Hello, it's your host Andrew here. If you're enjoying Send Me To Sleep so far and you'd like to help support the show, the best way to do that is Send Me To Sleep Premium. Over there you'll get ad-free episodes as well as access to all of our bonus episodes. You can find a link to a 7-day free trial in the description notes. Thanks so much for listening, and here's just a few ads before the show begins. Updating our tariffs to get you our best value, it's a smart tech that helps you take control of your energy future. We're here for whatever's next. Just one of the reasons why we're rated excellent on TrustPilot by our customers. Find out more about how we can help at yournext.com. Eligibility and T's and C's apply. TrustPilot February 2026. Hey, it's Andrew here, and I'm excited to share with you the newest show from Slumber Studios. It's called Sleepy History, and it's exactly what it sounds like. Intriguing stories, people, mysteries and events from history delivered in a supremely calming atmosphere. Explore the legend of El Dorado. See what life was like for Roman gladiators. Uncover the myths and mysteries of Stonehenge. You'll find interesting but relaxing episodes like these on Sleepy History, and the same great production quality you've come to know and love from Send Me to Sleep. So give it a listen, and perhaps you'll have another way to get a good night's rest. Just search Sleepy History in your preferred podcast player. Hey, it's Thomas here. I'm the host of Get Sleepy, another sleep-inducing podcast from the Slumber Studios Network. On Get Sleepy, you'll find hundreds of original bedtime stories and meditations to fall asleep to. Some of our listener favorites are our trips to the rainy day bakery, our Sleepy History series, and our adaptations of classic tales like Beauty and the Beast. Everything is designed with your sleep in mind. So if you're looking for another great way to ease into a restful night's slumber, then just search for Get Sleepy on your favorite podcast player. I'll see you there, my friends. So in 1910, prominent mathematicians and philosophers Bertrand Russell and Alfred North Whitehead published a book, and that book was called Princhiper Mathematica. And this book was their attempt to answer a seemingly innocuous question. Why is it that 1 plus 1 equals 2? And to hear that question for someone with a very base-level understanding of mathematics might sound quite silly. It may even sound too obvious to be asking seriously, especially as a prominent mathematician. But in asking this question and seriously attempting to answer it, Russell and Whitehead are tapping into something about not just numbers, but mathematics. That is quite mysterious. And that is the fact that in actuality, we very much take these concepts, such as 1 or 2 or 3 or even add or equals, very much for granted. I think when we all first begin to learn maths, well, 1 plus 1 equaling 2 is perhaps one of the first things that we learn. It is foundational to the idea of mathematics. And further still, I think that there might be something rather intuitive about the idea to people. It's easy to picture taking one thing, adding another thing, and calling those two things combined too. But Russell and Whitehead realized that it was actually a lot more difficult to formally, truly, and accurately describe what it meant for something to be one. What was this concept of oneness or tuness for that matter? You know, I could say that I have two hands. But what is it about these hands in front of me that give them their tuness? And in part, I know that Russell and Whitehead attempted to answer this question, partially using something called set theory, which is a very complex field of mathematics. And I won't pretend to know very much about it at all. But to my understanding, they were able to say something roughly like this. We could perhaps, when, for example, discussing the idea of two, use a metaphor, an analogy in collecting items of two. So as I've already said, I could say that I have two hands. And that's all well and good, but that doesn't really give us much of an idea of really what what two is. But perhaps if I then collected two apples and put them by there, by the side of my two hands, and we compared the similarities between these two examples of two, and then say, perhaps I were to go and get two pencils and put them by the side of my two hands and the two apples, we were to get two ducks, two tables, two chairs, two stars, two planets, and put them all next to each other equally. And ask ourselves, what is it that these sets of two things all share in common? And if we extended that out to all possible examples of what two things might be, and we tried to understand completely what it was that those sets of two things all shared, then we could call that tunas. We could collect all of those things, all of those pairs, into a set, and we would call that set the number two. And this is possibly to some listening now, an over-complicated way of describing two. Like I said before, it's something that feels very intuitive if we see two things, but of course that's two. But that is actually no definition of two at all, just because we can intuit something doesn't mean that it is necessarily logical or explained in of itself. And so this was one of the techniques that Russell and Whitehead attempted to use to describe what numbers were, as well as many other techniques in formal logic. And that really was what the Pritchaper Mathematica was, or at least attempted to be. It was primarily a work of philosophy. There was very little of what we might traditionally describe as mathematics in the book, despite its incredible length and detail and complexity, because you have to understand that these people were trying to write the rule book, as it were, for mathematics itself. And they were trying to prove that mathematics as a system was consistent in of itself. And it took them some