In Our Time - Pierre-Simon Laplace

Episode Date: April 8, 2021

Melvyn Bragg and guests discuss Laplace (1749-1827) who was a giant in the world of mathematics both before and after the French Revolution. He addressed one of the great questions of his age, raised but side-stepped by Newton: was the Solar System stable, or would the planets crash into the Sun, as it appeared Jupiter might, or even spin away like Saturn threatened to do? He advanced ideas on probability, long the preserve of card players, and expanded them out across science; he hypothesised why the planets rotate in the same direction; and he asked if the Universe was deterministic, so that if you knew everything about all the particles then you could predict the future. He also devised the metric system and reputedly came up with the name 'metre'. WithMarcus du Sautoy Simonyi Professor for the Public Understanding of Science and Professor of Mathematics at the University of OxfordTimothy Gowers Professor of Mathematics at the College de FranceAndColva Roney-Dougal Professor of Pure Mathematics at the University of St AndrewsProducer: Simon Tillotson

BBC Sounds, music, radio, podcasts. Thanks for downloading this episode of In Our Time. There's a reading list to go with it on our website, and you can get news about our programs if you follow us on Twitter at BBC In Our Time. I hope you enjoy the programs. Hello, Pierre Cémon Laplace was a giant in the world of mathematics, either side of the French Revolution. He addressed one of the great questions of his age,

raised but not answered by Newton. Was the solar system stable, or would the planets crash into? the sun or even spin away. He advanced ideas on probability, longed the preserve of gambling, and pushed them out across into science, and he posed his own startling question. If you knew how many particles there were in the universe, could you predict the future? With me to discuss Pierre Simon Laplace are Marcus Chesotoy, Simonier Professor for the Public Understanding of Science and Professor of Mathematics at the University of Oxford, Timothy Gowers, Professor of Mathematics

at the College of France, and Colvaroni Dougal, Professor of Pure Mathematics at the University of St. Andrews. Colva, can you tell us something at Leplasse's background? Sure. So he was born in 1749 in a little farming village in the Calvados region of France in Normandy called Beaumont-Anne-Age. It's kind of between Kahn and Le Havre. His father worked in the cider trade, possibly was mayor of the village, sources somewhat differ, and his mother's family farmed land just in the neighbouring village.

So he was from a... a middling background, let's say. Definitely deep rural stock and not much sign of anyone else in the family receiving a particularly high level of education. He went to the local village school, which was attached to the Benedictine Priory from the age of seven to the age of 16, and apparently attracted attention mostly for his prodigious feats of memory when he was there, rather than for any early mathematical prowess or anything like that. Now, most pupils at the school went on to either serving the church or the army, and Laplace chose the church. And so at the age of 16, went off to the nearest local university,

which was the University of Kahn, to study theology. Now, somewhere along the way at the University of Kahn, he attracted the attention of two of the maths professors there, who started taking a real interest in him. And partway through the degree, he decided that what he actually wanted to do was become a mathematician, so he never took his degree in theology, which is maybe just as well since he became an atheist not that long afterwards, and headed off to Paris to seek his fortune.

As far as we can tell, he essentially lived in Paris and its environs for the rest of his life after that. So he shook the country earth off his feet and became an urban man. Did he find his fortune? Yes, he did. So there's a nice story there. As he headed off to Paris, the person that one needed to meet was a certain Jean-Baptiste Le Hon d'Alembert, who was the most famous scientist in France, bar none, really, at the time, and had a huge amount of influence.

So Dallembert was permanent secretary to the important academic institution in France for scientists at the time, which was the Academy de France. He was a permanent secretary to that. He was a member of the Berlin Academy of Sciences and he was a fellow of the Royal Society as well. So Laplace's lecturer, Le Canoe,

had sent a letter of introduction with Laplace to Dallambert. And apparently, Dallumbert received the letter, but didn't answer. So Leplas, well, he wasn't. stupid. He thought for a while about what exactly should he do. And in the end, decided that the correct thing to do was to send Dallember a long letter on the mathematics of mechanics, on the general principles of mechanics. So that's the mathematics of how forces act on bodies, how that makes them move. It's how we might study planets orbiting the sun, for example. And upon receiving

this letter, Dallumbert changed his mind with a lacritary, invited him to come and meet. and said, you need no introduction, you have recommended yourself. Within a week of that, stories say, Laplace had been given a job teaching mathematics at the military school in Paris. And then it didn't take that long afterwards before Laplace was able to start presenting papers at the academy. Why was it so important that it should be part of the academy? In Paris at the time, in France at the time,

