Welcome to the Times podcast, I'm your host, the unveiling.
And today's topic is Number of Systems.
With me to discuss this topic is Dr. Alan Walker, a junior research fellow in pure mathematics at Trinity College, Cambridge.
Previously, he was a student at Modlin College, Oxford, where he began this very podcast series in twenty sixteen and which I am taking over now.
And this is sort of our hand episode.
He has also helped postdoctoral positions at McGill University, University of Montreal and at the Institute Matak Lefler in Stockholm.
And from January twenty twenty two, he will be a lecturer in pure mathematics at King's College London.
Our second guest is Ella, but she's a final year undergraduate in maths at Trinity College,
Oxford, and is more on the applied side of maths interested in networks and opinion dynamics.
And finally, we have Alberto Gonzalez, epaminondas from Spain.
He was a master students at St Peter's College, Oxford, and twenty one at twenty, twenty, twenty and twenty one.
And he just graduated and is now a PhD student in arithmetic geometry at the University of Warwick.
We talk about a number of systems today, which means different types of numbers, irrational numbers, whole numbers, natural numbers.
What are these different systems? How did they come about and why do we need them?
We talk about the history of numbers. We talk about the philosophical concept of a number.
So is it something that has always existed or is it something that man made up?
These are some of the things that we will discuss and try to illuminate a bit for you guys.
We also talk about infinity, meaning is there something bigger than infinity?
So if you say Infinity Times two, does that mean that is larger than an infinity?
We will find out. So stay tuned for that.
Also, please check out our show notes where we we can read more on what our guests prepared and so you can delve deeper into the material if you like.
So they did a great job that put a lot of effort in it. So, you know, just download the PDF document if you'd like.
Also subscribe to the podcast to get our next episodes.
We have the next one already produced and that episode will go up soon.
But for now, please enjoy this episode on Number Systems.
I want you to listen to me. I'm not just mom or dad.
Come on, just a second. Because he's no longer a follower of Marx, he's loving Engels instead.
Science is interesting because you don't agree. You can [INAUDIBLE] off.
It's just that I would just like to welcome you to the podcast and thank you so much for coming.
This is the first official episode of the in our Spare Time podcast.
And I have the pleasure of also having Elad as a previous host here.
And so we can talk about our today's topic is no systems.
I'm happy to ask you guys a few a number of systems, ask you guys a few questions on this, and you have all prepared notes for this.
And you said you shared them before. So we all were able to have a look.
And I found them really interesting and will be facing some of my questions on them.
I found that really helpful. But of course, you know, you guys are not expected to have read each other's notes.
It was more for interest. And if you saw any points that you would like to add on and the way I invested in vision this is that
I would ask certain questions and I would usually direct a question to one person in particular.
But then anybody can chime in and add to what has been said without, you know, the necessity of me having to direct the words to the person.
And sometimes I might just give opening questions. And, you know, we will try to approximate a regular conversation.
That said, I'm very happy to talk about a number of systems and you guys are all mathematicians,
which makes it really interesting and many people here listening to this will probably not be an expert in maths.
So this is a real opportunity to learn something.
And I'll let you in particular wrote show notes that even contained a short and brief story that I found really interesting.
And it was on Frank Nelson Cole. Could you share this story with our listeners?
So, yeah, this is such an anecdote that is being passed down through the mathematical community.
It does begin on the basis of truth. But it was embellished in this particular book written by.
But Ben in the 1950s.
But the stories that followed this, this maths professor, Frank Nelson Cole, at a meeting in 1983 of the American Mathematical Society.
So it's very esteemed medical society. And what did he do? He said nothing.
He just walked to the blackboard and carefully calculated to to the power.
Sixty seven minus one. So he took two and he did two times two and two.
Twenty times two. Has to do with how three. He did that sixty seven times and then subtracting one.
And that gives you a very long number. It's a number with twenty one digits.
OK, so he did that on one blackboard and then on an adjacent blackboard board he performed a long multiplication of two numbers.
He multiplied one hundred ninety three million, seven hundred and seven thousand seven hundred twenty one by the sixty one billion eight
hundred thirty eight million two hundred fifty seven thousand two hundred eighty seven.
OK, this took him most of the hour to do. And then the two final answers matched and thirty to twenty one digit numbers about the same.
And then you sit down. He sat down to rapturous applause. OK, the story survives because it's a kind of very theatrical story,
but also because it's really unusual that in general this is not what mathematicians do.
It's not what number 32 on a number of theories. It's not what I do all day, but it's.
It's I I told you the story almost as a contrasting anecdote, as what generally we don't mean when we're talking about numbers,
systems in general, we don't mean ever increasingly complicated calculations with the usual accounting numbers.
We tend to mean, as I'm sure we'll hear about in the rest of the podcast, expanding our usual notion of number two,
more abstract and richer notions of no and understanding what implications.
That's wonderful. Thank you so much for bringing in and breaking us into this topic.
Now, you used the word no.
There are quite a few times, and especially laypeople like me, when we hear, no, we usually don't think about this, we sort of take it for granted.
Now, this is one of the topics where if you look a little closer and try to look behind the facade of what the construct of number is,
you may face a few complications and questions that may be a little unsettling.
So that is why I would like to dig down deeper into what a number is.
L-A in your notes I found really interesting and also something I have to admit, something new for me, at least an idea of what a number is.
And also there's certain philosophical standpoints that you could understand a number from.
Could you give us a short explanation of what those are?
Yeah, absolutely. I thought this was a really interesting place to start, even though I'm far from a pure mathematician and I'm not a philosopher,
it's just a question that you can easily find different perspectives on and disagree on, even with people who aren't mathematicians at all.
The question, what is a number we kind of take for granted? We all come across them in school.
You learn to count, you find change at a supermarket. You know, you start learning about irrational numbers.
You work with PI in school. But what actually is a number? And do numbers exist without the human mind being that to use them and imagine them?
Do they say something?
About a wealth of a completely separate from that, do they arise because we have these physical things around us that we can measure?
Or is it just a game that we play? So the kind of.
