Part 47

In reality, however, this demand for certainty was extra-logical: it is not required for the purpose of analysing reasoning. For it is just as easy to reason from doubtful and probable premisses as from certainties, nor need the doubt in the reasoner’s mind affect the form of the reasoning. If, however, there is an imperative desire for certainty, it must be somehow gratified by logic. And there seemed to be no way of doing so except by ascribing absolute truth and certainty to the initial principles of science.

Of course it was covertly assumed that certainty could only be reached by starting from certainty, and that no possibility of a growth of assurance in the progress of the reasoning could be entertained. In a sense this assumption was correct (cf. §§ 27, 28), because it is true that the gradual verification of scientific truths does not render them absolute; but it led to neglect of all methods which appeared to start with premisses initially doubtful and hardening into certainties by gradual confirmation. No doubt it was not strictly impossible to reason from premisses not known to be true, but such reasoning was despised as ‘dialectical’, and no inquiry was made into the frequency of its occurrence in actual science. Why, then, waste time upon so unworthy a procedure, instead of fixing one’s whole attention upon the truly logical ideal, the absolute proof of absolute truth? Let us maintain, rather, the old Aristotelian conviction that the truly scientific syllogism proceeds from premisses that are true and underivative (because ‘self-evident’) and inerrant, and demonstrates its conclusion with ineluctable necessity! Thus the attainment of absolute truth was unobtrusively smuggled in as the aim of reasoning, and became an integral feature of the ideal of ‘demonstration’.

§ 11. From the standpoint of the scientific inquirer, however, this whole theory of proof is open to the gravest objections. He finds first that it is impracticable, being composed throughout of counsels of perfection with which he cannot comply, and then that, even if he could, they would be perfectly useless, and destructive of his aims.

(1) It strikes him at once that the Fixity of Terms is an obvious fiction. He will of course be aware, from his scientific experience, that fictions have their uses and are often indispensable; but he will know also that not all fictions are useful, and that the adoption of a fiction has in each case to be justified by its usefulness. Moreover, it is not so much its immediate and prospective use which justifies it, though this yields the usual motive for its adoption, as the ulterior uses ascertained ex post facto by experience.

He will ask, therefore, for evidence that an absolute fixity of terms is the vital necessity for logic it is declared to be. He will admit, of course, the familiar arguments for a certain stability of meanings which have come down from the days of Plato, but he will suggest that a relative fixity of terms is quite sufficient to content them. He will point out that in a progressive science any absolute fixity in its terms is precluded by the very progress of the science. For the terms in use must somehow manage to convey the growing knowledge they are employed to ‘fix’. The term ‘gas’, for example, must not be tied down to the meaning Van Helmont desired to convey when he invented it; it must incorporate all that physics has discovered about ‘gases’ ever since. Similarly, when Darwinism transforms the notion of ‘species’, and the discovery of radio-activity that of ‘atom’, these developments of meaning must be recognized as perfectly proper. To object to these conceptions as modern science uses them, on the ground that, because to Plato and Aristotle species were eternal and immutable, a ‘species’ that changes cannot be truly a species, or that because an ‘atom’ is etymologically ‘indivisible’, it becomes an impossible self-contradiction when it is made up out of ‘electrons’, will seem to him to reveal only the fatuous pedantry of an utterly unscientific mind.

§ 12. (2) If he is acquainted with psychology, he will perceive also that the fiction of the fixity of terms is subject to a further restriction. It is not only in science as such--for all sciences must be conceived as progressive--that the fixity of terms cannot be made absolute: a real fixity is strictly inconceivable for and in every human mind. For every term that is actually used to convey a meaning must be held to form part of a new truth, i.e. of a truth that was not previously in being. It is not a question of principle whether the truth is supposed to be new only to the person to whom it is addressed, or claims to be new to all, i.e. to science. For no judgement would be made unless it had something new to say. Hence every real judgement, as opposed to the verbal formulas which are called judgements in the logic-books, more or less modifies the meaning of its terms. If it succeeds in being a real judgement and a new truth, it establishes a new and previously unknown relation between its subject and its predicate. ‘S’ is henceforth an S-which-can-have-P-predicated-of-it, and ‘P’ a P-which-can-be-predicated-of-S. Thus both the psychological associations and the logical associates of S and P are changed. That logicians should not have noticed so obvious a fact can be attributed only to their inveterate habit of not using in their illustrations real judgements intended to cope with actual problems, but operating with their verbal skeletons, which are not being used by any one to convey his meaning, and so do not have any actual meaning.

