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Part I: The Transcendence of the Immanent

Lecture IV: Mathematics, Logic and History

The Cartesian movement had two main characteristics, which are closely interconnected. The first was the adoption of the individual consciousness as the starting-point of the whole process of thought, the other was the reliance upon “clear and distinct ideas”. The latter was necessitated by the former, for if the individual mind is to start from its own ideas alone, it can have confidence in its conclusions only if it can perceive these as necessary implicates in what it cannot doubt. There was indeed, as will shortly become apparent, no very great novelty in the special reliance upon clear and distinct ideas; but through the thought of Descartes it became more explicit, and was then, so far as I know, for the first time formulated. But while it was scarcely novel in principle, it was a necessary postulate in the Cartesian scheme in an altogether new sense. For if the isolated individual consciousness is the starting-point, there can be no external reference and then logical certainty is the only justifiable form of assurance.

“to obey the laws and customs of my country, adhering firmly to the Faith in which, by the grace of God, I had been educated from my childhood, and regulating my conduct in every other matter according to the most moderate opinions, and the farthest removed from extremes, which should happen to be adopted in practice with general consent of the most judicious of those among whom I might be living. For, as I had from that time begun to hold my own opinions for nought because I wished to subject them all to examination, I was convinced that I could not do better than follow in the meantime the opinions of the most judicious; and although there are some perhaps among the Persians and Chinese as judicious as among ourselves, expediency seemed to dictate that I should regulate my practice conformably to the opinions of those with whom I should have to live.”1

Of course it is true that all men have assurance on many subjects without reaching their conclusions by cogent inference from self-evident propositions. Such assurance mainly springs from sense-perception, or from tradition and the current opinions of our contemporaries. But these are both specifically mentioned by Descartes as unreliable. Descartes was fully aware of the necessity for accepting in practice much that he could not justify by his method. The second maxim of his “provisory code of Morals” was “to be as firm and resolute in my actions as I was able, and not to adhere less steadfastly to the most doubtful opinions, when once adopted, than if they had been highly certain”.2 But this maxim, difficult enough in any case, is not likely to be effectively obeyed if great emphasis is laid upon the exclusive right of “clear and distinct ideas” to be the objects or occasions of assurance. And it is significant that Descartes’ profession of attachment to the Catholic Faith is conspicuously lukewarm. It is dictated by his first maxim, namely:

Now if a man’s acceptance of Christianity in preference to Zoroastrianism, Mohammedanism or Confucianism is based on that consideration of expediency, it will be very hard for him to adhere to it as steadfastly as if it had been highly certain. He may on that basis practice mystical meditation and go very far in it, but he is not likely to come under the full force of definite religious faith. There can be no doubt about Descartes’ sincerity in his religion as elsewhere, but the wonder suggests itself whether he had any notion whatever of the true nature of religious faith—such faith, for example, as that of Francis or Luther or Pascal. His intention of conformity is like the intention of most of us to use our knife and fork in the conventional way; it has no suggestion of any submission to overwhelming Authority.3

Now it is certainly of great importance to distinguish between assurance which is logically well-grounded and assurance which is not thus grounded, and it may be necessary from time to time that this distinction should be enforced even at the cost of seeming to deprive of real justification all assurance which lacks proof. But that position has the value only of a temporary protest, for the area of experience in reference to which actual proof is possible is narrowly limited, and the most important of mental disciplines for almost all purposes is not that which distinguishes between certainty and probability, but that which leads to discrimination between the degrees of probability, and especially between degrees of justification attaching to unproved assurances. To such a discipline the Cartesian method contributes little or nothing. Having involved all convictions in artificial doubt, Descartes seeks a way of secure advance from his assurance of doubt to assurance of ascertained truth. There could be none such except the way of clear and distinct ideas. When the mind sees one idea to be essentially involved in another, it passes from the first to the second with absolute assurance. Even then its trust in the applicability of its conclusion to the real world reposes on faith in the veracity of God, who would be a deceiver if He had made our minds in such a way that they are bound to believe what may yet not be true. And if the veracity of God—or if God Himself—be not ascertainable by the way of clear and distinct ideas, these are themselves unreliable and we are left to choose between petitio principii, absolute scepticism and circularity. Thus European thought was launched on its labyrinthic course, which culminated in Kant’s attribution of our right to certainty to the function of the Understanding as itself ordering the experience of which it thereafter understands the order.

