Hello, thank you for that gracious introduction and for leaving your watch behind here.
And it's very, very helpful. Should I stay in time?
And thanks to Darrel Lombard as well for inviting me here, I'm delighted to be a speaker in this exciting programme.
It's particularly good to have the opportunity to talk about a pretty, a pretty simple, all encompassing theme.
The entire universe. That's what we going to be talking about tonight. Ladies and gentlemen,
I want to begin with an observation that Einstein made in a letter to an old undergraduate friend of his three years before Einstein died,
and it concerned what Einstein believed was a central miracle of.
Science, physical science, and that is the fact that we find ourselves in a universe that has a deep underlying order,
that order enables us to understand or have forged an understanding of the universe and do so in a mathematical way.
One hundred and fifty years ago, the peerless thinker about the natural world,
James Clerk Maxwell gave a lecture on the relationship between mathematics and physics.
Broadly speaking, the same topic as I'm talking about tonight.
Now I can't, of course, promise to bring you anything like his depth of depths of thought and experience.
But I would just note that at the beginning of that lecture,
he said that this relationship between the two subjects is far too magnificent for someone of his ability to discuss.
So I hope you feel very sorry for me trying to do justice to this subject from my relatively low, low, low perch on things.
But I will do my best and reflect on some of the things that some of the best thinking, I think is going on on this subject in this institute,
which is you'll, as you'll hear, has played a central role in the story in in the past few decades.
Let me begin by just giving a quick flyover of the of the topics I will be talking about.
You have to start somewhere in a really good place to start on this topic is that person you may have heard
of Isaac Newton and the torch of mathematics that he and others helped to shine on the natural world.
Theories of gravity, electricity, magnetism, atomic forces are based on mathematics,
and we'll be talking about the great successes that that arose from that observation.
But the path between physics and mathematics, the relationship with them hasn't always run smoothly.
And there was a what has been rather graphically noted by Freeman Dyson, a divorce between the subjects directly after World War Two.
But they did get back together. As you'll hear, Oxford has a lot to do with that,
and I'm going to be talking about the roles of the two foundational theories of the 20th century physics,
quantum mechanics and the basic theory of relativity as the basis of the engine, so to speak of that close relationship.
And I shall be asking whether it really is true that mathematics is today, broadly speaking,
leading physics astray, as many, many comets or several commentators have have argued.
And finally, just to embarrass myself and give you some amusement.
I'm going to I'm going to leave hostages to fortune by looking to the future of the subject.
And as you know, all predictions on about the future of science,
like the future of British politics, are completely worthless, but it's entertaining nonetheless.
All right. Let's begin with.
The event that happened on an unremarkable, otherwise unremarkable summer's day in sixteen eighty seven, the fifth of July.
Check the weather reports upon an ordinary day, nothing particularly going on. It was a Saturday.
And on that day in London, a three volume book was published for the first time.
And this was the book. The the mathematical principles of natural philosophy.
Now this book is quite famous, now we call it Newton's Principia.
It sold reasonably well. Never, actually. They actually tried. It was quite a difficult job to actually get to sell these books.
But it was.
Now many of you will know this has a status as the beginning of the mathematical approach to physical science, and it was written by Isaac Newton,
Cambridge professor, who who was if I may use the word genius once he is the presiding genius of this subject.
He was, by his own description, a natural philosopher. Remember what physicists did not?
He would never have used that word about himself. He was a natural philosopher thinker about the natural world.
The natural natural philosophy at that time was thought of as the rational study of God's creation.
Very important that God is in there as your Oxford historian Rob.
I live in his classic book Priest of Nature has demonstrated Newton regard himself more than anything else as a as a,
as a a man of God to use a term that would have he would have he would have used.
He was also a mathematician. Very important. And just to underline his his variety of skills, he was not just a great experiment,
he was a great thinker about the natural world and a mathematician. So he really was blessed with with with a terrific, we would now say, skill set.
In that book, he set out what we now pretty well, all of us, even even those who are not not versed in this subject,
know is the way you do mathematical science at a very basic level.
You have a basic mathematical structure. You make predictions about the natural world and you check that those predictions
against the most accurate observations one has if if the if they disagree,
then you feed back, you modify the theory and then put it back to nature.
You're trying to account for that order, so to speak, in the universe around us and in this monumental book, not easy to read.
Let's be clear, this was not. This was not J.K. Rowling. This was a tough read.
It was in. There was the law of gravity, which don't need to say what it is in detailed a mathematical description of the force between two masses,
any two masses at any distance apart, and also the laws of nature that that that that mathematically describe motion.
It worked. I don't need to go into the details for the point, the point of view of this talk.
It worked supremely well. You'll know great and justly celebrated observation perception that the force that pulls a piece of chalk or an apple
to the ground or here on Earth is exactly the same force as the force that holds the Moon in the Earth's orbit.
They all are a result of this of this law that is pervasive everywhere and maybe throughout all time.
Very important to notice, nonetheless, that although this was a a an absolutely astonishing achievement to have achieved that intellectual framework,
first of all, he didn't do it on his own. He was standing, as he said on the shoulder of giants Galileo, you're Kepler's,
right, the way back to the ancient Greeks, whom he revered more than anybody else.