Clearly, then, no science can interpret the fixity of terms quite literally. Or rather, it can only interpret it literally--as a matter of the literal integrity of the words that may convey a meaning. But in a scientific inquiry the convention of formal logic must be reversed; the fixity of terms must be understood not to be absolute, but to be merely ad hoc and sufficient to convey a definite meaning, which it is desired to develop. Accordingly it must always be assumed that the results of an inquiry are to modify its terms, and that it is permissible, and indeed inevitable, to develop their meaning, so long as they remain capable of expressing and conveying the new truth. We must come to every inquiry with a willingness to learn and to expand our terms. The Fixity of Terms, as it is tacitly presupposed in the traditional logic, is a scientific blunder of the gravest kind.

§ 13. (3) To renounce it, however, entails further consequences. It appears to undermine the whole notion of formal validity. For if we admit in principle that the meaning of terms depends vitally on that of the judgement in which they occur, how can we continue to rely absolutely on the mere verbal identity of its terms to hold together a syllogism? In any syllogism the middle term, M, may have one shade of meaning in relation to P, another in relation to S. It may be quite right to call M P in one connexion, and to call S M in another; and yet, when the two assertions are put together, they may lead to a conclusion which is an error or an absurdity. The man who (in his laboratory) would rightly declare that ‘all salt is soluble in water’ and (at his dinner table) as properly hold that ‘all Cerebos is salt’, could not combine these assertions to draw the conclusion that ‘all Cerebos is soluble in water’, without finding that the facts confuted his anticipation.

No doubt, when this had happened, he might explain it, ex post facto (if he knew logic), by alleging a hidden ‘ambiguity of the middle term’. We need not here discuss whether it is fair to treat as an inherent ambiguity what is really a juxtaposition of shades of meaning which were relative to different purposes and right in their original contexts, thus manufacturing a fallacy by selecting the premisses: the important thing is that the logician should be driven to admit that any middle term may become ambiguous in this way when a syllogism is constructed, and that this completely stultifies his assumption that the verbal identity of the middle guarantees the real identity of the objects to which it refers. If we call two things, which are and must be different if they are to be two, both ‘M’, we necessarily take the risk that the differences are irrelevant for the purpose of our argument. We may legitimately assume this, but if we do, our hypothesis has to be confirmed in fact; it is naïve to think that the verbal identity of the terms is quite enough. If, then, actual identity cannot be absolutely guaranteed, if there is always a possibility that the same term when put into a syllogism and used in reasoning may develop an ambiguity and become effectively two, it is evident that no amount of formal validity will safeguard the truth of a conclusion, even when the premisses are in themselves severally true. The syllogistic form is convicted of losing truth which it started from, and this is the very thing it boasted it could never do. Moreover, its coercive ‘cogency’ is exploded: whoever wishes to deny a ‘valid’ conclusion after admitting its premisses, has merely to suggest that by putting the premisses together a fatal ambiguity has been generated in the middle term.

§ 14. (4) The assumption that everything has been named rightly, and is what it is called, will scarcely commend itself to the scientific researcher. He will know from much painful experience that language only embodies the knowledge which has been acquired up to date, and too often is only a compendium of popular errors. Hence in any research which really breaks new ground the existing terminology will always prove inadequate, and new technical terms have usually to be devised in order to embody the new knowledge. The reason is obvious. Ex hypothesi we are inquiring farther into the subject, because our knowledge is felt to be insufficient. Accordingly the probable defects of the terminology we are initially forced to use must be borne in mind: we may expect it to omit what is unknown, to misdescribe and to classify wrongly what is partially known, putting together what does not belong together and separating what does, emphasizing the unimportant and slurring over the important, and generally failing to provide the mind with words that give it a real apprehension of the objects under inquiry. Hence the tacit assumption of Aristotelian logic that the terms reasoned with are fully known, that adequate notions are already extant, that truth has merely to be disentangled by a verbal criticism of existing opinions, and has not to be discovered outright, is false; nor can any argument from a verbal identity be taken as final.