Three most prominent, I would say, schools of thinking about this formalism, intuition is a larger system.
And just to give a brief overview, formalism is basically the idea that all these numbers and equations and symbols, they have no actual meaning.
They're just some sort of game. We have certain strings of syntax and we get to rearrange with them and mess with them according to certain rules.
So we've made this game where we have the number one, two, three, four. When you add two and three, you get the number five.
That's how we define this game. And with that, we can do lots of very difficult and interesting things.
Intuition ism is more the idea that numbers are just a mental exercise.
They're constructed by people in their minds and they don't reveal properties of the physical world around us.
But they're used by the human mind to analyse more complex mental constructs, to kind of analyse what the world around us is doing.
So in essence, the idea that numbers don't exist when they're human, they're not to think them up.
And then we also apply them to the world. So billions of years ago, numbers didn't exist.
There was no concept of, you know, there are three trees in this valley, 30 days since this extinction,
that those numbers weren't there because there were no humans to think of them. And a of them is roughly the idea that mass can be reduced to logic,
which was initialised by Frager and found some of its biggest proponents and Russell data after data and
concluded that natural numbers were reducible to sets and mappings so it could be reduced to logic.
Yeah, those are kind of the three biggest ones. And so that is fascinating.
And I also read something that you said I wrote about that sometimes that people think that numbers aren't just a construct,
but they're actually something real. And you mentioned the term Platonism.
Can you say something about that? Yeah. So Clintonism is the idea that numbers do exist in the real world.
We've kind of created a mental image of them. So numbers are kind of real non-physical things.
And what we mean by this, we say that numbers are real, meaning they exist outside of our minds,
so independent of a human being around to count the quantity or talk about it or think about it, it still exists.
And both the non-physical, you're not going to bump into the number twenty four in the street. You can't pick up the number five.
And in a sense, you can think of it as the logic of something has the potential to exist.
So it does exist. So, you know, millions of years ago there were ten dinosaurs on a hill and no one was around to count them.
But if someone were there, they could have been counted. That would have been exactly 10.
So that number 10 always existed because the potential for someone to describe that quantity existed.
And I really like this idea to have a kind of Segway into why that's a nice way of thinking,
to be it to help me accept that we can define things into existence unless it's kind of a weird jump.
When you go into undergrad, you start being told, you know,
let there be a group with the properties, X, Y, Z, let there be a no such that this is true.
And you think, well, it can't be true.
A nice example is at a level when you first meet complex numbers in which the square root of minus one and you think,
well, that's complete, that's completely crazy because that doesn't exist.
But accepting the potential for something to exist, I mean, that does help me kind of say, OK, let's go with it.
I can imagine a number that could do this so I can imagine some sort of thing that has this property.
And so it's real that sees it, if I might,
to chime in here on this very nice thought tripartite world that I would describe for us some of the the former lists,
the intuitionist and the the logicians, the the big battle of this was taking place in the late 19th century, early 20th century in mathematics.
And to kind of cut a long story short, the intuitionist lost in the sense that this is not say that intuitions don't exist anymore,
but the majority of professional working mathematicians would basically be formalised at heart,
albeit they won't write in a formal language, but they will go to sleep at night knowing the underlying what they're writing is
a formal basis based on logic and set theory that is underpinning what they do.
So this is the the Hilbert Brower controversy, which I believe actually is still its common idiom in German at least about something.
It's the equivalent of like a storm in a teacup in English.
There's like a German phrase like, oh, it's as irrelevant as the of controversy or something like this.
But your brow was intuitionist and Hilbert was the formalised and Hilbert one, basically.
So that is something interesting. You say Hilbert one as a as a layman again, how do you win such a fight?
Is it simply who believes, you know, who's the majority or can you disprove the other?
So the sense that I used it was in just a sense of academic culture, but as well as being an astonishingly productive.
But the. In his own right, had an enormous number of students,
and these students would go on to become just the leading luminaries of the first half
of the 20th century of mathematics and universities throughout Europe and in the US.
And so if you managed to influence enough people with your way of constructing mathematics and how you think mathematics should work,
then that's the way in which Hilbert won.
Hilbert had many more descendants than Browed did, mathematically, academically speaking, regarding his students.
For instance, the when we teach analysis one to First-Year undergraduates,
this is the logical underpinnings of calculus which we might come to discuss later in the podcast.
The methods that we use to do that,
Brower would not have accepted as valid because they're not valid in the intuitionist framework because of various uses of axioms of infinity.
However, the world over, we teach analysis one the way we do because help one umbrella lost.
I would like to add that sometimes things are not as simple as saying who won or lost,
because I have observed in different countries that have a different way of teaching mathematics.
So I think I think that there is overall a constant worry mathematics about how are
things defined and how some concepts are more useful to analyse certain problems or not.
And at the end, the ideas are more ingenuous, are the ones who stayed.
But your ideas also stayed sometimes in this like coexistence state, where two concepts can be used for solving different problems.
Interesting, you just said that different cultures or countries teach mathematics differently.
That is something I never thought about. Really. Do you have an example, maybe something that people can relate to?
Well, I think I started my undergrad in the in Spain and also I started high school in Spain.
And one of the things that struck me the most is that in the UK they do long division, different then differently than Spain.
They put the numbers in the other in the reverse direction.
So the first time, like I was in I can't remember which of course it was,
but the professor did a long division in the board and he was like, is everything clear?
And I couldn't see what he was doing because I was so used to it the other way that I really couldn't understand what it was.
So it is it is safe to say that there's much difference on how much is taught in some places in the world compared to others.
And of course, I feel like the UK and Spain are quite close.
So if we went to completely different countries, we could find things are even more striking.
Interesting. So and you guys are all mathematicians in.
So you've you've spent quite a bit of time doing maths in the academic context.
Would you say adieu even so nicely, explain the different some of some of the different philosophical approaches.
Do you guys ever think about this or is this really just more of an ivory tower type of thinking that is there?
But that's not really relevant today in today's academic mathematic culture anymore.
