Mathematics has proved to be one of the most powerful languages in order to understand the physical world around us.
It's helping us to tell where we come from, what's happened in the past.
We're back to the Big Bang. We started everything.
Even more impressively, mathematics seems to be able to tell us what might happen next.
It seems to be the best fortune teller we have.
And Galileo once famously summed up this kind of power of mathematics to navigate the world around us.
He wrote, The universe cannot be made until we learn the language and become familiar with the characters in which it is written.
It is written in mathematical language. The letters are triangles, circles and other geometric figures.
Which means it is humanly possible to comprehend a single word.
There are so many examples in the history of science where having a mathematical
perspective has allowed us to make new breakthroughs in one considers astronomy,
for example. It was in 1846 that we first observed the planets Neptune.
But how do we know when to look at it? That it wasn't a telescope which first found that universe mathematics,
its existence was predicted by reality because the way that the planets were moving when there had to be some other.
Was there some way of causing the planets to have the notion that by solving equations
we were able to know where to look in order to find the planets in that year.
And the French astronomer who said just like that, I discovered this planet was the point of his pan.
So it was mathematics which enabled us to predict the existence of Neptune and finally see it in the chemistry and the chemical world as well,
and made the land use a mathematical strategy in order to work out some sort of logic to the periodic table.
That logic then led us to certain gaps that were there, and he was able to use the mathematics of cotton searching,
something that should be something which fills that space in particular.
He predicted 2031, there should be an element and he was able to use these patterns to predict what is nature reading.
And sure enough, it took only a few more years,
1875 when we're trying to find the isolated atoms of the elements which eventually cooled down and I think
even the mon they will the LHC the reason that we're able to understand what's happening in the actually
being able to make predictions about things that we haven't been able to observe the next models will be that
because of mathematics with this language enabled us to to navigate and understand the world around us.
But actually in this lecture, I want to sort of go against that and to explore the regions of the physical world that are beyond the average.
And the interesting thing I want to explain to you is actually mathematics can sometimes be very powerful,
not only in telling us new things, but also helping us to realise the limitations of human knowledge.
And so I've chosen three different areas the irrational, the chaotic and the incomplete, which somehow shows things which we may never know.
So I'm going to solve the irrational, because it's my belief that mathematics is such a powerful language to describe
the world around us really grew out of the discoveries of the ancient Greeks.
Now you raise, in particular, Pythagoras, what matters.
Pythagoras discovered things which made me believe that the whole universe is run by mathematics.
One in particular is this discovery of the mathematics of musical harmony.
She's probably I missed this story, but the story goes that she was walking past a blacksmith one day.
Actually, this is a statue of Pythagoras.
Next to it was was based on the idea that since many years and there's even a town called Pythagoras in cells named after him,
and this is to the very famous right hand triangle that I have versus the elements.
There is an important point, I believe.
I think when I retire alone for a time, what a great place to live in was named after such a wonderful mathematician.
And that is the story of Pythagoras. One day I was walking past a blacksmith and he heard the and it was the mines.
And some of the names that will produce one angle sounded incredibly melodious together, and he was intrigued.
What was it about the physical properties of the at the angles of the hammers which was producing this harmony?
And also the fact is the slope that is around it, it seemed like the same coin hands.
It was a hands that were vibrating. We saw it was small. What was the relation between his hands to produce a beautiful harmonic relationship?
And he went home. He decided to experiment with a stringed instrument to try to reproduce these notes and then see what the lengths the strings were,
whether there was any secrets to what produced harmony.
And what he realised was, was hold on to ratios between the string things which produced notes that we responded to as harmonic.
So for example, this combination of this is the perfect fix.
This is a combination of things that we find most effectively pleasing, the perfect things you can produce by taking the strings, playing it out,
and then playing the string, which is two thirds the length of that string you will get to,
which have this perfect solution is a 2 to 3 relationship gives you a note.
I want you to take the string and then push, put it all, puts your finger on halfway down.
And you guys, I know it sounds like the note started with.
I'm cool at the octave and give you the same name of music because I sound so and I see the
whole of the musical scale is built up by taking these chaotic bits and building it out.
On top of that, we very naturally find that when you do 12 fits, you come back to the octave,
which is a mathematical reason why we break the scale up into 12 x examples of well,
we write divisible is this kind of strange effects that when you do 12 these suddenly seem to come out.
I know you started with, but it's my reason for the scale.
We break up into John and his idea of musical harmony being connected with these sort of ratios related
Pythagoras to the belief that the whole universe should be controlled by mathematics and his own operations.
Well, he was starting to get a bit of a shock, and it was his theorem about writing triangles, which eventually led to the discovery of the universe.
Isn't quite as simple as he thought it was. So you is a Pythagoras theorem writing triangles, the square.
The hypotenuse is equal to the some of the squares on the two short two sides.
So as you want to take a triangle which is taken out of square this I got a one by one square and cut out of the triangle with a diagonal across it.
So one side was one squared is two. So the length of that triangles, you should be the square root of two.
Now if I was going to need that, everything was holding on ratio. So he's a very natural length across the square.
This should be represented by something for number ratios but five of those and already so on to explore this number the square root of two.
This is a great tablet dating back almost 1800 BCE, maybe a little later.
And you can see the little notches there all the way the Babylonians write say numbers which are basically
unique using this cuneiform notation and able to estimate what the lengths across that swear is.
So they estimated the square to be 3547 divided by 21,600, which is pretty close.
It is square that fraction. You get quite close to two, but you don't get it exactly.
So it's a little too thick. Maybe it's not quite precise enough, another fraction which you do in advance, or you can go right a little bit closer.
But the discovery of the Siberians was actually however ball I would call you trying to approximate this.
We'll never be able to get a fraction which way you square.
It gives you the number two. It's a very useful proof and it ends up suppose you could have a fraction and you square that as equal to two.
Eventually you get to a fraction, you can cancel things.
And so one of the numbers is all in.
One is even as you work through the logic, you end up with the conclusion that all must be equal to even by the end of this error.
So the fact that if you find the right answer to this fraction, it would lead to a contradiction in mathematics.
So this is a discovery. The library is only half. The world isn't actually as simple as we thought it was.
It isn't made out of whole number. So what is the length of this volume for across the square?
I mean, what is it we can't write down as a fraction?
I got to write down an approximation right into the passage for the decimal.
But then even then the decimal has no cotton zero, never quite capture it.
So it was quite a shocking discovery, but it applied backwards.
Here was a length, so naturally construction is just across its length.
Yes, there seems to be no way to actually articulate what an X was.
I mean, we buried, I write is the square root of two. We seem to be happy, but what is that?
What is that number? Okay, so like for us, we can go back in a sample.
I can see what the Pythagoras group saw.
This was a limit to what we could do. We couldn't I could never write that number down.
We discovered subsequently beautiful formulas which see which shows some sort of passive, but again, they're only infinite formulas.
You have to have, I think the thing in the Balkans, we'll catch them.
So it was quite a shocking revelation by what he was able to discern on this one.
