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The Basic Principles of Science
On the nature and definition of Science there has long been, and will doubtless continue to be, much disputation, but on one characteristic at least of its practice, agreement is general - its unqualified devotion to the discovery of truth at whatever cost to its expectations and tentative assumptions. Its conception of 'truth', of course, may be limited — this again is a matter of controversy — but never qualified by compromise or expectancy of any kind. Within its own intellectual sphere, however that may be conceived, its disinterestedness has been regarded as absolute, and it has often been held up as a model for other human activities - political, theological, and what not - in which throughout history has been onlytoo evident the influence of prejudice and partisanship, from which science alone has kept itself free. Of the many expressions of this idea which may be found in the literature of the last few centuries, coming from both scientists and non-scientists, I select as a paradigm the following typical statement by the late Sir Henry Dale, O. M., a former President of the Royal Society, and one of the outstanding scientists and most universally respected representatives of his calling in this century:
And science, we should insist, better than any other discipline, can hold up to its students and followers an ideal of patient devotion to the search for objective truth, with vision unclouded by personal or political motive, not tolerating any lapse from precision or neglect of any anomaly,
It is not, of course, to be supposed that every scientist has on every occasion lived up to the counsel of perfection which this statement represents; far from it, although it is true that, on the whole, the history of science compares very favourably indeed with the history of most, if not all, other human activities.Nevertheless, there are examples enough of prejudices and preconceptions, on the part of both individual scientists and scientific organisations — it is sufficient to mention the general dismissal during the eighteenth century of authentic evidence for the reality of falls of meteorites, on the sole ground that such things could not happen: for belief in the inviolability of laws of nature was substituted belief in the inviolability of the existing conception of what those laws were. It would be a gross error to imagine that scientists, as a class, are inherently more honest in their thinking and actions than men in other classes — a fact evident enough when we compare their extra-scientific activities with those of others. They are human, all too human, neither better nor worse on the whole than politicians and theologians, than historians and business men, than artists and artisans. What makes scientists behave so much more consistently in accordance with their ideals is not a unique 'original virtue' but the nature of their job.
This, as I have said, is, in its fundamental essence, a matter of dispute and academic discussion, but, speaking in general terms— which is not to say false terms so far as they go - it may be said thatthe aim of science is to discover what actually exists in nature and to express the relations between natural phenomena in rational form, 'i.e. in statements which, when established by sufficient evidence and found to hold good over a sufficiently wide range of experience, we call laws of nature, and when less completely supported but still possessing some measure of plausibility, we call theories or hypotheses. The evidence is never complete, and experience is never exhaustive, so all these statements are subject to change, but, however tenaciously scientists may wish to retain those which they have learned to trust, there is a finality about both experience and reason that ultimately overrides all opposition and forces the scientist to acknowledge the error of his preconceptions, however reluctant he may be to do so. The historian may defend or condemn the execution of Charles I; the theologian may assert or deny justification by faith; and nothing that any of them can do can finally refute his opponent. But the astronomer who asserts the existence of seven planets and denies the possibility of more, is silenced when an eighth is discovered; the experience on which he relics for the reality of the seven must have the same validity with regard to the eighth, and he has no option but to yield. The mathematician, in whose calculations leading to what he has asserted to be a proof of a theory the accidental omission of a factor 2 is discovered, must likewise acknowledge his error: no matter how strong his belief in the theory may be, the demands of reason which he has trusted to establish it now demand its abandonment. Scientists must be honest in the long run because the nature of their occupation makes them so: experience and reason are irresistible.