there wasn't very much science going on at universities, where most of the science happened was at the academy. So this was an organisation that was founded in 1666 by Louis XIV. It had a strictly limited number of members. They were salaried. It existed to advise government on how to fill posts such as the job that Laplace got teaching at a military academy. And the only real scientific publications in France at the time

were the publications of the proceedings of the academy. So to be permitted to read papers to the academy, to have your papers in the proceedings of the academy was the only way to publish mathematics. Thank you very much. Tim, Tim Gars, before we get into detail, can you give us an overview of two of the questions facing Laplace

which he took up for the rest of his life? First, what was understood by probability when he picked it up? Well, probability had been developed to some extent by some pretty famous mathematicians like Fermat, Pascal, Bernoulli, de Mouvre. But the focus was very much on things like games of chance. So you could say probability was a collection of puzzles, some of them very difficult,

and mathematicians had worked out techniques for solving some of them. The place's main achievement in that sphere was really to turn probability from that into a real science that has applications way beyond games of chance. He also had various specific achievements within probability, such as rediscovering and developing Bayses theorem and also a famous result called the Central Limit theorem. Does he say, I wish I had the quote of my thinking is, but I don't, I apologise. Probability was a mark of our ignorance.

Yes, that is very much his view. So Laplace was a determinist, and you might think that probability doesn't have much of a role to play in a deterministic universe. But it does, because although determinism might be true in principle, according to Laplace, we can't know all there is to know in order to work out what is going to happen. So for example, if I toss a coin, it might be that physics will tell me exactly how the coin will pass through the air, but I can't know the initial conditions of the coin and everything about

the air molecules surrounding it and so on to sufficient accuracy to be able actually to calculate what will happen. So for Laplace, probability comes in because of what we don't know. So instead of being able to say exactly what will happen, we have to make probabilistic assertions. And the other big question that I headlined at the top was about the stability of the solar system. Can you tell us why he arrived, why he wanted to do something about that? What does the stability of the solar system mean? If we look at the planets, they seem to go in rather regular orbits. They go round around the sun and we know quite a lot about how that happens. But it might surprise people to know that we don't actually know how to prove that strange things

won't eventually happen like planets spiring in towards the sun or colliding with each other or being ejected from the solar system. In fact, the belief is that given enough time, these sorts of things will happen, but fortunately enough is millions or even billions of years, so it won't actually bother us. But this question of whether the solar system is stable or whether these kinds of phenomena might one day occur is one that has occupied people for for centuries. And it started, I think, with, well, you can jump in on the story. It has a very long history. We could start with Kepler, who looked at observations of Tycho Brahe and came to the conclusion that planets moved in ellipses round the sun. And this was a wonderful

theory because it fitted observation very well and was extremely simple. And then a huge major additional step was taken by Newton, who showed that a very simple law of gravitation was sufficient to explain why planets would move around the sun in elliptical orbits. But in order for Newton's conclusion to work, he had to ignore gravitational interactions between the planets and other planets and just consider the gravitational force exerted on planets by the sun. And there was a problem with the inequality of Saturn and Jupiter, which he spotted and which he later worked on. Indeed, so that that problem was that Saturn seemed to be slowing down and Jupiter speeding up, indeed that was the case. And so if that were to continue for long

enough, then Saturn would be ejected and Jupiter would spiral in towards the sun. Or if you extrapolate backwards in a simple-minded way, you would conclude that, as it turns out, six million years ago, Jupiter and Saturn would have been the same distance from the sun. So people started to worry about whether these kinds of things would happen. And Newton himself felt that his theory was perhaps not complete. And he even postulated that God gave planets a little nudge from time to time. Thank you very much. Marcus, would you

like to take that on? So how he got this question, which has been very clearly set out by Tim and brought God in at the end, which Newton did. How did Laplace approach this? The stability of the solar system. Well, it was a really big challenge because the prospect was that one of these

planets was going to fly off into space and the other one was going to whiz into the sun. And surely this probably would have happened already. So there need to be some explanation of this and was it nudges by God or Laplace sort of flirted with the idea that it might be the influence of comets. It was certainly going to be the influence of the other planets, which was the thing which hadn't really been solved. So that was his challenge, how to show that the influence of Jupiter on Saturn and Saturn on Jupiter might be able to save the solar system. And what he identified was a

a certain sort of planetary resonance. So there are sort of certain coincidences that happen. What do you mean by resonance? There's something rather acute, which happens, for example, with the moons around Jupiter. The three moons have orbits which are in a one to two to four relationship. So one goes around once whilst the other goes around twice and the other one four times.

