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Richard Elwes, "Huge Numbers: A Story of Counting Ambitiously, from 4 1/2 to Fish 7" (Basic Books, 2026)
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Richard Elwes
good, so good so good.
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Gregory McNiff
Welcome to the New Books Network welcome to the New Books Network. I'm your host Gregory McNiff and I'm thrilled to be joined by Richard Elwes, the author of Huge Numbers, a story of counting ambitiously from four and a Half to Fish seven. The book is published by Basic Books in April of 2026. Richard is an Associate professor of Mathematics at the University of Leeds in the United Kingdom, where he has taught subjects including logic, graph theory, computational mathematics, and the history of mathematics. His PhD is in mathematical logic and he has also published research on graph theory and related areas. He is a presenter on the YouTube channel number appeal and he is the author of six popular mathematics books, most recently Huge Numbers, which we'll discuss today. I selected Huge Numbers to interview Richard because it treats enormous numbers not as gimmicks, but as windows into the limits of human thought, language and abstraction, which you'll hear about in this interview. Specifically, I was fascinated by the way the book moves from ancient counting systems to modern computational theory, while all the time asking what happens when numbers become too large to meaningly visualize, physically represent or candidly just understand at a basic intuitive level what they represent. In addition, just being a book about the development of mathematics, this book is almost a guide map for expanding the human mind. Richard, thank you for joining me today to discuss your book.
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Richard Elwes
Well, thank you very much for having me, Richard.
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Gregory McNiff
I'll start with the question I ask all my authors. Why did you decide to write this book? And who is the target audience?
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Richard Elwes
Let me come to the second part of that first. I think that the target audience is anyone with an interest in numbers, really. I think you certainly don't have to have any mathematical expertise, but I think it is all about numbers. So you do have to have an interest, if not an education in numbers. So it's very much targeted at a general audience rather than specialists. Why did I come to write it? Yeah, it's an interesting question because it's quite a simple idea, Right. And in some sense I just had that idea. But it came to me from really spotting a similar theme from two different directions. It was from some courses I was teaching. I was teaching history of mathematics, as you mentioned, which is. So I'm not a historian, I think it's important to say that, but I'm a mathematician, but I have always had an interest in the history of the subject and I taught a course on it and I was teaching a section on ancient Egyptian mathematics. And the ancient Egyptians had. Well, they had two different numeral systems, actually, which we could get into the intricacies of the differences in a bit, maybe. But the important thing for this story is that they both just ran out of symbols at some point. So if you want to write down a very big number, if you want to write down 100,000, then that's easy enough because they've got a symbol for that, a hieroglyph. If you want to run down a million, they've got a hieroglyph for that. But if you want to write down anything bigger than that, you're really struggling because they just run out of symbols. And that's not something we experience with the numeral system we're used to. Of course, it doesn't have a sort of hard limit. It gets bogged down once you sort of start putting too many digits together. But it doesn't have a sort of ceiling, whereas the ancient Egyptian system really did. So that was one observation I made. And then the other observation was more out of my research in mathematical logic. I was reading about an area called reverse mathematics. So it's a technical area of mathematical logic. We can get into the weeds in a bit, if you're interested. But the key thing I noticed was that again, really enormous numbers and talking numbers on a totally different scale from Millions and billions at this point play an important role in testing the strength of logical systems. And I think that's not an obvious thing that would happen, but in fact it does so even in modern mathematical logic, if you get mathematical processes, which is generate numbers which get bigger and bigger and bigger so fast that the system can't actually cope with it. And so there were two really very different topics, but they had in common this idea of big numbers challenging the human invented system we have for handling numbers. And I thought that was quite an interesting theme. And I just sort of mulling it over for a little while before it eventually coalesced into a book proposal and then into a book.
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Gregory McNiff
Nice. I'm certainly glad it did. Did I want to hit on that idea of getting bigger and bigger numbers made me faster and faster, particularly in this age with AI and GPUs and computational power, which you discuss in the later half of the book. But just to follow up on the answer you just gave, what's nice or great about this book is it's not just a list of numbers and the dates when they were first acknowledged or recorded, but this is really a story of history and pushing, as you know, our conceptual limits. Do you think of this book as, you know, a combination of math, intellectual history or something broader about the human, the brain, the human mind?
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Richard Elwes
Yeah, it's something which has been sort of in the back of my mind for a long time. For a while I was sort of pushing back against it as being a maths book. Actually it's definitely a book about numbers. I mean, you can't get away from that. But I mean, mathematicians don't have exclusive rights to numbers. Right. Most people use numbers every day and most of those people are not mathematicians. So it's a book about numbers. And for most people throughout most of history, people doing things with numbers are not being mathematicians, pondering Fermat's Lewis Theorem or whatever your favorite pure mathematical conundrum is. There have been people doing completely practical things with it, buying and selling things, paying taxes and that sort of stuff. So it's really a history of numbers in that sense for most of the time. But then there are people who've just, at various different points around the world and throughout history have just sort of gone a bit nuts with it and said, right, what is the. Let's just go wild and just make some far bigger numbers than any practical requirement needs. So I mean, the sort of classic example which I discuss in the book is Archimedes, who did this amazing thought experiment about how many grains of sand would it take to fill up the universe. And that wasn't a practical question.
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Gregory McNiff
He wrote the king about that. Correct.