§ 15. (5) But of all the assumptions lurking in the theory of proof, the belief that reasoning can and should start from certainty will seem the falsest and most pernicious to the man of science. For it means that we are committed to a search for absolutely certain premisses as a preliminary to every inquiry, and proscribes consciously hypothetical, i.e. truly experimental, reasoning altogether, or at least condemns it as incapable of leading to certainty. This search, however, will either be perfunctory and uncritical, if it accepts false claims to certainty; or else vain, if it is conscientious. For every attempt to prove a conclusion absolutely demands two absolutely true premisses; hence the more we try to prove, the more we have to prove, and our search grows the more endless and futile, the longer it is continued. An immutable basis of absolutely certain truths, therefore, for reasoning to start from, is nowhere to be found. In no science is it possible to start with truths that are absolutely certain. In every science the initial ‘facts’ are doubtful; they are alleged, but not yet approved. They embody only unsystematic observation and prescientific experience of the subject, and so are probably the products of inaccurate observation, bad interpretation, false preconceptions, and popular superstitions. To acquire any considerable scientific value, such material has to be thoroughly revised and refined.

The validity of methods and the certainty of ‘principles’ are no more assured than the ‘facts’, initially. Every science has to work out its own appropriate methods experimentally; even if it borrows methods from another, it has to find out how and how far they apply to a new subject. Neither does a science acquire its principles by divine revelation; even if they fell from heaven ready-made, it would insist on testing the authenticity of the revelation. But philosophers have been extremely reluctant to admit that the certainty of principles is a gradual growth: for over 2,000 years they have been endeavouring to discover some way of securing an infallibility to principles which would render them independent of the working of the sciences which use them. But if their labours have proved anything, it is that no such way can be found.

(a) They have recognized many principles as ‘self-evident’, and equipped the mind with a variety of ‘faculties’, expressly invented to enable it to apprehend the ‘self-evident’ inerrantly. But they have not been able to agree upon a list of self-evident principles, nor even to find any truth whose claims to self-evidence have not been denied by competent critics. Nor have they been able to define their notion of ‘self-evidence’ itself; they cannot discriminate between the sound ‘logical’ self-evidence, which they conceived to guarantee truth, and its merely ‘psychological’ ‘mimic’, which is certainly much commoner, and becomes more intense and extensive the more unsound is the mind that ‘apprehends’ it. Hence an unprejudiced observer has no reason to put the ‘intuitions’ of philosophers and the ‘faculties’ which apprehend them on a higher cognitive level than those of women or even lunatics. They all impose themselves psychologically; but this proves nothing as to their logical value, and science has to test them just the same.

(b) The principles which are said to be necessary or logical ‘presuppositions’ all turn out to be hypothetical when they are examined. They are needed, no doubt, to solve the problem in hand, if the particular way it is formulated is taken for granted. But if either the order or the formulation of problems is altered, they cease to be either ‘necessary’ or ‘presuppositions’. For example, the ‘axiom of parallels’, alias ‘Euclid’s postulate’, is a necessary presupposition of geometry, if the existence of parallels is assumed. But if we prefer it, we can just as well (with Aristotle) make it our axiomatic ‘presupposition’ that the interior angles of a triangle are equal to two right angles, and can then deduce the existence of parallels. I.e. Euclid might have deduced what he assumed, and assumed what he deduced. If, moreover, we do not desire to construct a Euclidean geometry at all, we can deny both presuppositions, and proceed from alternative postulates, which lead to the various metageometries. The only things, in short, which all scientific principles presuppose are the desire to construct a science, and the desire to construct it in a particular way, which is simplest, or easiest, or most systematic, or most in accordance with the reigning prejudices. But these desires are the very things which the logician’s account of principles always omits to mention.

Again, the whole of Kant’s scheme of a priori presuppositions in the theory of knowledge rests upon an arbitrary assumption, viz. that mental data are to be conceived as originally discrete and are therefore in need of ‘synthesis’. But it is just as possible to conceive an analysis of knowledge which starts from the ‘presupposition’ of a continuum or flux, and proceeds to trace out the principles by means of which this continuum is broken up into a world of apparently distinct things and processes. Nor is it possible to say in advance of experience which of such ‘presuppositions’ is going to be more convenient and more conducive to scientific progress.