So this is a kind of resonance so that the gravitational effect is almost like pushing a swing where each time it goes around, and it sort of swings around in the same place, it'll give a little bit more energy to the moon. Now, sometimes these resonances can cause things to fly apart, and sometimes they can cause things to stabilise. We know that there are rings of Saturn which aren't there because of certain resonances which kicked particles out of those rings.

So was it one of these maybe that was explaining this inequality, the fact that one planet was accelerating, the other decelerating, and perhaps could, over a long distance of time, perhaps resolve itself by reversing this, slowing one down and accelerating the other. So we get a sort of periodic behavior. And there was a kind of near resonance for Saturn and Jupiter, because if Jupiter goes around five times, we find that Saturn goes around twice around the sun.

So the orbit of Saturn is 29 years. The orbit of Jupiter is 12 years. So over 60 years, you almost get a repeat. But not quite, and this not quite generally meant that it didn't really have much impact on things in the long term. But what Laplace realized is that previous analysis had kind of ignored some terms in trying to solve this, but actually one of these terms was going to be much more significant than they actually realized. And it was sort of because at some point in one of the equations you had to divide by this difference between the orbits of five times the

orbit of one planet and two times the orbit of the other. And this would suddenly cause one of the vibrations in the sort of variations of the planets to be much larger than anyone expected. And it turns out that over a 900-year period that actually the planets would periodically repeat this behaviour of speeding up, then slowing down and then sort of swapping over their behaviour that would just repeat itself to the end of time. And that's the stability you're wanting to look for. Colbert, thank you. Colver, can we stay out in space? What was Laplace's nebular hypothesis? The nebula hypothesis is Laplace's approach to answering another puzzle about the solar system. So as Marcus just said, there was puzzles to do with the speed at which various planets were going.

But another curious thing is that when you look at the solar system, all of the planets that were known then, and indeed all the ones that are known now, which is slightly more, were going around the sun in the same direction. It didn't seem to be any particular reason why they should all go around in the same direction. and all of the moons of all of the planets, of which they didn't know all of them then, but we know more now, were going around in the same direction as well. They were orbiting their own planets in the same way. That's actually not quite true for all of the moons, but it's possibly just as well that Laplace didn't know that,

because it might not have made him think about this question. So the answer was why? Why is everything going around the same way? Why is everything spinning in the same direction, both around the sun and around itself and around its planet if it happens to be a moon? Well, Newton had spotted this and as ever had said, well, clearly God set it up like this and left the matter at that. Laplace wasn't very happy about that explanation. So he initially calculated how likely was it that this had happened by chance. So if a planet was going by chance, it should be 50-50, whether it goes one way around the sun or the other way around the sun.

And he worked out he had 29 different coincidences. So the probability that it was happening by chance was less than one over 29 twos multiplied together, which is a very, very, very big number. So he said this is unlikely to be happening by chance. There should be some more likely explanation. And the one he came up with, it wasn't original to him, I should say. Emmanuel Kant amongst other people had suggested it beforehand, but Laplace didn't know about Kant's work,

so it was effectively original to him, was that at the birth of the solar system, there must have originally been a giant cloud of gas, hence the word nebula, so a big foggy mass. And that foggy mass must, for some reason, have been spinning just a little bit. So it would only have had to spin a tiny bit at the beginning. And because the mass of gas was so huge, it would slowly start to pull together under its own gravity. And then just like an ice skater pulling their arms in, that would mean that the gas would start spinning faster and faster. And the more it pulls together under gravity, the more it wants to pull together,

So it would eventually go through, he thought, a liquid state and form into a ball. And then it would continue spinning even more fast as it gets denser and denser and denser. And the ball around the equator of the way in which it was spinning would start to throw off lumps of matter, which would eventually become the planets, whilst the very centre of the ball would eventually become the sun. And so because the planets were being thrown off by this rotating ball of what he thought was liquid at the time, everything would wind up going in the same direction and everything would wind up spinning the same way. Well, he wasn't right, but he wasn't that wrong.

He was very clear to only describe it as a hypothesis, though. He said it wasn't based on any experimental evidence. It was just a mathematical description, which would seem more likely than God having put it like that. Thank you. Tim Gars, can we go back and look more closely at probability and this central limit theorem? It concerns what happens if you take a lot of, of observations and work out their average. So, for example, if you toss a coin and you work out

how many heads there have been, or you could divide by the number of times you've tossed it and say, what is this sort of average number of heads. And you then look at how that average is distributed. It turns out that for a large number of processes that you might do, that involve doing independent things over and over again, the way that the average is distributed after a while becomes the same. And what it becomes is a very famous distribution known as a normal distribution, which has a familiar, if you draw a graph of it,

it has a familiar bell curve shape. So there were indications that this result was true. De Moira had a version of it for specific contexts, but what Laplace did was to prove that this phenomenon held in very great generality. And this has been a fundamental result in probability in statistics ever since. Where does that take us?