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Richard Elwes
I know he wrote to the King about it, which is just this absolutely glorious thing to do. I just love the idea of someone, they're writing to the King with that thought experiment. Yeah. I mean, I'm fascinated by big numbers, but I haven't written to the King about it, I must say. Yeah. And then of course, once you get into that question, okay, let's just push it as far as we can. So let's leave all the practicalities aside and just go for the biggest numbers we can, then you're sort of automatically drawn into more technical mathematical considerations. So the last section of the book, I mean, I can't argue against it, it is definitely about more technical mathematics, specifically mathematical logic. That's where the biggest numbers arise. Because really, if you want to write down the biggest number you can, the sort of standard approach is to say, okay, it's the biggest number that it's possible to write down in some particular framework. And then, okay, you have to compare the different frameworks available and how powerful they are and so on. And that's all quite subtle, philosophically interesting, but also somewhat technical material. Yeah.
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Gregory McNiff
And I actually, I don't want to give the ending away, but you talk about Gram's number and Rao's number and I want to get into that towards the end because it's almost logical or self referential, particularly Rao's number. How much is mind blowing. But just to continue on this path, you know, reading the first, third, first quarter of this book, I was struck by how important language is, language and notation. And I guess that's not a great insight, but could you talk about the relationship between language and particularly notation in extending the limits of how we think about numbers, particularly in the early civilizations that you document and discuss.
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Richard Elwes
Yeah, absolutely, you're right. That is the central consideration at the early part of the book. And if you ask the question, what's the biggest number that some particular civilization had access to? Then the answer is entirely determined by their notational system. Or we could even talk about people without writing. So unwritten languages, you can ask the same question, what's the biggest number you have access to? And then it depends on the, the spoken numerals. And so maybe spoken numerals are a good place to start because the sort of most basic way of counting in some language is just to make up a new word for each successive number. So if you count it in English, 1, 2, 3, 4, 5. There's no pattern there. Right. And you just have to learn what those words mean. Whereas if I say 1234, that's not a word I've learned the meaning of. Right. There's a system working there. And different societies throughout history and around the world have developed different systems. And some of them are more powerful than others, some of them more efficient than others. I mean, an example I like to talk about is Roman numerals. And the reason I like to talk about it is because there's something a bit surprising that Rome was the biggest empire the world had ever seen. Absolutely enormous money flowing all directions, people paying taxes, a hugely complicated, enormous civilization. And yet it worked on Roman numerals, which are these just incredibly sort of clunky, hopelessly inefficient numeral systems. And think about trying to do long multiplication or long division with Roman numerals, I mean, it would be a complete nightmare. Right? So on the other hand, the Romans did have. They had a secret weapon which was they used Roman numerals for writing down numbers, but for actually calculating them. They had the abacus. That was what they used. And the abacus, if you just sort of look at it, it's actually. It's got a column. Okay. You've got some wires, but you can. It's almost like a numeral system itself. It's written in columns. And the system we're used to today is written in columns with. And the key observation is that the units column looks the same as the tens column. So in 77, if you write that down as a numeral, you got those two sevens. But they're playing slightly different roles. The two columns look identical, the two symbols look identical, but the position of the symbol is playing as important a role as the symbol itself. And that was actually an incredibly important discovery. So we call it place value notation. And it is remarkable that both the ancient Roman and ancient Greek worlds rose and fell without ever having that realization. Except arguably within the abacus, maybe. But it's not really a written numeral system.
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Gregory McNiff
No, no. I found that fascinating, too, because I think at one point you talk about how, you know, to some extent the Roman numeral system was a product of the. The empire's expansion, inertia and culture, and the military prowess more than, I guess, the most logical system. I guess I want to ask the obvious question. How did we, and by we, I mean, you know, the west, end up with today's Hindu Arabic numeral system instead of something like the Mesopotamia or Mayan systems, which were also interesting. Could you, could you talk a little bit about that path?
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Richard Elwes
Yeah, absolutely. I mean, it's interesting that you mentioned the Mesopotamian and Mayan systems because those two, unlike Roman numerals, are also place value notation. So that those were places where the same discovery was made of possibility of place value as it was in India, which is the source of our Hindu Arabic numeral system. And I mean, in the book I make the argument that there's nothing inherently superior for the Hindu Arabic system compared to, for example, the Mayan system. There's definitely something superior compared to Roman numerals. Right, so this is not some sort of everything's equal. Well, not a bit of it. The point is the same discovery was made in the Mayans, in the Mayan world and in ancient Mesopotamia. And I mean, one might imagine a modern high tech world built on either of those systems, but not on Roman numerals. So the answer to the question you ask is why do we use the Hindu Arabic system? That's a purely historical question. It's not a technical question because you could say Roman, Hindu Arabic would always win out over Roman numerals eventually because they're just better. Right, but compared to the Mayans or the ancient Mesopotamians, Hindu Arabic numeral systems are pretty comparable. Well, the chain of events is that this system developed in ancient India. It had various forbear systems. The Brahmi numerals were used about a couple of thousand years ago, slightly less, maybe they were those, those numerals were not place value. They were what's called a ciphered system, which is, and we could go into it if you like, but it's, it's a bit more, more efficient than the Roman numeral style, but it's not on the level of the, the place value system we have now. So it's similar to the ancient Greek system or the better of the two ancient Egyptian systems. And there's also, at the same time in ancient India, one sees a religious fascination with enormous numbers in several of their religions. So in Buddhism, for example, one finds a text called the Flower Garland Sutra where a good chunk of it is the Buddha. Just talking about enormous numbers, I mean really far bigger than anything practical, just for some sort of mystical appeal of them. And it wasn't just Buddhism. So one finds similar things in Hinduism and particularly the ancient Indian religion of Jainism. That's the place in the ancient world when one finds the biggest numbers of all. So one's got a pretty good numeral system in the Brahmin numerals and one's got a fascination with big numbers coming from various different religious standpoints. And it's certainly been argued that religious fascination with big numbers was a push towards the development of a place place value notation which is, which got, got going in, in India and then spread around, I mean fast forwarding over, you know, thousands of years of history, but it fast forwarded around the, the Islamic world and then, and then crossed, crossed over to Europe and now took over the world. Right. It's, it's just basically the stabst standard system for most, most of the world.