Was there a Pythagoras is there? So I'm sure that there are things that you can't actually capture as a novel.
In fact, it was such a challenge to the point of Nigerians. The story goes the point that remains resolved that they keep a secret.
He was to have a scientific revelation. And so they all get quite involved.
There's this number, this length, which you can't catch.
You know, eventually one of the Pythagoras blurted out, she discovered something we all right.
And that. But apparently the story evidence is that this mathematician was saying that.
Brown I think the public so they probably have a stake in such a famous role is very intriguing because, you know,
actually at school, I think we have universities learning to, you know, no one feels like it's always okay.
I might be able to write it down. But there is a measurement somewhere on that ruler where that square to actually kind of sits.
But I was kind of, you know, like, but you know, what is that number?
It's quite an abstract concept. It's not three squares.
That's how it's different now. It's interesting because actually we went through the issue, I think, on hating on things which seems impossible to do.
It's still we seem to accept these impossibilities and there's improvement possibilities into all subjects.
And I think, you know, one of the most interesting ones is a kind of revelation about taking another the square root of minus one.
Oh, well, you know, you could say the same things if there isn't a number which when you square it because you might as well.
Yeah.
The same way it wasn't a number which when you square it because you to I mean I can write it down but then we soon executive into the same degree
same for the square is minus one squared minus one sided appearing when we trying to solve equations quadratic equations accusing equations it.
Sometimes you just disappear and you get an answer which made some sort of sense.
But it took some time, centuries before Barry was the first one to kind of propose.
Well, maybe we should excuse this impossible number.
I mean, it was called an imaginary news day cause everybody's been rather derogatory terms, you know, you can believe it.
So there is a number which would be square is you minus one.
But, you know, sometimes it's a power by my sister to take a leap into the unknown and introduce something which seems to be impossible.
Now, of course,
the introduction of squared minus one opened up such extraordinary baseline six points to any control centre and the control that we want.
Using the mathematics of imaginary numbers and complex numbers, we will not be able to land on planes.
Also, it creates such efficiently good mathematics that we can use radars basically based on imaginary numbers.
You say we try to do the mathematics. Without it, the plane would crash was on its own and this has a kind of reality to it.
But actually it was in a picture, a picture that mathematicians came up with added in the 18th, 19th century.
You're able to see something we sort of field, we can see the square root of two on the root.
That's why we feel there should be a number there. It was introduction and some people would call.
But explain your diagram where all of our numbers are.
The standard measurements on a ruler extend out to one side, negative numbers on the other side.
Well, what about a new direction? This new direction opened up a two dimensional picture of numbers,
and once one started to play with the geometry of maximising picture realised it reflected the out of the algebra.
These numbers, that's really the moments that these numbers sort of got accepted into the mathematical books.
And so this sort of thing like we can see something sort of helps us to believe so.
But even it, it's, it's a kind of a strange picture but I guess, you know,
this institute is based on is named after somebody who approved the theorem about impossibility.
We had this equation that came up with Pythagoras.
His theorem explains why where you possess ways we can find numbers which when you square them out of them together,
are still squares three square at the scene where he was on his way.
But Fermat realised this equation when you take you've said it doesn't seem to be able to be solved.
And that's kind of one of the biggest challenges in math. Something mathematics is very good at is proving impossibility.
So it's very easy to find solutions, but to shoot throughout the infinite expanse of numbers, you will never find three numbers.
Say when you queue, two of them have them together. It's still a queue.
That's why it's an amazing challenge.
I think it's Fermat's Last Theorem is such an extraordinary theorem because although outward isn't particularly interesting,
that was the was a great idea but he's not having solutions but at the power of mathematics you know to prove
that you will never point solutions to something like this I think is one of mathematics great strengths.
And now actually ancient Greeks were very interested in some challenges which they believe should be possible.
But the challenge was, is it really possible or is it something implicit in the problem that makes it impossible?
For example, could you solve this equation or is it something about this equation?
Space impossible. They were trying to define geometric construction.
So basically. Challenges of taking the equipment compass compass direction,
but is making circles a straight edge and rule by the rule without any measurement of
what shapes we make out of the combination of this drawing circles and straight lines.
So, for example, they discovered that you can make positive reviews of this construction.
You take the circle, you may browse all straight lines between these things without any measurements at all.
I could divide equally circles into five equal units and create one of the shapes.
You can do that with the triangles, square pegs, a good hexagon.
But what about a seven figure? Can you make seven figure in the same way, using this as drawing circles, arcs of circles and straight edges?
And so I'm kind of fascinated by this challenge of all shapes, all possible.
What is it not? Which creates one of the classic challenges actually goes back to its with the end problem.
It was the residents.
It deals with somebody whose terrible play and I'm kind of intrigued to know what they should do in order to try and save themselves.
And what's Oracle? And ask the Oracle, Hey, what should you do?
The Oracle came back to the answer was, You must double the old.
What does that mean? And then it was interpreted that the old. It was a perfect cube.
And the challenge was, could you find a way to to double to create a cube is probably was double the cube that you had to solve.
So intriguingly, what this is asking you to do here is opening the cube.
So choose. So you go to take a cube roots of two. So the length of the new Q is the cube root, the two, all of you out of the first.
Q Is there a way to construct the cube, the two the square root moves are easy to square and just take the the diagonal across the square.
But is there a ways to construct the cube? Two, just using a straight edge and a compass.
And so that's what they try and fails to find his way.
And the thought was maybe the oracle was trying to distract them by deciding on doing math.
You I realised you were ill as a civil war going on and things like this.
But actually it turned out some of the mathematics have had a long, long time.
So the mechanics of symmetry and the madness in the 19th century, finally they sort out that this problem,
along with another principle, is another classical music, which is according to the lexicon, as something which is impossible.
Swearing is you saying that's like swearing. So it kind of means it's impossible.
Well, what does that expression mean? Again, it's a challenge that the Greeks have set themselves.
If you have a circle, how can you create out of that a square which has the same area?
So is there a simple construction like the one I do for the pencil, which produces a square?
And essentially this is I talk to you pi and it's Kubuntu and pi are both
examples of numbers which aren't the ones in the sense of the rational number.
Solutions are equal sort of the equation which also employs not only rational or transcendental,
but these sort of numbers, only numbers which are solutions that are very special equations.
It turns out with things that you could construct using a straight edge and compass.
But again, here's the power of mathematics. So you might say, Well, I'm trying to construct these things.
And I tell you, some people do, because like I said, that say, like, how do I have a whole box?
If you have any of them, I'll produce the box.
But it's very interesting that still people try and prove that it's possible to swim, circle all the way to try to set in the angle or to the keys.
But my madness has the power to show when things are impossible, which is good.
You don't waste your time doing things. But for me, I think, you know, KG, I swear to, you know,
I think despite the square root two is an irrational number called races and I think the decimal repeats itself.
It is a fraction actually. There's very little physics times.
It actually isn't square to even possible to represent physically because the discovery is in quantum physics
shows and in fact maybe on here it is actually not a continuous thing but is very much quantised even space,
it's believed, comes in quantised bits.