(c) It demands a high and rare degree of philosophic insight to perceive that very many principles are neither certain, necessary, nor probable, but simply methodological. Whether we think them true or not, we adopt them because of their eminent convenience. If they turn out to be false, candour compels us to call them methodological fictions; but they continue in use. Our belief in the trustworthiness of memory is a good example. For though we often find that our memory has played us tricks, we continue to accept as true what we ‘distinctly remember’. If no limitations to the truth-claim of such assumptions are discovered, enthusiasts will probably insist on promoting them to the rank of indisputable ‘axioms’, and hail them as absolute truths. But their scientific value is not thereby enhanced, and the cautious will eschew such exaggerations. For there is no real reason why the scientific rank of principles should not rest openly and entirely on their actual services, and why a ‘methodological assumption’ should not rank higher than a ‘self-evident truth’. For the latter is at most a fact of our mental organization which nothing has so far turned up in nature to set at naught, and as such a fact it is itself a thing to marvel at rather than an explanation of other things. The scientific spirit will always hesitate to acquiesce in the limits which are set to inquiry by sheer brute facts, and if the absolute truth of certain principles were merely an ultimate fact which could neither be impugned nor explained, this would go far to make these principles appear unintelligible and would be a constant challenge to dispense with them, or somehow to evade them. A principle, then, should always be prepared to state the reasons a science had for adopting it: only the reasons will appear from the actual working of the science. They will involve a reference forward to the facts it copes with, not back to higher principles or to any claim that proves itself by its self-assertion.

(d) Indisputable principles, then, are not consonant with the spirit of inquiry: it will gladly let them go, if it can attain truth and advance knowledge in other ways. It will not shrink even from repudiating the ideal of absolutely true and demonstrated truth, if it can be realized only by sacrificing the progressiveness of science; nor will it be dismayed to find that this ideal is unrealizable. For when the inquirer reflects upon his own procedure, he finds that it points to a radically different ideal, and that the existence of absolute truths would only be a hindrance and a restriction upon his endeavour (cf. § 28 (4)).

§ 16. Before, however, we attempt to delineate the logical ideal of the discoverer, it will be necessary to encounter a serious objection which protests on principle against such an undertaking, and urges that discovery by its very nature must elude logical treatment. It is contended, in the supposed interests of logic, that discovery is a process so inherently and incurably psychological that no logical account can ever be given of it. Discoveries are windfalls, and come as ‘happy thoughts’ to the gifted geniuses that make them, in a manner neither they nor any one else can account for or describe: they are therefore logically fortuitous, and to set forth the ideal of proof by which the truth of discoveries is tested is all that need, or can, be the concern of logic.

Certainly the great majority of deductive logicians have taken up some such attitude towards the process of discovery. Aristotle contents himself with a bare mention of ‘sagacity’ (ἀγχίνοια), which is defined as the instantaneous apprehension of the suitable middle term for constructing a demonstrative syllogism. When one recollects the weary centuries of painful effort and continual failure which elapsed while the élite of the human race were seeking for clues to, e.g., the mysteries of disease and of physical happenings, before they hit upon the notions of microbes and the mechanical theory, this naïve underestimate of the most difficult and essential of scientific procedures sounds like a mockery. Yet the whole Aristotelian school pass over the problem as lightly. They all seem to believe that, while it is merely low cunning to make a discovery, it is a real proof of mental capacity to arrange it ‘in logical order’ after it has been made, and to show how far short it falls of the logical ideal. Even the inductive logicians may be said to have participated in this attitude. For they were not more anxious to propound methods of discovery than to contend that their conclusions were just as rigidly proved and just as formally valid as those of syllogisms. They did not see that they were thereby accepting the demonstrative ideal of proof and giving away their own; what they should have shown was that this ideal was utterly nugatory, and that their own methods could never conduct to ‘proof’, but only to something vastly superior.