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Gregory McNiff
Now you touched on this and I actually wanted to ask. But India seems to have a special, I don't know if it's passion, respect or fascination with large numbers. Why do you think that is, Dan?
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Richard Elwes
If I, I know the answer of why it is. I think that it's, I mean you're certainly right. There is a, there is a fascination with large numbers one finds in India. Just much, much bigger number people contemplating much, much bigger numbers than you find anywhere else in the ancient world. And I mean, maybe I should give an example or two. I mean, so these are, these are far bigger than anything in one would require for any practical purposes. So, so maybe I'll give a quick example from Jainism. So they have the idea of a pit here. Pit here. So the word is a palaeopama and that's a year measured according to a pit. So what is the pit? It's a cubic pit about 10 km wide, deep and across. And you fill that cubic pit with lambswool and then you extract one strand of lambswool every century and then the pit year is the length of time it takes you to empty the pit. So that's just one example of a very big number from one ancient Indian religion. And there are plenty of other examples which I go through in the book as well. Now if you do a sort of back of the envelope calculation, which I did, that's something like 10 to the 31 solar years. That's one followed by 31 zeros. Now given that the age of the universe is 13 billion years or something, this is far bigger timescale than anyone has any practical reason to consider at all. So that's one example. And why did they have that interest? I don't know. I think one just has to say that there was a, you know, there had developed a, a culture of interest and fascination in pushing these things further. And then each generation just tried to push it a little bit further and there was a view that, you know, you were sort of glimpsing, glimpsing through the door out of the human realm. And these are such inhumanly big numbers, you're somehow seeing something vastly bigger than the human realm. And I think that's why there's a, as a sort of a religious, or could be a religious aspect to it.
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Gregory McNiff
That's interesting. And I know you talk about the Mayans as well. At some point, their grand long count was something like 43 octillion years. I mean, so not just specific to India, but I should say certain cultures just really seem to have this fascination with numbers that exceeds anything remotely relevant or practical then or even now.
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Richard Elwes
Absolutely. So the Mayans. And so you're quite right, the Mayans also had this fascination with big numbers. And it's not a coincidence, I'm sure that they were also one of the societies to develop place value notation. Because once you've got place value notation, you know, writing down enormously long numbers is just a matter of patience, really. I mean, if you could, you could write down, you write down 39s in a row and there you are, you've written some incomprehensibly enormous number. Whereas in other cultures without place value notation, you have to sort of really build your way up to that by inventing new notation or inventing new terminology. Whereas if you've got place value notation, actually the door is just open. You can just write down as big numbers as you want, really. And this was something that, yeah, they really took advantage of in both ancient India and ancient, ancient. The Mayan culture in Central America.
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Gregory McNiff
Fascinating.
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Gregory McNiff
I want to move on. You have a discussion on exponential growth notation and how we think about that. And I think you say in the book, I'm going to paraphrase here, but it is hard to overestimate the significance of exponential growth in human civilization. It seems to be underlying so much of our, at least modern notions of numbers and math. Could you talk a little bit about that, the impact and the role of exponential growth in our, I guess, our daily lives?
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Richard Elwes
Sure, yeah, absolutely. Well, we live in A high tech age, everyone knows that. And if rather than just thinking about, you know, the technology we have available, what's interesting to think about is how quickly that's changing. If you compare the technology we got now to what was available 20 years ago, 40 years ago, 60 years ago, 80 years ago, we live through a massively fast rate of change. And that technological change is one very visible example of exponential growth. So the sort of classic example of technological exponential growth is this law called Moore's Law. Moore's Law is the encapsulation of technological exponential growth. So Gordon Moore said that the number of transistors on a microchip would double every two years. And that has been happening continuously since the late 70s. Roundabout. And that and other similar examples of exponential growth in other bits of the technology. It's not just transistors or microchips. There's other things changing as well. But all of that exponential growth underwrites the change we see from the sort of 1970s computer to the computers we see today. And so, yeah, and you can't argue that that hasn't changed the world. Right? And that's. So that's one instance of exponential growth. Another exponential instance of exponential growth, which listeners will remember is the growth of a pandemic. So that in sort of classic pandemic modeling, you might have one initial infected person, and if that person infects two others, and each of those infect two others, and each of those infect two others, it's a surprisingly small number of generations needed before basically the whole population has been infected. So, I mean, those are two big matters that have been on everyone's mind for a long time. And there you find exponential growth. And it's something, I think exponential growth is something we just find at particularly dramatic moments of change. So whether that is a nuclear explosion, which again has got exponential growth in the actual nuclear chain reaction inside the bomb, whether it's technological, whether it's the growth of a pandemic, or whether it's the growth of the human population itself, which for many years was growing exponentially, or even maybe a rather rare thing, maybe actually faster than exponentially, but it has now. And so people are worried about overpopulation and so on, but it is true that human growth, while we are the global population, is still growing, but it has now slowed down. So it's no longer exponential, which is a source of comfort, while whilst it doesn't solve all the problems, it's a sort of necessary prerequisite to the problems being solved. Otherwise we would just simply explode beyond bounds of whatever resources we have available, which is a prediction made in the 19th century by Thomas Malthus. So, yeah, it is interesting that so many dramatic human changes. When you sort of dig beneath the surface, you do find exponential growth somewhere, because I think it is a mathematical expression of a loss of control, really. That's where it tends to rise.
