JBS Haldane
Published: Rationalist Annual, 1941;
Transcribed: for marxists.org in May, 2002.
THE PHRASE 'A law Of Nature' is probably rarer in modern scientific writing than was the case some generations ago. This is partly due to a very natural objection to the use of the word 'law' in two different senses. Human societies have laws. In primitive societies there is no distinction between law and custom. Some things are done, others are not. This is regarded as part of the nature of things, and generally as an unalterable fact. If customs change, the change is too slow to be observed. Later on kings and prophets could promulgate new laws, but there was no way of revoking old ones. Thus the unfortunate Jews, if orthodox, stagger under a burden of law which was increased over thousands of years by ingenious rabbis. The Greek democracies made the great and revolutionary discovery that a community could consciously make new laws and repeal old ones. So for us a human law is something which is valid only over a certain number of people for a certain period of time.
Some people also believe in Divine laws which hold for all men everywhere. The curious can consult a report, Kindred and Affinity as Impediments to Marriage (SPCK), by Anglican bishops and others who have tried to solve the fascinating problem of where human law ends and Divine law begins as regards marriage with relatives. God forbids you to marry your sister, it appears, but it is not so sure whether it is God or man who says you may not marry your niece. So many gods have issued so many different laws in the past that a study of history makes the theory of Divine law a little ridiculous. Just the same applies to the Stoic conception of a natural law incumbent on all men as men. Even if such laws existed they would not be eternal, for man has evolved and will evolve. Actually they turn out merely to hold for a particular stage of social and economic development.
Laws of Nature, however, are not commands but statements of fact. The use of the same word is unfortunate. It would be better to speak of uniformities of Nature. This would do away with the elementary fallacy that a law implies a law-giver. Incidentally, it might just as well imply a parliament or soviet of atoms. But the difference between the two uses of the word is fundamental. If a piece of matter does not obey a law of Nature it is not punished. On the contrary, we say that the law has been incorrectly stated. It is quite probable that every law of Nature so far stated has been stated incorrectly. Certainly many of them have. Nevertheless, these inaccurately stated laws are of immense practical and theoretical value.
They fall into two classes — qualitative laws such as 'All animals with feathers have beaks', and quantitative laws such as 'Mercury has 13*596 times the density of water' (at 0 C and 1 atmosphere's pressure). The first of these is a very good guide. But it was probably not true in the past. For many birds which were certainly feathered had teeth and may not have had beaks. And it is quite possibly not true today. There are about a hundred thousand million birds on our planet, and it may well be that two or three of them are freaks which have not developed a beak, but have lived long enough to grow feathers. It was thought to be a law of Nature that female mammals (defined as warm-blooded vertebrates with hair) had mammary glands, until Professor Crew of Edinburgh found that many congenitally hairless female mice lacked these organs, though they could bear young which other females could then foster.
And quantitative laws generally turn out to be inexact. Thus water is nothing definite. It is a mixture of at least six different substances. For in the molecule H2O one or both of the hydrogen atoms may be either light or heavy, and so may the oxygen atom. Similarly, mercury consists of several different types of atom. Thus the ratio of the densities of mercury and water is not fixed, though in the case of ordinary samples the variation is too small to be detected. But it can be detected if the water happens to have been taken from an accumulator which has been used for some time.
We have, I believe, gained a somewhat deeper knowledge of the meaning of natural laws from the work of two living English physicists — Jeffreys and Milne. In his Theory of Probability (Oxford, 1939) Jeffreys has something new to say about induction. Two contradictory theories are in vogue as to the laws of Nature. The older view is that they are absolute, though of course they may have been inaccurately formulated. The extreme positivistic view, enunciated by Vaihinger, is that we can only say that phenomena occur as if certain laws held. There is no sense in making any definite statements, though it is convenient to do so.
Now Jeffreys points out that, if a number of observations have been found to conform to a law, it is highly probable that the next one will do so whether the law is true or not. In Jeffrey's words: 'A well-verified hypothesis will probably continue to lead to correct inferences even if it is wrong.' This can be proved in detail if it is stated with sufficient exactitude, on the basis of some highly plausible assumptions. Thus we can use the 'as if' principle without denying the existence of natural laws. What is more remarkable, laws which ultimately turn out to be inexact are often far more exact than the data on which they are based. Thus Jeffreys remarks, speaking of gravitation, that 'when Einstein's modification was adopted the agreement of observation with Newton's law was three hundred times as good as Newton ever knew'.
Positivists and idealists have made great play with the fact that many laws of Nature, as formulated by scientists, have turned out to be inexact, and all may do so. But that is absolutely no reason for saying that there are no regularities in Nature to which our statements of natural law correspond. One might as well say that because no maps of England give its shape exactly it has no shape.
What is remarkable about the laws of Nature is the accuracy of simple approximations. One might see a hundred thousand men before finding an exception to the rule that all men have two ears, and the same is true for many of the laws of physics. In some cases we can see why. The universe is organized in aggregates with, in many cases, pretty wide gaps between them. Boyle's law that the density of a gas is proportional to its pressure, and Charles's law that the volume is proportional to the temperature, would be exact if gas molecules were points which had no volume and did not attract one another. These laws are very nearly true for gases at ordinary temperatures and pressures, because the molecules occupy only a small part of the space containing the gas, and are close enough to attract one another only during a very small part of any interval of time. Similarly, most of the stars are far enough apart to be treated as points without much error when we are considering their movements.