§ 17. In spite, however, of this wonderful consensus of logicians the above argument depends essentially on a confusion. It has confused two things which are perfectly distinct, the actual procedure of the individual discoverer, and the generalized description of the attitude of mind and procedures of discoverers, as they appear to subsequent logical reflection. Both present problems to the logician, but the problems are not the same. To anticipate the process of actual discovery may well be left to the prophets; it will transcend the powers of logic and indeed of any science, unless it be individual psychology, if it exists, or history, if it be a science. It may readily be admitted that anecdotes about the bath which fomented in the mind of Archimedes the idea of specific gravity, and the streets of Syracuse through which he ran and cried ‘Heureka!’, or about the apple-tree which shed its fruit upon Newton’s receptive head, and stimulated his brain to frame the law of universal gravitation, are beneath the dignity of science. Their narration belongs to history, which can go as deeply into their details as the scale of the history and the purpose of the historian demand; but the particular circumstances of a particular discovery may well be treated as ‘accidental’, and be smoothed out of the scientific record. But why does it follow that no common features can be traced in these histories of discovery, and that there cannot be compiled out of a sufficient number of them a generalized account of what appears to be the ‘essential’, i.e. really relevant, procedure of discoverers, which may serve as a guide and model to subsequent discoverers? Why should this be more difficult than to describe the method of lion-hunting from the records of lion hunts, or the treatment of a disease from the history of a number of cases? Indeed, it would seem that the thing has been done. Any discoverer may reflect upon his own discoveries, and, like Poincaré, formulate the method he has found successful. And if discoverers are not all perfectly unique in their methods, important uniformities will probably be found by comparing the methods of a number of discoverers.

Why again should it be assumed that the general account thus extracted from a retrospective study of discoveries must at once coincide with the logical ‘ideal of proof’? Why should it even point to this, or be related to it otherwise than by contrast? Surely the possibility should be discussed that there are two procedures for logic to consider, of which the one describes how human knowers, starting from what they believe themselves to know, set about it to fortify and extend their knowledge, while the other moves on a superhuman plane and describes, with Platonic fervour, how ideal demonstration, descending from absolutely certain principles, moulds into a closed and inexpugnable system all the truths which are deducible from these and alone intelligible. The two accounts must be distinct, for they have different starting-points and work upon different material. Nor need they ever have any point of contact. For it may well be that human knowing never attains to an absolute certainty and a completed system, while deductive proof never condescends to notice mundane fact.

This was certainly so in the first rapturous vision of a priori ‘proof’ which solaced Plato amid the elusiveness and opacity of the flow of happenings. The deduction of the intelligible order of the ideal ‘Forms’ from their supreme ground and (sole!) premiss in the ‘Idea of the Good’ stopped short of facts and events at the laws of minimum generality, and recognized in all the happenings of the sensible world an ineradicable taint of ‘not-being’ which rendered their stability impossible and their prediction vain. Aristotle similarly distinguished between the procedure which started from the notiora nobis, the apparent facts of perception, and that which began with the notiora naturae, the self-evident principles which could form the ultimate premisses of demonstrations. But that these two methods must somehow coincide was assumed rather than proved, in a way that should have discredited the doctrine. For Aristotle also was not able to explain how ‘science’, being of ‘universals’, could apply to particulars, which nevertheless he would not with Plato stigmatize as ‘unreal’, while the ascent from the sensible fact to the ‘universal’, which was called the ‘induction’ of the ‘principle’, is hardly validated by the naïve allegation of a mental faculty of ‘intuitive reason’ (νου̑ς) endowed with the special function of apprehending principles in their particular exemplifications. It is high time, therefore, that this whole assumption that a necessary congruity exists between the logic of discovery and of proof should be subjected to a thorough examination.

III

§ 18. Such an examination will speedily establish that the mental attitude of the discoverer is, and must be, quite different from that of the prover.