Yet at the outset there was a most welcome and profitable simplification. Scholastic Logic, like Ptolemaic Astronomy, had become more and more complex, as the attempt was pursued to fit into its framework the infinite subtleties of living thought. Refinement was added to refinement, just as Astronomy, in order to keep pace with the observed movements of the heavenly bodies, added epicycle to epicycle. It was most welcome, then, when Descartes made a fresh start:

“Instead of the great number of precepts of which Logic is composed, I believed that the four following would prove perfectly sufficient for me, provided I took the firm and unwavering resolution never in a single instance to fail in observing them.
“The first was never to accept anything for true which I did not clearly know to be such; that is to say, carefully to avoid precipitancy and prejudice, and to comprise nothing more in my judgment than what was presented to my mind so clearly and distinctly as to exclude all ground of doubt.
“The second, to divide each of the difficulties under examination into as many parts as possible, and as might be necessary for its adequate solution.
“The third, to conduct my thoughts in such order that, by commencing with objects the simplest and easiest to know, I might ascend by little and little, and, as it were, step by step, to the knowledge of the more complex; assigning in thought a certain order even to those objects which in their own nature do not stand in a relation of antecedence and sequence.

“And the last, in every case to make enumerations so complete, and reviews so general, that I might be assured that nothing was omitted.”4

It would be hard to conceive a more complete programme for the scientific era. Descartes lays down all its leading principles: here is set out the method of analysis which has carried us through molecules to atoms, through atoms to protons, electrons, and neutrons, and now threatens to dissolve these in mere measurements which are measurements of nothing; here too is that conviction that the simple contains the explanation of the complex, which leads to the denial of objective reality to aesthetic and moral qualities because these only appear at a stage of high development and advanced complexity.5

But it is noticeable that the first precept is ambiguous. That it should be so was inevitable, but it is also fatal. Descartes will never accept as true what he does not clearly know to be such; and this he paraphrases as meaning that he will never include anything in his affirmations which is not presented to his mind so clearly and distinctly that he could have no occasion to doubt it. But does this include Isaiah’s vision? or such a flaming apprehension of God as is recorded in Pascal’s celebrated fragment? And if not, why not? To the subjects of these experiences there was nothing about them either confused or questionable. Why is my perception that 2 + 2 = 4 to be regarded as either more clear and distinct, or more compelling of acceptance, than Isaiah’s perception of the Holiness of God? The truth is that in order to make his method work at all, Descartes was obliged to state its first principle in terms which covered much more than he intended; for if he had limited his terms to the scope of his intention, he would have been driven to assert a unique claim to truth on behalf of mathematics, and would have drawn upon himself the contradiction of all who dispute that claim. His position has plausibility because his fundamental precept is so stated as to be unexceptionable, but is then without notice so restricted in interpretation and application as to make the second and third precepts appropriate successors to it.

If we consider the Cartesian revolution as a moment in the history of thought, it appears as at once the repudiation of scholastic or formal Logic and the substitution for it of its own unrecognised principle. For the connexion between mathematics and formal Logic is intimate.

The influence of mathematics upon philosophy has been very close and very formative. It supplied an ideal for the search after knowledge, because it moves by necessary steps to a conclusion as certain as its starting-point. It offers an example of perfect cogency in its process and perfect certainty in its result. It was natural that philosophers should see in it the type of true knowledge. Plato, himself a mathematician, followed its lead; for him Physics was a science as purely mathematical as it is for Whitehead or Jeans; Whitehead is fully conscious of the close similarity between his cosmogony and that outlined in the Timaeus.6 But Plato’s Metaphysic showed the marks of the same influence. All the “propaedeutic studies” outlined in Book VII. of the Republic as a fitting preparation for that Dialectic which leads to apprehension of the Idea of Good, are mathematical; and the schematisation of Ideas to which the whole process points has many of the characteristics of a geometrical pattern.

