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This episode of rationally speaking is brought to you by Stripe Stripe builds economic infrastructure for the Internet. Their tools help online businesses with everything from incorporation and getting started to handling marketplace payments to preventing fraud. Stripe's Culture puts a special emphasis on rigorous thinking and intellectual curiosity. So if you enjoy podcasts like this one and you're interested in what Stripe does, I'd recommend you check them out. They're always hiring. Learn more at Stripe Dotcom. Welcome to, rationally speaking, the podcast, where we explore the borderlands between reason and nonsense, I'm your host, Julia Gillard, and I have two guests with me today, and they're both good friends of mine, Spencer Greenberg, who you might know because he was on a previous episode of this podcast.


He is a mathematician and entrepreneur running Spark Wave and Seth Cottrell, who is a physicist specialising in quantum information theory.


Hey, guys. Hey, it's great to be here. How are you doing? Let's just call and say hi. Hey, how are you doing? And Spencer, say hi.


Hey. It's great to be here. OK, so now you audience know what their voices sound like. So Spencer and Seth together run a website called Ask a Mathematician. Ask a physicist if that ask a mathematician.


Dotcom guys, does anyone call it AMAP?


Oh no. Well, I'll just blazed a trail then and end up doing something that that is the that is the Twitter handle.


But oh, I've only I've only read it, not heard it.


It's just a very satisfying word to say amap. I think so too. Yeah. So, so AMAP has been going for over nine years now which really astounded me. I feel very old and you guys have answered almost 500 questions on ask mathematician ranging mostly Sath to have done more than his fair share of the work.


It's true. Or so the questions.


They are all over the place they range from. What is quantum immortality to is it possible to eat all the ice cream in a bowl? Oh, that was yeah, that was a good one. And the website, as I said, is nine years old now. But just this year, I think this fall, Seth published a book called Do Colours Exist and Other Profound Physics Questions, which is full of some of the most entertaining and informative questions and answers from the blog.


Basically, if you have read the book, what if this is very much going to be up your alley so you can find it on Amazon? We'll put the link on the podcast website.


Oh, and and if you if you're looking for the e-book version, for some reason, they can't put that on Amazon. You can find that I was looking for it. Yeah, you can find that on the springer shop.


OK, we'll link to that, too. Thanks for letting me know. Thank you. So guys, to start off, why don't why don't we tell the listeners the origin story of AMAP? Where did this all begin?


Well, well, yeah. So so it it started at Burning Man. Spencer and I shared an office at in math school. And I think you're getting invited to Burning Man and we're looking for like an especially California friend of yours to go.


And and so so there's this this longstanding tradition of Burning Man. I'm actually not sure how many people adhere to it, but there's this long standing tradition of of giving back to the community, especially the first year you show up and we don't have. I mean, I can't speak for Spencer, but we don't have just a hell of a lot of skills that are applicable in the middle of a salt flat. So other than just kind of talking about our own research and science and what we're interested in, so we figured, what the hell, let's do that.


And got, I think, just a red bedsheet and some pieces of wood and nailed it to the ground and then just turned that into an ask a mathematician, ask a physicist booth. And honestly, I thought it was just going to be the two of us hanging out in a very small shaded structure, just kind of talking to each other for a few hours. But it really turned into a whole thing. Like every year we've done it, it's turned into kind of a throng.


I assume you had to expand your tent from that first year? We did.


Well, yeah, the first year was a little crowded. After that, we started basically going to barbecue supply and getting like big rain canopy sort of thing.


So we thought that maybe people want to know how the universe works now. So is that are those kind of questions? People would ask how the universe works? Because I've met Burner's and they they're all over the place.


They really are. We had a one point the like a large fraction of the undergraduate physics department from MIT showed up as we were setting up and asked this just bonkers question about like, how do you how do you physically solve certain energy problems? Like I said, you know, soap bubbles can be used to solve minimal surface problems. Are there any other physical phenomenon that solve any problems? I believe. Yeah, exactly.


I believe our exact answer was. You don't know what's the weirdest question, you've got a burning man. One of my favorites is this guy who just who came into our tent and said, why just fly? And and we try to respond to having this went on, he kept saying why over and over again? And we tried to ask him why, what now?


We just kept trying to trying to guess what he wanted as an answer and give him an explanation.


That was actually the first thing I asked was why? What, by the way, just to underscore how uncomfortable and weird this was, the guy was wearing like body paint and feathers and nothing else, I assumed.


What did you end up telling him? So finally, I think I said to him, nobody knows. And then he's like, That's right. That's an honest answer.


So. So you set up the you set up the website after coming back from Burning Man and just started. Were you answering questions that you got at Burning Man or did you just started taking questions from the Internet?


Well, both. At first we had a fair stockpile from from all the people who stopped by the booth. So I think like the first 20 or 30 posts or mostly from from the booth directly, that a lot of them since have been. But yeah, a hell of a lot of them trickled in from the Internet. I really didn't expect anywhere near as many people to be interested in the website as there have been.


We also did it on a Union Square as well. And yeah, I remember that. But honestly, the questions there were even weirder than they were at Burning Man.


You know, I believe that actually one one woman, one woman, we had to we had to ask her to repeat the question like five times to make sure that we really understood what she was asking and. She was the question, it was I think it was, how did DNA start the Big Bang? Oh, it was. It was. It was. How do scientists know that DNA caused the Big Bang or something like that? Wow.


