It was particularly Galileo who realised that mathematics provided the most effective terms in which to express physical observations, and it was he who contributed most to the introduction of those terms into science. The book of nature, he wrote, 'is written in the mathematical language'. But there are two things that should be said about this oft-quoted aphorism. The first is that 'nature', or 'the universe', as Galileo conceived it was a much more restricted concept than that which we hold and that with which modern science is concerned. It comprised only what we study in mechanics; all other phenomena — sights, sounds, smells, etc. — belonged in his view not to the external world but to the observing subject, and it was not at all his idea that mathematics played the all-comprehensive role in science that it is nowadays often assumed to do. Secondly, a language is a medium for expressing ideas, and it is just as capable of expressing false ideas as true ones. The fact, therefore, that something can be expressed with rigorous mathematical exactitude tells you nothing at all about its truth, i.e. about its relation to nature, or to what we can experience.
The most dangerous intellectual error of modern science, with which this book is concerned, lies in the fact that this has been overlooked. Mathematics is an immensely more powerful tool than the Aristotelian syllogism, and its use as a language in which to express the facts of experience has been so successful that the idea has crept unperceived into the minds of physicists that whatever it says must be true. This is openly expressed in the statement already quoted, that everything that is not mathematically forbidden is necessarily observable. Accordingly the habit has developed of assuming that a physical theory is necessarily sound if its mathematics is impeccable: the question whether there is anything in nature corresponding to that impeccable mathematics is not regarded as a question; it is taken for granted.The fact is, however, that mathematical truths are far more general than physical truths: that is to say, the symbols that compose a mathematical expression may, with equal mathematical correctness, correspond both to that which is observable and that which is purely imaginary or even unimaginable. If, therefore, we start with a mathematical expression, and infer that there must be something in nature corresponding to it, we do in principle just what the pre-scientific philosophers did when they assumed that nature must obey their axioms, but its immensely greater power for both good and evil makes the consequences of its misapplication immensely more serious.
There are so many instances, even in the most elementary uses of mathematics, in which its indications are obviously false, that it may seem strange that this fact is almost automatically overlooked in the more advanced uses of the tool. But there is a universal tendency, not only in science but in everyday life as well, to pay exaggerated attention to predictions that are realised and to ignore those that are not. If, on say three occasions in a week, we dream of something unusual which happens later to occur, there is a very strong pre-disposition to believe that the dreams and the occurrences are directly related, notwithstanding the thousands of instances of dreams, apparently of the same general type, that are not realised. In somewhat the same way, although almost all mathematical solutions of a physical problem give both true and false results, we habitually accept the former as valid and pay no attention at all to the latter, when we are working in fields of experience where our existing knowledge is sufficient to enable us to distinguish them at once. Here is an example which I gave in a broadcast talk a short time ago,4 to which I shall revert later:
Suppose we want to find the number of men required for a certain job under certain conditions. Every schoolboy knows such problems, and he knows that he must begin by saying: 'Let x = the number of men required.' But that substitution introduces a whole range of possibilities that the nature of the original problem excludes. The mathematical symbol, x, can be positive, negative, integral, fractional, irrational, imaginary, complex, zero, infinite, and whatever else the fertile brain of the mathematician may devise. The number of men, however, must be simply positive and integral. Consequently, when you say, 'Let x = the number of men required,' you are making a quite invalid substitution, and the result of the calculation, though entirely possible for the symbol, might be quite impossible for the men.
Every elementary algebra book contains such problems that lead to quadratic equations, and these have two solutions, which might be 8 and - 3, say.
We accept 8 as the answer and ignore - 3 because we know from experience that there are no such things as negative men, and the only alternative interpretation — that we could get the work done by subtracting three men from our gang — is obviously absurd. But what right have we to reject - 3? Clearly, none at all if we accept the substitution: 'Let x = the number of men required.' If we have proved that 8 is the answer, then with the same inevitability we have proved that - 3 is the answer; and if we have not proved that - 3 is the answer, then we have not proved that 8 is the answer. The two solutions stand or fall together as soon as we allow mathematical symbols to represent facts of experience. Yet the inexorable fact is that one answer is true and the other false.
Now in this example it is experience alone that distinguishes the true from the false solution. We cannot prove by pure reason that there cannot be creatures who, with regard to the qualities here considered, can be interpreted as negative men; we know from experience alone that they are as unreal as centaurs. If the problem had been one concerning charges of electricity, of which there are two kinds which we call positive and negative, it might have led to the same equation, and then both solutions would in all probability have been true. There is nothing intrinsically impossible in the existence of negative men, any more than in the existence of black swans: experience alone enables us to reject the solution - 3 as false.
But it is possible to obtain perfectly valid mathematical solutions of a problem which we can see without experience to be physically false because the physical interpretation requires what can be seen without experience to be impossible. Here is an example. Suppose we have a cubical vessel whose volume is 8 cubic feet, and we wish to find the length of one of its edges. Now physically what we are asking is the reading of a standard measuring rod when it is placed along the edge. But suppose there is no such rod handy. That does not matter, for we can solve the problem by mathematics. We let x be the required length, and all we have to do is to solve the equation, x3 = 8. But this equation has three solutions, viz. 2, √( - 3) - 1, - √( -3) - 1 — all having the same mathematical validity. But we know that the only one of these solutions that can possibly correspond to the reading of a measuring rod is 2, because of the necessary properties of measuring rods, which we should understand even if we had never made or seen one. We might one day discover negative men, but we cannot conceivably discover a standard measuring rod that can read √(- 3) - 1 because, owing to the accepted standards of measurement, such an object would not be a measuring rod. So we just ignore two of the mathematical solutions, and quite overlook the significance of that fact — namely, that in the language of mathematics we can tell lies as well as truths, and within the scope of mathematics itself there is no possible way of telling one from the other. We can distinguish them only by experience or by reasoning outside the mathematics, applied to the possible relation between the mathematical solution and its supposed physical correlate.