But Newton, I think it's fair to say he was a great synthesiser, as well as a great original, original thinker.
Now it's it's the fate of anyone in the field of science that their work is never finished.
Newton For all the tremendous success that his work had, he didn't completely finish that work.
It was still things he couldn't quite account for within the scope of his theoretical structure.
And it's very important to notice that, particularly on the continent,
people were very unhappy with his conception of gravity because he didn't give a mechanical origin for it.
What causes it? He didn't say. He said he he didn't know what it was.
He didn't feel it necessary to specify it. So on the continent, particularly in France, Newton was regarded as a great mathematician.
True enough. But as a thinker about nature, extremely eccentric.
And he finished his. He finished his life.
Poor, poor man.
Tortured by gout and not in the best state, but certainly revered by in Britain as the greatest natural philosopher the world had ever seen.
But he knew his work was unfinished.
He could not complete these calculations on the stability of the universe, aspects of putting planetary motion and what have you.
This is not in any way to to diminish his achievement, but just to say science is something that really doesn't have a,
you know, you don't go into science because you're going to solve everything you you set out to achieve.
What he certainly had done was to demonstrate in the words of his great rival,
LoopNet, that mathematics was a torch that you could shine on to nature.
It was a it was. It was. It was something that the abstract complex of ideas could be used to actually shed light on the world around us.
But it was it was something that was beyond the great Newton's can, for example, to give a description of electricity and indeed of magnetism.
That's one of many examples that needed to be sorted out.
And he was unable to do that with the tools at his command.
The Newtonian project was more or less completed, not in Britain, but on the continent, in particular in France,
and this is my candidate for the most important person in that completion, Seymour Le Plus, known as the French Newton, especially in France.
And he this gentleman was a superb mathematician, a devotee of Newton,
someone who who was not part of the naysayers that thought was not was not a deep thinker about the natural world, merely a mathematician.
And he with the person who was what we would now say supervisor and bear and all the continental mathematicians.
Claire, who the newly look Raj and others.
They collectively developed the the schema that Newton developed a whole new bunch of mathematical techniques
that that developed what we would now call the classical account of of mechanics and in cosmology.
Now. One of one of The Flash's achievements, and there were many is not nearly what enough known in our country,
in my opinion, was to write up in a multi volume masterpiece called Celestial Mechanics.
The the mathematical completion of Newton's account of of of the cosmos deployed with enormous
mathematical power that the place had one of those volumes was dedicated to his great friend and sponsor,
so to speak. Napoleon, who was a was a was a tremendously beneficial force, not for French science, in particular the study of of electricity.
Now such was the activity in in France that a take home message.
It's a bit rough and ready, but it will do it. Sometime said by historians that physics, as we now know it, broadly speaking,
is pretty much a French invention that took place in the early 19th century.
Now what do we mean by that? We mean but what we now regard as something that physicists do, the topics that they study?
They started to come into a sharp focus in and around the orbit of the planet at the turn of the 19th century.
The thing started to take shape before other other topics were allied to even psychology and things like that.
But the topic that we now talk about is physics that started to take shape in France in the early 19th century.
And the first school of mathematical physics was run by this gentleman here in his mansion in actually which doesn't exist anymore.
I'm afraid it south of Paris. And he had around him.
You held court with various students and other acolytes guiding their their
development of what we what we would we would regard as a Newtonian programme.
But it was a it was a more advanced programme mathematically. And in its range, people like bio ampere and others were there.
Now the very important to me totally revered Newton, but he moved things ahead of he developed the science that Newton had.
For one thing, particularly important is this this gentleman here was godless.
You may know the story about about Napoleon asking the place why God has no place in his books, and he said, I have no need of that hypothesis.
So he was not a natural philosopher in the mould of Newton, and he believed that in a deterministic universe with the present,
if you if you knew everything about the present, you could predict the future and go back to the past.
He was a chilly rationalist who believed he had great faith in the existence of natural laws that could describe the universe.
So we have shaping a quite a modern view of science. You go to his system of the world and you see that he's talking about phenomena all over
the map in physics and believing that you could talk about interactions between particles,
a more basic view of physics from which you could derive the results of experiments.
Again, that Newtonian insistence that you have to use your theories if you can call it that
and make predictions about the natural world that will be tested against nature.
So this this this man here was enormously influential set up, as I said, a standard way of doing physics at his his country mansion.
But by the time of the Battle of Waterloo in 1815, his trajectory fell.
But nonetheless, the French science continued to to flourish, although with different emphases.
And he's the author. Authoritarian way of running physics fell by the by the wayside.
One of the things that he and his way of doing looking at the natural world couldn't do was to describe electricity and magnetism.
The the the person who we now know most effectively shone the torch of mathematics onto that subject was this gentleman here,
James Clerk Maxwell, whom I mentioned at the beginning. Now he was my favourite Scot ever.
A wonderful, wonderful person. Very lovable human being, a Cambridge trained mathematician and a natural philosopher to his fingertips.
Early in the mid-19th century when he was being educated in in natural philosophy and in mathematics,
by then, physics really had taken shape and there were people studying optics,
material structures, heat engines, gases, the sort of things that we now study in school and in early, early years of of of college.