We have that same kind of energy that was on sort of continuous spectrum, but just temporary discrete energies that can see.
So the same may be true of space and even time that it comes in chunks,
it becomes a chance that actually can you actually construct the square root of two?
You have a slight. Which is why we didn't want you on either side then.
But let's suppose there is truly an irrational number.
Actually, things at one point, because I saw Pythagoras leave everything in his whole number ratios,
multiples of that basic one Planck length, which is the kind of indivisible unit of space.
So you could raise it to maybe all of these things are in many, if you can see them really still two things at once.
So the number is is full of primitive possibilities and numbers that possibly exist in the mind.
But what's interesting is in the 20th century, you also see that the interaction of mathematics and planets has been very powerful ever since.
NEWTON In being able to work out what the system is going, you know, how the system is set out, you know the equations that control it.
You shouldn't be able to run equations and work out what's going to happen in the future.
But then I find that in the 20th century made us realise actually when we made the mathematics,
actually modelling in the physical universe may not be as easy as we think and maybe nothing to do with quantum physics.
As the chaos theory already kind of says,
there are limitations actually to how powerful we can how much we can know about the universe in future using world.
So actually that all the Newton was a great belief amongst scientists and mathematicians but yeah the equations
Newton basically says that's we know how to set up all these equations and I think it works that way.
I want to read this way because it's a lovely quote like this segment of our class, which I sort of sums up as I believe at the time.
Post Newton already knows how powerful mathematics have become when you think of the present
state of the universe as the effects of its force and the course of its future arrangements,
which at certain moments we know all forces that are set in motion and all positions of all items at which nature is composed.
It seems like we're also also submitting these data to analysis.
It would embrace in a single formula the movies of the greatest qualities of the universe.
And there's that tiny atom. But it's such a danger that nothing would be answers in the future,
just like the past will be present before science and the good old physics of quantum physics machines to say,
actually, maybe things aren't as deterministic as we think.
There's still a question, actually,
whether quantum physics is just the current state of our the fact that we don't know where the electron will appear next.
But if we take it, all points will say, look, this seems to be the same as if you know how things are set in motion.
They're not completely right. It's just every will just works due to these equations.
This is the thing is, even if that is true, it isn't much.
This is the discovery made at the beginning of the 20th century that, you know,
what is the age where the good notes could acquire a computer and that it's actually any small area?
It doesn't you can't you have to know the things so precisely. Exactly.
In order to be able to do this, because any small error controls the system in and the discovery of the adventure of chaos.
The discovery which is on right on the right, was containing this box.
It was a mistake made in a paper which led to the discovery he was trying to solve,
a problem that had been set by the King of Sweden at the time, who love comparisons along the line, which is a challenge.
And he said so the challenge was to try and prove that the universe is stable.
All right. So the system is safe mixing with the US system.
So the question was and so does the sun with planets around and these raises predictable manner.
So people were able to solve that choose and were able to predict the eclipses and things will line up and it seems to be going around like clockwork,
these ellipses going around each other. But is that really true? There are some points in the future of the solar system like it's flyable.
Well, in the solar system just consist of two objects they're roughly Newtonian physics with already works out.
Yeah that we'll be saying it those two objects were just two ellipses around each other to the end of time and I is totally say what do you want?
I hope one way to solve it is if you just want extra objects inside there.
The stability of this system is is much harder to predict.
So you can set up a system where the three objects are doing something is repeating the pattern again.
And yes, there are other systems where this is a very small change in the initial condition, something can look very stable.
But what? The system is designed with slight error in its radio blaring, and the solar system has been fireable.
It looks like the central needs are passed in the next holding classes.
So what I'm doing is trying to prove this are also the system.
And he made some approximations in his angle, which is old school, rounding up of things to make too much difference.
And he submitted paper, won the prize, and then one of the referees put the papers in.
Oh, well, can you justify why the rounding up really doesn't cause any difference in the outcome?
And it was then that he realised actually, you know what, he really could.
You and I disagree that even a very small change could lead to a dramatic change in the outcome.
An interesting case of this situation is one side playing games, for example,
and I believe it went off and it's traced down to part of the title violations
and other variables and approximately the same direction and sure enough,
the path. But it's pretty close with the same balls that I thought of at the beginning.
That's all kind of feeding. Our intuition is if you make a small change, the effect on the outcome shouldn't be that surprising.
But if you change the shape of the table that you're playing days, for example,
if you have two straight edges than to say circles, in either end you can get a completely different path.
So the ball goes round and I shoot the ball with a very small change in the initial conditions.
The ball could do something completely different.
And this the discovery of the bomb, primary school or something probably heard this expression possible.
That's a very small change in initial conditions, but of course a completely different outcome.
So the example of the equations for the weather are we know what these equations all,
but they are very sensitive, these small changes as all the equations for the orbits of the planets.
So a very small change in the orbits. The planets triangles are the rounded outcomes, right?
So we run the models of the universe and as if you change very slightly the orbit of Mercury,
it's turned out there's a 5% chance and we don't want a precise location of mercury, as we would only within some standard reason an era.
But there are places where you can locate Mercury, where a very strange resonance happens with Jupiter, which pulls Mercury's orbit.
So she saw three intersect the orbit of Venus.
And there are examples where that's not only to say exactly what we actually it's Venus not seeing it now and then.
It's the whole we start to see some results to try to do exactly what they say.
Yeah, we already know that. Some very small changes in all current solar system.
There seems to be some, you know, massive numbers and it shows that she's having some very strong resonance could happen between Mercury and Jupiter,
but not long enough that we even have to worry about it. You could well in sunlight already, for example, but there are some of the points out,
some theories that sometimes you can have a very simple set or equation,
and still even that simplicity leads to a strange, extremely complex behaviour.
So one of the classic examples is a pendulum, so pendulums and that's being incredibly regulated.
So we used to the regularity pendulum in order to keep track of time, but I've got a slightly different pendulum here.
So this is in the classic pattern that we just one ball here, but it's a bit like a length.
So it's got a joints and it's called a double pendulum. And the behaviour of this double pendulum, we have a mathematical equation for it,
but still it's extremely difficult to predict what this does when you when it goes and it can sometimes just swing very simply like that loses,
energy goes. Or else it could sort of swing through.
I mean, I'm just saying this is a.
Boy down the side of the building.
You're. Oh, that's nice.
So there's an about the hangar of this thing. I mean, the interesting thing is, even if I try to start off in the same way,
so it's very slow and it's around, but sometimes it's extremely slow and.
So this thing, although the equations are very simple, it's incredibly difficult to predict and even to replicate the behaviour.
So we need to see which way this pain pendulum is going to go through next.
This is my favourite desktop toy. I spent ages playing this thing, so.
But the interesting thing is it's obviously did. It's even a simple system, but trying to replace it is very difficult to do.
Another muddy desktop toy and I have one that's here in the department system that some of you might have.
It's a pendulum. And so three months and do you set the pendulum more attractive towards the magnets?