This mathematical ideal of knowledge has often been valued for precisely those qualities which constitute its fatal defects: these are its indifference to Time and its precision. There is, of course, a great intellectual satisfaction in the elaboration of a chain of thought where every term is precisely understood and every step in the argument is appreciated as both necessary and secure. The intensity of that satisfaction led men to regard what occasioned it as the true type of knowledge. If our apprehension of the phenomenal and historical world never corresponds to it, so much the worse for that apprehension; but also—so much the worse for the phenomenal and historical world. For it was perceived that in the study of that world it was not our faculties but the subject-matter itself which refused to correspond to the ideal of knowledge. So there had to be devised a realm of entities apprehensible only by the mind, without any interposition of sense, which should be the counterpart of a true knowledge; and this must be regarded as the real world, or realm of true Being; for what is altogether real is what answers to knowledge most perfectly; to; pantelw¤~ o[n, pantelw¤~ gnwstovn.7 This realm of reality is changeless, not only in the sense in which a Law of Nature is an unchanging principle governing a process of change, but in the sense of having no reference to change or process at all.

The objects of pure mathematical “knowledge” answer to these requirements. The triangle is unchangingly what it is defined to be, and possesses unchangingly the qualities which it is proved to possess. Here is the field of study in which every term is fully understood and every step can be both necessary and secure; hence its deep fascination for those who are skilled in it. Hence also its immense and pervasive influence on all reflection concerning the nature of knowledge and the way to reach it. For traditional Logic is in form, and largely in substance, applied mathematics.

It is not easy to be sure what Aristotle took to be the real function of the discipline set out in the Prior Analytics, in which are given the rules for the Syllogism. What is quite clear is that that treatise is not his account of the way in which knowledge is reached. It gives the rules, not for seeking truth but for conducting argument. Sometimes those two are combined in one intellectual activity; but not all argument is controlled by a desire for truth. Desire to refute an antagonist is a more frequent motive. Among the Greeks, Argument was a fashionable game; it was necessary to have rules about it and a means of deciding who had won. For such a purpose the Prior Analytics is admirably designed. But whether in earnest or in fun, cogency in the process is, when attainable, of vital moment in argument. And cogency belongs to the regions of thought where every term is fully understood and each step can be both necessary and secure.

The key to necessary inference is the Universal, and therefore the history of Logic is largely the record of controversies concerning the Universal, from Plato’s Ideas and Aristotle’s Forms, through the conflict of Realists, Nominalists and Conceptualists, down to discussions of the concrete Universal in our own day. For purposes of that truly necessary inference, of which the syllogism is the type, a universal proposition is necessary. How is it to be obtained? There are only two ways of gaining the initial security, without which the whole process is insecure. One of these is “complete enumeration” of all instances comprised in or under the Universal. But this is seldom applicable; and if it is, no inference is necessary, for the observation in which it consists can detect the fact stated in the conclusion without going through the inferential process at all. The other way of reaching a secure Universal is to treat it as a definition; but then it can only be referred to experience hypothetically. When once the term triangle has been defined, the equality of its interior angles to two right angles can be proved—provided, of course, that the Euclidean postulates and axioms are admitted. But it does not follow that the interior angles of any wooden object that appears triangular are equal to two right angles. We can only say “If this is a triangle, then its interior angles are equal to two right angles”. As a matter of fact we know that it is not a triangle; for if it is made of wood, its sides are sure not to be geometrically straight, and if we really want to know the sum of the degrees of its interior angles—an unlikely yearning—there is nothing for it but to measure each and add up; and then, of course, the measurement will not be mathematically exact.

