The following are notes I took while reading the lectures.
1 The Law of Gravitation, an example of Physical Law
But people had seen in telescopes Jupiter’s satellites going around Jupiter, and it looked like a little solar system, as if the satellites were attracted to Jupiter. The moon is attracted to the earth and goes round the earth and is attracted in the same way. It looks as though everything is attracted to everything else, and so the next statement was to generalize this and to say that every object attracts every object. If so, the earth must be pulling on the moon, just as the sun pulls on the planet. But it is known that the earth is pulling on things – because you are all sitting tightly on your seats in spite of your desire to float into the air. The pull for objects on the earth was well known in the phenomena of gravitation, and it was Newton’s idea that maybe the gravitation that held the moon in orbit was the same gravitation that pulled the object towards the earth.
It is easy to figure out how far the moon falls in one second, because you know the size of the orbit, you know the moon takes a month to go around the earth, and if you figure out how far it goes in one second you can figure out how far the circle of the moon’s orbit has fallen below the straight line that it would have been in if it did not go the way it does go. This distance is one twentieth of an inch.
The moon is sixty times as far away from the earth’s centre as we are; we are 4,000 miles away from the centre, and the moon is 240,000 miles away from the centre, so if the law of inverse square is right, an object at the earth’s surface should fall in one second by 1/20 inch x 3,600 (the square of 60) because the force in getting out there to the moon, has been weakened by 60 x 60 by the inverse square law. 1/20 inch x 3,600 is about 16 feet, and it was known already from Galileo’s measurements that things fall in one second on the earth’s surface by 16 feet. So this meant that Newton was on the right track, there was no going back now, because a new fact which was completely independent previously, the period of the moon’s orbit and its distance from the earth, was connected to another fact, how long it takes something to fall in one second at the earth’s surface. This was a dramatic test that everything is all right.
As science developed and measurements were made more accurate, the tests of Newton’s Law became more stringent, and the first careful tests involved the moons of Jupiter. By accurate observations of the way they went around over long periods of time one could check that everything was according to Newton, and it turned out to be not the case. The moons of Jupiter appeared to get sometimes eight minutes ahead of time and sometimes eight minutes behind time, where the time is the calculated value according to Newton’s Laws. It was noticed that they were ahead of schedule when Jupiter was close to the earth and behind schedule when it was far away, a rather odd circumstance. Mr Roemer, having confidence in the Law of Gravitation, came to the interesting conclusion that it takes light some time to travel from the moons of Jupiter to the earth, and what we are looking at when we see the moons is not how they are now but how they were the time ago it took the light to get here. When Jupiter is near us it takes less time for the light to come, and when Jupiter is farther from us it takes longer time, so Roemer had to correct the observations for the differences in time and by the fact that they were this much early or that much late. In this way he was able to determine the velocity of light. This was the first demonstration that light was not an instantaneously propagating material.
Another problem came up – the planets should not really go in ellipses, because according to Newton’s Laws they are not only attracted by the sun but also they pull on each other a little – only a little, but that little is something, and will alter the motion a little bit. Jupiter, Saturn and Uranus were big planets that were known, and calculations were made about how slightly different from the perfect ellipses of Kepler the planets ought to be going by the pull of each on the others. And at the end of the calculations and observations it was noticed that Jupiter and Saturn went according to the calculations, but that Uranus was doing something funny. Another opportunity for Newton’s Laws to be found wanting; but take courage! Two men, Adams and Leverrier, who made these calculations independently and at almost exactly the same time, proposed that the motions of Uranus were due to an unseen planet, and they wrote letters to their respective observatories telling them – ‘Turn your telescope and look there and you will find a planet’. ‘How absurd,’ said one of the observatories, ‘some guy sitting with pieces of paper and pencils can tell us where to look to find some new planet.’ The other observatory was more … well, the administration was different, and they found Neptune!
But atypically the knowledge of the Laws of Gravitation has relatively few practical applications compared with the other laws of physics. This is one case where I have picked an atypical example. It is impossible, by the way, by picking one of anything to pick one that is not atypical in some sense. That is the wonder of the world. The only applications of the knowledge of the law that I can think of are in geophysical prospecting, in predicting the tides, and nowadays, more modernly, in working out the motions of the satellites and planet probes that we send up, and so on; and finally, also modernly, to calculate the predictions of the planets’ positions, which have great utility for astrologists who publish their predictions in horoscopes in the magazines. It is a strange world we live in – that all the new advances in understanding are used only to continue the nonsense which has existed for 2,000 years.