years to produce the Pritchaper Mathematica. I forget how many, but a lot of strain and effort went into it. I think, and I'm entirely sure, but I remember hearing that even to publish it, there was quite a financial strain. And it put a lot of stress on their personal relationships over the years that they attempted to produce it. And even once they had produced it, the Pritchaper Mathematica hurt it not. Fair very well, it did not release itself into the public to congratulations and fanfare. In fact, many of their contemporaries found the book to be extremely dense and overly complex and extremely difficult to follow. I myself have never attempted to read the Pritchaper Mathematica, but I have seen fragments of it explained on the Wikipedia page. And if that is any indication of the density of subject matter that they are trying to engage with, the complexity of the formal proof they are attempting to produce, then I don't think I would fair very well in opening the pages of the book itself. And when I think about this book and what a valiant attempt it really is, and in many ways a genius one, to realize that we take the idea of numbers and mathematics and these things that are seemingly basic as addition and subtraction as simply true in of themselves and to know that in actuality we don't have a way of knowing that those things are necessarily consistent and that mathematics will always work for us when we need it. Though seemingly it has so far. To understand that completely and then make an attempt at least at solving that so that we can have a stronger foundation and a stronger conviction that mathematics as a tool is actually useful to us at all. I think that's a wonderful aspiration. But it always makes me feel rather sad for Russell and Whitehead. Because it wasn't very long after they had published this work that they had toiled over that it was discovered that there was a way of not only falsifying some of the work that they had put into the book, but that in fact it was not at all possible to achieve the thing that they were attempting to achieve. And it was Godel and something called Godel's incompleteness theorems that was the nail in the coffin for the very idea in principle of the principle of Mathematica. And the incompleteness theorem again I am no expert on these things but I believe well let me see if there's a good way I can explain this based on my understanding. It essentially proved that for any consistent formal system such as mathematics there was always a way of creating inconsistencies within it as such that the system itself could not be used to prove its own consistency. And I believe someone once explained it to me like this. Godel was able to mathematically produce the equivalent of a sentence such as this sentence is false. Using mathematical notation and in what way I honestly am not sure he was able to produce an equivalent of that sentence and now of course when you think about that sentence this sentence is false. If you can take that sentence to be true then its own statement cannot be true. But then of course if you take the statement to be false then that in fact means that it is true which means once again the statement itself is inconsistent you begin to enter into these infinite loops of inconsistency. And the very fact that you can produce these sorts of things within mathematics means that we cannot know whether mathematics is actually completely consistent and an honestly useful tool for humanity in describing the universe. To a certain degree that is not quite correct because of course we use mathematics all the time to describe the universe and it seems to describe it very well. We even use it to describe theoretical other universes and dimensions in a way that works very well. So well in fact that we can use the same mathematics we have used to describe our observations to make predictions about future events that are accurate. And so of course it is a useful tool but I suppose what I am trying to get at is we don't really know if it is truly completely through and through correct until we have tried every possible variation of mathematics that can be done. And everything has come out consistent or perhaps until we have some kind of unifying theory of everything. Will we be closer to understanding or suspecting that mathematics is complete in itself but God all proved that at least using mathematics in itself it cannot be proven using any type of formal system perhaps. And I find that an incredible aspect of mathematics. You know I was quite lucky at school to have for a number of years a really great inspiring maths teacher and the reason they were so inspiring and good at what they did was they always seemed to have some other way of explaining a concept if you weren't quite understanding the gist of it. If you couldn't quite wrap your head around an algebraic equation they'd say okay well how about you look at it like this and we can draw it in a grid and use shapes and move them around or how about you look at it like this you can draw these these numbers and and these symbols on a timeline and we can draw lines between them to show the movement and how we're how we're arranging these things to to get a result we want and so on and so forth. And it made me realize that mathematics really in the way that we do it is of course formalized there is a formalized way of of doing mathematics as we are all taught in school but the concept itself is incredibly abstract you know there was no reason we necessarily needed these symbols and this formal way of writing one plus one equals two to complete mathematics in fact in the early days of mathematics there there were no symbols like this that was also quite a fascinating revelation for me when i discovered that and that when people like Pythagoras were doing mathematics in ancient Greece they were simply using shapes and geometry and describing the relationship between things if you