What do you guys think? Well, I'm interested to hear what Alaron Alvira have to say, but I can answer your question in a very concrete way,
which is that my latest paper was actually exactly on this issue, albeit it wasn't described as such.
But there's a notion in mathematics of an effective constant or an ineffective constant.
And if I just describe this or very briefly, an effective constant is one which could in principle be calculated and worked out.
So maybe in your maths paper, because doing so would be extremely complicated to do.
You don't actually do this calculation explicitly, but in principle you could do it because it's an effective constant.
But curiously, there are concerns that appear that are ineffective constants where actually the nature of the argument
means that there's no calculation that could be done to actually work out the value of this constant.
These things come up when you use the illogical law of the excluded middle.
So you're arguing by contradiction. There's not not a constant and therefore there is a constant, but it doesn't tell you exactly that.
And there are results all over mathematics all over no mine fields which are now known, but with ineffective constants.
And that's a limited knowledge.
You know, it tells you something, but it doesn't help you push further and further calculation because it's an ineffective result.
And so there are lots of people in the fields who go about their business trying
to prove effective results where previously only ineffective ones were used.
And then this comes right down to this difference between like intuition ism and formalism.
So an intuition is one of the central tenets is the law of the excluded.
Middle is not universally applicable. And so intuition isn't there are no ineffective constants, there are only effective constants,
whereas in formalism you can have ineffective constants because you're allowed to use the law of the excluded middle all the time.
OK, that's my long answer to the question, but I'm interested to hear it.
Yeah, I mean, I think I'm not at the I'm not in a position to discuss the academic world at my current stage in the finally undergraduate.
So my answer will be a lot less, a lot less technical than Allard's.
But it's something that I don't think about when I as part of my degree or when I do any sort of maths.
But it is something that I've just always found interesting, kind of as a as a side thought.
I remember I want to study maths for a long time.
And when I was 12 at some point and father of a friend of mine who is a philosopher who did philosophy at university,
asked me, oh, so interesting to study. So what is a number?
And I remember being absolutely terrified and sitting there going, oh my God, I actually don't know.
So yeah, it's just an interesting thing. An interesting thing to consider and needs.
Yeah. I think just is an interesting point of discussion for people who might otherwise really be terrified of maths,
which is a common reaction you get when you say you study maths. University first response is always I sucked in maths at school.
Terrible. So it's a nice, nice lead into that.
So I was going to say I'm just in the middle point between L.A.
I don't think about these kinds of, like, debates in my normal life.
But the more I get into the academic world,
the more often these kind of questions appear naturally when trying to solve problems and try to try to answer some questions about numbers.
Awesome, thank you. That's been really interesting, especially because you all have a slightly different take on it,
which may also be, you know, due to different different positions you guys are in right now.
But it may also be some different influences. So that is very, very exciting.
And also, I find it interesting that Ellen talked about how different professors have different PhD
students and they will sort of proselytise the gospel of their certain work and thinking,
because that is something that is also prevalent in the area of law.
And I wouldn't have thought that something as pure as mathematics.
But of course, I don't really know what what that means anyways.
Could has something similar. But it's fascinating to me.
Absolutely. And now we know that the topic of this whole thing is number of systems.
And so far we have. Been talking about some of the theoretical basis basis for this,
and I'm now curious about so when we talk about numbers, we have different types of numbers, as I am told.
And as I have begun to learn from your notes, I am sure I might have learnt in school as well, but I wouldn't count on that, no pun intended.
But but I found what you guys wrote really interesting.
And overall, can you maybe give us a brief overview of some of the most of the key number systems?
I would guess.
And maybe we can then open the discussion to the history of it, but maybe just for somebody who doesn't really know what types of numbers there are.
Can you show us around a little bit? Yeah, sure.
So, like, if we dismiss and ignore a discussion for a second, a number is something that we kind of all picture in our head.
Booty's is something that's cool, that is used to label, to count, to measure.
So I think around forty four million years ago, there were some bones that were found by archaeologist with some Tarle counts on them.
So those were the first instance of counting on how the numbers were.
Register the numbers that we know one, two, three,
four that we use for counting are what mathematicians usually called natural numbers because they have like this natural aspect to it.
But later it was discovered and it was used when numbers were used to count and we started getting people into that,
people that owe things to other people. We discovered the concept of negative numbers and we, of course, we discovered the concept of zero,
which to us it may sound it may seem natural right now, but in the people in the past, it was not a very natural number to consider.
So when we have the natural numbers, the zero and the negative numbers, that's what mathematicians call the integers in the middle.
But you said forty four million years ago, did you mean four million or when these pointless.
Oh, yeah, I think I meant forty four thousand. OK, so because there were manmade, yeah, there were manmade know fully four million sounds a lot.
Yeah. Interesting.
Awesome. So that is really helpful as a first as a first overview.
Allan, what do you before we go into the history of the individual ones, any any general things you want to add on, no different number of types.
So Avro just to get us going, has been talking about the the whole numbers.
I think once we start talking about the history, whole numbers come in sort of at the same time as fractions.
So we'll give them a different name soon.
But if you have a whole number and divide by a different whole number, that's something we call a rational number.
And this is another type of of number system. But we'll we'll talk about as well.
OK, interesting. So maybe then let's let's go into the historical development of individual numbers, a number of systems.
And I think your notes on this were magnificent and maybe you could start us off with that and also do not spare us or actually tell us about also,
please, if you could, the interesting story about how apparently finding out, I think,
negative or a complex number got somebody somebody killed in ancient Greece, that would be really interesting.
OK, so I'll come to that that attitude towards the towards the end of what I guess
what the the ancient Greeks thought about mathematics in a very geometric way.
And so a lot of their concepts seem strange right front to us because they thought less about numbers as a kind of one,
two, three, four, five, but more about Lents. So then numbers had a very physical manifestation.
So instead of the number one, they would talk about a unit line segment line segment, which you defined to have meant one.
Instead of the number five. You now have a line segment, which is, you know, five lots of the unit line segments laid end to end.