Can we make a direct test and not just wait to see whether the planets attract each other? A direct test was made by Cavendish on equipment which you see indicated in figure 6. The idea was to hang by a very very fine quartz fibre a rod with two balls, and then put two large lead balls in the positions indicated next to it on the side.
Because of the attraction of the balls there would be a slight twist to the fibre, and the gravitational force between ordinary things is very very tiny indeed. It was possible to measure the force between the two balls. Cavendish called his experiment ‘weighing the earth’. With pedantic and careful teaching today we would not let our students say that; we would have to say ‘measuring the mass of the earth’. By a direct experiment Cavendish was able to measure the force, the two masses and the distance, and thus determine the gravitational constant, G. Indirectly this experiment was the first determination of how heavy or massive is the ball on which we stand. It is an amazing achievement to find that out, and I think that is why Cavendish named his experiment ‘weighing the earth’, instead of ‘determining the constant in the gravitational equation’. He, incidentally, was weighing the sun and everything else at the same time, because the pull of the sun is known in the same manner.
Question : What is the ratio of the gravitational force to the electrical force? That is illustrated in figure 7.
The ratio of the gravitational attraction to electrical repulsion is given by a number with 42 digits tailing off. Now therein lies a very deep mystery. Where could such a tremendous number come from? If you ever had a theory from which both of these things are to come, how could they come in such disproportion? What equation has a solution which has for two kinds of forces an attraction and repulsion with that fantastic ratio? People have looked for such a large ratio in other places. They hope, for example, that there is another large number, and if you want a large number why not take the diameter of the Universe to the diameter of a proton – amazingly enough it also is a number with 42 digits. And so an interesting proposal is made that this ratio is the same as the ratio of the size of the Universe to the diameter of a proton. But the Universe is expanding with time and that means that the gravitational constant is changing with time, and although that is a possibility there is no evidence to indicate that it is a fact. There are several partial indications that the gravitational constant has not changed in that way. So this tremendous number remains a mystery.
To finish about the theory of gravitation, I must say two more things. One is that Einstein had to modify the Laws of Gravitation in accordance with his principles of relativity. The first of the principles was that ‘x’ cannot occur instantaneously, while Newton’s theory said that the force was instantaneous. He had to modify Newton’s laws. They have very small effects, these modifications. One of them is that all masses fall, light has energy and energy is equivalent to mass. So light falls and it means that light going near the sun is deflected; it is. Also the force of gravitation is slightly modified in Einstein’s theory, so that the law has changed very very slightly, and it is just the right amount to account for the slight discrepancy that was found in the movement of Mercury. Finally, in connection with the laws of physics on a small scale, we have found that the behaviour of matter on a small scale obeys laws very different from things on a large scale. So the question is, how does gravity look on a small scale? That is called the Quantum Theory of Gravity. There is no Quantum Theory of Gravity today. People have not succeeded completely in making a theory which is consistent with the uncertainty principles and the quantum mechanical principles.
In this lecture I would like to emphasize, just at the end, some characteristics that gravity has in common with the other laws that we mentioned as we passed along. First, it is mathematical in its expression; the others are that way too. Second, it is not exact; Einstein had to modify it, and we know it is not quite right yet, because we have still to put the quantum theory in. That is the same with all our other laws – they are not exact. There is always an edge of mystery, always a place where we have some fiddling around to do yet. This may or may not be a property of Nature, but it certainly is common to all the laws as we know them today. It may be only a lack of knowledge.
2 The Relation of Mathematics to Physics
The next question is whether, when trying to guess a new law, we should use the seat-of-the-pants feeling and philosophical principles – ‘I don’t like the minimum principle’, or ‘I do like the minimum principle’, ‘I don’t like action at a distance’, or ‘I do like action at a distance’. To what extent do models help? It is interesting that very often models do help, and most physics teachers try to teach how to use models and to get a good physical feel for how things are going to work out. But it always turns out that the greatest discoveries abstract away from the model and the model never does any good.
Maxwell’s discovery of electrodynamics was first made with a lot of imaginary wheels and idlers in space. But when you get rid of all the idlers and things in space the thing is O.K. Dirac discovered the correct laws for relativity quantum mechanics simply by guessing the equation. The method of guessing the equation seems to be a pretty effective way of guessing new laws. This shows again that mathematics is a deep way of expressing nature, and any attempt to express nature in philosophical principles, or in seat-of-the-pants mechanical feelings, is not an efficient way.