And everybody knew that what you aspired to with mathematical laws that could describe these various parts of of nature.
What he decided to do and he wrote to one of his two of a very famous thinker at the time,
William Thomson, that he was going straight after he got his undergraduate degree at Cambridge.
He was going to attack electricity. That's the word he used in his in his letter.
Now, this showed the extent of his ambition because many of the best thinkers about the natural world working on exactly that topic,
particularly on the continent. So what Maxwell decided decided to do was to study the relationship between electricity, magnetism and the Aether.
He just saw those as the package, right? No, it wasn't something he thought was a some kind of expressions.
It was central to his thinking. The really brilliant one of the really brilliant things that he did was he didn't start thinking
about it until he really had become acquainted with the best experimental work on that subject,
which was done by his hero, Michael Faraday,
a self-taught natural philosopher working in the basement of the royal institution who done some incredibly clever and thoughtful experiments there.
In addition to all the other things he did, one of the greatest scientists ever produced on should say,
scientists, natural philosophers on on on these shores. And it was.
Faraday, who was absolutely not a mathematician,
it's sometimes said that the most advanced mathematics Faraday ever did was balancing the books of the royal institution.
He was not a mathematician, but he came up with the absolutely central idea, right?
Which was one that we will learn about more less a junior school.
The idea of a magnet, for example, sitting there and it has a magnetic field around it,
a field that the space around is not completely empty, but is in some sense activity.
Some it could be acted upon by that, by that magnet.
And from that, the accounts see the English language now gravitational field, electric field.
That idea was born in a modern sense in the mind of Michael Faraday and the
much more mathematically sophisticated Maxwell picked up on that studied it.
And in the course of several years, amongst other things, he was a very, very but he was very, very versatile thinker.
He in the early 1850s came up with this theory of the electric electrons interaction with electricity, magnetism and the Aether,
and came up as a as a quite glorious prediction that out of the equations that he he principles that he used,
the theory that he predicted that light was was an electromagnetic disturbance, something that was only later verified.
Something that I want to stress very important about Maxwell is not only did he have the completely peerless feel for the way nature works,
it's very difficult to say what that means, incidentally. But there are some thinkers who just have this field.
They have very limited describe. They somehow have a feel for the way nature is built, and Maxwell really did have that.
But he also was a very good mathematician. Scrappy.
Yes. Not particularly rigorous, yes. But he was very interested in ideas right at the forefront of mathematics.
The theory of knots when you tie knots in your shoelaces.
That's the very primitive version of the thing that mathematicians now study in a very rarified way called knot theory.
The branch of geometry called projective geometry and in a mathematical object called Kotoni.
These were frontiers of mathematics in Maxwell's time,
and he was fascinated by them because he thought that those mathematical ideas could be used to shine a torch onto nature.
And in his great treatise on electromagnetism in 1873, you'll see references to that modern mathematics.
In there he was. He had become a very, very mathematical thinker, not always popular.
His friend and admirer William Thompson, was was distasteful of of of this Maxwell in inclination to mathematics.
Thompson described mathematics as the aethereal ization of common sense and was very insistent that mathematics should be the servant of physics,
not its guide. So Maxwell stood out and was, I say again a a a a a thinker of considerable depth on this relationship between physics and mathematics.
He died in 1879, far too young.
I think he was forty eight years old, not knowing whether that theory of electric magnetism was the right one.
There were other competitors on the continent, but his field theory of electron electromagnetism held out and is now seen,
in my opinion, quite right as the crowning glory of 19th century physics.
And by the time Maxwell was on his deathbed in Cambridge in 1879, that had flourished amongst all the specialisations in science,
a branch of science called theoretical physics, particularly in Germany,
where you had for the first time people who actually paid to do not to get their hands dirty on doing experiments,
but actually to think using mathematics about the natural world and to try to develop our understanding of the most basic laws of of nature.
They started very small, but this was the this was the discipline.
Who's who's most famous representative was was to be born a generation later.
This gentleman is the person I'm talking about Albert Einstein.
Einstein always described himself as a theoretical physicist, but as I shall argue, in my opinion,
he was about 50 percent a natural philosopher as well, which I do not mean in pejorative terms at all.
He was most definitely a philosophical thinker about the natural world, as well as a peerless theoretical physicist.
Contrary to popular myth, he was a bright child.
Inquisitive alright may not have been an outstanding genius when he was a young boy, but he was certainly very, very bright.
He was headstrong, very independent minded and intensely curious.
By the time he went to college, he was absorbed very much in the Maxwell's theory, which he which he said had the most impact on him.
That's a very important point. He saw this.
He had what you might call taste as a physicist, even as a young young student, and was skipping lectures, most particularly on mathematics.
He he really thought, and this is clearly on the record for people that spoke to him at the time,
that physics was essentially a subject that you needed only very basic mathematics to do.
You don't need to go into the pure maths department and study fancy structures that the real professionals are doing.
So when Einstein had his great year, his mirabilis that is justly celebrated.
If you look at those papers about the nature of light on the news, a modification of Newton's ideas on space and time,
which was which was the special theory of relativity and in Syria, Brownian motion.