And after a while stabilises and gets back, it's one of the ran.
And so I run three computer models of this system where someone gets the pendulum all through one corner of the table.
And I'm just trying to change a very small decimal place at a very, very small location where where I started engine.
And it resulted in a completely different outcome. So the first experiment was this plane that is a very small.
You can't see difference there. I end up with red light and another small change ends up in the it.
So again, this is great sensitivity to the initial conditions and think now I know what the equations are and I never know.
The physical universe is actually said. I'm going to apply those equations.
So how much is. I think it's useful in actually being able to navigate the physical world.
Actually such a small change. I just.
I'm never going to know. And even quantum physics sense kicks in beyond said things.
I don't get it. So basically that's my challenge exclusion that I'm never going to know.
It's that degree of accuracy.
Here's a computer model which tells you the physics, where the pendulum will end up, what colour the magnet will be, where it ends up.
So if you saw the pendulum over a yellow region, it means that it will end up in the yellow,
so that over a rays region the rate of course near the yellow that is involved as,
let's put it all in a wobble and stay on the other side of the river.
There are regions where I was starting the pendulum.
It was often this region where I wasn't quite able because this is an example of a fractal fractals, of the geometry of chaos.
If you look if you keep on zooming in, its retains its infinite complexity.
So even a very small change in shifted from the yellow region to Iraq region.
And so that's surprising to me.
There are regions here which I never know essentially enough about Angelman Syndrome, which is, I think which got a great surprise implicit.
And yet it doesn't help me to know. But it helps me to know that, again, we somehow strengthen our minds.
There are regions here where I know what's going to happen.
The regions here where the mathematics tells me this, okay, this is a region where you will know what will happen.
And I think it's apparent mathematics to be able to tell us sometimes when we can't know something.
And I think that surprised the weather. The weather is a chaotic system, is a non-linear equation.
But there are regions of the world where sometimes we can make long term predictions.
You know, I just spent some time in the desert in Jordan.
Well, yeah, we can be pretty clear when you run those equations.
It's like sitting in a den, you get a magnet. It's going to stay in the head regardless of how you change.
Conditions will change. Wind speed is not going to alter things.
And so you say there are we know there are reasons why they gather region,
red region and consequently all but other regions like the weather in England as well.
Got that bottom line. Could we change wind speed enough that if you change what I said,
one of the locations I want to work with many of these stars is to take the data or as much as they can,
but then they have to run models and they vary that data and see what the effect is.
And I believe those in the mantle because I've seen them models and it's amazing
they might 1000 models after five days they get pretty similar from there.
But after ten days it just seems like a spaghetti mess of predictions and just small changes totally changed radically,
which is so fascinating that the mathematics of chaos can help us and know how.
How much can we know what is really beyond what we know?
As in one of the examples that was discovered,
there was a guy who ran some of these program programs of the weather and he he run the data that he wanted to rewrite.
And he looked at all the the input was rewrite it and he came out with a completely different graph.
Couldn't well, I thought I run the same equations, the same numbers.
What I realise is a computer was doing some internal approximation so well that puts in points five,
six, 27 it was outputting the data that would be put in.
5 to 6 and you can see less on started over similarly but in complete divergence.
What's happening.
This is like three decimal places and the thing is it's always the chaos and the equation that passes, controls and makes them all fizzle.
And after a while, one place where it's nice has an effect.
It's an interesting population dynamics. Here's a question.
What was the question to which you think there's an animal which is believed very some don't ever commit suicide every four years.
So I'm going to take a vote in the audience. Which animal using has this kind of crazy?
Okay, so who's going to vote for the musket?
So you think muskrats. I'm sorry, a couple of muskrats.
Three, four. Yeah. Okay. What about polls?
I had on the news this morning, the poll of the disappearing races and losing is always the French calls.
It's go one very few. Those who think it's little lemmings that are breaking towns and cities.
Oh, look, it actually has a bunch of animals.
Yeah, absolutely. Right. So there is a general belief that's a learning population that has been observed, collapses about every four years.
And people kind of like explanation for why and what was happening to these leaders every four years.
They just seem to lose this kid. And his theory it's the lemmings.
So that was ever played out before years and this is intriguingly this kind of
hypothesis seems to me and confirms when a nature film called the White Wilderness went
out and managed to record footage of these extraordinary events where these lemmings
were going to the edge of the cliff and then throwing themselves over the cliff.
So it seems we finally confirmed that it was documented evidence of these animals.
They had, in one way, images of Larry Summers and then falling over themselves, oversized lunch, drowned in the water.
So this theory finally seems to be that now, for copyright reasons, I'm not allowed to show this on film because these are really litigious.
So I'm not saying that they don't mind me saying this, but I did want to show you a bit of the movie where this happens.
I said this is this is an accident, not the last chance to.
But I believe that race. This one really does want to.
Where you are. The roads.
Few years ago, the cameramen who filmed that equals came clean.
And a few years ago, the cameramen who filmed this documentary Clean those lemmings, they filmed everything from not wanting to go through the cracks.
It turned out that they set off a spinning wheel, which you could see off camera.
And there was some of the crew praising the lemmings on this spinning, spinning wheel and then having to be shot.
Absolutely. They said, well, I was very curious.
So it wasn't for many, it wasn't a suicide pact and, you know, shooting the camera for a long time every four years.
So what was it? Yeah, because still, we had come up with some theories why lemmings are disappearing every four years.
It turns out that it's mathematics that was killing this many students at school like I knew it was.
There's a lot of ways in which controls how the formation of lemmings varies from one season to the next.
And what I want to do is to show you how this equation sometimes leads to stability.
That sometimes leads to very chaotic behaviour.
So I'm going to run the experiment this week to form a slightly simplified version of this will help you be available.
So, so this point, we're going to run an experiment.
So where we're going to start with a population of eggs and then we're going to double that population.
But then there's enough resources for all of the lemmings to survive that season.
So the formula tells you how many lemmings out of that population do not survive.
So what you do is the previous generation multiply by the current generation by 20, and then that would be the number as well.
Okay. So we're going to really well, we're going to have split. So you're going to play the math.
So we're just a little over two eggs, 90 to 1 inches. But he's got call and say he's healthy.
I saw him go up there. Great. So you go the first one and you come down onto the stage.
I guess we need a second level. I think we need one.
Sorry. Sorry. Woman Here, you can. Come on now.
That's good. See, I think we need a mammal.
This is a massive problem in this whole process.
So you just call this. I say he's our first generation.
We're talk to that. So they're going to double up the next generation.
You have seen more so. So now you can pop, right?
So we have and have a second lemon, please. So it generates up to four landings so you can have another volunteer.
Excellent. Come down. You come. And so now to work out it this a new generation, not all of you are going to survive.
So we got to go for eggs. Excellent. So what does it mean?
As I remember, I said, you take the number of that is this generation, multiply it 100.
And so the previous generation, that's four times to write about ten.
So that's only one is small and too small for doing that.