To summarize, I would use the words of Jeans, who said that ‘the Great Architect seems to be a mathematician‘. To those who do not know mathematics it is difficult to get across a real feeling as to the beauty, the deepest beauty, of nature. C. P. Snow talked about two cultures. I really think that those two cultures separate people who have and people who have not had this experience of understanding mathematics well enough to appreciate nature once.
3 The Great Conservation Principles
If I have something moving, for example a ball rolling along at a constant height, then it will stop on account of friction. What happened to the kinetic energy of the ball ? The answer is that the energy of the motion of the ball has gone into the energy of the jiggling of the atoms in the floor and in the ball. The world that we see on a large scale looks like a nice round ball when we polish it, but it is really quite complicated when looked at on a little scale; billions of tiny atoms, with all kinds of irregular shapes. It is like a very rough boulder when looked at finely enough, because it is made out of these little balls. The floor is the same, a bumpy business made out of balls. When you roll this monster boulder over the magnified floor you can see that the little atoms are going to go snap-jiggle, snap-jiggle. After the thing has rolled across, the ones that are left behind are still shaking a little from the pushing and snapping that they went through; so there is left in the floor a jiggling motion, or thermal energy.
Einstein understood gravitation as being generated by energy. Energy and mass are equivalent, and so Newton’s interpretation that the mass is what produces gravity has been modified to the statement that the energy produces the gravity.
4 Symmetry in Physical Law
As time went on, new laws were discovered after Newton, among them the laws of electricity discovered by Maxwell. One of the consequences of the laws of electricity was that there should be waves, electromagnetic waves – light is an example – which should go at 186,000 miles a second, flat. I mean by that 186,000 miles a second, come what may. So then it was easy to tell where rest was, because the law that light goes at 186,000 miles a second is certainly not (at first sight) one which will permit one to move without some effect. It is evident, is it not, that if you are in a space ship going at 100,000 miles a second in some direction, while I am standing still, and I shoot a light beam at 186,000 miles a second through a little hole in your ship, then, as it goes through your ship, since you are going at 100,000 miles per second and the light is going at 186,000, the light is only going to look to you as if it is passing at 86,000 miles a second. But it turns out that if you do this experiment it looks to you as if it is going at 186,000 miles a second past you, and to me as if it is going 186,000 miles a second past me!
The facts of nature are not so easy to understand, and the fact of the experiment was so obviously counter to commonsense, that there are some people who still do not believe the result! But time after time experiments indicated that the speed is 186,000 miles a second no matter how fast you are moving. The question now is how that could be. Einstein realized, and Poincare too, that the only possible way in which a person moving and a person standing still could measure the speed to be the same was that their sense of time and their sense of space are not the same, that the clocks inside the space ship are ticking at a different speed from those on the ground, and so forth. You might say, ‘Ah, but if the clock is ticking and I look at the clock in the space ship, then I can see that it is going slow’. No, your brain is going slow too! So by making sure that everything went just so inside the space ship, it was possible to cook up a system by which in the space ship it would look like 186,000 space-ship miles per space-ship second, whereas here it would look like 186,000 my miles per my second. That is a very ingenious thing to be able to do, and it turns out, remarkably enough, to be possible.
It is possible to tell that the earth is rotating by a pendulum or by a gyroscope, and you are probably aware that various observatories and museums have so-called Foucault pendulums that prove that the earth is rotating, without looking at the stars. It is possible to tell that we are going around at a uniform angular velocity on the earth without looking outside, because the laws of physics are not unchanged by such a motion. Many people have proposed that really the earth is rotating relative to the galaxies, and that if we were to turn the galaxies too it would not make any difference. Well, I do not know what would happen if you were to turn the whole universe, and we have at the moment no way to tell. Nor, at the moment, do we have any theory which describes the influence of a galaxy on things here so that it comes out of this theory – in a straightforward way, and not by cheating or forcing – that the inertia for rotation, the effect of rotation, the fact that a spinning bucket of water has a concave surface, is the result of a force from the objects around. It is not known whether this is true. That it should be the case is known as Mach’s principle, but that it is the case has not yet been demonstrated. The more direct experimental question is whether, if we are rotating at a uniform velocity relative to the nebulae, we see any effect. The answer is yes. If we are moving in a space ship at a uniform velocity in a straight line relative to the nebulae, do we see any effect? The answer is no. Two different things. We cannot say that all motion is relative. That is not the content of relativity. Relativity says that uniform velocity in a straight line relative to the nebulae is undetectable.