You look at those simply wonderful papers and you will see that the mathematics that Einstein uses.
All right. It might be beyond most of us, but it was. It was certainly not especially advanced by the standards of professional mathematician.
And I would I would just make the point that Laplace, if he were looking down from heaven or wherever right,
would have had absolutely no problem understanding that mathematics.
So Einstein was someone who I think it's fair to say, disdained advanced mathematics.
All this changed, and it changed quite quickly at around this picture of time.
This picture was taken in 1912. By then, he was thinking deeply and he did.
I don't think he thought in any other way.
He was thinking about gravity and the need to is he saw it to set up a new understanding of the of of Newton's theory of gravity,
which could not account for a very small number of observations, for example, on the motion of of of Mercury.
He believed on theoretical grounds that that that theory of Newton's had to be superseded.
And he came up with a new principle of principle of equivalence on which he based is based his thinking.
But he found it extremely difficult to realise that in mathematical terms, using his assumption that everything would be simple and straightforward,
it was only when he spoke to close friends in in mathematician friends in in Switzerland in 1912 that he started to see that the clue,
the clue to solving the problem of actually setting out that theory was to use something called differential geometry,
which almost certainly he'd skipped as an undergraduate.
And he then saw it was it was experience that never left him.
He saw that in that advanced mathematics was that was the key to solving his problem of finding a successor to Newton's law of gravity.
This was extraordinary. Let me just come up with an analogy.
Whenever a physicist or anybody else goes into a new field, write it.
It's because they do not really know where they're going.
They have a sense vaguely what they're trying to do, and they don't know how to solve the problem.
It's this is a tough field to be in. This is like heading out into the desert, right?
Not a not knowing how you're going to make it to the next oasis and then finding on the side the perfect survival kit.
Jeanette, just engineered for you. That's what differential geometry did for him.
It was sitting there waiting for him and was perfectly suited for the development of his new theory of gravity.
He delivered that that theory in November 1915, having nearly been given the biggest shock of his life.
Why? Because he found it an enormous struggle to complete that theory, but not just the mathematics, but the interpretation.
It was it was an extremely difficult problem, and he found it very, very difficult to finish off.
He actually came very close to the answer about the time we took this picture, but he couldn't finish it off.
And he very bravely, I think, went down to Göttingen and spoke to the mathematicians there about his incomplete theory.
And the great mathematician David Hilbert, who was then past his best as a mathematician,
saw that this Einstein was was a tyro in mathematics and was right on his tail breathing hot breath down Einstein's neck.
He was literally days behind him, and Einstein did finally beat Hilbert to the to the goal and set out that new law of gravity in November 1915.
Now what is important here is that Einstein, by his recollection, got to that goal.
Guided by mathematics that when he found the right natural structure, then he then the problems he'd been wrestling with fell away,
and this became the most important guide to him about the way to do theoretical physics.
You trust mathematics to deliver it for you.
Very important to say that excellent scholars Juergen Ren and his colleagues have looked at
Einstein's notes on this and shown that that is a very partial way of looking at that research.
In fact, he was looking at both what what we call a two pronged strategy looking at the experimental side,
the need to be consistent with Newton's law and what have you and the mathematics alternately?
Right? Whereas Einstein had a very partial recollection of this, that it was the mathematics that delivered.
Now Einstein was the only person there, I guess. Right.
But he this became his credo. Right. That mathematics would lead you to develop your your your theories.
Him having had this colossal success of producing a brand new theory of gravity, and let's face it,
you have Newton's gravity that almost everything and you produce a successor that reproduce everything Newton did and had made additional predictions.
Einstein wanted to go one step further. He wanted to take that theory of gravity and extend it so that it included electricity and magnetism.
Now, how do you do that? What most people would do is look for experimental clues.
That's the Orthodox way of doing it. Art experiment to give you a guide.
Now, Einstein's view taking that philosophy that I just spoke to you about was not to do that because there weren't any clues,
but instead to generalise the mathematics of his theory to focus on the mathematics.
This was seen as a very eccentric way of proceeding.
Let's be clear, there were very few people that took this seriously, and he was ridiculed by his frenemy, Wolfgang Paoli, for full, as he put it.
Moving over to pure mathematics of playing with what he called a completely squeezed out lemon, which was the general theory of relativity,
it was seen as a completely foolhardy activity,
and he didn't actually really go public with his philosophy of this mathematical philosophy for some years.
He finally did so. In Oxford at Rhodes House, a nice building now.
I think it's a very good wedding venue. But but theoretical physicists, but ones that know about Einstein,
think of this is where Einstein actually went public with his, his philosophy, his way of doing theoretical physics.
And this was in 1933. Saturday, the 10th of June, just after or around exam time.
This was a societal highlight of the season.
Einstein about to flee to the United States of America, his life having been made impossible by the rise of the Nazis.
He was staying in Christchurch, and this was to be the first lecture that Einstein gave purely in English, and he was seeking accessibility.
He knew that there'd be people in that audience who didn't know anything about relativity or atoms or anything like that.
So he wanted to speak in plain English, and he gave a talk there that a title, even the title is significant.