So now we're going to decide where we're going to play a game of musical lemmings.
So I got three chairs there and four of you.
Well, I'm going to help you, basically, to take over this thing I have on here.
This is a new conference. So let's do this.
So there's going to be some visual arts for musicians and a cute learning space.
Saving lives. Let's see.
How do they. Oh, oh, oh.
He's letting us into my I'm going to kick her over the end to the great great great, great.
Now. Next generation, 3 to 6.
So I need three more volunteers. That's a lady right there.
You're going to find out. And one more.
Excellent. So we got six lemmings in this generation.
But again, for your your that not all of them are going to survive.
So that is way out there. So previous generations, 3.6, it seems like by chance I talked to this and all the chat.
So is the chance. Six lemmings.
That's why you're on a petition on the looking glass. Nice symmetry of this.
But so too is allowing.
Back to the lighthouse. And.
And it.
Oh, goodness. I was eating slices.
The drawings are going to take a few names designers and I talked to eight so I full mobile in all last.
So it all starts 1 to 2 all you can show off your computer that you don't see right
before I sit at three right by some results and I'll name my friend all picked on.
Exactly. Just me. One more. And if you have one German pick.
Okay, excellent. Let me. Let me. Anyway, I'm sorry.
You've just been. Right.
So that's why I got out this time.
And we had four in the previous generation or more, say 470 divided by ten, 73, two guys, 300 employees are going to be appointed.
So another chairman will consider it.
So I hear you union has run out of room.
Okay, so you the living is it? Yeah, I after.
Hey, man. Christmas three.
And, you know, 583, how many of our original members you got off?
Yes. You were one of the first, if you want to say anything at all,
because actually what the interesting thing is what happens out, because five marriages will happen through five for $8.
Up to ten. Five times ten times ten.
50 divided by 5 to 5 and so on. So ten out of the final ten, so on.
So what happens now is that's well, that the population is declining with this formula.
Actually, we have a stability that happens.
And from now on, if I keep repeating this, we will just have five people remaining here and we get this sudden stability.
So this qualified remaining money we can. Magically disappear in gadgets.
So the interesting thing is that when we doubled the population based on the data, the landings actually say.
So, I mean, what happens if we change the rate at which they are treated?
You see, they triple each season something slightly different. So in terms of what we ran is again.
And so they're really more productive actually because they went from global athletes
to having been in office for six months and fortifying 4.8 days in that time.
And now it is tripling. You get actually not this ability, but it's growing how you get where we're sort of pinballs between two different values.
And if I slightly dial up the rates at which the population of the lemmings changes in one missions and that's a 3.5,
we see this behaviour where every four years the population suddenly drastically drops.
And so this is what's going on with the Islamic, is that the rate at which they reproduce is controlled by time up to 3.5.
And then you see this pattern emerging where every fourth in generation levels collapses.
But the interesting thing is there's no pattern there, but I got a little bit further and then is thought to quadruple every season.
Then suddenly those patterns disappear and chaos ensues.
This now has such sensitivity to light pollution.
This one extra learning is a population of thousands.
The is the result of these equations could cause one year to have been an overnight that he saw, but it was one more time to collapse the equation.
So it's interesting. It's again, it's very important to know we are in a region where you can make predictions.
But the paradox, again, is if you don't have anything up your own fault,
you reach the chaotic region and then you won't be able to know what's going to happen next.
I'm going to finish you with another area which is more about mathematics and the limits within mathematics itself to know everything.
I mean, there's not a general feeling that it's easy to make a statement about numbers
and go past conjecture that every even number is a solid to prime numbers.
That's what true results. It's either an even number which card is up to by numbers or whether you even have this kind of feeling.
I think every occasion breaks as that statement like that.
You either be able to prove that it's true or eventually prove that counters and [INAUDIBLE] stay until the right mathematicians.
The late 19th and early 20th century can't express the feeling of outside positions that really make take some time.
We may have to come up with ingenious arguments and wiles, and everybody wants something.
That's all there 350 years of that world.
But the very school that a mathematician there is no ignore all.
And in my opinion not at all for that reason for some reason why no one is in finding out.
So what it is, in my opinion, there is no in in the machinery of this office.
We must move. We shall.
And I think that a lot of leading mathematicians, as we went into the 20th century, that we should be able to eventually be able to prove that.
I'm not saying that this is true or false, but in the 1930s I was taking fertilisers and it was,
if I say it was receiving the honour of becoming a citizen of his hometown of Königsberg.
At the same time,
conferences going on where an Austrian magician was proving that they're all trying to prove what is now known as the incompleteness,
since there are statements about numbers that we will never proof.
If you have an axiomatic system which captures statements about numbers, there will be statements inside which are true.
But you won't all be able to deduce from those those actions for that number system.
So an extraordinary there showed that there is a gap between truth and proof, an amazingly good will.
And they managed to do it by using sort of one of those sort of linguistic kernels.
It's the Chinese, this linguistic paradox about a mathematical tool,
which they don't currently have enough to be able to prove a difference between what's true within and on the system,
what can be proved, and then all of the linguistic hypotheses to exploit it.
Which is why is like this statement is false.
So what does that mean? Doesn't have any. Well, if they say if that's true.
Okay, well, that's what they say. It is false.
If that's true, it means the thing is false. Oh, so they call me.
They call the true case. They must be false. So they say this is false.
Evil same is false. Is saying this whole. That is it's true. Oh,
that is so but so they basically and linguistically we know that we can have producing really don't
have any means this doesn't have a meaning that we can sort of think of any type but mathematics.
So we don't expect to find these kind of linguistic paradoxes.
So actually, but it's a that we can actually exploit this idea when I'm going to use it for this segment is also going to choose the statement.
This statement is not true. Okay, well, fine.
But that's still a wasted statement. But they're claiming the penalty and they are actually used.
Some of my favourite numbers right now, you want to be able to do this.
He would do something called the girl code. He was able to take the statements.
It was about same as saying this statement does not have a proof within this axiomatic
system he was able to use their coding to change is interesting about numbers.
So statements about numbers really is he made some statements about numbers that same numbers must either be true or is going to be false.
That's not like is you going to say about numbers which has a meaning and therefore now I mean, for example,
somebody being provable might be about the girl who the one sentence being debateable by their great impotence, of course.
So you might say, well, ace is going, carrying on this number, the visual encoding, all the actions.
So now we had it because it was translated as English statement about numbers.
It was either going to be true or it's going to be false. I wanted to exclude the possibilities.
Well, suppose it was false. Okay.
Now the reason is, is this whole thing about is this reason the masses is saying is false.
So actually, what does that mean? It means that both of these statements is provable.
Now, in the same words is provable. It means it's true.
So in fact, it can't be false because we get the same really.
The logic say logical, a spiral that is false.
Turns out that we conclude it, which means it must be true. Okay.
So maybe it is in a logical parallel to you. Okay. What if it's true?
Oh, hold on. What? I could be true. I mean, the only alternative is to say homicide, undesirables, or it's true.
We certainly can't be both because we have a logical science.