The familiar illustration is a syllogism opening with the proposition—All men are mortal. This certainly cannot be established by complete enumeration, for not all even of men already born have as yet died. If approached in that way, the proposition is a generalisation which has no irresistible claim to universality. It can only acquire universality by becoming a definition or part of one—Man is mortal. That could be justified if it could be shown that essential elements in human nature include or supply the causes of mortality. But even so we should not have certainty that any particular man now living is mortal, for it always might be that in him evolution had produced a specimen of a new species in which the cause of mortality is automatically counteracted by new organic adjustments. If it be urged that he would not be a man, I must reply that this is to save logical consistency by the sacrifice of rational interest.

And if these methods fail to give us secure knowledge, certainly Induction is no better. No one ever supposes that the rules of inductive inference as formulated by Mill or by anyone else can give us certainty. Mill indeed has the distinction of having affirmed that essentially real inference is from particulars to particulars; and Bosanquet echoes this in his famous definition: “Inference is the indirect reference to reality of differences within a universal by means of the exhibition of this universal in differences directly referred to reality”.8 But Bosanquet’s “universal” is not a term or proposition at all; it is a systematised apprehension of a group of interrelated facts. We need have no hesitation in assenting to the summary criticism of traditional Logic which declares that Deduction has no right to its starting-point and Induction has no right to its conclusion.

But of course this only holds if we suppose that knowledge is concerned with the world of actual experience. And we find ourselves confronted with this paradox: if knowledge is of actual experience it lacks cogency; if it has cogency it is not concerned with actual experience.

In the traditional Logic there is much which is of permanent value concerning Terms and Judgements. For here we are in the realm of direct apprehension, and the manifold variety of this apprehension is recognised and articulated by the traditional Logic, as, for example, most conspicuously by Aristotle in his doctrine of Categories. But when the traditional Logic deals with inference, its concern for cogency leads it to treat all inference according to the mathematical model, and in effect to reduce all thinking to mathematics. For its laws of inference take terms (to use its own language) in extension only. Its laws with regard to Excluded Middle, Illicit Major, Illicit Minor, all deal with extension only. The fact that such a heresy as Hamilton’s Quantification of the Predicate was even possible goes far to prove this contention. These rules treat a Universal as a class or an area, within which or without which the particular instances in question fall. In its essence it is either arithmetical or geometrical. It is obscurely yet truly an application of the science of Quantity; and the name of that science is Mathematics.

That this was bound to be so we may perceive by approaching the matter from another side. As we study Greek philosophy and the continuation of one strand in it into Scholasticism, we become aware that, except in relation to Ethics and Politics, it is all in fact, though not in intention, more intimately a study of cogent mental process than of empirical reality. The Astronomy which Plato includes among the propaedeutic studies is to dispense with all reference to the observed movements of the heavenly bodies; it is a purely notional discipline. And if it be true that transcendent entities were conceived by Plato to exist in correspondence with the perfectly pure notions thus studied, this was not because they were actually experienced,9 but because “what is perfectly real is perfectly knowable”, and the counterpart of perfect intellection is perfect reality.

Aristotle did not accept the doctrine of self-existent Ideas. But he did believe in Real Kinds—gevnh—and this belief persisted into the Scholastic period. There is very much in Aristotle himself which represents another line of thought, and to these elements in his whole body of teaching we must give attention shortly. But the notion of Real Kinds, finding their common characteristic in a Universal, held the field for centuries, so that the great logical controversies concerned the nature of Universals. And here the traditional Realism of the Church was indispensable to any belief in the validity of thought. The test of a doctrine was not that it should be authenticated by experience, but that it should be established by inevitable process from unquestioned premises. That was part of the trouble about Copernicus and Galileo. The dispute arose because the conclusion reached was novel; but in the background of thought was the apprehension that this novel conclusion represented a new mentality—a mentality that would discard the whole established apparatus of thought, as it did when Descartes induced in himself a doubt of all but his doubt.