And most men manage to protect themselves from injury so far as is needed to keep both ears. Whereas trees cannot protect themselves from the loss of branches. It is very rare to see a completely unmutilated, and therefore completely regular, tree. Mendel's laws, according to which two types occur in a ratio of 1:1 in some cases and 3:1 in others, are theoretically true if the processes of division of cell nuclei are quite regular, and if neither type is unfit so as to die off before counts are made. The first condition never holds, and the second probably never does. But the exceptions to the first condition are very rare. In one particular case a critical division goes wrong about once in ten thousand times. The effect of this on a 1:1 ratio or 3:1 ratio could be detected only by counting several hundred million plants or animals. Differences in relative fitness are more important. But even so the Mendelian ratios are sometimes fulfilled with extreme accuracy, and are generally a good rough guide.
Jeffreys points out that in such cases it is often much better to stick to the theoretical law rather than the observed data. For example, if you are breeding silver foxes and a new colour variety occurs which, if crossed to the normal, gives 13 normal and 10 of the new colour, you are much more likely to get a ratio of about 1:1 than 13:10 if you go on with such matings, even though if you breed many thousands the 1:1 ratio will not hold exactly. The mathematical theory which Jeffreys has developed concerning such cases is particularly beautiful, but can hardly be summarized here.
Milne's theories are extremely revolutionary. He starts off with very simple postulates. He assumes some geometrical axioms — for example, that space has three dimensions — but does not assume Euclidean geometry. He also assumes what he calls the principle of cosmological relativity — namely, that observers anywhere in the universe would see much the same things. There is no favoured point or centre, no limit beyond which there is no more matter, and no direction in which matter progressively thins out. This is an assumption, but it is only the natural extension of Copernicus's theory that the earth is not the centre of the universe but just one star among others.
He then imagines observers on different stars communicating by light signals. This is, of course, unrealistic. But I have little doubt that, if his cosmological views prove valuable, later workers will be able to replace it by a more realistic hypothesis. Given this possibility of signalling, and clocks, he shows how the observers can graduate their clocks and establish a geometry. There is nothing very surprising in this. What is remarkable is that Milne claims that he can deduce some physical laws as necessary consequences of his basic assumptions. In particular, he deduces a law of gravitation which reduces to Newton's at 'small' distances measurable in units less than light-years.
This does not seem impossible. The law that the angle in a semicircle is a right angle was first observed as being at least very nearly true. Then twenty-five centuries ago Thales opened a new era in human thought by proving that it must be true. Milne may be a new Thales. Of course, later mathematicians showed that Thales, and Euclid too, had made a number of concealed assumptions. The proof was not as simple as they thought. And even if Milne's theories meet with no stronger criticism, they will doubtless meet with this one.
Milne claims that some, and perhaps all, physical laws are inevitably and rationally linked. He accuses those who say that laws might be otherwise of using 'magical', not rational, thinking. Dirac goes even further, and suggests that there is nothing chancy about the distribution of the matter in the universe, and that an all-wise mathematician could deduce this too from a few postulates. I must say I find this much harder to swallow. Laplace's theory, that given a full knowledge of the universe at one time one could deduce its state at all times past and future, was difficult enough to believe. This is worse. But in so far as any elements in these theories are accepted, this will be a signal triumph for Rationalism as against theories which recognize an irrational element in the universe.
However, if Milne simplifies natural laws with one hand, he complicates them with the other. Lengths may be defined in two ways. They may be referred to a material object, such as the standard metre, or to a wave-length of light which has the merit that it can be reproduced anywhere. If all the standard metres were lost, they could be reproduced with an accuracy of about one in thirty million by reference to known wave-lengths such as that of the red cadmium line derived from spectroscopic observations. One result of Milne's calculations is that the length of the metre, measured in standard wave-lengths, is increasing by about one part in twenty million per year. If you like, you may say that the universe, including the standard metre, is expanding. But it is simpler for most purposes to say that atoms are vibrating quicker. It makes not the slightest difference to any observable phenomenon which of these statements you choose. in fact, on this theory, and indeed on several others which have been worked out in less detail, many of the laws of Nature are changing. There is nothing arbitrary or haphazard about this change, but simply an increase in certain physical constants with the time.
This has important philosophical consequences. If true, it rules out any theories of a cyclical or recurrent universe. At a sufficiently early date the properties of matter were so different, and in particular chemical processes so sluggish, that life must have been impossible, or, to be accurate, material systems similar to any existing organisms could not have lived. Thus we can see why, even if the universe had no beginning, life has not got very far yet. And in the far future life will also be impossible for beings constituted like ourselves, though it may be that our descendants could keep up with changes in the laws of Nature by natural or artificial evolution.
Once again, I am sure that Milne's theories, even if they are partially correct, will turn out to be too simple for the immense complexity of the real world. But they give us at least an inkling of how posterity will think of natural laws. So far from being laid down by the arbitrary word of a creator, they may prove to be a system as intimately and rationally knit together as the propositions of geometry, and yet changing and evolving with time like the forms of plants and animals.