It enables you to do calculations concerning very large numbers of events that would be hopelessly difficult to do otherwise. So it plays a huge simplifying role in many calculations dealing with data of all sorts of kinds. So one other example perhaps would be if you want to know what a planet is doing and how it's moving, you have to rely on observations. And any observation is going to come with errors, particularly if you rely also on historical observations, which you often want to do because of the large timescule. scales involved. So then the question is, how do you use the fact that observations come with errors to work out what's actually going on underneath, what's actually causing the observations? And again, probability comes in and the central limit theorem comes very much in, because if you then take the average of all these observations and assume that the errors

are somewhat independent, you can then say that the way that the errors are distributed, or the way their average is distributed will be, again, normally distributed. It'll have this bell-shaped curve, and that enables one to make calculations that would otherwise have been extremely difficult. Thank you. Marcus, Marcus, you so joy. What is Laplace's demon? This thing called Laplace's demon.

Yeah. A wonderful idea, and it's really at the heart of the idea of determinism that somehow the present is going to predict everything that will happen in the future and can explain everything that happens in the past. And it relates, in fact, to what Tim mentioned about Laplace's interest in probability because he realized that we mere scientists cannot know everything and therefore we have to use some probabilistic methods in order to make some predictions. But in the book that he wrote about probability,

he speculated about, well, what if we actually did have complete knowledge? And he writes, and I think it's worth quoting, because it's such a wonderful statement. He says, assume an intelligence. that at a given moment knows all the forces that animate nature, as well as the positions of all things of which the universe consists, and further that it is sufficiently powerful to perform a calculation based on this data. It would embrace in a single formula the movements of the great bodies of the universe and those of the tiniest atom. For such an intellect, nothing would be uncertain, and the future, just like the past, will be present before its eyes. And this is actually, I think, a little bit of consequence of the fact that Laplace had shown that the universal laws of gravitation that Newton had come up with, which people were beginning to worry may not really explain the universe because there were these anomalies like Saturn and Jupiter running at the wrong speeds.

He'd managed to explain this. And in some ways, he said all the things that we thought were going to disrupt Newton's ideas have actually shown that Newton was right. And I think this was the moment when he said, okay, if Newton is right, then these equations that Newton came up with, provided we know where everything is in the universe, how it's moving now, a snapshot, that snapshot can be run using Newton's equations into the future and back into the past. And this leads to the idea of the universe being clockwork, deterministic. And this idea of, I mean, Laplace never calls it Laplace's demon. This is an idea which came a little bit. later the idea of a demon having this kind of complete information. He talks about it as an intelligence, but he believed if you know everything about the universe, and it does need to be

everything, because chaos theory says a small error in your knowledge can result in a completely different outcome of where the universe is going. But if you know everything, that intelligence will be able to just run the equations into the future and back into the past to know both the future and the past just from a snapshot of the present. Colver, can we try, can you tell us something about his role in the development of the metric system? We must remember that he was in French revolutionary times. He survived the revolutionary, although one of his scientific friends were executed. He then survived Napoleon and he survived the Bourbonne. So he was quite canny. But the metric system came out of this. What finger did he have in that pie? So Laplace was really

important in the founding of the metric system and indeed it seems quite likely that he was the person that named the meter the meter and that's how kind of central he was. So the metric system was invented to solve two problems but also in a spirit of idealism. So the two problems that

the French had were that they unlike England which had just one unit of measurement since the time of Magna Carta France had different units of measurement in different bits of the country so it was quite a nightmare trying to convert a weight or a distance in one bit of France to a weight or a distance in another bit of France.

They also had a system a bit like the imperial system where some things go in 12s, like inches to feasts and something's going 14s, like pounds to stone. So they wanted to devise, and here's the idealism, a system that would work for all people at all time, and it should only use powers of 10. So it should use tens and hundreds and thousands and so forth, but it shouldn't use any other numbers. So the Academy created a commission three weeks before the fall of the Bastille to look at creating a new system of weights and measures and put Laplace very squarely centrally on it. So they were torn between two possible measures for what the length could be. One was should it be the length of a pendulum which beats once per second? So how long a piece of string with a rock

on the end would you need to have it swing from one side to the other in a second? That was eventually discarded because it's not constant all around the earth. You get different measures in different places. So how did they come up with the metric system? So their eventual decision was that it was going to be one 10 millionth of the distance from the equator to the pole passing through Paris. So they sent out a team of surveyors to measure it, made a whole load of sample meters that were looking like they were roughly the right length that was going to be measured. And when the measurements came back in in the 1798, the length of the meter was fixed. And from that, we got the entire metric system.