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Gregory McNiff
That's interesting. When do these numbers really start to, I should say, go from our understanding conceptually to just a very long time or a very large amount? And you talk about the universe being 13.8 billion years old. I think the latest estimate is trillions of galaxies at what point? And maybe I should have started the interview by asking what's the distinction of four, you know, between four and five or four and a half and you know, how far? Yeah, maybe if you could talk about that. And then when do. When do the billions, trillions and octillions start to blur? Like conceptually, how far can we think about it in terms of numbers versus some abstraction of a very large amount?
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Richard Elwes
Yeah, it's a. It's a great question and much of the book is really about probing that, because I don't think there is one answer. I think it all depends on the tools you've got available. So, you know, you're talking about, in that question, you're talking about billions and trillions. Those are numbers. I mean, I think a million used to be 100 years ago. A million was in an example of just a very, very big number. People used it to mean just an enormous number. Whereas nowadays millions are pretty run of the mill. Right. You're reasonably wealthy if you're a millionaire, but you're not one of the richest people in the world being a millionaire. So if you're interested in finance, for example, millions are just not big enough. Now you need billions. And if you want to talk about the sort of the global economy, you need trillions. So the point I'm making is that this does change. If you talk to someone. A hundred years ago, a billion would have been. Seemed incomprehensibly big. But now these are numbers which fly around quite a lot and we sort of get used to them. So it's partly about. Just about getting used to them. And it's also about the tools we've got available. So we've got a spoken numeral system which can handle these numbers pretty efficiently because, well, we've got these numbers million, billion, trillion, quadrillion quintillion, which it helps to know a little bit of Latin, but there is a system there, right? And you can push it quite far. Although really you don't want to rely on those words for too far, I think. But the point I'm making, it all depends on the tools available. And we've got very powerful linguistic tools. So we here now are much more comfortable with much larger numbers than people at various different times throughout the world in the past. So you ask me about the four and a half. So the subtitle of the book is from four and a half to fish seven. We can get to fish seven later if you like. But so four and a half, what's that about? With my tongue slightly my cheek, I. I argue that that's the first big number. Why so? Because that's the first place where a human system for grappling with numbers breaks down. And that system is the most innate one we have. So it's one which precedes any sort of language, either spoken or written. And it's just instant recognition. So if I put know three chocolate bars on the table and say, greg, how many chocolate bars are there? You will instantly spot that there's three, right? You don't have to stop and count. Whereas if I put, you know, empty a box of 12 on the table and say, how many are there? Well, if you say, give me an immediate answer, you'll probably get it wrong. If you want to get it right, then you've got to employ some more sophisticated technology, namely you've got to count, probably using words. So this instant recognition is something which goes way, way back in our evolutionary history. So it long predates any sort of language because once. And the reason we know that is we see the same thing in a wide variety of other species. So even baby chickens, for example, can do this. Okay, so their limit is not four and a half. They've got a lower limit of three. But nevertheless they can do it. They can spot the difference between two and three, just. Whereas if you try and challenge them between four and five, they totally have no clue. They pick randomly, but between two and three they can tell the difference. So that's the most innate number system we have. And four and a half is the place it breaks down because. So why four and a half? Well, it comes from the experiment that originally spotted this, which was done in the 19th century by a guy called William Stanley Jevons. And he was chucking black beans into a box in his room and he wanted to just see how many he could immediately identify without stopping and counting, without allowing himself to correct it. And if it was four, he always got it right. And if it was five, he pretty much mostly got it right. He almost always got it right. But he made occasional mistakes. And when he plotted the graph, the cutoff looked like four and a half. So we say four and a half's the cutoff for this. Subitizing is the word people use for it. It means instant recognition. So that's the starting point. So when do numbers start to become sort of blurry and confusing? Well, if all you've got is subitizing, if you haven't got any sort of language, then beyond that limit is where that starts to happen. Right. Because if you want to identify numbers precisely beyond that limit, then you need some more sophisticated technology, which probably means some kind of language with words which you can use to associate specific quantities. And that's something which only humans really do,
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Gregory McNiff
I think. So I want to follow up on that. We're all familiar with the notation around exponents, but that almost breaks down when we start talking about numbers or concepts like grants numbers and then you introduce new up arrows. Could you talk about how you've moved beyond or we've moved beyond to standard exponents to, like, mountains and towers and advanced notation for these concepts?