In the first place, the discoverer is not in possession of the knowledge he covets. It is for him a desire, an aspiration, an aim to be attained. Proof, on the other hand, presupposes knowledge. Not only must the demonstrator know the assured truths he uses as premisses, not only must he have a supply of absolutely certain truths if his proof is not to remain hypothetical (§ 9), but he must already know the conclusion he exhibits. He cannot be ignorant, like the discoverer, of the result he is to arrive at. He is not engaged in discovering new truth, he is only showing how it follows from old truths. His retrospective contemplation has merely to retrace the history of its attainment, or rather to rearrange it in the more pleasing order which he calls ‘logical’. This order is not that in which it was discovered, nor even that in which it could be discovered. For there are such things as necessary errors, indispensable artifices, and indefensible fictions, and the way to a truth often lies through them. Thus from time immemorial mathematicians have represented the continuous by the discrete, quantities by numbers, knowing full well what fictions their practice involved. Again, mathematical calculation of shapes, areas, and motions necessarily presupposes the fictions that bodies have the ideal and regular forms to which they ‘approximate’, and that their ‘mass’ is concentrated at their (ideal) ‘centre of gravity’. It is more than doubtful whether the notion of an ‘evolution’ of species could ever have been reached, except by starting from the false notion of the fixity of species, or whether the true nature of the mobility and development of meanings could have been understood except by correcting the Platonic theory of immutable and eternal ‘universals’. To ‘proof’ all these incidents and accidents of the history of discovery are irrelevant; all that has to be done is to show that the new truth can be deduced from the old, and that a ‘logical connexion’ exists between them.

§ 19. Not only is this much easier to do than to make the discovery, but it is very much easier to follow. Any one can see the connexion once the data have been arranged in logical order. Hence the assumption that this order somehow represents the actual process in a perfected form is natural enough. But it leads to contempt for the procedure of discovery. The discovery is made to look so easy that it becomes impossible to appreciate its difficulty and its merit, and it seems astonishing that no one made it long before. For did not the ‘facts’ all but force it upon the dullest mind? Who could have failed to see that fossils must be (at least) as old as the rocks in which they are embedded, that obviously worked flints, similarly, attest the antiquity of man, that northern Europe is scratched all over with the marks of a gigantic glaciation? It is forgotten that these ‘facts’ were not there until there came a mind prepared to notice them. Hence none of these discoveries were in fact easy to make, and they were preceded by a long struggle of the human mind with false preconceptions and the illusory ‘facts’ which they had engendered.

Nor are discoveries easy to get recognized when they have been made. The persecutions to which discoverers of new truth are subjected always and everywhere (more or less) form as discreditable a chapter of human history as the persecution of moral reformers. Those may count themselves fortunate who are simply ignored. Hence everything has to be ‘discovered’ over and over again. Nothing new ever enters the world, just as nothing old ever passes away, without infinite pains and after a protracted struggle. One curious result of this inertia which deserves to rank among the great fundamental ‘laws’ of nature, is that when a discovery has finally won tardy recognition, it is usually found to have been anticipated, often with cogent reasons and in great detail. Darwinism, e.g., may be traced back through the ages to Heraclitus and Anaximander. Thus it is true that there is ‘nothing new under the sun’; but only because when a new truth first appears it does not prevail: when after a hundred repetitions it is at length recognized, it is no longer strictly new. Accordingly, the ‘discovery’ of a truth is only the beginning of its career, the first step by which it makes its way in the world, and still very distant from the crowning ‘proof’ with which logic complacently adorns it ex post facto, when it has ‘arrived’. The slowness and difficulty, then, with which the human race makes discoveries, and its blindness to the most obvious facts, if it happens to be unprepared or unwilling to see them, should suffice to show that there is something gravely wrong about the logician’s account of discovery.

§ 20. Quite apart from the difficulties which the psychological constitution and social organization of man put in the way of innovators, the making of a new truth which formulates a new ‘fact’ is also intrinsically anxious work. It is not merely that its maker can have no assurance that his enterprise will succeed, that he cannot start with a feeling of certainty from established truths, and be wafted by an irresistible wave of logical necessity to the safe haven of a predestined conclusion. He must start with a consciousness of ignorance and an all-pervading feeling of doubt about every step of his inquiry. This doubt he should not, moreover, endeavour to disregard or to suppress; for it is the best guarantee that no way to the truth will be passed by in his explorations. Doubt, therefore, should be recognized on principle, and equipped with a technique of testing and experimentation: the inquirer should be proud that he has to feel his way in fear and trembling to the very end.