Thank you. Tim Garz, can you tell us about what's known as Bayes theorem and how Laplace worked on that? Yes, so Bayes' theorem was discovered, at least a special case of it, by Thomas Bayes in the 1740s. It concerns what is sometimes called inverse probability. So direct probability is where you know what the causes of something or some random process are, and then you have to work out probabilities of various effects. Inverse probability turns that on its head.

You have the effects, you're given the effects, in the form of some sort of data, and you want to work out what the causes are, or at least you want to work out what the probabilities associated with various possible causes are. And Laplace, in 1774, rediscovered Bays' rule, or Bays' theorem, and in fact in a form that's closer to what we now call Bayes' theorem.

So there's actually some debate about whether Bayes really deserves to have his name associated with the theorem. And he then used it. And again, this ties in with his interest in celestial mechanics because, well, Colver gave a nice example of this inverse probability earlier on. She was talking about whether it could be thought of as a coincidence that all the planets go around in the same direction. So there's an example where you have the effects. You've got the data in the observations of the planets.

And then you're trying to reason about the causes of those effects, which led Laplace to postulate the Nebula hypothesis. And so it was really Laplace who took Bays's theorem. He then, in 1781, it was actually pointed out to Laplace that Bays had discovered a serum in a slightly different form. But it was really Laplace who took Bays' theorem and showed how important it is. it was. And these days, more recently, it's become extremely important. It's absolutely central

to machine learning and data science, which is obviously a very hot topic now. Marcus, I mentioned earlier that Laplace covered a wide area. We've talked about the big ones. Can you talk about briefly his work on, say, heat with La Boisier? Yes, I think this was quite a surprise to me to see this mathematician working on chemistry. But it shows a little bit Laplace's kind of of wheeler-dealer character because he saw Levoisier as a slightly senior member of the academy and might be quite useful to work with him. And this is why he ended up working with Levoisier on heat. And he actually brought a very mathematical view to chemistry. And so we see qualitative things turning into quantitative ideas. And one of the reasons they could sort of do that is that

they develop this thing called an ice colorimeter, which measures the amount of heat coming off things. So, for example, they were interested in, if you have a reaction of chemicals, how much heat does that produce? And they would measure it by the amount of ice that would be melted. So they just basically measured it by how much water was dripping off this, the thing that had the reaction happening inside it. And one of the things they discovered, for example, and they wrote this wonderful memoir on heat in 1780, is that the amount of heat that's given off by a reaction, if you try and reverse the reaction, you've got to put the same amount of heat in in order to reverse the sort of chemical equation. And one of the other rather cute things is that they put a guinea pig

inside this calarimeter for 10 hours and looked at the fact that it just was breathing away there and it was melting the ice. And so it's actually one of the reasons they realize that life is just souped up version of combustion. The man who's working with La Boisier. who claimed to have discovered oxygen, was executed in the French Revolution. How did Le Plas escape censure from three different governments? So Le Plas has often been described as quite a smooth political operator. I think he was very much attempting to keep his own skin together. So he survived both the French Revolution.

Four days after the storming of the Bastille, He was reading a paper to the academy on the inclination of the ecliptic, which is the kind of plane that the sun appears to take around the earth. He was in some danger during the French Revolution, because he was a very well-known public intellectual. He was vilified by Morat as part of his pamphlet Le Channoté Montaigne. And Le Plas, for some of the early 1790s, decided that the correct solution was to stop going into the academy

before it was suppressed and to take his wife and children and go and live nearby in the countryside. for a bit. But by 1794, he was back to Paris. By 1795, he had been elected as vice-president of the new Institute of France, which was taking over in place of the academy. Somehow he got through that bit really comparatively unscathed. Things got more interesting for him, though, with the next major event to happen in French politics. So all the way back in 1785, he had been the examiner at a French military academy of a young gentleman called Napoleon Bonaparte. and they had stayed in touch. Napoleon was very, very interested in science. In fact, in 1797, General Napoleon

had been elected to membership of this new Institute of France, which had taken over from the academy, and Laplace had been the person who had hosted the ceremony. When Napoleon came to power, he made Laplace Minister of the Interior. I can't think of many other situations of such a famous mathematician, having such a serious post in government. He only had it for six weeks. Napoleon afterwards said that Laplace had brought the spirit of the infinitely small into administration, which was perhaps a little bit mean. But he kicked him out to replace him with his brother, so we can make of that what we will. He sailed through Napoleon, and then we come to the Bourbons, and he sailed through the Bourbons too.

