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.
Now an interesting question comes up. Is there a place to begin to deduce the whole works? Is there some particular pattern or order in nature by which we can understand that one set of statements is more fundamental and one set of statements more consequential? There are two kinds of ways of looking at mathematics, which for the purpose of this lecture I will call the Babylonian tradition and the Greek tradition. In Babylonian schools in mathematics the student would learn something by doing a large number of examples until he caught on to the general rule. Also he would know a large amount of geometry, a lot of the properties of circles, the theorem of Pythagoras, formulae for the areas of cubes and triangles; in addition, some degree of argument was available to go from one thing to another. Tables of numerical quantities were available so that they could solve elaborate equations. Everything was prepared for calculating things out. But Euclid discovered that there was a way in which all of the theorems of geometry could be ordered from a set of axioms that were particularly simple. The Babylonian attitude – or what I call Babylonian mathematics – is that you know all of the various theorems and many of the connections in between, but you have never fully realized that it could all come up from a bunch of axioms. The most modern mathematics concentrates on axioms and demonstrations within a very definite framework of conventions of what is acceptable and what is not acceptable as axioms. Modern geometry takes something like Euclid’s axioms, modified to be more perfect, and then shows the deduction of the system. For instance, it would not be expected that a theorem like Pythagoras’s (that the sum of the areas of squares put on two sides of a right-angled triangle is equal to the area of the square on the hypotenuse) should be an axiom. On the other hand, from another point of view of geometry, that of Descartes, the Pythagorean theorem is an axiom.
So the first thing we have to accept is that even in mathematics you can start in different places. If all these various theorems are interconnected by reasoning there is no real way to say ‘These are the most fundamental axioms’, because if you were told something different instead you could also run the reasoning the other way. It is like a bridge with lots of members, and it is over-connected; if pieces have dropped out you can reconnect it another way. The mathematical tradition of today is to start with some particular ideas which are chosen by some kind of convention to be axioms, and then to build up the structure from there. What I have called the Babylonian idea is to say, ‘I happen to know this, and I happen to know that, and maybe I know that; and I work everything out from there. Tomorrow I may forget that this is true, but remember that something else is true, so I can reconstruct it all again. I am never quite sure of where I am supposed to begin or where I am supposed to end. I just remember enough all the time so that as the memory fades and some of the pieces fall out I can put the thing back together again every day’.
The method of always starting from the axioms is not very efficient in obtaining theorems. In working something out in geometry you are not very efficient if each time you have to start back at the axioms. If you have to remember a few things in geometry you can always get somewhere else, but it is much more efficient to do it the other way. To decide which are the best axioms is not necessarily the most efficient way of getting around in the territory. In physics we need the Babylonian method, and not the Euclidean or Greek method. I would like to explain why.
One of the consequences of this is as follows. Imagine a lot of stars falling together to form a nebula, or galaxy. At first they are very far out, on long radii from the centre, moving slowly and allowing a small amount of area to be generated. As they come closer the distances to the centre will shorten, and when they are very far in the radii will be very small, so in order to produce the same area per second they will have to move a great deal faster. You will see then that as the stars come in they will swing and swirl around faster and faster, and thus we can roughly understand the qualitative shape of the spiral nebulae. In the same way we can understand how a skater spins. He starts with his leg out, moving slowly, and as he pulls his leg in he spins faster. When the leg is out it is contributing a certain amount of area per second, and then when he brings his leg in he has to spin much faster to produce the same amount of area. But I did not prove it for the skater: the skater uses muscle force, and gravity is a different force. Yet it is true for the skater.
Now we have a problem. We can deduce often from one part of physics, like the Law of Gravitation, a principle which turns out to be much more valid than the derivation. This does not happen in mathematics; theorems do not come out in places where they are not supposed to be. In other words, if we were to say that the postulate of physics was the equal area law of gravitation, then we could deduce the conservation of angular momentum, but only for gravitation. Yet we discover experimentally that the conservation of angular momentum is a much wider thing. Newton had other postulates by which he could get the more general conservation law of angular momentum. But these Newtonian laws were wrong. There are no forces, it is all a lot of boloney, the particles do not have orbits, and so on. Yet the analogue, the exact transformation of this principle about the areas and the conservation of angular momentum, is true. It works for atomic motions in quantum mechanics, and, as far as we can tell, it is still exact today. We have these wide principles which sweep across the different laws, and if we take the derivation too seriously, and feel that one is only valid because another is valid, then we cannot understand the interconnections of the different branches of physics. Some day, when physics is complete and we know all the laws, we may be able to start with some axioms, and no doubt somebody will figure out a particular way of doing it so that everything else can be deduced. But while we do not know all the laws, we can use some to make guesses at theorems which extend beyond the proof. In order to understand physics one must always have a neat balance, and contain in one’s head all of the various propositions and their interrelationships, because the laws often extend beyond the range of their deductions. This will only have no importance when all the laws are known.