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Richard Elwes
Absolutely. So this is where we really leave behind numbers which are of any practical use at all. Right. So the only people who are interested in this are either pure mathematicians or just sort of curious travelers. Okay, so it comes from the observation that. Let's just think about multiplication. What is four times five? Well, it's repeated addition. It's five plus five plus five plus five. Right. So multiplication is defined as repeated addition and then exponentiation 5 to the power 4 is repeated multiplication. So 5 to the power 4 is 5 times 5 times 5 times 5. So 4 times. So 5 multiplied by itself 4 times or 5 fours multiplied together. Sorry, 4 fives multiplied together. So once you've made that observation that multiplication is repeated addition and exponentiation is repeated multiplication, then you can push the same idea further. And the next thing that people call tetration is basically repeated exponentiation. So for 5 to the power 5 to the power 5 to the Power 5. So we write. If we write that down, that'll look like a tower of four fives, each a superscript of the last. And that is a number which is now just blowing away any number that anyone needs for practical purposes, which is why tetration is just not something that arises even in physics. To be honest, even physicists contemplating the extremes of the universe don't need numbers this big, but mathematicians love this stuff. So once you've thought of this, then you can just repeat it again. The general terminology we use is at least, and there's various different ways of describing numbers on this sort of scale. But the one that I like the best and the one I use in the book are called Knuth arrows, after the computer scientist Donald Knuth, who invented them. And basically we use a single Knuth arrow to mean exponentiation, and then a double Knuth arrow means tetration. So five, double arrow four will mean five to the power five to the power five to the power five, which is the same as tetration. And then each subsequent Knuth arrow is defined as the previous one repeated. So triple Knuth arrows are repetitions of the double knuth arrows. So five, triple arrow four is five, triple double arrow five, double arrow five, double arrow five and double arrow five. And at this point, so you asked me a little bit earlier, where do our intuitions for big numbers run out? And we've certainly reached the point now where any sort of intuition is gone, right? We're far, far, far past that, because we can't even write that as a conventional Tower of Powers. So if you wanted to sort of just try and write down a massive number, you might just say five to the power five to the power five to the power five to the power five to the power five and so on. But the numbers I've just described there, I would have to be repeating that phrase to the power five to the power five to the power five, sort of beyond the end of the universe and still not get anywhere close. Right? So these Knuth arrow notation is incredibly powerful because with just a few symbols, it very, very rapidly takes us into essentially utterly alien numerical realms. These are just realms that no one has had ever any cause to visit. The people who got closest were, again, the ancient Jains. Actually, they did actually do some extraordinary thought experiments which got them into the realms of the double Knuth arrow. So that's tetration and possibly even triple Knuth arrows. And that's very remarkable that a thousand years ago, people were approaching these numbers, but there was really only them. So they were the record holders for most of history. But nowadays, armed with Knuth arrows, we can push it further. So you can then talk about having a large number of Knuth arrows, right? So now we know how to build Knuth arrows. You could talk about five arrows, five, where the number of arrows is itself a very, very big number. And that's how one can Always. And that's the thing with big numbers, is that you could always push it further. And when I say push it further, I don't just mean a bit further. You could always push it much, much further. Each step sort of completely eclipsing everything that's gone before. And that's the. So by iterating large numbers of Knuth arrows, one gets into the realm of this number called Graham's number, which was in the Guinness Book of Records for a number of years as the biggest number that ever featured in a mathematical proof. And if you want to describe it, you need to essentially iteratively describe the number of arrows in a slightly elaborate way. So it gets a bit complicated, particularly, particularly on a podcast without a written medium. But this stuff is relevant to pure mathematics, even though for most people it's a completely alien realm. Mathematicians do have use for this stuff.
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Gregory McNiff
Yeah, that's fascinating. I mean, it's so beyond the everyday experience of, I'm sure, even engineers or. And people who do have a mathematical literacy, but absolutely fascinating. Richard, maybe this is a good point to ask you. What exactly is Fish number seven?
- [39:34]
Richard Elwes
Okay, well, let's just firstly explain what the name is. Fish number seven. So the fish is the pseudonym of a Japanese googologist who writes under the name Fish. So googologist, by the way, is the term for someone who's interested and studies enormous numbers. And so Fish is a Japanese guy, and it's his seventh number. So he wrote down. He wrote down seven numbers. This is his seventh and his biggest. So that's just what the words mean. And what it is was essentially his attempt to write down in a nice concise way. And these are relative terms at this point, by the way, essentially the biggest number anyone had ever come up with. And it's. I cannot give you a very short, short account of it because it's a little bit. It's a little bit technical. But let me. Let me try and give a flavor of it. So the way. If you just want to write, if you want to describe the biggest numbers possible, then essentially the most efficient thing to do is to. To give yourself some sort of intellectual framework, some language, and say, it's the biggest number. The number I'm describing is the biggest number accessible in this framework. And then you need to put some limitation, like the number of symbols, for example, that you can use. So you could do that in just. With the ordinary. Suppose you've got nothing but the ordinary digits. What's the biggest number you can write in 10 symbols? Well, it's just 10 nines in a row. Right. If you're allowed to build superscripts, then you could do nine to the nine to the nine to the nine. So it all depends on the rules. So the rule book that is in play once we're up at the level of Fish 7 is a rulebook called set theory, which is a super powerful framework from modern mathematical logic. And set theory very, very quickly takes you into. You can describe enormously big numbers very, very quickly. And there was a big number Competition held in MIT in the year 2007 between two philosophers, Augustin Rao and Adam Elgar. And they just challenge each other who can write down who can describe the biggest number essentially. And the eventual winning entry became known as Rao's number. And essentially what that is is it's the biggest number describable in the. I'm simplifying a bit, but it's essentially the biggest number describable in the language Set theory in a Google symbols where the number Google is 10 to the power 100. Okay. And so that for a long time was just viewed as the biggest number people had ever come up with. But of course Google just liked to push things further. And Fish's number seven was his attempt to push that further. And he essentially did that by strengthening the language in a slightly complicated, self referential way. We can discuss if you're really interested, but maybe it's easy to let people read the book, but the point is it's a way to sort of superpower an already super powerful language and to just describe. Yeah, to just basically essentially attempt to describe a bigger number than anyone else had ever managed.