OK, but in doing so, they can construct all the whole numbers, at least all the positive numbers,
but it means that some of the No3 arguments that they wrote that come down to us just look really strange.
So. The Euclid's proof that there are infinitely many prime numbers,
which is now a very staple little three life proof that that's given high school and university,
if you look in how you collect in the third century, B.C. actually went to write it.
It looks so strange because it's all about geometry, because instead of numbers, they're using lens.
But anyway, so that's a kind introduction to how the Greeks thought. And so when they're dealing with proportions between different lengths,
they're dealing with fractions of a line segment of length five, another line segment of length seven.
Then the proportion between one and the other is like the fraction five divided by seven.
So that's how rational numbers came into that, that mathematics and notice of the word ratio survives in this number.
No rational in the nomenclature.
So that's kind of why we call them rational numbers, because they're to do with these ratios between different lengths.
The the thing that you mentioned in your your question is to do with the problem between what we,
the Greeks would call commensurable lence and incommensurable.
And so what this means is that if you have two different lengths,
which are both common multiples of a common unit line segment, we call them the Greeks call commensurable Lents.
And because they're a fraction. Right. So, again, what it's five times the length.
One is seven times the length. You get a rational number five divided by seven.
But the problem is that not all ratios are commensurable.
There are some that are incommensurable and this is what you get if you take a square and you take the ratio
between the length of the diagonal of a square and the length of the side length of that same square.
It turns out that that's an incommensurable ratio. Now,
there is an anecdote which comes down to us about a member of the Pythagoreans school who was the first to discover that incommensurable ratios exist,
in particular this ratio exists and that this discovery was so Diabolik with such
a problematic discovery to great mathematics that he was thrown off the boat,
which the whole concept of travelling. This is a much repeated anecdote.
It's almost certainly false because almost all the well, all of the sources we have about Pythagoras's life come from Roman.
Times come from first century A.D. and later, whereas he was living at his school,
was living in about the sixth century B.C. And so it should always take with a massive pinch of salt,
anything that said with any detail about the the wider culture of the figurines, I mean,
that they recently existed is recounted and it becomes almost mythic figure throughout later Greek civilisation.
But yes, it's a great story and probably a false one, unfortunately.
But the final thing I'll say, I've been hogging the floor for a long time is that the Greeks did work out how to deal with a theory of proportion
involving incommensurable ratios that this was the great work of the greatest mathematician you've ever heard of.
Could you, DOCSIS, who in my opinion is every bit as great as Archimedes, who is much more well known,
but he constructed a theory of proportion, which is a startlingly modern idea and won't have time to describe it in this forum.
But he enables incommensurable ratios to be brought into the same type of mathematics as commensurable ratios, as rational numbers.
And so this constructed something we now call the real numbers, which perhaps are let out of our continue the story from that.
Wonderful. So I can tell you also enjoy myth busting after you have ruined The Imitation Game for us.
Eliot, I'm sorry. No problem.
But the truth always wins in the end. So thank you.
It's been very helpful lot.
Do you want to continue talking about the real numbers as a as a next step?
And you know what, I would actually leave that to. Sure.
I think you guys have much better notes on this. Sure. Um, I just.
Yeah, I try to space it out. Ibro, what's your what's your take on real numbers?
Well, real numbers, if it's something that informally is quite hard to describe,
because most of the descriptions of real numbers are kind of formal, are like proof, very mathematical.
But for our listeners, I would suggest when they think about real numbers, think about anything like everything that we know for sure,
like the probability they will rain tomorrow, the height of the queen of England, I don't know.
Anything that you can think of is kind of a real number in those real numbers.
We have the rational numbers, which are the fractions, and we also have the irrational numbers,
which are the ones that has gold incommensurable but that we generally called irrational, such as, for example, the square root of two E or PI.
Can you can you explain what E and I exactly is?
So he is well.
How how can this plane I'm sorry, isn't asked something like this, I don't know how complicated that is.
Maybe if it's if it's something insane, like if I ask what a number is.
So I didn't want to make it too complicated. Oh, no, no, no.
So, OK, I just need to think about it. It is a very important constant that appears in everything that's related to growth,
in fact is what gives no gives name to to the term exponential growth.
OK, when we mentioned exponential growth, what it is behind is that we are elevating something to a power of tea.
And high, on the other hand, is the ratio between the circumference of the circle or the length of a circle on its diameter.
That's Pierret. Yeah, and Hind I.
Oh, I is not a real number. OK. Oh, I can explain it like you said hi earlier.
And as I said, I'm sorry. I am sorry.
Yeah. No, I shouldn't do it. Paya's that I that I should know.
Fantastic. Great. So we have now discussed some of the real numbers.
And would you guys say that the next logical step to expand our ever increasing knowledge of number of systems would be complex numbers.
Is that a step? The next thing we should think about?
Yes, I think that that that's a good next step. Mm hmm. Would you lead us on Elitch?
Well, inhabited by the ALA has a night oh, sure, I don't yeah,
I don't want to put anyone on the spot and if you if you'd like to say something about that.
Yeah, absolutely, and so complex numbers are the next step, it's the next thing that we learn about in school,
most of us who did a level Abitur will have, we've come across complex numbers.
And they're also known as imaginary numbers,
and they got their name from Descartes who called them imaginary because he believed they were wrong in some way and they didn't exist,
which is an interesting idea and something that I think that's something really interesting to talk about.
If you don't mind if you don't mind me taking us on another Segway into something else.
I've read this in my notes as the leading question, are complex numbers real, which is, of course,
terrible phrasing because we've just established complex numbers and real numbers of different things.
But I suppose the question that I really want to ask is, do complex numbers exist?
And the initial answer is always.
No, right, you look at these in school, you've just learnt about real numbers, you've learnt that when you multiply to negative numbers together,
we square a negative number, becomes positive, a positive number, it stays positive.
There's no way you can take the square root of a negative number and have that.
Exist and be real. So complex, so a complex number we define in terms of AI, which is the square root of minus one.
So I squared is going to equal minus one. Why is it AI is that is that random or imaginary?