This is an example of the wide range of beautiful ways of describing nature. When people say that nature must have causality, you can use Newton’s law; or if they say that nature must be stated in terms of a minimum principle, you talk about it this last way; or if they insist that nature must have a local field – sure, you can do that. The question is: which one is right? If these various alternatives are not exactly equivalent mathematically, if for certain ones there will be different consequences than for others, then all we have to do is to experiment to find out which way nature actually chooses to do it. People may come along and argue philosophically that they like one better than another; but we have learned from much experience that all philosophical intuitions about what nature is going to do fail. One just has to work out all the possibilities, and try all the alternatives. But in the particular case I am talking about the theories are exactly equivalent. Mathematically each of the three different formulations, Newton’s law, the local field method and the minimum principle, gives exactly the same consequences. What do we do then? You will read in all the books that we cannot decide scientifically on one or the other. That is true. They are equivalent scientifically. It is impossible to make a decision, because there is no experimental way to distinguish between them if all the consequences are the same. But psychologically they are very different in two ways. First, philosophically you like them or do not like them; and training is the only way to beat that disease. Second, psychologically they are different because they are completely unequivalent when you are trying to guess new laws.
As long as physics is incomplete, and we are trying to understand the other laws, then the different possible formulations may give clues about what might happen in other circumstances. In that case they are no longer equivalent, psychologically, in suggesting to us guesses about what the laws may look like in a wider situation. To give an example, Einstein realized that electrical signals could not propagate faster than the speed of light. He guessed that it was a general principle. (This is the same guessing game as taking the angular momentum and extending it from one case where you have proved it, to the rest of the phenomena of the universe.) He guessed that it was true of everything, and he guessed that it would be true of gravitation. If signals cannot go any faster than the speed of light, then it turns out that the method of describing the forces instantaneously is very poor. So in Einstein’s generalization of gravitation Newton’s method of describing physics is hopelessly inadequate and enormously complicated, whereas the field method is neat and simple, and so is the minimum principle. We have not decided between the last two yet.
What is necessary ‘for the very existence of science’, and what the characteristics of nature are, are not to be determined by pompous preconditions, they are determined always by the material with which we work, by nature herself. We look, and we see what we find, and we cannot say ahead of time successfully what it is going to look like. The most reasonable possibilities often turn out not to be the situation. If science is to progress, what we need is the ability to experiment, honesty in reporting results — the results must be reported without somebody saying what they would like the results to have been — and finally — an important thing — the intelligence to interpret the results. An important point about this intelligence is that it should not be sure ahead of time what must be. It can be prejudiced, and say ‘That is very unlikely; I don’t like that’. Prejudice is different from absolute certainty. I do not mean absolute prejudice — just bias. As long as you are only biased it does not make any difference, because if your bias is wrong a perpetual accumulation of experiments will perpetually annoy you until they cannot be disregarded any longer. They can only be disregarded if you are absolutely sure ahead of time of some precondition that science has to have. In fact it is necessary for the very existence of science that minds exist which do not allow that nature must satisfy some preconceived conditions, like those of our philosopher.
Suppose you have two theories, A and B, which look completely different psychologically, with different ideas in them and so on, but that all the consequences that are computed from each are exactly the same, and both agree with experiment. The two theories, although they sound different at the beginning, have all consequences the same, which is usually easy to prove mathematically by showing that the logic from A and B will always give corresponding consequences. Suppose we have two such theories, how are we going to decide which one is right? There is no way by science, because they both agree with experiment to the same extent. So two theories, although they may have deeply different ideas behind them, may be mathematically identical, and then there is no scientific way to distinguish them.
However, for psychological reasons, in order to guess new theories, these two things may be very far from equivalent, because one gives a man different ideas from the other. By putting the theory in a certain kind of framework you get an idea of what to change. There will be something, for instance, in theory A that talks about something, and you will say, ‘I’ll change that idea in here’. But to find out what the corresponding thing is that you are going to change in B may be very complicated — it may not be a simple idea at all. In other words, although they are identical before they are changed, there are certain ways of changing one which looks natural which will not look natural in the other. Therefore psychologically we must keep all the theories in our heads, and every theoretical physicist who is any good knows six or seven different theoretical representations for exactly the same physics. He knows that they are all equivalent, and that nobody is ever going to be able to decide which one is right at that level, but he keeps them in his head, hoping that they will give him different ideas for guessing.