yourself said a squared plus b squared equals c squared to Pythagoras he wouldn't understand what you meant by this because that particular way of formalizing mathematics and that particular way of learning the length of a hypotenuse was not formalized by Pythagoras himself though the way in which we do it was he simply described it using words and shapes it's very interesting i think i also remember somewhere i can't quite remember where hearing about much later than this but still long ago enough before we had our particular formal system of mathematics mathematicians would write poetry about the things they had discovered as a means of explaining certain concepts and i think that's a really nice idea the idea that you could turn something that feels so formal and perhaps dry and boring and difficult to understand into a piece of heart something that might capture your soul as well as your mind as you're having some mathematical idea explained to you it's a beautiful way of bringing mathematics and literature together and i would love to see more of that i think i don't know if perhaps something like that exists already maybe you already have an example in your head i'd love to hear about it if you are if you have time to let me know in the comments part but this idea of doing away with the formal symbols of mathematics and and exploring it in a more visceral and perhaps artistic way really appeals to me because there's so much wonder in numbers they can be so so very small smaller than we can conceive of and they can be so so very big utterly mind-bogglingly larger than we can conceive of whenever i think of the biggest numbers i'm always reminded of a very famous mathematical large number known as graham's number now exactly what graham's number is i frequently forget as much as i do know is that it has something to do with counting the sides or perhaps vertices or is it connections between vertices in four-dimensional shapes or could be in dimensions higher than four it enters into a place within mathematics that goes way beyond my understanding but there was a mathematician called graham who was in this field of high dimensions and attempting to work out the minimum number of connections between vertices in a hyper-dimensional shape and unable to find the minimum i believe he attempted to find the maximum number of connections and just came up with this phenomenally almost unhelpfully large number that has come to be known as graham's number the popular youtuber Vsauce Michael Stevens even has a wonderful visceral way of understanding just how large graham's number is and he explains it now i won't be able to remember precisely the the steps that he goes through but it's something like if you take one drop of water out of the ocean in an attempt to empty the ocean and walk around the earth in its entirety and call that one then take another drop and walk around the earth again in its entirety and call that two and you continue to do this until the sea is empty i think i've even got this wrong i think you do you do these steps and it's a measure of time that's right you are counting the minutes it would take you to do this i believe but you continue this process until the sea is empty and then you are something like one hundred millionth of the way through the number and so you must repeatedly do this although i am absolutely doing this explanation zero justice because i believe there were in fact more steps than this and as much as i can remember is that listening to him describe precisely what needed to be done and how long it would in fact take to do these things only to be a third of the way through or however it might have been described i completely boggled my mind i believe he explains this in an episode of the rest is science and i think that episode may be about large numbers or perhaps just interesting numbers i would highly recommend it if you are interested in that sort of thing something else michael stevens has spoken about before in this vein and something that i always love to ponder on is the idea of infinity and how some infinities are larger than others which is quite an unintuitive idea really you hear that sentence and you think well surely infinity is infinity how many different kinds can you have if something goes on forever well then surely it just goes on forever infinity being more of a concept than it is a number as people often mistakenly claim it reminds me of being at school in primary school that's quite a common recurring argument probably extremely predictable for any of the teachers or caretakers listening the children would have including myself at the time where perhaps you having some sort of argument a childish argument about who had the most pretend money or chocolates or any amount of thing and of course you'd start with the imaginable numbers i've got 10 i've got 100 i've got a thousand i've got a million perhaps we would stretch to a billion that was within our mindset but unable to move much further past that in terminology years away from ever hearing the term google plex inevitably it would quickly descend into i've got infinity money and then sure enough the child would come in and say well i've got infinity in one which is such a beautiful and childish thing to say because of course it completely misunderstands the concept of infinity and that one added to infinity is still infinity but going back to this concept of smaller or larger infinities when you begin to think about it for a short period of time it really does make quite a lot of sense so for example if you took the standard integers one two three four kept going and going and going you would never reach an end of the integers they exist as such that they will go on and on forever and the concept of this continuous stream of never ending integers is what we traditionally think of as infinity but then of course we have decimal numbers between one and two we have one and a