So I do believe AI stands for imaginary numbers. Yeah, of course. Imaginary. OK.
So do these exist, how do they come about? Well, we have this beautiful no system we filled up and we say, OK,
we have real numbers, describe everything in the world, but then we come to a problem.
There's a theorem called the Fundamental Theorem of algebra.
And I'll just start with this is one problem where where complex numbers start coming up and being necessary to you.
And so the fundamental theorem of algebra proven by Gauss in 1799 states that.
Any non constant polynomial with complex coefficients has a root in the complex numbers,
which sounds really complicated, but you can basically take it to mean if we have a polynomial equation.
So something like X squared plus one is zero or X cubed plus three, X is seven.
Anything like that, anything with powers of X in it that will exist. At least one solution for X and immediately.
You see now we have an issue because X squared plus one is zero. That means X squared is equal to minus one.
Then X would have to be the square root of minus one, but that doesn't have a solution in the real,
the solution would have to be a square root of minus one. And every root of a polynomial can be expressed as some real plus some complex component.
So. Is the square root of minus one less valid or less real than integers and rationals rationals real's in general?
Well, the thing is, we use polynomials and other equations all the time in maths and physics and engineering that ridiculously useful.
There are situations where by using code numbers, you can find real numbers, solutions. They just pop up in the world all the time.
And so, for example, if you imagine you have a pendulum,
you can model its swinging motion using equations that describe the forces acting on this pendulum.
And we want to find a solution to these equations,
so an equation that says it tells you X the distance from the equilibrium point as some function of time, that's a solution.
Then we have a way of seeing where this pendulum is every point. Right. You put in a time, I tell you where it is.
So we have a few different options for what happens.
The first possibility is the system is oviduct, which means that the system tends to the equilibrium position into infinity and doesn't cross again.
So you can imagine if we have like a really rusty hinge and you left this pendulum slightly
to the side and you drop it and it just kind of really slowly moves back to the middle.
In this case, the solution to the equation of motion is real.
We've just got a real equation, but as is most often the case, you can have under motion.
So the pendulum swings back and forth until eventually it comes to rest at the equilibrium point.
In this case, if we solve these equations, we get complex solutions, but that describes a real phenomenon,
so complex numbers arise all the time in situations to describe motion.
They appear in situations to simple equations of motion and can be reformulated to yield real results that tell us something about the physical world.
No, and so why does this mean that they're real, does does it give them any more weight?
I think there's a really interesting comparison to be made here to negative numbers, because I think,
as I discussed earlier, if you imagine you're living in ancient Greece where numbers are considered in terms of units.
So the number five is five times the length, seven to seven times the length and everything is considered geometrically.
And in that context, negative numbers don't make any sense.
The Greeks don't really accept negative numbers, much like they didn't accept irrational numbers.
And this holds for centuries.
So they call, who lived in the 17th century, also rejected negative routes of equations force because they represented numbers less than nothing.
If you think about it, if you live in a world where you have to count your cattle and how many bags of grain you have,
how many fingers you have, the negative numbers don't really make any sense. What's less than nothing?
Less to less than nothing. But we learn about negative numbers so early in school,
and that's useful all over the place and equations describe motion when we look at factors and negative distances,
when you're losing something, when you're in debt, you say of minus seventy points or something like that.
So we've come to accept negative numbers as, you know, useful. And they exist because they appear in equations we use to describe nature all
the time and exactly the same way complex numbers appear in natural equations,
in the solutions that ridiculously helpful.
So you can kind of take that same jump with negative numbers and say, look, they appear everywhere and we use them and they're important.
And so in some sense they exist in some sense that I can't use the word real.
But yeah, in some sense they exist. And I will come to the end of that long ramble that.
OK, fascinating.
Thank you so much, and that was really, really interesting and it helped it helped me at least understand it from a more direct point.
So I so I'm assuming to get some some sort of progress under way so that no systems.
Where would you guys say these number of systems or different types of numbers, do you think?
Are they discovered sort of they are there or do you think man creates them?
And maybe that this is this also goes a little bit into what we discussed at the beginning,
if it's whether it's real or not, I guess in that point, what do you what do you guys think?
Well, I'm. Maybe a grown old in the tooth now, but I very much think that their man's creations, that that's a way to.
Yes, avoid some of the philosophical issues about the difference between Platonism or
Etonians and whether you're considering those objects exist in some higher realm,
the platonic forms, and somehow you're accessing the forms, or whether this is the way I think you're playing a formal game and constructing
these imaginary numbers is just solutions to equations in a formal sense.
So I'm very much of the second view. And again, I would say that.
What's the 19th and 20th century of mathematical development, more broadly speaking, a move away from even worrying about,
you know, do these objects exist in some sense towards what we can construct them, a manmade form of objects or manipulate them?
And that will be enough. So. So that's my view.
But it's not the only. I agree in the sense that I think the numbers are mostly manmade, but at the same time,
sometimes I think about what if aliens came one day to the Earth?
Like, I wouldn't be able to believe that they didn't know about natural numbers or complex numbers.
Like for me, so hard to think that anyone who has dealt with mathematical problems have not used the same tools as we did.
And so in that sense, I kinda believe that we also discovered numbers, not only we created them.
Well, I should chime in, but there are plenty of languages of the world that have no words for any no higher than two,
and these are not unsophisticated languages that have many, many words for many of things.
But the ability to differentiate between that being for things or seven things would have
just not being an important differentiating idea in the culture where this language grew.
And so there's no word for any number three.
That doesn't mean that it's another facet of culture, but it's a very different way of conceiving the world.
I find that very interesting. Yeah, that is probably a depends on on the environment, as you said, so for example,
if you have many different tribes in in arid regions, they probably won't have a word for snow because they never encountered snow.
They didn't have to.
And so maybe if, you know, if you have cultures that didn't need the construct of a complicated maths, then they would also not create this.
That's probably Soza Languages is deeply ingrained in culture, and the question is how far maths?