Suppose that we were in telephone conversation with a Martian, or an Arcturian, and we wished to describe things on earth to him. First of all, how is he going to understand our words? That question has been studied intensively by Professor Morrison at Cornell, and he has pointed out that one way would be to start by saying ‘tick, one: tick, tick, two: tick, tick, tick, three:’ and so on. Pretty soon the guy would catch on to the numbers. Once he understood your number system, you could write a whole sequence of numbers that represent the weights, the proportional weights, of the different atoms in succession, and then say ‘hydrogen, 1`008’, then deuterium, helium, and so on. After he had sat down with these numbers for a while he would discover that the mathematical ratios were the same as the ratios for the weights of the elements, and that therefore those names must be the names of the elements. Gradually in this way you could build up a common language. Now comes the problem. Suppose, after you get familiar with him, he says, ‘You fellows, you’re very nice. I’d like to know what you look like’. You start, ‘We’re about six feet tall’, and he says, ‘Six feet – how big is a foot?’ That is very easy: ‘Six feet tall is seventeen thousand million hydrogen atoms high’. That is not a joke – it is a possible way of describing six feet to someone who has no measure – assuming that we cannot send him any samples, nor can we both look at the same objects. If we wish to tell him how big we are we can do it. That is because the laws of physics are not unchanged under a scale change, so we can use that fact to determine the scale. We can go on describing ourselves – we are six feet tall, and we are so-and-so bilateral on the outside, and we look like this, and there are these prongs sticking out, etc.
5 The Distinction of Past and Future
So when things have become all of the same temperature, there is no more energy available to do anything. The principle of irreversibility is that if things are at different temperatures and are left to themselves, as time goes on they become more and more at the same temperature, and the availability of energy is perpetually decreasing. This is another name for what is called the entropy law, which says the entropy is always increasing. But never mind the words; stated the other way, the availability of energy is always decreasing. And that is a characteristic of the world, in the sense that it is due to the chaos of molecular irregular motions. Things of different temperature, if left to themselves, tend to become of the same temperature. If you have two things at the same temperature, like water on an ordinary stove without a fire under it, the water is not going to freeze and the stove get hot. But if you have a hot stove with ice, it goes the other way. So the one-way-ness is always to the loss of the availability of energy.
In fact, although we have been talking in these lectures about the fundaments of the physical laws, I must say immediately that one does not, by knowing all the fundamental laws as we know them today, immediately obtain an understanding of anything much. It takes a while, and even then it is only partial. Nature, as a matter of fact, seems to be so designed that the most important things in the real world appear to be a kind of complicated accidental result of a lot of laws.
6 Probability and Uncertainty — the Quantum Mechanical view of Nature
There was a time when the newspapers said that only twelve men understood the theory of relativity. I do not believe there ever was such a time. There might have been a time when only one man did, because he was the only guy who caught on, before he wrote his paper. But after people read the paper a lot of people understood the theory of relativity in some way or other, certainly more than twelve. On the other hand, I think I can safely say that nobody understands quantum mechanics. So do not take the lecture too seriously, feeling that you really have to understand in terms of some model what I am going to describe, but just relax and enjoy it. I am going to tell you what nature behaves like. If you will simply admit that maybe she does behave like this, you will find her a delightful, entrancing thing. Do not keep saying to yourself, if you can possibly avoid it, ‘But how can it be like that?’ because you will get ‘down the drain’, into a blind alley from which nobody has yet escaped. Nobody knows how it can be like that.
There is always the possibility of proving any definite theory wrong; but notice that we can never prove it right. Suppose that you invent a good guess, calculate the consequences, and discover every time that the consequences you have calculated agree with experiment. The theory is then right? No, it is simply not proved wrong. In the future you could compute a wider range of consequences, there could be a wider range of experiments, and you might then discover that the thing is wrong. That is why laws like Newton’s laws for the motion of planets last such a long time. He guessed the law of gravitation, calculated all kinds of consequences for the system and so on, compared them with experiment – and it took several hundred years before the slight error of the motion of Mercury was observed. During all that time the theory had not been proved wrong, and could be taken temporarily to be right. But it could never be proved right, because tomorrow’s experiment might succeed in proving wrong what you thought was right. We never are definitely right, we can only be sure we are wrong. However, it is rather remarkable how we can have some ideas which will last so long. One of the ways of stopping science would be only to do experiments in the region where you know the law. But experimenters search most diligently, and with the greatest effort, in exactly those places where it seems most likely that we can prove our theories wrong. In other words we are trying to prove ourselves wrong as quickly as possible, because only in that way can we find progress.