The method of theoretical physics note is not a method, not my method, the method.
And it was a method almost nobody else was using. This is fake news.
This is a mocked up picture of Einstein in Rhodes House.
All right. Pretty much what those people would have seen when Einstein was giving that lecture
thanks to people to the my friends at Rhodes House for allowing me to take that picture.
He was. He presented in a very friendly way to to his audience.
I don't think there's a single mathematical symbol in the lecture about his project
of generalising his theory of gravity so that it included electricity and magnetism.
And as I said, not focussing on experimental clues but focussing on mathematical structure, he said in the crucial line of that lecture,
and I've had it multiply translated by German scholars,
and they all say the same thing that the creative principle of the theory is in the mathematics.
He went, he was that strong. He believed that's where the creativity lied.
I say again very, very controversial and singular way of looking at developing theories.
The embarrassing thing was, and I again, I'd be very respectful when I say this.
This had got him nowhere. He was making very little progress on on on, on this subject.
One of the reasons why was because the new discovery of quantum mechanics, which was the theory of motion on the atomic scale,
showed that nature was was granular and had these discontinuities at a small distances that he's think what Einstein is thinking could not encompass.
So Einstein at that Einstein was not going to be deflected.
He was extremely stubborn person. I'm sure he'd be very proud to hear him describe like that because that's what he that's what he was.
But he knew that the new generation was coming up and the new kids on the
block who were taking over from him as the pre-eminent physicist at the time.
And these people were almost all the new, the young quantum theorists.
This is a famous picture of the Solvay conference in 1927,
where many of the pioneers of of quantum mechanics are sitting round the the person who knew his place in their businesses,
in the community, Einstein sitting front and centre.
And there are many of the pioneers of this most revolutionary theory of the 20th century, much more revolutionary to the general theory of relativity.
Why? Well, Einstein always said that the general theory of relativity was meant to be a generalisation of the Newton theory.
He did not see it as revolutionary. He said that more than once, quantum theory was was seemed to be a sharp break with the theory that went before
it in order to understand this bizarre behaviour of matter on the atomic scale.
Now, the wall I want to concentrate on is the person who was quite proud to have his picture taken here.
And this gentleman's name is Paul Dirac. And you could see him there sitting on Einstein's right shoulder now.
Dirac was very modest person. Right, right.
But Mike, he was not too proud to send a copy of that picture to a school.
He was very proud of sitting right behind his hero, someone who turned him on to theoretical physics when he was a teenager.
Now, the Iraq did not invent quantum mechanics on his own. By any means he was.
It was two other people Heisenberg and Schrodinger, who got the ball rolling.
Dirac then moved in and became one of the pioneers of the subject.
But what distinguished Dirac was his particularly mathematical approach to the subject, and that's why he interest us in this lecture.
Particularly Dirac had come as an outsider into theoretical physics.
He he'd been he'd done an undergraduate degree in engineering,
learning how to fill in tax forms, learning about aircraft, motor cars and what have you.
He then did a degree in mathematics, both in Bristol, and then he went to do a Ph.D. in Cambridge.
Right. Having having had no formal training in theoretical physics.
When he read Heisenberg's paper, he was brought into that field and then shot to the to the front line and became a peer of those people.
Durack had this knack of taking projects in quantum mechanics,
a study of this motion at the atomic level and finding and linking it to modern mathematics,
I might say not doing it on purpose, but he found himself doing modern mathematics.
Give you an example of that.
When he read that first Heisenberg paper in his parents house in Bristol, he saw something at just a line you can go home and read tonight,
whereby in Heisenberg's way of looking at motion,
you take to observe objects and be and when you multiply them together, the order of multiplication matters.
Everyone here knows it. Six times nine is the same as nine times six. That 10 times four is the same as four times 10.
In quantum mechanics, you take these two operators and the order actually matters.
Now, the horrible word direct, unfortunately coined for this, was long called mutation.
It means the order of multiplication matters. Heisenberg, by his own admission, said, Well, this is probably this looks a bit embarrassing.
We'll sort this out. Dirac saw that as key non-call mutation is a key theme of 20th century mathematics and
was absolutely central to the way that Dirac developed his his version of that subject.
What the Iraqis most famous for in his is coming up for coming up with an equation for the first fundamental particle to
be discovered the electron and combining special relativity and quantum mechanics into a description of the electron.
And it brought him into the mathematics of spinners that describe the behaviour of the electron spinners.
We discovered a few decades before in Germany describing the motion of a spinning top, and Dirac was a hit on that and hit on spin a geometry,
which is a very important theme in 20th century mathematics and in his for my money, there's hardly a personal thing.
His greatest paper, which is on how quantum theory leads naturally to a theory of a magnetic monopole.
Everyone knows that magnets come in north and south. But in quantum mechanics, it's quite natural to have a monopole, a north alone or south alone.
And in doing that, he was driven to be doing creative research in the field of topology,
which is about the properties of shapely shape things that don't change when they're stretched or not torn, but otherwise distorted.
So Dirac, those are three good examples.
There are others. He kept finding that physics took him to mathematics,
and the mathematics then guided him in his approach to physics in a wonderful phrase that he coined in nineteen seventy five.