There is, however, one striking difference between the refutation of a hypothesis by experience and by reason which we must acknowledge, though it may be left to the psychologist to explain. Experience — i.e. observation or experiment — usually carries much greater conviction than reason, though ultimately they have equal authority. When a hypothesis is used to predict a certain experimental result, and the relevant experiment when performed yields the opposite result, there is generally no further discussion; the hypothesis is dismissed, or at least changed. But when the reasoning involved in a hypothesis is disputed (the example just given of an accidental mathematical error is of course a very special case, though it differs only in degree from reasoning processes much less obviously erroneous) there is usually no such general agreement on the truth of the matter, although, according to the strict principles of science, there should be.Newton, albeit with an ill grace, acknowledged an error in his reasoning concerning falling bodies which was detected by Hooke; he did not insist on an experimental test. But there has frequently been less readiness to abandon a cherished idea on rational than on experimental or observational grounds. In the nineteenth-century controversy on the age of the Earth between the geologists and the physicists, both sides had all the available evidence before them, and the difference in their conclusions arose wholly from the reasoning which they applied to it. We can see now, not only why equally intelligent reasoners reached widely different conclusions from the same evidence, but also that a stricter regard for the difference between necessary and probable conclusions from the evidence would have enabled a disinterested adjudicator to form a single judgement even then. It was, of course, a matter which could not be tested by experiment; had it been so, the dispute would have ended. In the event, only further experience, not then available but the possibility of which might have been anticipated, led to an agreed conclusion.
This greater degree of conviction which experience provides has had an important consequence in the progress of science.It has led to a relaxing of the demand that scientific hypotheses shall be strictly rational and a greater reliance on the ultimate verdict of experiment. This is not merely a development of the chief characteristic of the modern scientific movement that began in the seventeenth century and was marked by an exchange of interest solely in a priori reasoning prevalent in the Middle Ages for an interest primarily in experience. The pioneers of that movement - Galileo and Newton in particular - indeed insisted on the primacy of experience, but they relied no less than the Schoolmen on faithful obedience to the demands of reason in their ordering of experience and their deductions from what it revealed. Galileo has been criticised for his reasoning from 'thought experiments', and not only were these 'experiments', which were a novelty at that time, but also they involved rational thought and permitted nothing that violated the strict rules of reasoning. Newton, though he declared, in a famous phrase, that he did not make hypotheses, and in fact did make numerous experiments, nevertheless also laid down 'Rules of Reasoning in Philosophy', and did not hesitate to use what many today would regard as hypotheses. The scientific movement of the seventeenth century was a blend of experience and reasoning, in which both were essential but the reasoning was confined to what was derived from experience, and everything that was derived from 'principles' that had no justification except that they seemed necessary or good to those who adopted them, was firmly eschewed. But what gradually developed later, as a result of the greater degree of conviction that an experimental result brought with it, was a permissiveness in the framing of hypotheses, arising from the certainty that, if they were wrong, experiment would inevitably reveal that fact, and there was always a chance that, however improbable they might seem, they might turn out to be right.
There is much that can be said in defence of this - or at least there was - so long as the hypotheses are recognised for what they are -namely, a means of arriving at truth and not truth itself. Anything imaginable might be true - there are more things in heaven and earth than are dreamed of in our philosophy - and a few more dreams, which are not accepted as reality until waking experience confirms them, can do no harm, apart from a possible waste of time and money in a good cause, and might lead to the discovery of truths that would otherwise remain hidden. This indeed has happened not once or twice in the history of science. But it is attended by two dangers.The first, which was evident many years ago, is that the dreams shall be substituted for reality and accepted as true, not only before experience verifies them but wholly in their own right, regardless of whether experience verifies them or not. The second danger, which is relatively new, and demands far more urgent attention, is one reason why this book has had to be written: it lies in the fact that the experimental testings of the hypotheses of modern physics are attended by such possibly catastrophic results if the hypotheses are wrong, that the preliminary confirmation, that they are not necessarily wrong through violating the laws of reasoning, becomes imperative. Anything imaginable might be true: what is not imaginable — such as that Hitler both is and is not dead, or, to take a requirement of the hypothesis with which this book is chiefly concerned, that one clock can work steadily both faster and slower than another - cannot be true, and experiments based on the assumption that it is, are bound to lead, sooner or later, to anomalous results.