It must be true. So this is a statement about numbers which is true, which is it's not proven.
By using this trick,
he was able to actually capture a statement about numbers which by its very nature was true but with no provable within the system.
It was kind of logical triggering,
but actually turned out that is a very fundamental questions about numbers turned out to be this kind of rather strange area of like,
whoa, is it true or false? Is it provable? And so this concerns the mathematics of infinity financing infinity.
For many years, infinity was always something which my brain could capture.
But the mathematician George cancel at the end of the 19th century, and they want us to construct a mathematics.
They allowed us to understand infinity, that it was an infinity, but help us realise there are many different sorts of infinity.
It's all the same as we're trying. We have a right to try it and that the value of the wealth is judged by the number of chickens you have.
So you have a normal system. I'm not going to count 1 to 5 chickens and I don't pay for it.
But there are. Suppose you or I do. I didn't have another system of going free.
So you want to bring in that is lots of that. So one, two, three and then you have lots.
But even if you've got lots of chickens, you can check whether your lots of chickens is more or less than your neighbours.
Lots of chickens. How do you do that? Well, for examples, why don't you three lots of chickens.
So here's somebody who's also chickens as well. So you've got to see the plots like infinity is I can't count the 3.3.
I've just got ten chickens, lots of chickens.
Let's see, I could compare apples if I can look up.
I find the the white also has a smaller and not something that I do on the top.
So is this idea that you could if you got something which is don't count you survive as compared so that obviously we have one,
two, three, four, five but we can't count infinity but so maybe we have comparative.
So cancel that. Yeah. Okay. Suppose you had chickens on my chickens and many chickens and they all have on this.
One, two, three, four is as you know, obviously we've never had chickens and he's not even I call it the same chickens.
Why? Because we compare them the we've got the same results for every chicken that I produce.
He just produces the chicken, which is double has double the number.
And so actually even numbers is the same amount in infinity as all the left personalise.
It perhaps is twice as much, but I mean that I think they could be compared.
So you sort of you would then it's easier just the same size and you need a guy who has a huge treat is a room number with fractions wrong.
So yeah you see why is it that this guy has as many chickens as you could handle it that way?
Well, if you want to do it, you sort of find a logical system to pan chickens out.
So classic because between one and two there is many fractions.
How does going to be able to compare my one, two, three, four or five?
But after three days of all of these chickens and fractions.
But they don't. Right. What you do is you you line up for the fractional chickens integrates so it has an
infinite land and great so I put all the chickens that had one on the bottom of that
fractional the first right and on the second row of all the chickens which have two
on the bottom of the second and the third row and have all ones with three overall.
So now you can see this, somebody is actually the chicken which has one over two is also the same chicken as 1 to 4.
But I don't like it anymore. Chicken.
So so every chicken, every single fraction and me and I put it in the freezer of eighth column is where I changed the occurs.
And then here was kind of, I have a cheap trick.
The way to compare the chickens is to okay I got my first chicken is going to go to the chicken in the top left hand corner.
My second chicken is going to go. Currently starring roles in I Have a Chicken, which can be paired up with every single chicken,
but it isn't that great because I just sneak through a single line and that single line tells me the order in which I put my chickens,
which is half full numbers, so you can actually count the fractions when you're able to do this little bit of
mathematics and tell you what hundreds fraction is inside air and 100 chicken.
That's me. This is handle snake and we use it to power up fractions and ornaments.
So why not say okay, everything is the same.
But no, he's a guy with a big chicken.
He's going to have every possible irrational numbers in a strange coincidence.
So there's pi inside every one of our over there.
And this is where we choose Dad's old ratio. And I think, oh, I know you think of it's not foreign.
It's pretty rare that you can match up all infinities in the same size.
So you start saying today, well, that's what you should wrong.
And she won with pi two to square two.
So I to find some way to pair up all of my games which are all of these graphs,
numbers and how I want to try to always find them and she can actually manage.
It is still left over because there is the prime and as you can say, I'm always going to be able to guarantee you cheating.
You haven't found it because it's not however you can.
So you have one with some arrangement of the two, the second, one, three, three, one.
And now I'm going to produce a chicken.
I'm not going to make sure this chicken is not one of the ones you can already, because it's the first decimal base.
And my answer is a decimal place which is different from the first base pair with the first chicken.
What about second decimal base? Last difference of the second decimal places, the chicken that you paired with the second of your audience.
And so what you do is you keep on changing the decimal by setting the hundredth decimal place of is missing or asking
chicken will be different from the hundredth decimal place of the chicken that is matched up with chicken number 100.
So and why wondering apparently this chicken is nowhere on my in my attempts to pair these up
do I myself what is the minimum what will the million unit in once chicken pay up with my
chicken which if you look at the minimum once decimal place this music is different now one is
not achieving the goal here I'm going to be so the way he's constructed is missing or has what?
I'm sure that it's nowhere on this list.
You may want to have this shop more. Only add this one to the end.
But you know, I can see more by the century there's always going to be chickens that are out,
which are as far as far as hard as you try to shut down, you never be able to count the chickens.
Skaggs was able to create this for the frenzy of all the original numbers.
There's no reason why that is discounted, starting with I swear to do the numbers.
Rules are lined up in a room and the numbers we can use to measure with the things that you see all day is numbers.
That's a bigger to infinity than just the whole number crunching numbers.
That's why I thought the universe might. Okay, here's the question.
Okay. So that's what we want to be one of these arrangement of this.
And it's one infinity, which is the counting numbers. Okay.
Is there a sense of numbers in between?
Can we find a neighbour who is genuinely richer than I am in my whole numbers, but generally poorer than the guy with all the you were asking?
It's a bad question. There is that there are there is that you would say an only which is always my thought is infinity was just some sort of sorcery.
Nobody said, no, no, this is the most beautiful I have. Actually, I think, you know, there was a test going on in the mathematics.
I had to leave all my choice of my eight discs of mathematics.
So it's just a few just beautiful, sort of finite arguments on several of these different affinities.
And I totally agree with David Hilbert said this is the most astounding product of mathematical thought,
one of the most beautiful realisations of human activity in the domain of a purely intelligible.
No one should expect the paradox which had all created for us.
But it did raise this question, which turned out to be one of these questions that Campbell had raised in their statements about numbers,
which we can't kind of decide whether they're true or false. So the question is going to continue.
My thought is that David HUME would show you that he had 23 problems and he
challenged positions of the 20th century with the Riemann hypothesis was on there.
But the first question you asked was about the Continuum hypothesis.
No. These are the numbers you used to measure. Will surely there should be an answer.
The question is a set of numbers. Strictly data.
And the whole numbers is smaller than the original ones.
But following on from that, as it discovered that actually either answer is true,
it turns out that the ask in yes and you create a system of mathematics which is
totally consistent where there is a set between there is a kind of the system.
So the answer is no. I mean, this is kind of mathematics question you want in an exam where you really talk about the first question, correct.
But I think it's a kind of a this really is really the magic of mathematics.