Now all this, as we said, was inevitable, because the human mental capacity was fully developed, but its store of systematised observation was very scanty. The great strength of Greek thought is in the realms of Politics and Ethics—or in other words in the study of the field of human action; if to this is added Logic, that is still a study of a sphere of human activity. Physics and astronomy were in a rudimentary state, so that Mathematics had not its point of contact with sense-experience which has in our time enabled it to appear as the master-science of reality. In other words, the available material on which the human mind had to work consisted mainly of the processes and activities of the human mind itself. It was this part of the tradition which was able to establish itself. And inasmuch as in other departments there was no possibility of testing thought to any great extent by external reference, it was inevitable that it should be tested by its own standard of internal coherence. The result of its processes was transferred to the external world by means of the dogma which correlated perfect reality with perfect intellection.

That was not the whole, nor the profoundest element, of Greek philosophy. Plato’s combination of wide range and penetrating insight cannot be summarised in terms of scholastic or formal Logic; but still more important is the fact that Aristotle added to the treatise in which he laid the foundations of that Logic another in which he set forth the proper procedure for a mind bent on the acquisition of knowledge. The Posterior Analytics is a far more difficult work than the Prior Analytics and exerted far less influence. But whereas the easier work is mainly concerned to give rules for conducting a process of inference from established or assumed knowledge, the harder work is concerned with the more vitally important stage of attaining to the knowledge from which inferences could be drawn. It would be out of place to discuss here the Aristotelian scheme; it is enough for our purpose to call attention to the stages by which such knowledge is to be reached. They are five: Sensation, Memory, Experience, “Induction”, Reason: ai[sqhsi~, mnhvmh, ejpeiriva, ejpagwvgh, nou¤~.10

Plainly this is not a technical analysis of inductive inference such as we find in Mill’s celebrated Methods.

It is in fact something much more valuable, and the vagueness of some of its terms is part of its merit; for what it describes is not a cut-and-dried procedure according to rule, but the activity of living thought with all the elasticity and delicate adjustment of response which is characteristic of life. The process begins with sense-perception—the mind’s first apprehension of its data. But these must be stored in the mind. If the mind is only conscious of them as they pass, they may initiate processes of association but not of reflection or scientific ratiocination. So in addition to sensation there must be Memory, But this again will only clog the mind if it is unsystematic; here, therefore, Aristotle introduces an element vital to such thought as he is describing but quite incapable of direction by precise formulae; this is “experience” or, to give the suggestion of the word more fully, aptitude due to familiarity. By means of such aptitude the student is able to enter on “Induction”, or, more closely, Adduction of Relevant Instances. And then Reason—by an unregulated intuition—universalises the result, or detects in the relevant instances the universal which is to be the mainspring of subsequent deduction. The apprehension of principles by Reason is always intuitive.

How modern that is! It is hardly distinguishable from the method of scientific advance outlined by Poincaré in his Science and Hypothesis. The three last stages are specially significant. Experience: it is only the man familiar with the subject who is qualified to judge the instances provided by Sense-Perception and stored in Memory, and to say which are relevant and worthy of consideration. Only the Historian can securely estimate historical evidence; only the lover of Art can select the pictures on which may fairly be founded a generalisation concerning Italian or Flemish painters; only the wise man—Aristotle’s frovnimo~—can say what is the rule by which action in given circumstances should be regulated or judged. And then, when Experience has facilitated Adduction of Relevant Instances, Reason attains to the comprehension of the Essence of what is under consideration by right of its own inherent faculty for universalisation. So Knowledge (ejpisthvmh) is reached; what it apprehends is Essence (oujsiva); the formulation of the essence so apprehended is a Definition (oJrismov~); and from the definition Demonstration (ajpovdeixi~) may proceed. And so the merry game of the Prior Analytics and of Scholastic Logic is set agoing.