Really, really laying in some premises into that question there.


Provocative. Yeah. How did you answer? I just give it to south, you know? Yeah, yeah. No wonder he's got many more questions than you in the website.


You're the physicist. You take us here, you kind of have to pick that one apart. It's like, well, not a lot of scientists think that or think I started with what? Where did you hear that exactly?


Good. Good. So what would you say out of all the questions, including just the ones submitted online, what would you say has been the most controversial question in the sense of, you know, generating the most heated debates in the comment section or angry, you know, feedback from readers?


I think it's the basic math questions. Interesting. People get real mad about that. Yeah.


For me, I think it was when I wrote on, what, a zero to the zeroth power equal.


And what was your answer? Well, I basically said that zero zero three equals one because Mateschitz say so, which is which is, I think, the correct answer. But I tried to explain in detail why why that's the case. And so kind of to give you the intuition for it, why you might think it doesn't equal one is so. So imagine you have like x rays, the x power, right? Mm hmm. Well, if you think of that as like, you know, zero rates, any powers or zero.


Right. You know, people people think because, you know, zero square two zero zero zero zero cubed zero zero zero, which is zero. So people have this situation that zero to the X power, regardless of what X is, you should always get zero. Mm hmm. Right. So some people say zero zero zero four zero. On the other hand, you can think of it a differently and you can say, well, if I have x rays, the zeroth power, that's always one.


Right? If I have to the zeroth power, that's one five three zero one zero zero zero power should be one. And so you kind of had that debate. Yeah.


Like about an unstoppable force meets an immovable object.


Exactly. Exactly. And yeah, kind of. And the funny thing about it is mathematicians tend to use your power equals one. And, you know, it's that kind of I think that bothers some people at y you know, how do you justify it? But the irony is it sort of justified on pragmatic grounds because it makes formulas simpler and it actually doesn't it doesn't really matter which way you define it. You can kind of work around it either way.


And that really bugs the hell out of people because people think of math as like completely precise and objective. But it turns out when we say X to the X power, there's actually an ambiguity in what we mean. And that's so. So we think of math is totally precise, but sometimes it's not. And in that case, we're actually pointing at an ambiguous statement and math. And that's why that question doesn't really have an answer. And you could send it to one because it makes your formula simpler.


And and, you know, everything works out fine.


I can kind of see how that would be like existentially horrifying to people, kind of like, you know, finding a glitch in the Matrix or something. Like if I pull on the thread all of my you know, all the logic and sense meaning of the universe as I understand it will start to unravel or something like that. Yeah.


And you see you see this ambiguity with, for example, integrals as well. You know, if you write an integral, people think of an integral as like the area underneath the curve. Right. But an integral actually has multiple different definitions. There's a reason integral to the big integral and they're defined differently. And when you write an integral, you usually are not really clear which when you mean. And mostly you don't worry about it because they can't get the same answer.


But what if they don't know you're actually talking about. Yeah, yeah.


Those those kind of a glitch in the Matrix things. I mean, as soon as you tell somebody that you can't do something like you can't divide by zero, you can't you can't have a particular answer for, you know, zero to the zero power. And of course, in physics, you you can't go faster than light. You cannot predict when a radioactive animal decay, that sort of thing. I mean, these are these are facts. They are like nailed down.


But man to people go after each other. If you just throw that out, it's like throwing meat to to to a pack of dogs.


Is it just like it it inspires the revel in them that you know. Oh yeah. Well watch me.


Yeah, exactly. As soon as you tell somebody you can't do something, there's a physical law that says you can't do something like.


Well, I mean I, I, I spent like most of my seventh grade free time in science class trying to invent a perpetual motion machine because they told me I couldn't. So I would I would fall under this under this property myself.


Yeah. Perpetual motion. Yeah, exactly. I mean, I totally sympathize.


I spent three years trying to figure out how to to just mathematically break RSA encryption and just totally failed, but learned a hell of a lot doing it.


Well it is that. Is that literally impossible or is it just extremely difficult. So you should treat it like it's impossible. It is well, there's there's now a quantum algorithm that if we ever get a computer capable of running, it will can can break it. But in a nutshell, it's you have a very big number and you want to factor it into its factors. So if I just give you, like, a 50 digit number and say, all right, this is what times what, you're kind of a bad place.


So there are a hell of a lot of people who've spent a hell of a lot of time being very clever about this. And they found special situations. But by and large, it's it's there's no proof that says that it's hard, but it's hard.


That's well put. Actually, speaking of controversial questions, I was rereading some of your posts on quantum theory this week, and it seemed like the commenters got pretty angry about a lot of it, for example, of the many worlds interpretation, like they would do the thing where they tried to shame you for, know, promoting post truth like they would say, I'm so disappointed in you. You call yourself a physicist. Am I right that quantum theory seems to make people angry?


And if so, why?


Yeah, it does. It does it. Resignation and weariness.


Well, I mean, haters going to hate, as they say, studying quantum physics.


One of the big things that's done for me is it's given me a lot of sympathy for people who believe completely crazy things and are absolutely convinced of it, but can't explain it to their friends because that's basically what you believe.


That's what I believe. Like everyone. I'll see somebody, you know, living here in New York.