It's cultural, but again, as you told me earlier, there's quite a cultural component to maths and how it develops and what's accepted and what isn't.
So I think that it's a it's a complex field and just to to localise ourselves.
So I've talked about Gousse.
That's last. So and you said something about 1799.
So complex numbers is that when complex numbers were sorry if I have to ask again,
were discovered, as it were, or what's and what's the next step then.
No, it wasn't actually then I kind of used that theorem that was proven proven by Gauss as
an example of where you see that these complex numbers have to arise at some point.
I believe it was in the 16th century. And I'm sure I will have much, much more detail on this.
But it was almost a sport at the time to solve more and more difficult polynomials.
And as we've established by giving the example, the fundamental theorem algebra, you will sometimes get complex solutions to those.
So that was an awareness that these numbers were around and they were used in the
calculations also sometimes when they were real solutions to the polynomial. But as to the rigorous definition connected.
Yeah, you could say the discovery. What can I just quickly ask I don't know if you define it at the beginning, but I as the embarrassing as it sounds,
I had to look up what polynomial was, but also because I think it's different in German.
I hope so. Can you briefly explain to the layperson what what do you mean by polynomial?
Oh, yeah, of course. So in this context, what I'm talking about when I say polynomial is just a an equation that has an unknown in it.
So classically you'll see that X that involves and that can also involve powers of that unknown.
So I'm talking about a polynomial. I mean, an equation like if I the most with an expression like X squared plus three X
plus seven for X to the power of ninety eight minus five X the power of three,
that's also one a.m. So anything like that.
And when you make it into an equation so you have an equal sign and then you say X squared plus three X is equal to 15,
then those you can solve to find X. And that was what was of interest to many other mathematicians around the 16th century.
But I woke entirely to find out is entirely right.
And it's been saying I think the only thing all of this has been recounting,
these numbers first came in almost by accident from the solving algorithms for polynomial equations in the 16th century.
And then it took like 200 years. You know, this theorem, the Gauss also mentioned at the very end of the 18th century.
Now, that's Turing. And 50 years after complex numbers were first introduced in a rather indirect way.
And this was like two centuries of confusion in the mathematical world.
And people were trying to figure out how to deal with these objects in a rigorous way,
how to integrate them with the other kinds of arithmetic that they've been doing for millennia.
And it wasn't really until the beginning of the 19th century in which period complex numbers began to be manipulated on a rigorous footing.
I see. Thank you. So now after that.
So I have looked at some of the other number systems that came after that.
What is what is the the progress, what the mathematicians construct after the complex numbers,
or is there anything else or what was what were other discoveries that came after what Ella has described?
I can say a little bit and then also defer to Howard, as I was saying, there's this thing called the fundamental theorem of algebra.
And just to repeat what that says, if you have a polynomial and it has coefficients coming from the complex numbers,
just so you have five X squared, minus three X plus two equals zero.
But instead of five, three and two, you have some complex numbers.
This theorem says that you get solutions that are also complex models.
And so this process that we've been describing where you have to include more and more symbols,
you have to include a square root of minus one interior mathematics in order to find solutions.
This stops with the complex numbers. We have a highfalutin phrase for this.
We say it's an algebraically closed field.
So this whole sequence of expanding from whole numbers to rational numbers to real numbers to complex numbers,
that does stop with the complex numbers. But you can look within the complex numbers and find some collection subsets of
particular kinds of complex numbers that enjoy their own kind of rich structure.
And this is what a lot of number theory from Gauss onwards is is about.
So there are things that we have algebraic numbers, algebraic integers.
These are two examples. And I feel I've been hogging the floor a bit.
So I wonder if I ever want to talk a bit about algebraic numbers.
Yeah, so first, I would like to add this comment on how basically what we have been building in this podcast
is like this kind of Yaki where we had we started historically with natural numbers.
Then we move to integers, then we move to Rationals, then to irrational and rational, something that is real numbers and then to the complex numbers.
So is this kind of like. Kind of ladder.
We are building, we are climbing and like each one, each set contains the previous ones.
So like all the complex numbers contains all the randomness and such.
And then I think, like once we arrive of the complex numbers, we are basically almost in the top of the ladder.
So now it's just other numbers. No systems appear to us inside the complex numbers.
And I think that's where where most of the numbers that we are going to talk about today now are are found and going back to the underbite numbers.
So, as I mentioned, we have this kind of polynomial equations.
And if we consider those polynomial equations with who has integer coefficients, then we have a very interesting kind of number,
which are the algebraic numbers, which are numbers that can be found, other solutions of these polynomial equations on.
Yep, I don't know how much I should I talk about on the back numbers.
Because that's not really my thing. Now, that's really helpful.
That gives us a good overview. Do you want to add anything valid?
Well, I can talk more about the notes I wrote on this, that's the subject, if you if you like.
So as Alvarez defined for us, algebraic numbers are.
And we numbers that we come across a lot in school, algebraic numbers like the square root of two.
That's an algebraic number because it's a solution to the polynomial equation.
X squared minus two equals zero. And indeed,
so say these Italian mathematicians of the 16th century who were solving a higher degree
polynomial equations where you have X cubed cubic equations or X to the full Quantic equations,
they were also constructing algebraic numbers.
But and all the numbers that they were constructing could be formulated in terms of square roots, like the square root of two,
but also with, say, cube roots, if you have a degree three, polynomial would also involve cubits.
So the cube root of five is some number where if I multiply it by itself and multiply it by itself again, I get five.
But in the early 19th century,
a number of different repetitions coming from different angles realised that once you had degree five polynomials or more,
then you could no longer describe algebraic numbers.
In this way, they could no longer be purely described in terms of square roots and cube roots and fifth roots and so on.
If you had a degree five polynomial that wasn't enough and far beyond the scope of this podcast to talk about the methods involved in that.
But it turned out that the key was understanding the symmetry of the solutions to these polynomial
equations and creating a mathematical language to talk about the symmetry of the solutions.
And so a huge number of different fields in both mathematics and science today in
which having a rigorous way of talking about transformations that preserve structure,
symmetries in an object all that began by thinking about algebraic numbers in the early 19th century.