But it's a very resonant one is that mathematics will lead you by the hand and take you to new paths, a new territory where you can take stock.
Now that's a critical phrase, ladies and gentlemen, right?
He's saying that if you trust in mathematics, it will take you to places that will enable you to make progress.
Now, one thing that even his worst enemy wouldn't say about Dirac, right?
No one would ever call him a philosopher. He was an A. philosopher. So he must have been quite surprised at the end of nineteen thirty eight.
This is what 12 years after this picture was taken to be invited in Edinburgh to give a talk on the philosophy of physics.
This is something that Dirac had no interest there at all in.
But he decided to take up the topic and gave a talk on the relationship between mathematics and physics,
which was a topic remember that Maxwell spoke about in 1870 to remember.
This is one of the great lectures of of of direct oeuvre and indeed one of the great lectures, in my opinion, public lectures in theoretical physics.
You can go and read it online. Anyone can read it. It contains no mathematical symbols at all, too far as I can recall.
This is more fake news. This is another mock up of Dirac standing outside the Royal Society of Edinburgh in central Edinburgh.
And this is what it might have looked like on the sick.
The February 1939 storm clouds gathering very close to the beginning of World War Two and Dirac gave a lecture that lasted an hour and.
He reviewed right from the most basic level, the cleaner in that organisation could have listened to the opening of that lecture.
Right. He talked about how the universe or the world around us has a mathematical quality
that is not obvious to people until they begin to study it right at that level.
And then he goes right up to the application of quantum theory relativity, no using no mathematics,
but thinking about how we could develop or how he and his peers could develop their thinking about the subject.
It's most famous amongst his amongst many people today,
being the place where he coined his principle of mathematical beauty, which is the statement of extremely bold.
Some people would say overly bold and perhaps even foolish,
statement that physics advances fundamental advances from one theory to the next or the next.
Through increasing mathematical beauty, you start off with something that is relatively ugly,
and the theories gradually become more mathematically beautiful.
Now, even if you're not a professionally trained philosopher, you don't need to see there's a big hole in this argument, right?
I mean, without being disrespectful to direct. How do you define what beauty is?
People have different views on this. Dirac had no time for that at all.
He was. He was concerned in mathematics. You really don't do know. But what beauty is?
And he stuck by his, his guns. And it was, as he said, in later life, almost a religion.
He believed that mathematics beautiful mathematics would take you to to the correctly to the next stage.
Now these are three statements that I just want you to look at and their statements that where I wish people,
people of direct stature could all write like this with with, in my opinion, truths of great beauty themselves that are very, very simple.
Look at that. Look at this for a statement. The mathematician plays a game in which he himself, sorry about the sexism,
invents the rules while the physicist plays the game in which the rules are provided by nature.
That's it, in a nutshell, that's why physics and mathematics are different.
They are not the same thing. Very important to realise that this is even more germane.
In my opinion, the rules that the mathematician finds interesting are the same as those that nature has chosen.
Now, that's not obvious. Why should it be right that that.
But mathematician pure mathematicians looking at these patterns of ideas are finding themselves
on the same territory that state of the art theoretical physicists are working on as well.
That is not at all obvious, and in this is this is almost wild.
This lecture at certain points. I mean, Dirac, a very reserved, very quiet spoken person.
He wild in the sense of he really does talk in very bold sense about the future of the subject makes one observation, which I love,
which he was saying that philosophically, the most interesting thing about theoretical physics was not quantum theory of relativity.
It was cosmology. I wasn't expecting to see that. Now, today that looks extremely modern, right?
That's what he says in passing in this lecture. It's an amazing lecture. And he also says possibly mathematics and physics will one day unify.
Now, that's also a very bold, bold statement, but one it is coming back into fashion may not be correct,
but it's something that's starting to look very modern, as will will.
We'll talk about now that unity certainly didn't come after the Second World War,
as Freeman Dyson has explained in his many talks with me at the Institute for Advanced Study.
Freeman began life as a as a I think it's fair to say, a mathematical prodigy,
a number theorist trained at the feet of G.H. Hardy at Cambridge University.
And by Freeman's account, although he was a totally gifted mathematician,
he found that the mathematics he was doing at Cambridge was not the stuff that was really cutting edge at the time he was there,
which is in the Second World War. And just and shortly afterwards, the pace setters were known as Bourbaki,
which is an organisation of French mathematicians and anonymously grouped up under that name of Bach.
You first met in 1934 with the ambition to rewrite the whole of mathematics set out
of all of those axioms to make plain the unity and logical structure of mathematics.
The one thing it wasn't interested in was applications to the real world had no interest to them at all.
Philip Freeman told me that he heard the big, bulky agenda at Cambridge referred to as a French disease.
I didn't know that was a synonym for syphilis, but it is. They certainly had no time for Bourbaki at all.
Freeman chose to concentrate on problem solving,
and that's what led him to his triumph in in in in setting out a quantum theory of electricity and light.
But his observation that he ran home to me many times is that at that time after the war,
there was a divorce between physics and mathematics, where the two went their own separate ways.