I'll see somebody walking down the street talking about, you know, well, the Illuminati in the tinfoil hats, they'll be like, yeah, you and me both.


So, I mean, I hesitate to say many worlds because that that kind of evokes an image of, you know, Slider's in some episodes of Star Trek and that sort of thing. And that's that's not the exact interpretation I hear, too, is called relational quantum physics. And it basically says it's basically the idea that you take the laws of quantum physics as established and then apply them in general. And then you ask the question, well, is there anything wrong with that?


And the appalling answer is there doesn't seem to be.


Well, is this actually different from many worlds or is it a rebranding of many worlds?


It's it's it's so similar that by and large, most people can't tell the difference. The many worlds interpretation, as it's generally described, is the universe is puttering along and then there's some kind of quantum event, whatever that means, there's some kind of event. And then suddenly the universe just pops into these these different events. As a technical term. You're limited in physics, just like the sound effect.


It suddenly splits into these these divergent branches that never meet again in the universe becomes from one becomes many. And that is unfortunately also not it doesn't exactly jive with the physical laws I've been listening to. I've been listening to your show. And one of the things you do is you ask people if they have a recommendation for reading. And unfortunately, I don't usually. But I've read a paper a couple of years ago that I was thinking, if you have some physicists who listen, they might be jazzed about this.


There's this paper called A Bell Inequality in Position in Time. The Nobel Inequality, a bell experiment is how you demonstrate that things are in many states. And the bell and equality in position and time demonstrated that not only does the universe in some sense end up in multiple states in the future, but you can demonstrate that it is literally in multiple states in the past.


So the idea of the universe going forward and kind of continuously splitting it, it doesn't jive with the laws in the first place. And secondly, it doesn't really jive with the fact that it seems to be just as split up in the past as it is in the future. That makes sense.


It's not like a big tree that's branching out, out, out.


It's sort of always been branched in a sense that it's kind of it's it's there are branches in the past that kind of interact to create what we've seen now and then there are. So all you can really say is reality seems to be the way it is right now and anything that can happen does. And if anything that could have happened did leading up to this particular moment and. I don't know, I get I get nervous saying many worlds because I don't know how much of this you're going to have to edit out.


I get nervous saying many worlds because all of this is based on on. Actual physical desktop experiments that you can do so if you do like the double slit experiment, you can you can demonstrate pretty conclusive well conclusively that photon is going through both slits at the same time. And when you sit back and say, well, what what does the photon experience? The photon doesn't literally experience going through both slits. But there are kind of two versions of the photo, each of which feels very confident that it's going through one slit and definitely not the other.


So in some sense, the different versions of the photon are in different worlds. But at the same time, it's all happening on your desk right now, like you can point out and say it's right there. So it doesn't seem fair to say that these are other worlds. It's just that the one world we live in is weird.


Hmm. Yeah. I mean, it is it's true that when people talk about, like, their, you know, the copies of me and in other worlds or something, they are kind of implicitly imagining, like far away, like unimaginably far away. There is another me or, you know, infinite muse or something like that which which I guess you are. What did you call your framing of the theory previously known as many worlds, what they called.


Well, many worlds are still around the theory known elsewhere as many worlds. Yeah. Relational quantum mechanics.


And how widespread is that? Did you come up with that name? No, no, no, no, it was it's been around since the late 90s. I think I just don't get around. It's I mean, it's it's it's very FUBU for for quantum theorist, I suppose. It's it's kind of the idea that, yeah, there's nothing particularly special about quantum particles, and in exactly the same way there's nothing particularly special about the experimenters doing the experiment.


But it. It's not it's not about kind of the observer being on high influencing the particle with their observation, it's about what is the nature of the interaction between the these systems, right.


Yeah, I really have to wonder whether that like how dependent the reception of the theories was. Like if we had started with that instead of with many worlds, would would the acceptance by people, maybe even by physicists, have been different? I don't know. No idea, it's. You can you can get you can successfully get through a couple of physics degrees and never really have to get into the nitty gritty of this stuff, right?


Yeah, I kind of imagine it. I approach it from the standpoint of of kind of like the debate between the heliocentric theory and the geocentric theory, heliocentric theory. We go around the sun on the geocentric theory of the Earth is definitely sitting still and the planets are doing goofy, weird things. So so you look at Newton's laws and they they seem to work just fine here on Earth. And then you look up at the sky and say, well, the planets don't seem to be following Newton's laws.


They're doing loop to loops and they're being pushed along an epicycles and whatnot. And the question that you should be asking is, all right, assuming that Newton's laws apply in general and assuming that everything is moving around, obeying conservation of momentum and gravity and whatnot, assume that all that is true. Would you notice it would feel like you're sitting still? And the answer is a very unintuitive. Yes. Like we are. I think we're orbiting, I think, 30 kilometers per second, which is just ludicrously fast.


But you don't notice it and very much. Well. Philosophically similar, you look at the laws of quantum mechanics and you say, all right, well, what does an interaction between quantum systems look like? And from the outside, it looks like entanglement. And from the inside, it looks like wavefunction collapses. Looks like a Kopenhagen interpretation. Cool.