Awesome, really helpful. So we are nearing the one hour mark,
and at this point I wanted to only close on one thing that you mentioned earlier and then I would open the floor for any points you would
like to raise or would have that we didn't get to and let you said something earlier about calculus and that we should that we would get to.
That is what it is. That's still something you would like to discuss.
Well, maybe very briefly, I think in things that I said earlier, it's essentially come up, albeit we didn't actually perhaps use the word calculus.
So the the real numbers have a very useful property.
Which is that if I take a sequence of real numbers, which gets closer and closer together, here's an example.
If I take a half and then a third and then a quarter and then a fifth,
and so these numbers are getting smaller and smaller, getting closer and closer together.
We can perhaps see intuitively that they get closer and closer to zero.
OK, we call zero the limit of the sequence.
Now, the real numbers have a useful property that no matter which sequence I take, as long as they get closer and closer together, this limit exists.
And the rational numbers, for instance, don't have this property. We say the rational numbers are incomplete, but the real numbers are a complete.
And this limiting process turns out to be a very convenient way to talk about infinitesimal processes,
to talk about things like speed, which is sort of the infinitesimal distance divided by time calculation.
And what how fast is a car going at this instant while it's going to be some limits of average speeds over shorter and shorter intervals anyway.
So this limit process of the real numbers turns out to be a very convenient property to build up notions of calculus,
notions of taking instant speeds of or moving objects.
And that's one of the reasons why the real numbers are extremely useful.
Thank you. Wonderful. One thing I just just came to my mind,
because we've talked about we've talked about a lot about we've talked a lot about ancient Greece and Western mathematics in that sense.
And I think Avro at the beginning talked about I think it was you about zero.
And I don't I don't remember if I read this in one of your notes or on Wikipedia or somewhere that the zero was maybe an Indian concept.
So I don't want this to be all to Western focus. But is mathematics at least, you know, the traditional mathematics to maybe the 21st of 20th century,
is that mostly a Western innovation or is it actually an Eastern innovation that we appropriated,
as it were, or is it a harmonious confluence of brilliant minds that created a beautiful whole?
What is your your take on that? Again, I open the floor to anyone.
You been speaking quite a lot so you can cut all this out now, just get the thing going.
We shouldn't forget that the fact that Greek mathematics even survives into Renaissance Europe in the 14th
and 15th centuries A.D. is because it was preserved often in Arabic translation by the Islamic empire,
particularly the scholarship that happened in Baghdad between roughly eight hundred and eleven hundred twelve hundred.
So even expanding the story that much, it's not just the case.
The classical European mathematics survived and that's become the the antecedents of modern mathematics done done today.
That's just not true at all. I think that I should admit that I know much less about is what communication there was between the
Indian schools of Propagator in the 7th century A.D. and also the long history of Chinese mathematics,
how much Silkroad communication that was from those centres of learning into Europe at various periods of time.
I don't know if that Rovira know something about that. I should admit, I just really don't.
No, I don't know about it,
but my guess is that they developed kind of independently and then with time with commerce on how people were travelling from one place to another.
Eventually they all came into contact.
I know that when the Arabic came to the to Spain in the past, they brought great developments in mathematics and science with them.
So I imagine that that's how in many other places was as well.
The cultures came into contact and the best mathematical advances were implemented more for practical reasons than maybe for theoretical or like.
Interesting reasons. Well, maybe, I mean, the the symbols we know for the numbers, one, two, three, four, five, that Arabic symbols.
But, you know, there are things that were known in India in the 7th century ad that were not known in Britain, say, until the 17th century.
It's like things like Taylor series, the sign and cosine.
OK, perhaps this is beyond my ability to explain without visuals, but it turns out that these trigonometric functions.
This is where you take a ratio of the various lengths of a right angle triangle.
OK, you can express these as these ratios, but they have another life as actually an infinite sum of polynomial expressions.
This is a link between what we've been talking about as well.
And these things are now called Taylor series, because Taylor in the late 17th century created a kind of general theory for these theories,
but certain special cases were known for millennia before in India.
And I'm not sure that until a 20th century scholarship, these things were connected.
So I think it's been a complicated story. Which ideas got transmitted and which didn't?
Yeah, no, definitely. I feel like most of it got lost in the sense that the practical things probably it was transmitted.
But if people were not experts or were not keen to the concept of tailless areas, they probably didn't share it.
Audience some interesting to share it with other cultures or maybe like to be appropriated by other cultures.
Sometimes it gets you know, some of these contexts were not like specific contacts, like violent ones.
I guess one thing I wonder about sometimes is that we always speak about ancient Greece rather than any Roman maths.
Or maybe maybe that's just me, because I don't know a lot. But I just know the only thing about it,
about mathematics and Rome is basically that we had to abandon Roman numerals because they didn't really work for calculation.
Is that is that accurate, that that the Romans didn't really improve much upon mathematics or did they actually contribute?
Have you ever heard about anything where they did some important stuff? I have to admit that I haven't I don't see that they were great engineers,
but they did not seem to inherit any of the the Greeks interest in the logical underpinnings of mathematical proof.
And they move it. Archimedes. So that's against Thierer. Against the ledger.
Yeah, true. Interesting.
Well, awesome. So this is basically what I had planned directly,
but I also realised that we haven't discovered discussed any everything that you guys put down in your notes.
So this is why I would just invite you all to just let me know what you would have like to talk about, which we didn't get round to,
where I maybe veered off course too much and we didn't get to talk about an interesting thing and maybe L-A.
What what do you think? Is there anything you would have like to cover or talk about?
So I write way too much in my mind, so there's no point going to be the kind of paddocks and stuff,
maybe one facts that I think just more people should know about.
I don't know whether this can be worked in early discussion.
Is the fact that. Various aspects of mathematical developments that we usually think of as Greek ideas were actually foretold,
like the idea of the Pythagoras's was actually probably the last person in the Mediterranean to learn of Pythagoras's there,
because we have the clay tablets from eighteen hundred B.C. in Babylon,
which has a list of Pythagorean triples for what we now call Pythagoras and triples a squared plus B's quite equals C squared.