All the power. It's all magic. It's all this is the idea that we can turn mathematics in itself, using products to understand the systems.
Actually, we can realise notation. We don't realise it's something like a question, fundamental question like well,
is there a set of numbers which is getting bigger and smaller than these things, true or not, that we can actually realise?
Well, there are some things that we never know.
And so sometimes to questions or answers.
And you do that. So we have some points for you.
So you might ask questions. Stand up.
Good to talk to you, sir. So do you think the continued medical science is true or false or neither?
So this is a very good point.
And you might say, well, I have as I said, this is a meaningless question because you've got two systems where one is it's true.
One is it's false, but it's still I think the question has a lot of relevance because, you know,
the point is we're trying to capture numbers to be simply we have these numbers that measure three legs en route on rules.
So surely there should be an answer to it.
Should we should be introducing an axiom which says that, yeah, no, I don't mean these numbers, which is terribly consistent.
We don't have a set between 19 these ones and then that's the numbers.
So I think they're all these are all to this idea of the continuum rivals as being sort of an unsolvable problem that,
you know, we decided to make it as not for many years we will do this sort of next year.
And so that means it's a bit like non-Euclidean geometry.
You can either have geometry, which includes the axiom that there are all kind of lines are points,
or you can have a dormitory where those objects, maybe you just have different sort of not systems of numbers,
but but there's been a bit more recently saying I think we should be writing down which number system we work in,
where we whether we, we're talking about the set of real numbers.
And I say that or I think armies in which it was related sort of to the physical world, you know, these numbers sort of finds it.
We try to actually with these numbers to express our feeling about physical lengths as well.
So perhaps one of the we should be a meeting of the axiom for the computer models or not, which one we really think we do.
But as I say, there is a move to sort of displace these segments to actually say, no, I think we're left with it.
This is not what we mean by the real number.
But very often when we're doing mathematics, you will find we will make it say, I assume the continuum hypothesis.
It means I work in Islam as a system, which assumes you continue my hypothesis or it will say it is.
You know, there are some very curious examples where you can prove that there is a theory
amount of numbers which rather bizarrely one and the proof goes with it.
You see the continuum hypothesis, you argue this way and you prove the thing about numbers.
But in the other numbers system where you see the continuum of offices is false, you can prove a different thing.
You probably saw the same answer about five numbers. Now, is that the proof of the statement that five numbers, because it's sort of, you know,
you're going to make an assumption in both cases, but you don't you don't know yet.
So the combined effect, of course, is either it's true or it's. So it's sort of a win win situation.
So my question that's really free for researchers.
So I'm going to remain on that's. Unless you're down there in my coming to my house.
I was wondering if you think there's an infinite number of outcomes or whether you
could ever predict the future if you could predict every single possible outcome?
Uh huh. Well. Uh. That's a very difficult question to answer because no one's going to get into questions of quantum physics.
I mean, quantum physics. I mean, suppose the universe is infinite, for example, and, uh, and is it's going to actually be one.
There's a very analysis you can do that everything is possible, will happen.
And it will happen in theory often, but it won't matter most continuous factor.
And then, I mean, certainly I think I think the the conclusion is that, you know,
within a continuous maximum, certainly infinitely many possibilities with infinite data inputs.
But but it's there are arguments by people that infinity doesn't exist as a physical object.
And it's, uh, which seems crazy to me, but the idea is that the universe might just.
I mean, it's certainly true, if you think about it,
that there will certainly be a finite number that nobody ever will think about because, you know, the advanced a lot.
And and we, you know, the final number of people on our number,
the amount of time you'll always have something at 8000 ends and heat death in a horrible sieve slows down.
And, you know, we we know that there will be a number which has never, ever been thought so.
So I think that maybe infinity, it's just a purely practical concept.
There are some people who sort of think this the universe might be a bit like a computer which has the kind of limits to its capacity.
And when you hit that, it kind of resets.
TIME So I think it's sort of that, you know, Mother Earth to like that you've had up to 12 and then I think that's why you have a mathematics machine.
Whereas, um, and so, so I think it's a very interesting factor and I think it is that tension which goes right back to facts,
the sovereignty of Pythagoras. That's where to I mean, to say that's a physical place.
But, you know, what does that physically really have to something like infinity?
Because it's easy, physically impossible because of the quantised nature of space.
I mean, so I think it's a very interesting tension and one which I hope is flexible down, which is the relationship to the world around magic.
So what exists in the mind and and the world, the world,
the physical universe and my mind grew out of trying to decide that the universe is interesting quite whether
they they do sort of match or whether there's any difference that's played on that I've been working on.
We on in October it's all about this tension actually.
Yeah. Question Here we get microphone. Grace the.
So you talk about imaginary and real numbers. As for me, a two dimensional space.
And it's easy to imagine that we have three dimensional space and many more dimensions.
So are there systems that are imaginary than three dimensional imaginary information?
So on. Is there a simple way to think about that, like the square root of negative one or no?
Yeah, it's a great question. One of the first discoveries about these imaginary numbers, squared minus one.
What about four, three, two, minus one? Perhaps that's really the new, new anomaly.
After I had a new direction,
I one of the first great discoveries about these imaginary numbers where it seems to me the square root of minus one, it's incredibly powerful.
So they have combinations of squared minus one and real numbers.
So it's two lines or graph of numbers represents, you know, what's the direction, the real stars in the imaginary line.
So now we can solve all polynomial equations as soon as we've added in squares minus one.
So you will challenge that will come generates some new fighting between the square root of the square in minus one that's inside that.
Also we can express that actually in terms of just the squared minus one.
But that doesn't mean actually. Okay, fair enough.
So this is a very powerful analysis. So all quiet on the equations that might surprise us, but maybe they're all you know what?
If you added a new direction to the numbers and even trying to do this and they went from 2 to 3 dimensions,
those actual and what they just couldn't produce consistent.
I mean, it was very careful that it is that idea that, you know, the square root of minus one, you can only make these imagine leaves divided.
It doesn't collapse all of mathematics and the discovery was noted.
But that's a serious point. You're going to make new things provided that in a way on the continuum makes a
set an axiom which says there is a set between these two infinities and also
exactly which just isn't that of them are consistent they don't produce contradictions
so squared minus one and that mean and it didn't produce contradictions.
It produces richer mathematics.
So the challenge is, okay, can you throw something else in and produce a rich produce something which is genuinely new and which didn't collapse?
The system always is actually going to make a set from 2 to 4.
So they're really simple. Hamiltonians discovered by an Irishman position.
HAMILTON Right. Did you ever get to stop in? There's a breach.
And this is where he was alleged to have had some revelation about how to cook up a new number system.
Where is it? So we have the same line that squared minus one.
He introduced all three things. I think we all have discovered the easiest way to mind as one.
But then he was interested in what relationship he would use.
The relationship between the ideas and the case and how they multiplied and any kind of was easy.
Famously, he's going to call these into the side of the bridge with this mathematically.
So these are all things like Apple that they use a lot in computer graphics, for example, as a system to model the way that computer graphics works.