I like that analogy. I have a question for both of you. It occurred to me that having both a mathematician and a physicist on the episode with me is kind of a perfect opportunity to talk about a question that's come up a couple of times in the history of rationally speaking, perhaps most recently with in my episode with Sabina Hassenfeld. She's a physicist who wrote a book called Lost in Math about how physicists, at least in some subfield the physics, are too prone to evaluate theories based on their beauty.


Like is this is a theory simple? Is it natural instead of, you know, what does the evidence suggest is most likely to be true? Do you guys have an opinion about whether mathematically beautiful theories in physics are more likely to be true?


Yeah, I think it's a really interesting question. And the first thing I would say about it is like, so you can split people into like people who believe that there's a God that made the universe and the laws of physics and those that think that there is no intelligent being that made them. If you think an intelligent being made the universe like why exactly what you think that that that intelligent being would use simple laws, like why not use complex laws?


You know, I just it's God's way smarter than we are. All right. And again, if you also if you think that there is no intelligent beings in the universe, then just the universe is again, why would you really expect the laws to be inherently simple? Like, why should it be that us beings that eventually evolved will be, you know, smart enough to understand them? So. So I don't really see, like, really strong apriority reasons to think that the laws will be super simple.


But that being said, I think there are some reasons why we might find that that some of our theories end up being simple and being preferrable to simple theories. And just to give one example of that, imagine that you have some phenomena that you don't realize are connected to each other. So at first you think, oh, these are just different things going on. But then later you eventually recognize, oh, wait, they're all manifestations of the same phenomena.


So, for example, it may be totally unobvious that like the reason that you fall down after you jump is related to the reason the planets move in certain motions. Right. But eventually you kind of unify those things. And that actually can really simplify your theory a lot rather than trying to explain everything using different ideas. So that's just one example why simplicity can appear even if has nothing to do with like the universe is fundamentally simple.


Hmm. Yeah, that's a that's a good word. Yeah, that's a good way to put it. The laws of the universe are very they're very holistic. All of them apply all the time. It's just that some of them kind of don't matter, like the fluid, the the laws of fluid dynamics apply in deep space. There's just no fluids around for them to apply to. So you don't notice them so much, I think. Well well, we wrote a post about this, and Spencer made a point that over time has been sinking in about just just how very profound it was.


I think you pointed out that the equations tend to look pretty because the ideas that we find ourselves using over and over again, being naturally lazy beings, we use more and more succinct notation for it.


Oh, like using short words, like words like end or though or he or she. They're short because we use them so often, not the other way around. Or is that a good analogy. Yeah, I didn't say it. Well hopefully you got it.


No, no, no. Yeah, exactly right. Yeah, actually, exactly right. Like for for example, there's a mathematical operation called the determinant that you apply to matrices to to square grids of numbers, and in some sense, it's about as complicated as it can get. If you if you know somebody who's taken linear algebra recently and you say determinant near them, they'll just huddle up into a ball and start shivering. I mean, nobody likes to do these things.


And yet the notation is just, you know, if it's a matrix, A, it looks like absolute value obey. I mean, it's the cutest little notation in the world. That's a really good example.


Yeah. I mean, and you look at the you look at, say, the wave equation when you describe the philosophy behind how waves work, it it kind of makes sense if you imagine a jump rope when it's when it's like if you're imagining people playing double Dutch, if you imagine the jump rope when it's up and kind of concave down in a minute, it's going to be pulled back downwards. And when it's at the bottom and it's concave up, it's going to be pulled upwards in a second.


And that's basically what the wave equation is. But when you actually have to write it down using math, you've got you're equating second derivatives with each other. And then he said, well, what exactly do you mean by second derivatives? And then some of the notations are actually well, is the sum of second derivatives in all three dimensions. And then pretty soon you realize that this this very compact notation is really talking about a hell of a lot.


Right. Even just to find the second derivative you have to define in terms of the first rather than the first derivative usually defined in terms of limits. And so, like, if you were to write out all all that stuff for someone who had never heard of these ideas, you're like, wow, that seems really complicated now.


Hmm. Yeah. Well, I mean, there was this one aspect of beauty that Subpena was talking about, which doesn't seem captured by what you guys are talking about now is called naturalness. And the way Sabena defined it was, I'm quoting here, a theory should not appear like it has been handmaid. It should not have some conspiracy's among the numbers that they've turned out to be just right. For example, in particle physics, she said, we have a big gap between several energy scales, one energy scales called the Planck mass.


That's a pretty large energy scale. It's about 15 orders of magnitude larger than the heaviest particles that we know, for example, the Higgs boson. And that brings up the question, where does this large ratio come from? This is what we call unnatural. And she was saying that there isn't really good justification for using naturalness as a criterion to to, you know, lean towards some theories and away from others. And I've actually talked to some other physicists who think there is I mean, of course they do, because she was arguing against them.


But I, you know, met them. Who do Seth or Spencer, do you agree with the beina about naturalness? Because that seems to be more about like the values of the of the the, you know, coefficients in the equations as opposed to how we you know, how complicated does the equation seem to us?


Well, one thing I think you can say about that is that sometimes we humans, we're trying to force the theory to work. And so we'll like stick in the thing, just try to make it fit, you know, and like a classic example of this would be like the epicycles trying to model the movement of the stars when you haven't yet realized that, like, it's better to move the center of the solar system to the sun instead of the earth.