- [43:09]
Gregory McNiff
Yeah. And I should point out, I think Rael was a philosopher, which I thought is interesting and gets into this conversation. It's like math and philosophy are almost converging at this point. I want to talk a little bit about the computational aspect and specifically Turing and Busy Beaver numbers. Could you, can you briefly talk about what Busy Bieber numbers are and why they're quote, uncomputable?
- [43:31]
Richard Elwes
Sure, yeah. So setting aside set theory, which is a very esoteric and maybe it's fair to say, slightly controversial branch of mathematical logic, leaving that aside, the biggest numbers people have come up with come out of theoretical computer science and they come out of these. There are these things called the busy beavers and those are built from Turing machines. So people probably heard of a Turing machine. What is a Turing machine? It's essentially Alan Turing's answer to the question what mathematical processes are computable and which aren't okay, so the. And nowadays, of course, we've got computers, so we know we have got a better idea of what it means to be computable. So which mathematical processes could you write a computer program to carry out and which couldn't. Couldn't you? And I mean, it's actually quite easy to come up with ones. You can't, as soon as you just add a little bit of randomness. Right? So if I suppose I. I'm going to write down a sequence of zeros and ones, and I'm going to generate the zeros and ones by tossing a coin. So if I get a tail, I write down zero. If I toss a head, I get down one. And suppose I just keep doing that, and I'll just spend the rest of forever doing that. Okay, we'll get a sequence of zeros and ones. That sequence of zeros and ones has got no pattern, no order in it, and you would not be able to write a computer program which could generate that sequence. It's an uncomputable sequence. Okay, so Turing was posing that question, and which processes are computable and which aren't. And he came up with essentially a model of computation. And so what Turing machines are are these little theoretical gadgets. They're not physical things, by the way. No one's ever actually tried to build or use one, except once as an art project. But there are theoretical machines which buzz up and down a ticker tape, reading and writing zeros and ones, depending on what they see working. And Turing's insight is that any. And this is actually incredibly important realization, far more important than it sounds, which is that any sequence of zeros and ones which is computable by any means is computable by an appropriate Turing machine. Right? In other words, they completely capture the idea of what it means for something to be computable or not computable. Okay, so that's what Turing machines are. And then, okay, so for people who are maybe a little bit lost, you can essentially think about them as computer programs. That's really what they are. It's just a particular programming paradigm, I guess. And with any computer program or Turing machine, one clear division one can make is does it stop or does it run forever? So it's very easy as soon, if you're a beginner computer programmer, you only have to have had, like, probably an hour's lesson in whatever language you're learning to be able to write computer programs of those. Both of those kinds, one which does. Does its thing and then stops or one which just loops forever. It's easy to generate both of those kinds of programs. So then there's this mathematical question called the halting problem, which is, if I present you with a Turing machine or equivalently a computer program, can you. Is there some way you can tell in advance whether it's eventually going to halt or whether it's eventually going to stop? And that sounds like a very arcane thing to be interested in, but actually, if you could do that, you'd be able to answer all sorts of mathematical questions immediately, because there are lots of mathematical questions we don't know the answer to, which you could easily write a computer program to probe, right? So there's this one example. It's a thing called the Goldbach Conjecture, which says that any two even numbers you can write as two primes added together. So, for example, eight, an even number is written as five plus three, two primes added together. Okay, so is that true for all even numbers? Well, we think it probably is, but we don't know for sure. But you could easily write a computer program which would go through and check, right? Check even numbers one at a time. Now, if I had the superpower to know whether or not that program would stop in advance, then I would know whether the Gelbach contractor is true. And the same goes for a whole load of other mathematical problems. So that being able to predict in advance whether computer programs stop or carry on forever is incredibly important. But that's an uncomputable problem. So there is no computer program you could write which would interrogate other computer programs and reliably tell you whether they stop or not eventually. And the Busy Beaver numbers? Sorry. I realized building my way up to the Busy Beaver numbers. You asked me to introduce them and doing my best, but it's a few steps to get there, but I think we're nearly there. So the Busy Beaver numbers, what they are is the stopping times of the slowest Turing machines, which do actually halt. So any Turing machine, which, again, is just a computer program, either carries on forever or it eventually stops. So let's just forget the ones which carry on forever. We'll just abandon them. And all the ones that stop, you can then say, okay, which are the slowest ones? And so there's a sort of basic measure of the complexity of the machine, which is its number of internal states, which is something like, you know, it's equivalent to the number of lines of code or something. And then you. So you can talk about the slowest halting Turing machine with one state, the slowest Halting Turing machine with two states, with three states, with four states, and so on. And those numbers I've just described are the Busy Beaver numbers. And that is an incredibly fast growing sequence. It grows out of control very, very quickly. So, I mean, actually the first few are not that intimidating. The first one is just one. The second one is six. The third one is 21. The fourth one is 107. And then a couple of years ago, in 2024, we discovered, a team of mathematicians discovered the fifth one, which is just over 47 million. But that's probably as far as we're going to get because we don't know what the sixth one is. But we do know that you're going to need at least three Knuth arrows to describe it. Right. So that's putting us sort of far beyond anything which could be testable on any real world computer. So these Busy Beaver numbers, they sort of demarcate. There's the sequence of enormous numbers which really demarcate the computable world from the uncomputable world.