Now it is this latter kind of reasoning that — according to the argument outlined in the Introduction, to which I can get no answer and which seems to me plainly unanswerable — invalidates the special theory of relativity. The problem here is to find the relation between the rates of two exactly similar standard clocks, A and B, of which one is moving uniformly with respect to the other, on the assumption that the motion is indeed truly relative, i.e. that there is no justification for ascribing it to one rather than to the other. Now this is a problem that can be solved mathematically, and we find that there are two solutions, known technically as the 'Galilean transformation' and the 'Lorentz transformation'. According to the first the clocks work at the same rate, and according to the second they work at different rates. The special theory of relativity regards the second as true and the first as false; the usual expression is that 'a moving clock runs slow'. But, as we have said, it is a condition of the problem that either clock can be regarded as the 'moving' one, so this second solution (subject, of course, to the truth of the postulate that the motion is truly relative) requires equally that A works faster than B and that B works faster than A, and just as we know enough about measuring rods to know that they cannot read √(-3) - 1, so we know enough about clocks to know that one cannot work steadily both faster and slower than another. Hence, without in the least rejecting the Lorentz transformation as a mathematical solution of the problem, we can say at once that it is not a possible physical solution. Nevertheless, in modern physics it is universally assumed to be so, on the sole ground of its mathematical validity.
How such an obvious error could have occurred and escaped immediate recognition is explained in Part Two, but it may be said at once that the apparently simplest way of exposing it — by setting two clocks in relative motion and observing their rates — is impracticable because the difference which the theory requires is too small to be detected except at velocities far too high to be yet attainable. Experiments have been made in which elementary electrically charged particles (conceptual bodies, such as electrons, protons, etc.) have been used instead of clocks, and observations of what have been regarded as their 'rates' have been made, and these have shown that such 'rates' differ for particles which, according to electromagnetic theory, have vastly different velocities. These observations have been held to constitute an experimental proof that the Lorentz transformation is a physically valid solution of our problem. But there are two reasons why this argument fails. In the first place, even if it be fully granted, it shows only that one 'clock' works more slowly than the other — which would be quite possible if the motion of each was absolute, as Lorentz showed before Einstein's special relativity theory appeared. If the motion is relative, however, and the Lorentz transformation is a valid solution, then also the second 'clock' must work more slowly than the first — and this, it need hardly be said, has been left unproved. The second reason for the failure of the argument is that the interpretation of the particles as 'clocks' and of the observed phenomena as their 'rates', and the assumption that they move with velocities, ascribed to them (it is, of course, quite impossible to observe them; their existence and properties have all to be inferred on theoretical grounds) depend on the truth of a theory that itself depends on the truth of the Lorentz transformation (this is explained in Part Two), so the argument is circular: the observation proves the physical truth of the Lorentz transformation only if we first accept a theory which itself requires that transformation to be physically true.
An experimental test of this requirement of the special relativity theory is therefore at present impracticable, and the claims often advanced that such a test has been made are spurious. But surely, one does not need an experiment to prove that one clock cannot at the same time work both faster and slower than another. And this brings me to the most serious aspect of this whole matter. How is it possible that such an obvious absurdity should not only have ever been believed but should have been maintained and made the basis of almost the whole of modern physics for more than half a century; and that, even when pointed out, its recognition should have been universally and strenuously resisted, in defiance of all reason and all the traditions and principles of science expressed by Sir Henry Dale in the statement quoted at the beginning of this chapter?
This question has two aspects, an intellectual and a moral one. Both are astonishing, but of their reality and profound importance there can be no question. The former is the less difficult to understand, though it needs a careful survey of the history of the subject to make it credible: this I attempt in Part Two — necessarily less completely than is desirable, but sufficiently, I hope, to show that what appears patently absurd in one context may present quite a different semblance in another, and to explain how the special relativity theory came to be accepted in spite of its contradictions (disguised as 'paradoxes') in the early decades of this century. After all, it was not so very long ago that men of the highest intelligence believed that Moses wrote the account of his own death recorded in the Pentateuch. But the more serious lapse is the moral one, not only because of the intrinsically greater seriousness of a moral as compared with an intellectual fault, but also because the nature of science itself does not ensure its eventual correction as it does when the mistake is intellectual. When Dale wrote of the unflinching fidelity of science to the answers which nature gives to its questions, he took it for granted that those answers would, in the long run, be unmistakable, and the contribution that science had to offer to civilisation lay in the moral sphere, in its acceptance and publication of those answers, at whatever cost to expectancy and without prejudice or preconception of any kind. It is in the failure of present-day science to live up to Dale's ideal in this respect that, notwithstanding the incalculable physical danger involved in the intellectual error, lies the ultimate offence. That is so, not only because fidelity to truth for its own sake is ultimately more compulsory than that for the sake of physical well-being (if that is disputed I shall not argue the question), but also because the loyalty of science to truth has a far wider relevance than that exhibited in the matter of special relativity alone, wide though that is. In an age in which science has begun to play a dominant role, quite beyond the control or even the comprehension of the non-scientific citizen, the whole future of civilisation is dependent on the absolute unqualified fulfilment by scientists of their moral obligations.
Science At the Crossroads
Herbert Dingle