I want to discuss now the art of guessing nature’s laws. It is an art. How is it done ? One way you might suggest is to look at history to see how the other guys did it. So we look at history.
- We must start with Newton. He had a situation where he had incomplete knowledge, and he was able to guess the laws by putting together ideas which were all relatively close to experiment; there was not a great distance between the observations and the tests. That was the first way, but today it does not work so well.
- The next guy who did something great was Maxwell, who obtained the laws of electricity and magnetism. What he did was this. He put together all the laws of electricity, due to Faraday and other people who came before him, and he looked at them and realized that they were mathematically inconsistent. In order to straighten it out he had to add one term to an equation. He did this by inventing for himself a model of idler wheels and gears and so on in space. He found what the new law was – but nobody paid much attention because they did not believe in the idler wheels. We do not believe in the idler wheels today, but the equations that he obtained were correct. So the logic may be wrong but the answer right.
- In the case of relativity the discovery was completely different. There was an accumulation of paradoxes; the known laws gave inconsistent results. This was a new kind of thinking, a thinking in terms of discussing the possible symmetries of laws. It was especially difficult, because for the first time it was realized how long something like Newton’s laws could seem right, and still ultimately be wrong. Also it was difficult to accept that ordinary ideas of time and space, which seemed so instinctive, could be wrong.
- Quantum mechanics was discovered in two independent ways – which is a lesson. There again, and even more so, an enormous number of paradoxes were discovered experimentally, things that absolutely could not be explained in any way by what was known. It was not that the knowledge was incomplete, but that the knowledge was too complete. Your prediction was that this should happen – it did not. The two different routes were one by Schrodinger, who guessed the equation, the other by Heisenberg, who argued that you must analyze what is measurable. These two different philosophical methods led to the same discovery in the end.
- More recently, the discovery of the laws of the weak decay I spoke of, when a neutron disintegrates into a proton, an electron and an anti-neutrino – which are still only partly known – add up to a somewhat different situation. This time it was a case of incomplete knowledge, and only the equation was guessed. The special difficulty this time was that the experiments were all wrong. How can you guess the right answer if, when you calculate the result, it disagrees with experiment? You need courage to say the experiments must be wrong. I will explain where that courage comes from later.
Today we have no paradoxes – maybe. We have this infinity that comes in when we put all the laws together, but the people sweeping the dirt under the rug are so clever that one sometimes thinks this is not a serious paradox. Again, the fact that we have found all these particles does not tell us anything except that our knowledge is incomplete. I am sure that history does not repeat itself in physics, as you can tell from looking at the examples I have given. The reason is this. Any schemes – such as ‘think of symmetry laws’, or ‘put the information in mathematical form’, or ‘guess equations’ – are known to everybody now, and they are all tried all the time. When you are stuck, the answer cannot be one of these, because you will have tried these right away. There must be another way next time. Each time we get into this log-jam of too much trouble, too many problems, it is because the methods that we are using are just like the ones we have used before. The next scheme, the new discovery, is going to be made in a completely different way. So history does not help us much.
We must, and we should, and we always do, extend as far as we can beyond what we already know, beyond those ideas that we have already obtained. Dangerous? Yes. Uncertain ? Yes. But it is the only way to make progress. Although it is uncertain, it is necessary to make science useful. Science is only useful if it tells you about some experiment that has not been done; it is no good if it only tells you what just went on. It is necessary to extend the ideas beyond where they have been tested. For example, in the law of gravitation, which was developed to understand the motion of planets, it would have been no use if Newton had simply said, ‘I now understand the planets’, and had not felt able to try to compare it with the earth’s pull on the moon, and for later men to say ‘Maybe what holds the galaxies together is gravitation’. We must try that. You could say, ‘When you get to the size of the galaxies, since you know nothing about it, anything can happen’. I know, but there is no science in accepting this type of limitation. There is no ultimate understanding of the galaxies. On the other hand, if you assume that the entire behaviour is due only to known laws, this assumption is very limited and definite and easily broken by experiment. What we are looking for is just such hypotheses, very definite and easy to compare with experiment. The fact is that the way the galaxies behave so far does not seem to be against the proposition.