Something that was confirmed by a Princeton colleague, Karen Allen Beck, winner of the Abel Prise this year,
who had exactly the same experience or very similar experience 20 years later.
And where that where where the gradual coming together of physics and mathematics was reviewed was viewed as something to be as deeply suspicious.
Now, I'm afraid I don't have to go through that far too quickly.
The second part of my talk here, but you'll forgive me if I go quite a gallop here where I come in as a spear carrier was in this era 1970, right?
And this is when I became a student. Now this, of course, is the sexier side of that time.
In fact, it was pretty a miserable time.
As I remember these wars in Southeast Asia and the Middle East, a UK economy plummeting, known then as a sick man of Europe.
Nothing much changed there then. But for me and for far more eminent people than I could ever aspire to be.
This was a time of huge excitement and that the reason for that was because of the activity in understanding the subatomic world,
the subject of particle physics and also on the macroscopic scale Astro Astrophysics.
And I must admit that not being particularly mathematically inclined myself,
I was very suspicious to see these incursions of mathematics made at that time
when it seemed to me that the real action was in accounting for experiments.
Now, with the benefit of hindsight,
that the critical thing that was that was starting to take shape now then and I urge this is the take home message tonight,
because it's still true today, is that the theories that that if you want to produce a fundamental theory,
then as Dirac had alluded to in that lecture, those theories must be consistent with the special theory of relativity theory
of motion that Einstein and others had put out in 1995 and quantum mechanics.
Nobody has ever found not done an experiment that has contradicted those two theories.
So it's an incredibly strong empirical base.
And what physicists have done much of what they have done in basic physics has been jamming those two theories together.
Now it's worse. It really is worth stressing that that is extremely difficult to do.
Einstein thought that he was not alone.
It was impossible to take theories that look so different with such different mathematics and find a way of jamming them together,
such that they produce just about a theory that works. And it did because it produced the first theory of what we call quantum electrodynamics,
which is possibly the best verified physical theory in in nature.
And it also triumphed in astrophysics as well.
I remember when I was a twenty five year old graduate student going to a lecture, not here, of course, but in Oxford,
given by the great Dennis Sharma when he was talking about the way Stephen Hawking had had shocked his community
by putting forward this idea that you could actually bring in quantum mechanics to the study of black holes,
which is a classical subject, and come up with this physical notion that black holes ain't so black but actually continually emit radiation.
Something that was, I saw people saying, is completely nuts at the time.
Now it's part of the textbooks. But he did it with a tour de force calculation that did not combine the rigorous rigorously in one fell swoop,
but brilliantly brought the theories together in a way that made the calculation extremely robust.
I still remember that talk of still remember Dennis's admiration and excitement for the people in the audience with whom I was one.
What we were seeing was a different jamming together of quantum mechanics and relativity,
the theory at the gauge theories of the forces that govern the atom.
Now this theory was we now call quantum field theory governing the basic interactions all together the nucleus and the atom.
I say again, it's a it's amazing that the theory even exists, right?
But it does. Right. And like very careful study of the symmetries we leading theoretical physicists found what we
now call the standard model that has had nothing but takes for nature from nature for decades.
So this is an extremely powerful way of doing theoretical theoretical physics.
The I want to illustrate one particular meeting that really brought in my opinion, and it's central to my book here.
That was the central moment, I think,
when a central moment where mathematics and physics really did start to come together and the divorce was became a thing of the past.
It was the meeting between the great mathematician for many years here at Oxford, Michael Attia and below Edward Witten, the American mathematician.
Attia was a great geometry without question one of the great mathematicians of the 20th century.
And he had become interested in these gauge theories that govern atomic atomic physics that, as I described.
And he and in the early 1970s, he and his he described to me, he began to morph into a quasar.
A physicist and the person who he was most close to was Edward Witten.
Many years his junior, a complete singularity, an outsider like Dirac,
an undergraduate in modern languages and history, and briefly studied economics.
And by by the time he switched, I think it was the twenty three he switched into into theoretical physics.
And they met in the fall of nineteen seventy six in the in an office in the Massachusetts Institute of Technology.
Microsoft has described that meeting to me a dozen times. He was there.
There was about twelve people sitting around a small office in that there,
and he said one person knew more than everybody else put together, and that was the recently graduated Ph.D., Edward Witten.
Together they were.
Edward was had begun work just but not just. I shouldn't use that word.
We were looking at data closely looking at what one could derive from it.
But with encouraged by a TIA and with an enormous mathematical ability of his own,
they started to bring those two subjects together and analysed ever more closely the mathematical structures of those gauge theories.
I can't resist it is a tribute to Michael, who died a few months ago, his favourite image that it was like having two intersecting tunnels,
one being dug by gauge theories, the other way being dug by mathematicians.
They were oblivious of what each others were doing, and at a certain point they intersected and found they were working on each other.
Subjects that intersection, ladies and gentlemen, took place at the Mathematical Institute of the University of Oxford.
That is when I would say those two things intersected. And Frank Yang, who also has a claim for being part of this, that is where it took place.
And in 1977, seven A and several other very eminent scientists,
West were working on that common territory between pure mathematics and theoretical physics.
Stunning monopoles things called instant ones, which are events in space and time and a plot.