The first danger - the substitution of imagination for experience - was, as I say, realised long ago; this is how I exemplified it in a book published in 1931:2
I will give three quotations from representative scientists, covering the period from Newton to the present time and separated by roughly equal intervals. The first is from Newton himself (1687):
Since that was written the process has gone even further. Not only are hypotheses held to contain the 'real truth'; it is now claimed that any (mathematical) hypothesis is necessarily true. In a recent paper, two physicists, 0.Bilaniuk and E.C.G.Sudarshan, write:3 'There is an unwritten precept in modern physics... which states that inphysics "anything which is not prohibited is compulsory". Guided by this sort of argument we have made a number of remarkable discoveries, from neutrinos to radio galaxies.' 'We', of course, means scientists in general, and it is evident from the context that 'prohibited' means mathematically impossible. The statement that neutrinos and radio galaxies, or anything else, were so discovered is, of course, nonsense, but the statement is taken seriously and has instigated experiments directed towards the observation of 'tachyons' — hypothetical particles that travel faster than light - and stimulated a serious discussion in Nature on whether an effect can precede its cause. The relation between mathematics and physics is discussed in Chapter 6; in the meantime, this is sufficient to indicate how far we have gone along the path that started with the recognition that hypotheses might assist in the discovery of phenomena: first phenomena became 'useful tools' for the creation of hypotheses, and now hypotheses themselves are enthroned as necessary phenomena.
But it is the second danger that calls for immediate attention and, as indicated in the Introduction, has made this book necessary: it arises from the fact that modern physical experiments are such that the unexpected results which they produce might be catastrophic. Ironically enough, it is the very safeguard against the first - the certainty that experiment will ultimately show up the falsity of bad reasoning - that constitutes the essence of the second. We can contemplate with equanimity a temporary disregarding of truth, for we know that truth is great and will prevail, but the means by which its triumph is achieved may now ensure that there shall be no one left to care whether it prevail or not. When Rutherford's early experiments with atoms produced a result quite impossible if atoms were as he had conceived them, he declared that he was as surprised as if he had fired a bullet at a piece of tissue paper and it had rebounded and hit him. Similar misconceptions today, when chain reactions may occur that were not possible in Rutherford's experiments, may cause unimaginably great disasters, and the necessity that the hypotheses on which modern physical experiments are planned shall be scrutinised with the utmost care and freedom from prejudice is thus paramount. In fact, as later chapters will show, it is ignored.All unconsciously, scientists have allowed themselves to relapse into the mental state which science is usually regarded as having displaced — that of imagining how nature ought to behave and then assuming that she does so, instead of examining nature with an open mind and then expressing her observed behaviour in rational terms.
The factor that has made this possible, if one may use metaphorical terms to express the idea more vividly, isthe exchange by reason of the cloak of Aristotelian logic for that of mathematics. Both begin with so-called 'axioms' which are conceived in the mind without reference to experience, and their implications are developed into extended systems of thought which necessarily follow from the axioms but may or may not correspond to what can be observed in nature. For example, it was a mediaeval axiom that all celestial bodies moved in circles or in orbits that could be analysed into circular movements. This had nothing to do with observation: it was assumed before any regard was paid to observation of the actual movements of the bodies, and when those movements were observed it was regarded as a necessity to analyse them into circles of which their obviously quite different paths were the resultants. The essence of the scientific approach, applied to this particular example, consisted in taking the observed movements as the starting point, and expressing them in the simplest terms, without restriction to any preconceived notions of what those terms should be.
I shall consider in more detail in Chapter 6 the relation between mathematics and physics, but the matter is so fundamental for our present considerations that some preliminary remarks on it are desirable here. 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, v( —3) — 1, — v( —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 v(- 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 thatthere 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 v(—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, andthese 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.
That, I repeat, is why this book has become necessary. It is evident to me, in the fact that the simple question that I have put has remained unanswered while experiments continue on the assumption that the single sentence required to answer it can be withheld with impunity, that science has failed to accept nature's answers humbly and with courage and to give-them to the world with unflinching fidelity. However, I cannot rest content with my own judgment in such a matter. I shall simply relate the course of events, asserting nothing for which I have not complete objective evidence. If on occasion it seems necessary to insert comments of my own, it will be perfectly clear that they represent my own judgment and not objective facts. I then leave the reader to judge for himself what conclusions are to be drawn from the facts. Whatever they may be, I think it is unquestionable that the public has a right to be informed of what is actually occurring in a matter that concerns it so vitally, and, as will be seen, this is my only means of informing it. I begin, then, with the moral aspect of the matter, presenting it in narrative form, and necessarily, the story being so long and involved, omitting many minor details which do not modify the general import. The intellectual problem is reserved for Part Two.