half and between one and a half and one we have one and a quarter and between one and a quarter and one i'm now beginning to show my lack of math skills and if you can work that out and you are still listening i don't want you to comment it now but if you keep that in your mind and use it as a mantra to help you fall asleep i'd love for you to in the morning come back and let me know what that middle point between one and one and a quarter is well you can see where i'm going with this because we have something known as irrational numbers the concept of decimals that go on and on forever pi of course being a very famous one phi or the golden ratio being another well we can in fact keep going and going and splitting this difference for infinity going down and down this time instead of up and up and then you can consider how many potential decimals they are or even how many potential decimals there are between just one and two how many possible infinite excuse me how many possible decimals are there between one and two well the answer is they are infinite and yet curiously in this instance we have an infinity of numbers limited in a very finite space simply between the distance of one and two the smallest numbers we can conceive of save zero though i suppose there is some philosophical contention as to whether zero is in fact a number or yet another concept such as infinity though both no doubt are still useful in mathematics there i say necessary you know i love thinking about special numbers there are so many within mathematics but then come to think of it there's so many within life you know numbers aren't always strictly mathematical sometimes they just behave as symbols that represent something beautiful and meaningful in our lives you know 21 will always be an important number to me because 21 is where i grew up my house number where i lived for years and years my friend my oldest friend is particularly fond of the number 17 he's never truly explained where it first started but he has this theory that it seems to crop up everywhere in his life and i have to say i am a converted believer in his theory ever since i was once had a roulette table with him and had the fortunate luck to be present when he bet twice on the number 17 and one not long after having explained this special number in his life and it's funny how when someone says something like that and you start to take it on a little bit how you'll start seeing the number 17 everywhere as well just seeming to crop up i'm glad it's 17 that he chose being a prime prime number i don't know why it's special to me that it should be a prime number but prime numbers are particularly special and for those of you who don't quite know prime numbers are simply numbers that are only divisible by one and themselves so you cannot equally divide 17 into anything without creating a decimal just one and itself you'll never find 17 in any time's tables apart from perhaps the 17 times table and prime numbers are very interesting and unuseful to mathematicians and also people trying to create codes or securities honestly i'm not entirely sure how it is that they they use prime numbers to to do this but i do know that they're important and prime numbers are also difficult to find because there is no mathematical way of predicting prime numbers there's no regular formula for discovering them the largest prime numbers we have found are found purely through great computing power alone numbers being crunched constantly until one is found and it's often very valuable to find new ones and it's almost mind-boggling sometimes when you see some of these larger prime numbers strings of digits do you think to yourself truly can a number that large really have nothing divisible within it outside of itself and one yes it's something very magical about prime numbers my favorite number has always been the number six in my head it always appears purple now i don't know if that means i have some kind of synesthesia or something i have a lot of associations of color with numbers and letters which seem to remain consistent over time i'm not sure that's quite synesthesia i think that might be a relatively common experience of people you might be able to tell me different but that purple color of six gives it this sort of magical and mystical feeling i think there's something symmetrical something that feels rounded about the number six as well something that feels natural about six i think this is why i've always enjoyed hexagons as well so they are sort of a geometric representation of six and can be found all over nature in honeycomb in rock formations in flowers in the molecular or atomic structures of certain material six is my go-to number i think i'm fairly confident that if i asked you now to think of a number that meant something to you perhaps your favorite number or a number that excites you in some way that evokes something bubbling inside of you in a positive way i think you'd be able to think of something i'm sure without thinking too hard a number already presented itself to you in your head and before we finish for this evening i'd like you to hold that number in your mind's eye i'd like you to focus on its representation what does it look like this is a number floating in the void is it written in a particular font is it engraved in a particular place i want you to focus in on the memories that that number evokes for you what is the first time in your life that is represented by that number for you why did it become so special in your mind than your heart what people are associated with those memories and those feelings for that number and how and what way has it been present in your life more recently and as those memories and feelings are gently revealing themselves to you i want you to count from one to your number and when you are finished let your mind drift into infinity um do You you you you you you you you you you you you you you you you