I can give you another example, even more interesting and important. Probably the most powerful single assumption that contributes most to the progress of biology is the assumption that everything animals do the atoms can do, that the things that are seen in the biological world are the results of the behaviour of physical and chemical phenomena, with no ‘extra something’. You could always say, ‘When you come to living things, anything can happen’. If you accept that you will never understand living things. It is very hard to believe that the wiggling of the tentacle of the octopus is nothing but some fooling around of atoms according to the known physical laws. But when it is investigated with this hypothesis one is able to make guesses quite accurately about how it works. In this way one makes great progress in understanding. So far the tentacle has not been cut off – it has not been found that this idea is wrong. It is not unscientific to make a guess, although many people who are not in science think it is.
Some years ago I had a conversation with a layman about flying saucers – because I am scientific I know all about flying saucers! I said ‘I don’t think there are flying saucers’. So my antagonist said, ‘Is it impossible that there are flying saucers ? Can you prove that it’s impossible?’ ‘No’, I said, ‘I can’t prove it’s impossible. It’s just very unlikely’. At that he said, ‘You are very unscientific. If you can’t prove it impossible then how can you say that it’s unlikely?’ But that is the way that is scientific. It is scientific only to say what is more likely and what less likely, and not to be proving all the time the possible and impossible. To define what I mean, I might have said to him, ‘Listen, I mean that from my knowledge of the world that I see around me, I think that it is much more likely that the reports of flying saucers are the results of the known irrational characteristics of terrestrial intelligence than of the unknown rational efforts of extra-terrestrial intelligence’. It is just more likely, that is all. It is a good guess. And we always try to guess the most likely explanation, keeping in the back of the mind the fact that if it does not work we must discuss the other possibilities.
For instance, Newton’s ideas about space and time agreed with experiment very well, but in order to get the correct motion of the orbit of Mercury, which was a tiny, tiny difference, the difference in the character of the theory needed was enormous. The reason is that Newton’s laws were so simple and so perfect, and they produced definite results. In order to get something that would produce a slightly different result it had to be completely different. In stating a new law you cannot make imperfections on a perfect thing; you have to have another perfect thing. So the differences in philosophical ideas between Newton’s and Einstein’s theories of gravitation are enormous. What are these philosophies ? They are really tricky ways to compute consequences quickly. A philosophy, which is sometimes called an understanding of the law, is simply a way that a person holds the laws in his mind in order to guess quickly at consequences. Some people have said, and it is true in cases like Maxwell’s equations, ‘Never mind the philosophy, never mind anything of this kind, just guess the equations. The problem is only to compute the answers so that they agree with experiment, and it is not necessary to have a philosophy, or argument, or words, about the equation’.
That is good in the sense that if you only guess the equation you are not prejudicing yourself, and you will guess better. On the other hand, maybe the philosophy helps you to guess. It is very hard to say. For those people who insist that the only thing that is important is that the theory agrees with experiment, I would like to imagine a discussion between a Mayan astronomer and his student. The Mayans were able to calculate with great precision predictions, for example, for eclipses and for the position of the moon in the sky, the position of Venus, etc. It was all done by arithmetic. They counted a certain number and subtracted some numbers, and so on. There was no discussion of what the moon was. There was no discussion even of the idea that it went around. They just calculated the time when there would be an eclipse, or when the moon would rise at the full, and so on. Suppose that a young man went to the astronomer and said, ‘I have an idea. Maybe those things are going around, and there are balls of something like rocks out there, and we could calculate how they move in a completely different way from just calculating what time they appear in the sky’. ‘Yes’, says the astronomer, ‘and how accurately can you predict eclipses?’ He says, ‘I haven’t developed the thing very far yet’. Then says the astronomer, ‘Well, we can calculate eclipses more accurately than you can with your model, so you must not pay any attention to your idea because obviously the mathematical scheme is better’. There is a very strong tendency, when someone comes up with an idea and says, ‘Let’s suppose that the world is this way’, for people to say to him, ‘What would you get for the answer to such and such a problem?’ And he says, ‘I haven’t developed it far enough’. And they say, ‘Well, we have already developed it much further, and we can get the answers very accurately’. So it is a problem whether or not to worry about philosophies behind ideas. Another way of working, of course, is to guess new principles. In Einstein’s theory of gravitation he guessed, on top of all the other principles, the principle that corresponded to the idea that the forces are always proportional to the masses. He guessed the principle that if you are in an accelerating car you cannot distinguish that from being in a gravitational field, and by adding that principle to all the other principles, he was able to deduce the correct laws of gravitation.