And so you're kind of putting in these awkward circles within circles, within circles to try to fit the curves. And so sometimes there's things like power. Humans are trying to, like, jam a theory into fitting. And then as soon as you say, no way, we need to move everything to the the sun is in the center, like it just suddenly gets rid of all that, like cruft. And even my understanding is that even once we move to thinking of the sun as being at the center, it still didn't quite fit because we had, I think, the idea of like a circular orbit.


So it's still a little awkward and people are trying to make this kind of ad hoc adjustments and then eventually is like, oh, wait, no, wait there ellipses, elliptical orbit. And then we got rid of that crap. So there's something there maybe about how humans can, like, put awkward kludges in because the theory doesn't quite work.


Hmm. Yeah, no, I yeah, I totally agree with the the naturalness thing.


I mean it's kind of but it is a good criterion. Naturalness.


Yeah. Yeah, yeah. I mean it's not, not immediately. It helps a lot to see what's been done before to get to get kind of a sense of the smell, the character of natural law. It's it's, it's a little bit like reading bad computer code. Some, you know, you make an error and instead of kind of stepping back and looking at looking at the code overall, you say, well, I'll just write a little patch for that.


And then pretty soon things don't line up. So you have to write a patch for that. And then pretty soon I need to really stress the metaphor. Instead of having a nice, smooth blanket, you've got this quote that's just nothing. But, you know, patches all stuck together, that software. So so, yeah, when you when you read when you read a description of physical law that says, well, I've got the. This idea that explains this particular thing.


But it doesn't really explain anything else, and I don't know why it works, then you start to get suspicious. So like the epicycles, they really did a pretty good job of explaining the movement of the planets, but they didn't answer questions like what's going on? Why do they exist at all? Like, I think the biggest users they had actually kind of naturalism, they went well, circles are very natural.


So circles and circles, of course, sir.


But yeah, why were why were they the exact sizes they were and why were they lined up the way they were sort of thing. They didn't have an answer for that. But but Newton's law of gravity, it's very just kind of everything works like this. Not just the planets, but just the Cavendish experiment demonstrates it's not just planets, but like rocks sitting next to each other about the exact same laws of gravitation. So it's it's a simple thing that explains a hell of a lot.


I'd say that physics. In the early to mid 20th century, they started to swing wide when it came to to particle physics and they got some remarkably simple philosophies about how particles should interact and how they should work. They kind of they tied in the basic ideas of of quantum mechanics with relativity, got these ideas of how particles should exist and how they should interact. That was just just gorgeous.


It's just absolutely beautiful.


And they just they started coming out like, you know what? I bet there are like all of these particles out there. And I bet they weighed this much and then they went to their their particle accelerators and dammit, if they weren't exactly right.


And it has a way of of causing you to become a little bit cocky about the power of math to to spend, they're going to take that lying down to turn off its computer and tell.


Well, it's basically the. Here's the thing, math on its own owes nothing to physical reality. Indeed, math, not even true. I mean, no, I mean, it's not like it's not like physicists are going to are going to go to their particle texted. People at CERN are not going to suddenly discover new math. They'll discover new facts about the universe. That may imply some new rules that we could write using math, but they they won't be, you know, discovering the next biggest problem or something or they're not going to discover a new math rule that we didn't know about.


The rules of math are, for lack of a better word, they're made up. They are exactly as made up as as language. But that's not to say that that we they don't mean things like the words that we're using right now are made up, but they still mean something.


OK, but OK, Spencer, at the beginning of the episode, let's get some let's get some reality check. Our getting really controversial here. Yeah.


OK, so at the beginning of the episode, we are talking about how zero to the year with power equals one to sort of by decree of mathematicians.


You didn't know. You didn't know mathematicians. No, no. They can just declare it.


And yes. But what if different mathematicians declare different things, then argue about it forever and a series of papers or something.


OK, not each other.


So. So here's I actually. I actually. That one did no one did bother me, I, I will find in AMAP yeah.


Are you trying to find I went out and tried to find an example where it was used and defined to be something other than one and I didn't.


But there you go.


Zero zero equals one by a convergence of all mathematicians who disagree are disappeared. They're no longer mathematicians. They're right. They haven't been heard from since way back, way back in the day.


A bunch of mathematicians got together and they decided to write down just a list of all of the underpinnings of mathematics just so that they could all agree and what they were talking about. And that gave rise eventually to like Gödel coming along and saying, well, you know, computing is just incompleteness theorem and kind of throwing everybody under the bus, mathematically speaking, but it forcing everybody to write down exactly what the axioms are, what are the basic rules.


Force them to also admit that there are kind of mathematical dialects, like whether or not you want to include the continuum hypothesis, continuum hypothesis or the axiom of choice.


These are things that are they seem to be true. They seem to work, but we can't necessarily prove them. So they're kind of thrown onto the list of axioms, as actually I think I think the axiom of choice has been proven to be impossible to prove as well as proven to be impossible to disprove. So I that falls away from the axiom category.


Yeah, I think that's a really important point that most people don't know about math, which is that, you know, math started with just people trying to do things in the world. Right. Like they're counting how many sheep there are, something like that. Right. And eventually after doing that in many different things, you know, counting sheep can carrot's can children, you know, you start to generalize and instead of thinking of of numbers is like adjectives.