Beginning to apply it to other subjects as well. Luckily, we now know the physics of solids I saw to my dismay,
I have to say because I was I wasn't anything like a sophisticated mathematically as I what needed to be.
I thought this would go away. I thought that nature couldn't be that complicated, right?
But it has stayed the subject. Mathematical complexion has endured,
and particularly through the next generalisation of quantum mechanics and relativity coming together in something called string theory,
which is witness, said wisely, My view is the only interesting idea that is a development of both general relativity and the gauge theories of atoms.
String theory began to take shape properly in in the summer of 1984,
when we had a viable account of of not just gravity, but the the the three fundamental forces that hold together atoms.
And for the first time ever, it gave a reason why gravity must exist, Witten said.
To me, that is the main reason why he was interested in string theory, and it became, this is a key point for us.
This became a Klondike for mathematicians. Even today as we speak.
I think it fair to say that if that string theory has above all,
been of enormous benefit to two mathematicians and there been other developments of two by physicists,
the something called end theory, which is a generalisation I further development of string theory and two years later,
one male to same is linking between string theory and gauge theory.
It appears to people working in this field that these mathematical connexions demonstrate that physics is on the right track.
It is. It would be idle to pretend that there aren't people who oppose that view, and they are calling card.
Mainly the biggest one is that string theory has yet to make a direct connexion with experiment by
by virtue of a new prediction that can be that could be tested in laboratory that cannot be denied.
That hasn't happened, but string theory libs it has applied, which most naturally applies at very, very high energies.
So it's going to be very difficult to test, which is not to say it's impossible, but nobody has actually contradicted it.
But there are without doubt there's reasons to be concerned about about the interpretation of it.
And I might say as well it has its connexion with ordinary space.
The one that we live in is also problematic. That cannot be denied.
But what I would like to say is that the application of string theory by it is such
that people who the top the best experts see that still as the most promising theory,
they'll all people who disagree with that. It's a free country. They could work on those other topics.
But I think it's fair and reasonable to say that these other competitors a minority sport.
They may disagree. Of course, they're free to do that.
So let me just look, if I may, to the to the to what we now call the doubly unreasonable effectiveness.
Mathematics, of course, is incredibly on a credibly effective in physics.
But physics now we know more than ever is extremely effective in mathematics.
And I've given several examples of here which are on and I'm not going because of time.
I'm not going to go through them, forgive me, but I covered them in detail in my book.
There is absolutely no doubt that that physics and mathematics are joined at the hip in fundamental physics today.
So what do I what's my view of looking to the future?
Well, my view is that there is good reason to believe that we are on the right track,
guided by quantum mechanics and special relativity, but that we definitely need to be patient.
My generation, if I can call it that was used to the results coming every few years.
Major new major new results that were rich pickings from from from Mother Nature.
Today they come much more slowly. And I think we're going to have to be in physics.
We're going to have to get used to the mathematical timescale where things are done much more, much more slowly.
This is these are my predictions all silly, right? Well, they're not intended to be silly, but they probably will.
But I want to show why. I think this is how I think this what we're aspiring to here.
First of all, I predict that the quantum field theories that are now used to describe atoms which
are extremely difficult to calculate that is ripe for a completely new understanding.
Quantum field theory will be reformulated in a way where these horrendous calculations will be much simplified.
There are already evidence to get us started on that. I predict that will happen.
I also predict that string theory. Yes, I'm sure there are things that need to be radically revised in the theory.
I'm sure that maybe even revolutions around the corner.
But I believe that this is not wasted effort that this is this will be seen as a perfectly worthy step on the on on the path to the future.
They understand the understanding that the senior gave us of gravity in terms of a equivalence between a string theory and gauge theory,
I believe will be seen to be that B b be applicable to the real world.
Most important, this is a particularly big thing that I can't explain in detail,
but I talk about in my book and you can read elsewhere space time developed most famously by Einstein that is going to be superseded.
I don't know of any leading theoretical physicist now who believes that the Einstein in view of time is is it is complete,
it is going to be superseded and we're being guided on that track.
I say again by quantum mechanics and special relativity,
and we're seeing I think this is very heartening a trend towards unity, particle physics, cosmology,
condensed matter that solids information theory where there is a coming together of all areas, all across fundamental physics,
with mathematics being the glue between them and mathematics providing a guide to help us understand those relationships.
So I think we're going to have to get used to something that surprising new insights
from experiments and observations will start to become few and far between,
right? I know it's a bit depressing, but they're not going to come every few years.
Nature is getting harder and harder to question constructively about these things.
So I think physics tells us that it's going to be a long, a longer term thing that it seemed when I was starting out during those slow decades.
And I think we're in one now.
We will physicists will have to focus, I think, on mathematics and and do, as Einstein said and Dirac said and others are separate those two.
I think the point is that by looking at the mathematical content,
you will be able to advance the subject so that it's ready for the next generation of of experimenters.
Now it seems a harsh thing to say, but I believe we're going to have to take this.
While it's difficult to get clues out of nature, we can console ourselves that nature doesn't speak to us just in terms of observations.
It also speaks to us in numbers. Thank you very much.