Somehow he manages to keep on comparatively good sides with everyone, yes. So he was out of town the day that the Senate voted to kick out Napoleon, which was maybe wise of him. He was unavoidably detained that day. And so when the Bourbons came back in, Napoleon had already made him a count. Louis XIII made him a Marquist and appointed him to the new body which replaced the Senate. So the upper house, which was called the Chamber of Peers. So Laplace got criticised by many people for being a monarchist before the revolution,

then being a revolutionary during the revolutionary years, then being pro-Napoleon in the Napoleon years and finally becoming a monarchist again. but I think what he really cared about was being able to do his science. He obviously enjoyed the name and title, but he carried on doing science throughout. I'm staying alive. We probably should also include the wonderful story of Napoleon being rather surprised that God had been written out of Laplace's book on celestial mechanics and Laplace saying, well, I just didn't need that hypothesis.

But Colvo, didn't Napoleon have some nice retort to that as well? No, it was Lagrange that had the nice retort. So apparently Napoleon was amused by this and repeated it to LaGrange. And Lagrange said, ah, but it's such a nice hypothesis. It explains so much. Oh, lovely. Gosh, I'm quite surprised at Lagrange. Tim, it seems there are so many ideas associated with Laplace.

Can you tell us about the Laplacean? Yes, so Laplacean is a quantity that you associate with another quantity. So supposing you have a quantity that varies in space. So, for example, if you have a body that has different temperatures in different parts, then that quantity might be the temperature at a particular place. Although I think the example that first interested Laplace was something called gravitational potential, which is connected with gravitational force. And the Laplacian is a measure of, the Laplacein at a point,

is a measure of how the value of the quantity at that point differs from the average value in a very, very tiny neighbourhood around that point. It has a more mathematically precise definition than I've just given using partial differentiation, but that'll do, I think, as an intuitive definition. And partial differential equations come up all over physics, and the Laplasian comes up in many, many partial differential equations. So, for example, if you want to understand how heat works, you need the Laplacein. If you want to understand how fluids behave, you need the Laplasian.

If you want to understand how the gravitational potential works, you need the Laplacean. So it's just a... Is it a sort of key, then? It's just a very, very fundamental concept within partial differential equations. And partial differential equations themselves are absolutely fundamental in physics and many other disciplines.

So it's perhaps one of the most... It's called a partial differential operator. It's one of the most important of those. Marcus, Marcus, you're sorry to it. Let's go back to Laplace's demon. didn't quantum theory do something to damage this? Well, yes. I mean, there are certainly some very modern kind of challenges to Laplace's demon.

I mean, quantum physics, of course, immediately is a challenge to this because it says if you set up an experiment in the same way, there are probabilities. And, you know, Laplace's theory of probability and beyond is now an integral part of how we make observations in quantum physics. And it says, you know, that electron which can go through one slit or the other, it's not determined in quantum physics by the way you set up the experiment. There seems to be some genuine randomness. And that's one challenge. But there are other challenges as well, which you don't even have to come to quantum physics to challenge the fact that Laplace's demon may not be able to predict the future or the past.

How has that stood up to interrogation in later centuries? Well, we've already indicated that quantum physics is a major challenge to Laplace's demon, that the present doesn't predict the future in terms of the world is quantum. But even if we stay within classical mechanics and Newton's world, the extraordinary thing is that in the 1990s, it was discovered that you could set five planets in a position such that the gravitational forces on them would, in a finite amount of time, send one of those planets off to infinite speed and so would leave basically the universe. You wouldn't be able to predict what happens after that finite amount of time.

Even more interestingly, if you go in actually if this had happened in the past and you tried to look at the time now and go backwards, there'd be no way to explain how suddenly this fifth planet came into our system at infinite speed slowed down and joined the other four. there's even a very simpler version of this. If I roll a ball up a hill and just at the right speed such that it comes to the top and it is still perfectly balanced at the top of the hill. If I then go into the future and try and see what happened in the past,

I look back and this thing is still on the top of the mountain and then at some point it starts going down the mountain. And there seems to be no way to predict that moment from the classic. equations. So there are many challenges to this. What would this intelligence be like, for example? There are, the idea of Turing already shows that we can't tell whether certain programs stop in a finite amount of time. This intelligence wouldn't be able to tell that. There are so many challenges now to Laplace's demon and this idea of determinism without even needing to go to quantum physics. But you still think it had an impact on, sorry, impact on the philosophy.