They're referring to sheep or something like that. You start thinking of numbers is nouns. You kind of take this flip in perspective and you say, oh, we can just think of them as being three and not just three sheep or three children, but just three. And then we can talk about what properties they have. Right. But but really, math started as a thing that's useful. And people used it for a long time doing all sorts of things from counting to areas of plots of land.


And then eventually, after a very long time, people eventually were like, hey, what the heck is this thing we're trying to do? Can we, like, put it on a firm foundation? And that's where it really starts are going crazy. And a lot of people think that math is on a firm foundation. They think that someone has proven that, you know, one plus one equals two. But but really, if you if you really get into the core of it, you start to realize stuff is talking about these different flavors of math that depend on what axioms you choose.


And math is just don't agree on inlike zero that zeroth power. They don't agree what axios to use. And the probably the most famous example of the continuum hypothesis, which is a very, very fun kind of question, which is I say, you know, the integers have their infinite number of integers. Right. And there's also an infinite number of real numbers. And the continuum hypothesis is about is there an infinite set? Those are just any infinite set that's bigger than the integers in a very precise sense of like being a bigger infinite set.


But but smaller than the real numbers, you know, is there an infinity between those two? And basically it turns out to be also independent of the actions of math. And so you can either add it as an axiom or you can add it's opposite of an axiom and it's totally arbitrary. And so we get different flavors of math and there's no way to decide between them except to say, well, if you like a more complex, interesting math.


Yeah. Or you want a tidier cleaner the other way.


What where do you end what sort of math you like to it. I don't, I don't know. I'm confused and alarmed by the question. Well where do you end?


I don't have opinion because it's never come up in any problem I want. Now, coming up now, I don't know.


But I think I think it's really interesting. I think about like what is is math, really. And I agree with Seth that math is a language, but it's different than most other languages because it's extremely precise. And and I think of it as a language of patterns. So it's a language we invented. It allows you to describe sort of any type of pattern, at least that we know how to describe can be described with math. And so it's not that crazy to say, well, OK, that it's less about kind of math being the thing that is inherently provable and defensible and more about that.


It's this thing that allows us to do all these different things by describing sort of any sort of pattern we come across.


Hmmm. You know, it it becomes very frustrating. I get well we get we get a lot of people writing in who have, you know, figured out a theory of everything or. Figured out all of math sort of thing, by and large, they haven't chocolate, but yeah, they I have a question.


Which one of you gets weirder mail? Like, I know mathematicians and physicists who are at all in the public eye tend to get weird mail, which now Seth runs a site.


So he gets all the weird mail. But but I would say I think the physics mail tends to be weirder than the math now.


And so by and large, by and large, honestly, it's it's a lot of the same people will be sending a lot of theories and posts. We've got a lot of polymaths in that sense. What will get people will write in and have like a very particular idea about like, oh, I've figured out how to divide by zero or something like that. And it's very frustrating for them when I come back with one. All right. So do it.


That's fine.


You're like no skin off my nose.


Yeah, mostly because it's like it's like it's like you got to it's like you got a chessboard and somebody says, well, I want to play checkers. Well, that's that's fine. Go ahead and play checkers. I mean, we have it when you ask and ask and answer questions about math, you're always talking about these are the the big ideas that we've all agreed upon, you know, to is bigger than one. There are numbers in between one and two, that sort of thing.


These are these are axioms that we've we've made up. And if you disagree with them. It's not. Explicitly, that you're wrong. I mean, you may be self-contradictory and therefore wrong, but it means that you're not playing the same game as everybody else. But you may not necessarily be incorrect. Right.


People sometimes get bugged out when you do things like you make up a new operator and you're like, I'm defining a new operator called Plus. It's not the same as the plus operator, you know, and it has to be that one plus one equals one and so on. And people are like, whoa, what are you talking about? And you're like, oh, no, no, this is a math is a language. And we can define things the way we want.


And I can have this plus operator that has these different properties. And actually some officials have done funny things like this where they'll redefine addition, multiplication, exponents and so on, and then they try to redo the theories of math in these new weird this new weird world where all the operators mean different things and they can actually create kind kind of very interesting, beautiful, consistent theory is really kind of cool.


Is there a use for that or is that just like a fun party trick? Sometimes they're sometimes there is a use because, I mean, you know, the way I think about applied math is basically you look around at the world and you notice patterns and you notice repeated elements. Right. And then you then you go back and you say, can we describe these patterns with math? And you write down the math to try to describe them. And once you've done that, what's cool then is you can then forget about the world for a while and just look at the math and study what properties it has and derive things about the math that you didn't know before.


And then later, once you've done that, you can take those new things you've learned about the math and then go make predictions about the world or check that the world and you often will find 008 The World has that property that I predicted it would because I correctly found this pattern in this pattern according to the not implied these other things. Right. And so insofar as you can find parts of the world that correspond just just like the way you can find parts of the world to correspond to the regular plus operator, like, you know, I take three balls, put them in a bag, add two balls in the bag, and at five.


Right. You can find things in the world that also kind of correspond to some other kinds of weird operators that are kind of like pseudo ed. operators or whatever. And then that math will describe those parts of the world. Right.


I can see, however, how it would be maddening to a certain kind of person to be told like, yes, you're completely correct in your own math.