- [50:39]
Gregory McNiff
That's a great explanation. And now is probably the time to ask you the obligatory AI question. In this case, coupled with proof checking software, combining the two, do you think they will eventually push mathematical exploration beyond what A, we're capable of, or B capable of comprehending?
- [51:00]
Richard Elwes
It's a big question, isn't it? No, no, it's okay. It's okay. It's a good. It's a good question. And who wrote the book?
- [51:10]
Gregory McNiff
You're the expert, Richard.
- [51:12]
Richard Elwes
The book's about big questions. I like big questions. I mean, of course, this is something which is moving very, very rapidly right now, and if it continues to move at the current rate of pace that it's moving at now, then I think, yes, I think eventually, and don't pin me down on when eventually is, but I think eventually AI mathematicians will overtake human mathematicians. Yeah. So I think maybe, though, we should just be a bit more precise about what we mean by AI mathematicians, because there's two different things you could mean, and you actually pointed to both of them. So when you talk about AI these days, what we mostly talk about is large language models and things related to them, and those are rapidly getting better at maths. So if you ask them, you know, a few generations ago, if you ask them any sort of mathematical question, when I say generation, I mean AI generation, not human generation. A few years ago, ask any mathematical question, they were pretty useless. Nowadays they're getting better and better and better and they can write down mathematical proofs proper, rigorous mathematical proofs. At the same time, they are very much not reliable. So yes, they can write down proper mathematical proofs, but they can also get things wrong. And they can write down something which looks. They can also write down something which looks like mathematical proof and actually is just when you probe it is actually not right. So they're fallible. So we can't rely on those. I don't think we're ever going to be at the point where that technology becomes completely reliable and we can just outsource mathematics to that. But then there is a whole other approach to, I don't know if you want to call it AI, but it's certainly computational mathematics, which is the phrase you mentioned, is proof checking software. And what proof checking software does is it's actually much more traditional computing in a way. It's much more step by step. And essentially what it does is you take a mathematical proof. What is a mathematical proof? Well, it's a sequence of logical deductions, basically, and potentially a very, very long sequence of mathematical, of logical deductions. So a proof of a big mathematical theorem might have sort of many millions of individual logical deductions packed into it. And what people have been doing is essentially unpacking that and putting it into special software called proof checking software, which will basically go through line by line and certify that each of those logical deductions is correct. Okay. And what you can start to see now is that those two approaches are coming together because you could ask your large language model to produce a proof of some theorem. And okay, and then I read it, and maybe I'm not a big expert in the area. I can't really tell if this is right or not. But if I could then also ask the AI to, okay, now translate that proof into the sequence of individual logical deductions in a way that proof checking software can verify, and go ahead and do that verification form, feed it to a proof checker. Then we might, we'll actually get a verification out of that. So then we'll have a sort of stamp of a stamp saying this is a valid proof. And at the moment, you know, humans are very much involved in that process. The AI will not, will not do all of that. But you can, I think you can start to see those two sides coming together where you may be able to, may be able to plug, plug the two together. This is years away. This is years away. But once you've got to that point, that will be quite a moment because it won't just be that the AIs are coming up with proofs, maybe in collaboration with humans, but they're actually certifying them in a way which humans will have to accept, even if we don't understand how the hell proof works. Because all you have to check to trust the verified proof. All you have to trust is the verifier, and we pretty well trust those. So if the verifier gives you the tick, then maybe we have to accept it. So you can see this coming across the horizon. I think it's not there yet and I don't think it's going to be there soon. But that will be certainly a very interesting moment. But it's all about, you know, exploration and finding out it's not just proving difficult things for its own sake is not that interesting. Right. We also want to prove things which tell us things. So I think still there's going to be a role for humans to guide the whole process and then explore different areas. Supervise. Maybe it's been more of a sort of supervisory role. No.
- [56:18]
Gregory McNiff
That is fascinating. I want to circle back my final question there about pushing the boundaries of the universe and our thought. But two quick. The questions before that, or I think they're quick, but go. His incompleteness theorem. You have a fascinating insight here. You wrote, quote, his stroke of genius with arithmetization, I. E. Encoding everything in sight into numbers. Could. Could you briefly describe why you say that?
