn more about how mathematicians think about beauty and what we can do, particularly to think better about beauty, math, and science. Are there any references you might direct our viewers and listeners to get a better sense of some things we've been talking about today?

Carlo: Well, the book by David Bohm, as I mentioned before, is interesting. He was one of the most philosophically-oriented. When I was younger, I understood a lot about science and also math, I understand by reading Maritain. Maritain has some good books on the philosophy of nature, philosophy of science. My three ideas on the subject, I once wrote one article, I think it's in the Church Life Journal, if you Google my name. I wrote an article on beauty and science, more or less. That's most of what I have to say, I'm afraid. But it's right there if you want to take it.

Brandon: Yeah, we'll put the links in our show notes. Thanks again, Carlo.

Carlo: Yeah, you're very welcome. Thanks for having me.

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