Yeah. Nobody likes it. Everybody hates it. Yeah. We really we really do have this because there are things that you can say about the universe that are hard and fast and definitely true. You may have different ways of describing them. You may have different ways of writing them out. But they are are definitely true. They are facts. But in math, it really is a language. It is it is kind of up in the air what you decide to use or not use.


But that being said, there are some constraints, right? Like if you have inconsistencies, your whole system will fall apart and it won't be useful because can't you just invent a new math?


And which inconsistency is allowed? Well, you could do that. You can have inconsistent maths, like, look, I'm not going to deprive anyone of a consistent math, but the problem in your own backyard, you just get a very uninteresting, boring thing where you can prove any statement, right?


Yeah, we actually had somebody show up at the booth pretty young. I think there were like 15 years old, came up and they were like, I want to I want to break math. What if what if five was equal to zero? Was like, well, that's that's fine. That's that's modular arithmetic. And we got into this whole thing about how incredibly useful modular arithmetic is. And they were so excited. Like, I tried to break math and it just kept going well.


I was worried he would be mad because he wanted something to rebel against. And you just deprived him know the most bitter ending.


You know, one thing that's funny I've observed when people want to kind of like challenge physics or challenge math and have their own theory, which I think, you know, surprisingly, no people want to rebel against it is that almost always the theory they come up with are less weird than the reality, which, you know, maybe the universe is more creative than humans are.


I don't know. But but, you know, you think about relativity. A lot of people like who come up with their own theories, they want to say, oh, no, Einstein was wrong. But then they have this theory that's just like way more normal and plausible than sounding the relativity. It just happens to be totally untrue. And relativity happens to be right.


Relativity is really bonkers, like a really bizarre like saying that that's, you know, space and time are not objective and can be warped by, you know, by motion and so on. And you assume quantum mechanics, I'm even crazier. It's like the ratings of a madman. But it just happens to be true.


But I think was I think it was I mean, do you think it was Michio Kaku who said the only thing that quantum mechanics is going forward is that it works literally the only thing I have time for, like maybe one more question.


And I think I wanted to I want to use it to ask Seth about something in one of his posts. I think this is the post on one of the posts on quantum entanglement. And in that post, you tell this fascinating story of how some, like Frend parapsychologists, proposed an experiment using entanglement that ended up changing physics. I think they were called they're called the fundamental physics group. But it was spelled weird. Do you know what I'm talking about?


The fundamental physics group. Yeah.


Can you are you do you remember that story well enough to recount it to our audience, Fred, Alan, Wolf and Serfaty, maybe they've come up a number of times. Oh, God, yes. No, I do know so.


So so if you ever read about entanglement in popular media, there's kind of a a very common refrain that goes along with it.


You know, entangled particles, parentheses, entangled particles are particles where if you. Do something to one, it instantly affects the other, even if it's on the other side of the universe, right. So. To be absolutely clear, no, in no way whatsoever, entanglement is a lot like correlation. So if if if if if Spencer had set up a pair of quarters, put them into sealed envelopes and given one to each of us and the US, and he can be trusted and assured us that they are both either heads up or both tails up, then we could go to opposite sides of the universe.


And if I looked at my coin, I would know what your coin was, no matter how far away it is. But I'm not actually influencing it.




So if you understand correlation that way, then you understand about 80 or 90 percent of what's going on. And there are subtleties because it's quantum mechanics and it's complicated or whatever. But that's that's the big idea. The correlation is really the more important piece of it.


Anyway, these guys, they were the kind of acid, the fundamental physics group I found the physics that spelled F y f ICS, fundamental physics group.


Just context, you can say.




By the way, by the way, in all seriousness, if you're going to do a Google search for them, also do an image search just to get a sense.


I didn't do that. I don't have to do that.


You really sure those guys were a whole thing anyway? So they had this this whole idea. They were not they were not slouches. They were actual well, they still are actual genuine physicists. They just like a lot of physicists, they were wrong about some stuff. And they they set out this idea for how to communicate faster than light. And it was all right. You and somebody else get a pair of entangled particles and then both of you copy them a bunch and then you kind of do different measurements.


And if you do your measurement this way, then the guy on the other end will have to do their measurement this way anyway. It's subtle, but it it would have worked. It would have allowed people to communicate at any speed over any distance.


And on a cursory read through, it's not obvious what is wrong with that, and yet at the same time, relativity and kind of the basic structure of the universe has some things to say about that. For example, if you if you're communicating faster than light, it turns out that you should also be able to communicate back in time. So this kind of raised some red flags. And anyway, actually, it it was reasonable enough that unlike a lot of kind of bonkers theories that actually caught the attention of some some physicists who knew what they were doing.


And they comb through very carefully and found a couple of assumptions that they'd made, one of which was that you can copy your particles and then measure that was the copying stage that it turns out can't be done. And that then became the no cloning theorem. So these guys kind of it's very self is very subtle. And these guys are kind of. By kicking over anthills and coming up with these these wild theories actually helped to advance the science through through.


No fault of their own. Wow. Food like an accidental reductio ad absurdum of some of these physics assumptions. Yeah, it's kind of like like, I don't know, the Big Bang or shouldn't Schrodinger's cat, which were like attempted reductio ad absurdum rights, but it turned out to actually work except this time.