One of the most important things in this ‘guess – compute consequences – compare with experiment’ business is to know when you are right. It is possible to know when you are right way ahead of checking all the consequences. You can recognize truth by its beauty and simplicity. It is always easy when you have made a guess, and done two or three little calculations to make sure that it is not obviously wrong, to know that it is right. When you get it right, it is obvious that it is right – at least if you have any experience – because usually what happens is that more comes out than goes in. Your guess is, in fact, that something is very simple. If you cannot see immediately that it is wrong, and it is simpler than it was before, then it is right. The inexperienced, and crackpots, and people like that, make guesses that are simple, but you can immediately see that they are wrong, so that does not count. Others, the inexperienced students, make guesses that are very complicated, and it sort of looks as if it is all right, but I know it is not true because the truth always turns out to be simpler than you thought. What we need is imagination, but imagination in a terrible strait-jacket. We have to find a new view of the world that has to agree with everything that is known, but disagree in its predictions somewhere, otherwise it is not interesting. And in that disagreement it must agree with nature. If you can find any other view of the world which agrees over the entire range where things have already been observed, but disagrees somewhere else, you have made a great discovery. It is very nearly impossible, but not quite, to find any theory which agrees with experiments over the entire range in which all theories have been checked, and yet gives different consequences in some other range, even a theory whose different consequences do not turn out to agree with nature. A new idea is extremely difficult to think of. It takes a fantastic imagination.
What of the future of this adventure ? What will happen ultimately ? We are going along guessing the laws; how many laws are we going to have to guess ? I do not know. Some of my colleagues say that this fundamental aspect of our science will go on; but I think there will certainly not be perpetual novelty, say for a thousand years. This thing cannot keep on going so that we are always going to discover more and more new laws. If we do, it will become boring that there are so many levels one underneath the other. It seems to me that what can happen in the future is either that all the laws become known – that is, if you had enough laws you could compute consequences and they would always agree with experiment, which would be the end of the line – or it may happen that the experiments get harder and harder to make, more and more expensive, so you get 99.9 per cent of the phenomena, but there is always some phenomenon which has just been discovered, which is very hard to measure, and which disagrees; and as soon as you have the explanation of that one there is always another one, and it gets slower and slower and more and more uninteresting.
That is another way it may end. But I think it has to end in one way or another. We are very lucky to live in an age in which we are still making discoveries. It is like the discovery of America – you only discover it once. The age in which we live is the age in which we are discovering the fundamental laws of nature, and that day will never come again. It is very exciting, it is marvelous, but this excitement will have to go. Of course in the future there will be other interests. There will be the interest of the connection of one level of phenomena to another – phenomena in biology and so on, or, if you are talking about exploration, exploring other planets, but there will not still be the same things that we are doing now. Another thing that will happen is that ultimately, if it turns out that all is known, or it gets very dull, the vigorous philosophy and the careful attention to all these things that I have been talking-about will gradually disappear. The philosophers who are always on the outside making stupid remarks will be able to close in, because we cannot push them away by saying, ‘If you were right we would be able to guess all the rest of the laws’, because when the laws are all there they will have an explanation for them. For instance, there are always explanations about why the world is three-dimensional. Well, there is only one world, and it is hard to tell if that explanation is right or not, so that if everything were known there would be some explanation about why those were the right laws. But that explanation would be in a frame that we cannot criticize by arguing that that type of reasoning will not permit us to go further.
There will be a degeneration of ideas, just like the degeneration that great explorers feel is occurring when tourists begin moving in on a territory. In this age people are experiencing a delight, the tremendous delight that you get when you guess how nature will work in a new situation never seen before. From experiments and information in a certain range you can guess what is going to happen in a region where no one has ever explored before. It is a little different from regular exploration in that there are enough clues on the land discovered to guess what the land that has not been discovered is going to look like. These guesses, incidentally, are often very different from what you have already seen – they take a lot of thought.
What is it about nature that lets this happen, that it is possible to guess from one part what the rest is going to do? That is an unscientific question: I do not know how to answer it, and therefore I am going to give an unscientific answer. I think it is because nature has a simplicity and therefore a great beauty.