- [56:45]
Richard Elwes
Yes. So that was. I don't think this is a quick question, but very, very happy to tackle it. Very happy to tackle the. So Godel's incompleteness theorem is a theorem about. It's a meta mathematical theorem. It's about which statements in mathematics can be proved and which can't. And the key insight that Godel had which allowed him to prove his theorem, which is essentially that there are some statements which are unprovable in whatever proof system you've got set up, is that you can encode the whole of that logical proof system inside numbers itself. And once you've done that, so you've got a logical proof system which is set up in order to analyze proofs of mathematical statements. But if you can encode that whole proof system inside the language of numbers itself, then that suddenly gives you a way for that proof system to talk about itself. Essentially it becomes self referential. That was Godel's great insight. So that's what we mean by arithmeticization. So it's that he took that proof system and then arithmeticized it in the sense of translating it into the language of numbers. And then, in fact, Alan Turing later, we've been talking about Turing machines, he did a very, very similar thing. So he took these sort of theoretical computer programs, theoretical models of computing, and he arithmeticized them as well. In other words, he translated all of them, essentially extracting their source code as numerical data, and then that allows them to interrogate themselves as well. So this idea of self reference is what arithmeticization is about. And it turns out that numbers are the domain where you can really do this in a precise way. You can get numerical systems which can talk about themselves because you can encode that whole numerical system in numbers. Yeah, okay. So that's what arithmeticization is. Was there a second part of your question I forgot?
- [58:53]
Gregory McNiff
That was great. We're almost in the home stretch. The, I guess, penultimate question is my own personal favorite. I love Ludwig Boltzmann. I just think he's a fascinating character. I'm not going to ask you to go through his. All his entire history of contributions, which you do very ably in the book, but you do conclude, quote, that I am almost certainly am a Boltzmann brain. Why do you think?
- [59:16]
Richard Elwes
Why am I almost certainly a Boltzmann brain? Yes. Okay, so what is a Boltzmann brain? The idea is that, so we now imagine the impossibly distant future, sort of maybe more than 10 to the 10 to the 10 years in the future or something, where the universe has completely dissolved and there's nothing left except just individual particles flying around. So we've passed heat death. Okay, so the idea is that in that extreme state, things will still occasionally happen just out of the random nature of quantum reality, essentially. So quantum theory tells us that nature is fundamentally at the bottom level, random. So things just randomly happen. And among those things which will randomly happen could Ludwig Boltzmann argued, or at least people following Ludwig Boltzmann argued, could be just the spontaneous appearance in the post heat death universe of some highly, highly complex structure which is just insanely hallucinating that it's a human being on a planet called Earth, Maybe in conversation on a thing called a podcast. And how can any of us know that we really are who we think we are, rather than these hallucinating Boltzmann brains? And the place where big numbers sort of get involved is because, okay, there's lots of questions you can do are just philosophical uncertainty, Right? How do I know I and Descartes did it? Right? How do I know I am who I think I am, rather than some crazy future scientist just tormenting a brain in a lab or something? But the point with Boltzmann brains is that big numbers really make the problem worse, because we've got essentially indefinite amounts of time to play with. So these Boltzmann brains won't just pop into existence out of random fluke once. It'll happen again and again. Again. Okay, it's enormously unlikely, but we've got more than enormous amounts of time to play with. So it'll keep happening on big enough timescales, and eventually the total population of Baltimore brains who's ever been and gone will exceed the total number of human beings who've ever been and gone. Right. And then you have to sort of. Well, you don't have to, but the argument is that just the sort of laws of probability mean that I'm more likely to be one of them than one of us, because there's just much, much more Boltzmann brains that have been and gone than there were human beings that have been and gone. So I must, on the laws of probability, say I'm much more likely to be a Boltzmann brain. It's a mischievous argument, but that's the argument.
- [62:05]
Gregory McNiff
Yeah, I always find that fascinating. So, final question here, and Richard, you're probably the best person to answer this, given that we've talked obviously about huge numbers, language, computational ability, mathematical thought, as well as in the book, you trace the sort of boundaries of the universe, how it began, how it ended. So, question, do you see the universe as fundamentally mathematical, or is math the best language we have to describe the universe and explore the limits of it?
- [62:37]
Richard Elwes
Ooh, you are throwing me some tough questions today, Greg. I mean, I definitely believe the latter, so I certainly believe, and I think it's not even that controversial. I mean, the language of mathematics has proved itself just incredibly powerful for describing the universe on different scales, you know, from the biggest scale to the smallest subatomic scale. If you write one, write down the laws that make the universe tick as far as we understand them today, you're automatically guided into writing things in mathematics. There's no other language which we can approximate it. I definitely believe that. True. Now, does that mean that the universe is itself a mathematical object, or is that merely the sort of best approximation we can come up with to it? I'm afraid I'm going to bottle it. I think that's just too big a philosophical question to address in that. I'm not sure there's anything we could do to ever enable us to answer that question in either direction. I just can't imagine any kind of experiment which would settle it. So I definitely believe that mathematics is the best way we have of describing and approximating the universe on all the different scales. Is there ultimately something else behind it, or is it all mathematics all the way down? I just can't think of a way we could ever tell, so I'm afraid I'm just going to have to leave it hanging.
- [64:13]
Gregory McNiff
No, maybe that's the next book in that note. Richard, thank you so much for joining us today. I enjoyed the book and the conversation, I think as the audience clearly grasped the book is more than just a list or accounting of huge numbers, but uses them to explore these deeper questions around language, abstraction, and the limits of human thought. Thanks Richard. It was a great conversation.
- [64:36]
Richard Elwes
Thanks so much for having me. It's been really, really fun.
- [64:38]
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