Yeah, well, yeah. Schrodinger's cat, he was Schrodinger came up with the the story of it literally to to underscore how completely insane quantum physics is and how it can't possibly work like that. And everybody laughed about it. And then pretty soon, you know, the 90s rolled around and we started being able to actually do these physical experiments and test them out.


And then pretty soon the story became, well, maybe Schrodinger was serious with you alive when when it became clear that his bonkers experiment, thought experiment was actually possible.


I don't I don't think so, too. In the in the in the early age of in the early age of quantum mechanics, like the early 30s or so when when Schrodinger and in boring all them were running around. They were they were kind of they were getting some some subtle hints from the mathematics that things kind of need to be in multiple states. But they were all classical physicists, so they were all used to this idea. Well, you know, maybe you don't know what's going on, but there's something is going on like, you know, before we look at the coins, we don't know if it's heads or tails, but it is one or the other.


We just don't happen to know. And it wasn't until I think nineteen sixty four that the first proofs came out that said that no, if the results of these quantum quantum experience are literally incompatible with the assumption that things are in a definite state. Those were the belt tests, right, and I don't think Schrodinger lived to see that too bad.


I would have loved some reaction shots. So before we wrap up that, you've already kind of anticipated my question at the end and answered it. I was going to ask about a work that has influenced you or change your thinking about something. And what was the name of the book that you mentioned or was it an article about what it was?


It's a it's a paper. It's called the it's the forensic experiment. And it was a bell. Inequality in position in time. That'll be exciting for four physicists, physicists. If you're listening and you know and you hear, fine, don't get intimidated by it. It's an interesting paper for anybody else. I've been reading through the origin of the Species and just enjoying the hell out of it. Yeah, it's. It's. Darwin really thought about this stuff a lot and and did a really good job of anticipating people's arguments against evolution.


Everything I've ever heard, every every remotely cogent argument I've ever heard against evolution, he addresses I mean, the guy really had this stuff nailed down. He's a little wordy. He's a little namedropping. He's super into pigeons.


But it is really boring to cats. So I don't think that's fair. Yeah, that's fair. Or write an article about what happens if you can you do the double slit experiment with the cat canon.


So it's right if you fire cats instead of light. Yeah, OK.


The answer to that one, no one wants to claim credit for the Cat Cannon thought experiment. I think that was your question. I'm not you know, I really wasn't.


I think maybe it arose out of like a drunken night of speculation and it's disappeared from my mind. As I as you say, that it starts to sound vaguely familiar, like it's the thing I once asked you about. I had no memory of that. I can't believe it's not my fault after all these years. So full of bad cat puns. Now I feel bad.


It's in the book, by the way. Oh, excellent.


Yeah. So you as it turned into a great a great discussion of of. The experiment, the very basic math behind it and how if you were to attempt to do it and kind of like why you don't notice quantum effects on the scale where we are. So I think I think it turns out that if you were to do it with actual cats. Even assuming little mascot's, which is the kittens you'd need, you'd need a double slit apparatus, something like 30, 300 quadrillion light years long, which is just a how many longer universes?




Before I let Spencer answer the question about a resource that influenced his thinking, I just want to note that while physicists may be obsessed with cats, it has been my experience that mathematicians, specifically this mathematician, is obsessed with ducks, because every thought experiment that he gives me, it seems, involves ducks. And the example that I think I shared on Twitter recently was I even forget what the original topic of conversation was, of something abstract. He was trying to explain to me and I didn't understand.


And so I said, can you just give me a real world example of this, Spencer? And he said, sure. And he thought for a minute and he said, OK, so imagine you have an infinite number of boxes with ducks in them.


This is your real world example, Spencer.


Anyway, so Spencer not have outed you as being obsessed with ducks. What book or or resource would you name is something that's influenced your thinking?


Well, unfortunately, I don't have any duck related books to recommend that would have been so great if your pick was duck related. I would have been so vindicated. What do you got? Well, one book I think is really cool. It's not an easy book, but it's just cool and interesting is this book called The Discoveries, where each chapter is about a great discovery in science. But what makes it so cool is they actually have a part of the original paper that proves the thing.


And you can actually. Yeah. After you read the chapter and learn a bit about it, you can then go peek at the original paper like an excerpt of the original paper and read what the original scientists wrote. And I just think it's so neat to see it, kind of how it was first presented, you know, Einstein presenting his theory and so on. And so just a fun thing to prove is. Yeah, it's really great because I unless I'm like really comfortable with the field, I usually don't want to just go straight to the primary source.


But then when I'm reading the secondary source, I'm always wondering, like, what am I missing? So it's kind of nice to have your cake and eat it, too.


Think it's a rare glimpse into kind of the original science, but in a way that's a bit more accessible than trying to go right there as well.


Right, exactly. All right, well, I guess we we unfortunately have to wrap up, but this has been a blast. It's so great to have you guys on the show after all these years. To you. And I just want to I want to put in one more plug for Seth's book.


Do Colores exist? You can order it on Amazon in time for the holidays. Hint, hint. It's it's full of fascinating and informative questions like the ones we've been talking about today. And just to give you a peek, I looked at the table of contents and it's divided into four sections. I love the Seth. The sections are big things, small things in between things and not things so strongly recommended.


And also make sure to check out ask a mathematician, NORCOM for, you know, lots more questions and answers.


So this concludes another episode of Rationally Speaking. Join us next time for more explorations on the borderlands between reason and nonsense.