The sum of the squares of the two sides of the rectangle triangle is the length of the hypotenuse.
This is over a millennia before Pythagoras left. These things were known about in the Babylonian.
OK, well, one thing I don't know, this might be the last time I have a collection of brilliant mathematicians right before me.
And I don't know if this fits here or not, but I read some somewhere in one of the Simon Singh books that there are different types of infinity.
And I forgot already why that is. Is that anything has that anything to do with what we discussed?
And does it make sense to discuss this? This is actually one of my one of my favourite topics.
I have derailed a lot of podcast discussions. Yeah, the idea here is that.
Well, we know that there are infinitely many natural numbers, for example,
there, and certainly infinitely many prime numbers and many rational numbers,
logically, everyone would tend to agree because you can just keep counting one, two, three.
That's that's definitely an infinity. So the question is, are all infinities the same size?
Well, how on earth do you determine whether to infinities all the same size?
What is infinity plus infinity? How would you define such a thing if anyone is interested in that?
Hilbert's Hilbert's Hotel, I believe it's called, yes,
is a really interesting and interesting explanation of why infinity plus infinity is infinity and in fact,
all three of those are the same size, but maybe something not to get into here.
The idea is that if we can match up, if we have two different lists and both of them are infinitely long and we can match
up one element from the first list to an element from the second list and again,
the next emphasis on the next element of the second list.
And we have a system of doing that so that every single element in each list has been matched with one from the other.
Less than clearly, those two must be the same size. So if we take kind of as a baseline that the natural numbers are.
And infinity, we have infinitely many natural numbers, then we say, OK, let's find what other.
What size are other infinities by seeing if we can match them up one to one to the natural numbers one,
two, three, four, five, et cetera, and we find that. Actually,
there are the same number of natural limits of the size of the infinity of natural numbers is the same as the size of the infinity of integers,
which is counterintuitive. Sounds crazy. How can you know?
Zero, one, two, three, four, five have exactly that list.
Have exactly the same number of elements as. You know, every all the negative numbers, minus five, minus four, minus three, minus two,
minus one zero, one, two, three, four, five, and all the positive numbers as well.
But there is a way of matching them up so that if you get one element from that first list of
just the positive numbers matched to an element from the list of negative and positive numbers,
and you will in the end of matched up every single one from the first to the second list goes to are the same infinity.
We find that rational numbers are also the same infinity. So rational numbers are numbers you can write as a fraction, which is really interesting.
And extremely counterintuitive, and if you're interested in that matter,
that's also something that's probably best Googled because that's a nice visual for it.
But we find that irrational numbers cannot be listed so that you can match them one to one to the natural numbers.
That must always be more if you say here I have a complete list.
So it's a proof by contradiction. If you say here I have a complete list of all the irrational numbers and I've listed them up, so I have the first,
the second, third, etc. so that I can match them to the number one, two, three, four, five, six.
So I'm trying to match all the natural numbers to all the rational numbers.
You can then always construct yourself another irrational number that is not in your
original list and so that infinity must be bigger than the infinity of natural numbers.
So we call this infinity. The smaller infinity, the infinity of large numbers called Alaf no alev from the letter and the larger is called Alpha1.
And that's I also think that's a really fun proof that you call this the irrational numbers like that.
Interesting. Amazing. Yeah, wonderful. That is sort of what I bit of what I remembered.
Quite a bit better, explained Eliot, if you want to add onto this.
Also, very little to add to Ali's wonderful explanation, which is to put a bit of historical context.
So this way of thinking about sex was introduced by Cantor in the final two and a half decades of the 19th century,
and it was extremely controversial in the mathematical community.
So the two leading politicians of the day were Hilbert's and Poincaré.
Hilbert's is already in our story. Perenjori was the French politician of the age,
and they had almost diametrically opposed views about whether this constitutes a valid mathematical argument, a worthwhile endeavour.
And that was very positive. He thought this was an astonishing new insight into real numbers and into constructions.
I think he said it. Countless ideas were like a nightmarish aberration, but he flipped.
Mathematicians would soon wake up from something like that. So frankly, I was not a fan, as you can see.
Yeah, Hilbert won this one as well. We emerged first year no courses in the UK.
We would teach Cantrell's Dacula argument, which is just described.
What Crocker also didn't like what Cantor had to say, Crocker is the one who said God created the insurgence or else is the work of man,
and he didn't mean to say he felt these arguing with infinite sets, with a diabolical thing to do as a vital thing.
I'll say bring us more up to date to the 20th century.
Allah talked about Allah snort and one this size of the natural numbers and the size of the real numbers.
OK, but the question that begs, is there some infinite sets with a sigh strictly in between,
with it's a bigger infinity than the natural numbers of smaller infinity than the real numbers?
This was called the continuum hypothesis. And some people thought it was true, some people thought it was false.
The actual veracity of the thing was more astonishing than I think anyone could have guessed,
it turns out that this proposition is independent of all the other axioms of mathematics.
So that's an astonishing claim and basically that there's an entirely consistent mathematical world in which there is
an infinity between these two infinities and as a completely consistent mathematical world in which there is not.
So this astonishing theorem of pure logic was proved by Paul Cohen building on the work of Cutco in the mid 1960s and what the field metal,
which is the highest price in mathematics. So today, you know, that's just a few historical bits on top of Alan's explanation.
Super. All right, well, that was really interesting and you guys have truly made mathematics more approachable, I think, to the general audience.
That's sort of what we aimed here at what we aimed at.
So thank you so much. Yeah, so thank you so much, guys.
Awesome, awesome. Yeah, thank you, Elliot, for creating this in the first place.
And yeah. So all right then. I wish you guys a great Sunday and enjoy and have a good start.
And we thank you for hosting. This was great. I look forward to hearing the final thing.
Now, thanks to you for organising it. So thank you guys for listening to this episode of In Our Spare Time, I hope you enjoyed it.
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