of science? Well, I think it does because there's still the question of, I think there's a big divide now between whether you really believe the world is probabilistic and quantum or whether actually if you set the equations up and run, then that you can make predictions into the future. So, you know, the determinism actually is a real challenge to the idea of free will, of course. We don't have free will if Laplace is correct because all our decisions in the future. are determined by our current setup. So I think a lot of philosophers really wanted Laplace's demon to fail

because it's a real challenge to the idea of there being free will. Colva, can we tax you on the calendar in the French Revolution, the attempt to revolutionise the calendar as they revolutionised measurement? Absolutely. So in addition to setting up the metric system, the plan was to revolutionise time itself.

So firstly, the clock, and then the calendar. So for the clock, they aim to make 10 hours in a day, each comprised of 100 minutes, and each minute comprised of 100 seconds. So seconds would be a bit shorter than they are now, but minutes and hours would be considerably longer than they are now. And that was actually implemented.

It lasted about a year and a half. It was very unpopular and caused problems with the international sale of clocks, because of course none of the time was comparable. What lasted somewhat longer was the French Republican year, so the calendar. They stuck with 12 months, but they decided that each month should be 30 days long. It should be comprised of three, 10-day weeks. So this real obsession with everything being in 10s, even when it doesn't fit. Now, of course, there's not 360 days in the year.

They were well aware of that. Laplace was an expert in such things. So the idea was that you would add five or six what they called complementary days to the end of the year, which would be public holidays. Now, this was where they kind of ran into problems. They wanted the first day of the year to be the autumn equinox, but they also wanted to say that every four years, we should have a leap year,

except once a century when we shouldn't, because it doesn't add upright otherwise. And both of those definitions were in the original decree. Now, the problem is that equinoxes don't come around quite as precisely regularly as that. So they said a new year zero from the beginning of the French Revolution, from the day when the Republic was declared. And Laplace enthusiastically used this new calendar. In his big popular book, Exposition of the System of the World,

he explains both the clock system and the calendar system. I was reading it last night. Napoleon got rid of the calendar. Nobody really liked the new names for the months, and people hated the 10-day week because they only got a half-day off on day five and a full day off on day 10, and they much preferred having one day and seven.

So later editions of Laplace's book explain why the Gregorian calendar, which is the one we now use, is considerably more convenient. But I find it a fascinating contrast between the metric system, which was very much designed for all people and all time, and this new calendar, which was very firmly tied to French revolutionary politics. Thank you very much. Tim, there's not much time left, but are there any other equations or interests associated with Laplace that it's worth mentioning at this stage? Very briefly, I think one couldn't end without mentioning the Laplace transform, which plays a huge role in many, many disciplines, including engineering, which maybe in just the time I have I would just describe as an almost magic method of converting hard problems into easier problems.

And so naturally it is very useful. So, for example, it turns differential equations which are difficult to solve into algebraic equations, which are considerably easier to solve. and it has therefore many applications in physics and other disciplines. In fact, even in pure mathematics in number theory, de Laplace transform plays an important role in understanding prime numbers, for example. Marcus, what would you say his legacy was, the legacy of Laplace? Well, I think we can judge his legacy in some sense by the number of things which are named after him.

Laplace transform, we've just heard Laplace's equation, Laplace is coefficient in heat, Laplace resonance, Laplace distribution, I mean, the list goes on. And I think he can really justifiably be celebrated as a kind of French Newton. But I think one has to say that he wasn't always so great at giving credit to the other people who were involved in a lot of the discoveries that were made. This is really not a time when one individual is making progress. It's many people working together.

And I think Laplace sometimes was a little bit naughty in not giving credit where it was due. And just finally to you, Tim, and then to Colver, what do you think the key impact of Laplace has been? I would say that it's not a single thing. It's just if you do an undergraduate degree in mathematics, and as Marcus has said, his name just comes up all over the place. So he was somebody who had a huge impact on 19th century mathematics, well, 18th and 19th century mathematics.

And that is part of the foundation for later mathematics. So he's just become ubiquitous in a way that I think nowadays is not possible because the subject has become so much bigger and we have to specialise rather more than he did. Colver? I think two main important consequences. One was this confidence that we could explain the entire universe with a smallish number of mathematical laws, if only we could properly get our hands on them. And so he's showing all of these things are a consequence of Newton's laws. And the other was his introduction into science of these more abstractions. abstract concepts such as the potential that Tim was talking about before.