And so there are a number of systems beyond the Hamiltonian.
The Octane is, but then is going to go to 16.
But there's a marvellous reason why you can't have a very strange I mean, you think or any one has a two,
you should be able to just analyse the system and then you can point to minimisation where if you go any further you do sort of classicism.
So your intuition is one is exactly what happens if we can do this, why can't we do something again and see what happens?
And that's how we have to solve it on the systems.
Beyond that, there are different oppositions hamiltonians were such that they a massive of all the worlds by numbers.
The numbers don't matter which one you do. Okay, so you get a daily news disruption.
Other issues. Yes.
Because you can hear.
In theory, a used formula for that is showing that you can essentially produce certain point variables no longer say or create more energy and chaos.
So wondering, can we apply that to to any stable system and predict when it will become chaos or by the very nature of chaos is made possible?
Is it because the the earlier illustration showed of the two planets safely overseas and you introduced third I think is crazy made me think.
Are we able to calculate how much changing gravity we can put into space with all the satellites in mind?
Use changing gravity. Could that destroy the solar system?
Might make us fly. And so many things said show that.
Well, I you could kind of like the point at which comes out say, well, so can we have the landmass?
We can put up a little bit of that as well.
I think that's a very good question.
And, you know, certainly it is this idea of only the systems are very much involved in trying to find those regions which have irregular behaviour.
The moment you can set up systems, all three bodies which are stable and have a certain stability even with small changes.
But I think that's the challenge of each particular system that you've got is to understand in a way that changes the pendulum.
If you understand that diagram, the description of that space, they think they know anything.
And although of course they keep on sticking satellites up because I'm still in the small generation,
but some days there'll be some point where you'll naturally you go on sort of chaotic if someone is actually in I mean,
actually one of the great unsolved problems in the 20th century 23 for the beginning of this millennium,
we were challenged with seven points through the millennium.
So one of the reasons that solved the equations is because one of the Riemann hypothesis primes, which we still don't understand,
but one of them is called an Audi A6 equations,
and these are equations assemblies and these are equations that we we do not yet have a supply of solutions for.
And the challenges alone understand the behaviour of these equations with you really dollars if
you do and they control the air behind an aeroplane or the way fluids travels through a point.
And again, you define these critical switching,
the names which are responsible for this change in behaviour and it's very important to know where they are.
You can calculate the case of the of the feedback equations of the population.
Exactly. The numbers in the dial where we get is from chaos.
I don't have to convince you that turbulence,
in fact is one of those extraordinary free kicks that was ever taken in the history of football that you remember.
The match between Britain and France and boxers was in gold and romantic holidays,
but the readings go near all the people in this world and authors of football.
What do you believe is called super young and go for a short golf course, although he's also been flying through the air.
I mean, there's a wonderful shot at the Academy,
shows what it is like going off into the crowd and being lost and then somebody them off for the swans in the air, it goes it all goes away.
And as you what was involved with this is because when you hit the ball very fast,
the the turbulence on board is Celtic and the Celtic sentiments and doesn't cause very much frightful surprises.
And it's like it's the same effect that happens in golf.
It was the closing being carried through the air because the speed of the ball in your face and there's
something that is critical moment where that the chaos kind of gets switched off and it just becomes laminar,
becomes much more regular. And that's like a Soviet break being puts on the football and it slows down,
which by the speed of the ball is able to have much more effect on the trajectory.
And that's when suddenly the ball swerved in the back of the net.
So it's that change in from has to to non healthy behaviour which was responsible for now we're going to pause maybe after that.
Have you said it's all very.
You refer solely to a lot of amazing people who are mathematicians in yourself as a mathematician.
Do you feel that academician mathematician has narrowed more recently,
and do you think it is beneficial to the field to widen or narrow the definition?
Well, that's it's very interesting because I thought of what you can challenge,
and I had a little bit of quizzing and it is the history of mathematics.
And so some great examples of women. Yes,
but we are in a we are in a period where if you look at history and also you have
many more women on board and it's really a complete mix of this kind of games.
And I think one has to be very careful about these 30 inches all competitions.
Maybe that's because you I to you also have this. But we have all but actually rise to the challenge is that I think very often recently
a lot of subjects are quite narrow minded and gets wise in the way that they think.
And one of the beautiful things about mathematics and language is how the interactions with other subjects.
So here at the University of Oxford Mathematics and a lot of time talking to biology,
which has a different way of thinking and it takes some time to get its mindset.
The way I think is in some ways the mathematician some, but the things that are mentioned to be very exciting.
And we both have different problems,
different ways and different ways combining all disciplines which I think are leading to really exciting development.
So, so I think there is actually rice is one shouldn't get stuck in thinking of mathematicians as just one particular thing.
And the more multifaceted we think of this not that we have individual ambition that the more exciting mechanics we.
Now we have time. Okay. And I.
Whoa, whoa, whoa, whoa.
Oh, great. You are awesome. So let's do you got to take a quick break here.
You're to read. Interestingly, our sound engineer knows some people listen, and if you're not queasy, they love it.
I possible to go back to the limits of science.
Looking forward, do you think Max will continue to make this great progress?
Do you think there's a sort of limit to what the human mind can do?
You're going to hear all of them, quite honestly.
That's a really interesting question, because I don't want it to be said about logical impossibilities,
limiting what is beyond even those of us in the future.
But what about the human brain? The human brain has a finite, finite number of neurones, new connections.
Maybe there maybe sensing that there are some logical arguments within mathematics
just beyond the capacity of one human brain to navigate this as it is.
There's no way that we will ever be able to go through all that whole process again.
So there are so many things, almost just from a biological level, which I think are limits to mathematics.
The amazing thing for me is how often these challenges like Fermat's Last Theorem
or other theorems we've managed to lose upon grade rejection rejections.
Quite often when you apply to solving these problems, you just get this depressing feeling.
Yeah, it's just beyond anyone's right.
And as I said, it is a kind of human limitation.
There's also the limitation girls can get wrong worried about.
Okay, no. Maybe the golden conjecture is a true statement about numbers which can't be proved within not existing, maybe.
And there's a wonderful novel by Apple's so-called Pixel Supernova conjecture,
which features a mathematician who then about as well and has a whole crisis that
he's dedicated his life to something that actually turned out to be untrue.
But I think more interesting is the question of the human biology and, you know, all the limitations, just the capacity of what our brain can do.
So so maybe the Riemann hypothesis, maybe the proof of that, you know, has a complexity to it,
at least that we all get on a certain level, and that the complexity so much that the human brain will never be able to navigate.
The amazing thing for me is that time and again, these things which look like they're just beyond the capacity of human psyche,
somehow find some way to simplify the problem or some new way in.
And we're able to capture we are finding a little bit of breaking up here, even things as complex as handles idea of comparing infinities.
So I still am a believer in, you know, this piece of equipment up here to be able to navigate into the future.
Just amazing searches of mathematics. But I think we do have to recognise that there will be things that by their nature we will never know.
So thank you. Very, very.