So we don't just have to do science in terms of things that we can see and feel around us. We can invent mathematical objects, which might well turn out to describe the universe better than the knowledge of our own senses. And that led, for example, Lord Kelvin in Victorian era saying that mathematics is the only sound metaphysics. So I think that confidence that released Victorian science was probably his biggest legacy. Well, thanks to all of you very much. Thanks, Colbaroni Dougal and Tim Garde and Marcus Hussotoy and to our studio engineer, Jackie Maghram. Next week is Arianism, the Christianity of the Goths when they sacked Rome. Thank you for listening. And the In Our Time podcast gets some extra time now with a few minutes of bonus material from Melvin and his guests. What didn't you say that you'd like to have said? That's a good kick-off.

La Grange. Legrange. why wasn't he mentioned? You know, I think this is a product of Laplace's incredible ability to sort of make everyone feel like it's him that did everything. And I think, you know, when it comes to celestial mechanics, and perhaps that was my fault for not binging up the fact that LaGrange, as well as Laplace, had done a lot.

And actually for us mathematicians, I think LaGrange is much more mathematically minded than Laplace, who is really just interested in describing the physical universe and doesn't really care necessarily. I mean, he had these, he always say, well, this bit's clear and it would be just him washing over a really complicated explanation. I love the description of intelligence as a perfect sledgehammer, just forcing its way through all the problems with no elegance at all at times. Yeah, he's really a physicist's scientist rather than the mathematicians, I think.

Yeah, so we've got to do a program on Lagrange. I had a really nice example I didn't get time to do because it would probably have taken too long about how you can use a Laplace transform to go between the two main measures that we hear about COVID at the moment. So there's the R rate, which is how many people, on average, each infected person is going to go on to infect. And there's the growth rate, which we're hearing a lot about at the moment, which is at the moment something like minus three or four percent a day, plus or minus some error bars to use some central limit theorem language. And actually, the Laplace transform lets you move between one of those measures and the other one. Well, I did have something

but it wasn't that point. It was that I could say tiny bit. tiny bit more about why the Laplacean comes in when we're talking about heat. Because I said that the Laplacein measures how much a quantity differs from the value of that quantity on average in a tiny little neighbourhood. So if you want to know how heat changes in a body that's being warmed up, then if you've got a point where it's a little bit colder than the average around it, it will tend to warm up at that point. And if it's a little bit bigger, then it will tend to cool down. So the Laplacein is telling you when you've got a body what the rate of change of the temperature at a particular point will be. So that's just one example. And so the heat equation is,

actually that was mainly Fourier who did that analysis. But Laplacean plays an extremely important role in that analysis. I haven't, I didn't get any idea I didn't ask. So what sort of a man he was? I heard later on he had two children and went off to the country for safety, but he allowed himself to be called a count and such. What sort of a person was he around Paris at the time? He was quite unpopular for the kind of telling anyone who would listen that he was the best mathematician in France. But conversely, he did have lots of very long-lasting and deep friendships.

And he was very generous later on in life when he'd made it and was a Marquist and all the rest of it. He helped lots of young mathematicians and young physicists get their start in their careers by helping them get published, helping them get jobs, even holding back on publishing things had done himself so that they could get in there first and make their own progress. Yeah, Arbal described him as a very jolly fellow.

I think the other important thing that perhaps we missed was that he was quite good at giving accounts that people outside of the scientific realm enjoyed reading. So actually he was a good, in some sense, populariser of science, even though they were still quite complicated, he stripped several accounts of any equations such that they could be read by the public. So you could say you'd already taken on Hawking's idea

that one equation halves your audience. So I think actually people knew about science very much because of what Laplace was writing for people outside of the academy. I was going to say the same applies to his treatise on probability. Quite a lot of that is in prose rather than in equations, and it's quite pleasant to read. Do we know much about what he added to the French army when he was a lecturer down at the military academy in his first years? The only thing I've heard of that was memorable was the Napoleon thing.

He had a big important influence on French education later on, in that he helped set up the École Polytechnique and the Ecotel Normale, both in their initial form immediately after the French Revolution when the university. were abolished and later on when Napoleon tried to create the Ecole Normale Superior, which was going to be better than the original version of the Ecole Normale. So he didn't teach very much at them, but he helped set the syllabus and he helped set the exams. So coming up with the notion of what a modern scientific education should be was substantially down to him. Just one of a small point that might be worth mentioning is just that part of the reason that we know rather little about his early background

is that there were the sort of records of his family all got burnt, I think, and some others got lost. So in principle one could have known more, but unfortunately the evidence has been destroyed. Well, thank you all very much. In Our Time with Melvin Bragg is produced by Simon Tillotson. And if you're looking for more mathematicians

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