Now Aristotle does not seem ever to have decided clearly what is the relation of the Essence apprehended by Reason to the initial data of sense-perception. He rejected the doctrine that the Essence is a transcendent entity apprehensible by pure mind; yet he certainly did not regard it is an adjectival quality of the objects of sensation. Perhaps it would not be unfair to say that having refused to exalt the Essence as a transcendent entity, he reached the same result by depressing that in which it is found as a mere substrate—uJpokeivmenon. He did once at least offer the suggestion that knowledge (ejpisthvmh) is a potentiality of which the realised activity is contemplation (qewriva)11—a doctrine which anticipates Spinoza’s cognitio tertii generis which is scientia intuitiva.12 But he did not follow this through to the thought of the apprehension of the sensible world in all its manifold particularity by an intelligence able to grasp all the universal principles which that would illustrate. On the contrary the perfect intelligence of God was supposed to have no apprehension of the material world just because this will not perfectly fit into intellectual forms; this world is not a proper object of knowledge; God, being perfect intellection, therefore knows nothing about it, and has for His occupation the thinking of thought.13 Thus at the critical points Aristotle dropped out of his pattern the thread that connects him most closely with modern science and philosophy. If experience is in any way recalcitrant to the forms of pure thought, it must be ignored, and pure thought is left in possession of the field.

But even pure thought is not thought about nothing at all. It is thought about the aspects, functions, elements of reality which are intellectually apprehensible with perfect clearness and are capable of definition. Experience presents us with a multitude of men and another multitude of dogs. It is evident that these have something in common with each other, which may be called Animal Life; it is also evident that the members of each multitude have something in common with one another which they do not share with members of the other multitude—human nature in the one case, canine nature in the other. There is therefore ground for setting up two Real Kinds—Man and Dog—and considering the characteristics of these. But whereas the definition of a Triangle states the whole nature of Triangle as such, no definition of Man states the whole nature of Humanity as such. A doctrine of Real Kinds, with definition of essences by genus and differentia, is not going to help living thought to make much reliable progress with regard to such a Kind as mankind.

And yet the Universal is the key to cogent reasoning. Can we get further if we pick up the thread that Aristotle dropped and follow that? This will be in effect the method of the Concrete Universal, which has often been described as the distinctive contribution of modern thought to Logic. But at once a distinction must be made. For the unfortunate modern philosopher can never for a moment ignore the problem of Time or Process. The Scholastic, having distinguished his Real Kind by definition of its Essence, treated it as one of the unchanging constituents of eternal reality. Though there was a process by which he reached his knowledge there was none within that knowledge itself and none within the subject-matter of that knowledge. Mathematics supplied the norm. There is process in the discovery that equilateral triangles are equiangular: but there is no process in the triangle, and when once the discovery is made there is no process in the knowledge of it. That knowledge is secure and static. Its only defect is in relevance to anything besides its own regulations.

But the Cartesians altered all that. For them the natural world was the centre of interest. For them Geometry was no longer a study of figures precisely corresponding to their definitions, but the articulation of the spatiality of the world as apprehended in experience. And the scientific process which has its very life in this constant reference to experience has led us to conceive the world as perpetually changing. Even in Physics, the object studied is a process; Space is no longer considered in isolation from Time, for Motion, which involves them both, is the initial fact. Pure Mathematics alone is occupied with the Timeless. Elsewhere the Real Kind with its changeless Essence is gone, and its place is taken by the evolutionary Species.

As far as I know, the momentous consequences for Logic which are implicit in the idea of Evolution were never appreciated until attention was called to them by Mr. Michael Foster in an article contributed by him to Mind in January 1931.14 Mr. Foster begins by pointing out that the Universal as conceived by Aristotle determines its own particularisation. Thus it is inherent in the general notion of Triangle that triangles should be distinguished as equilateral, isosceles and scalene; for if there are three sides, then all three may be equal, or two may be equal, or none may be equal. But this illustration of the principle illustrates also its difficulty; for this differentiation does not carry us over from the Universal to the Particular of actual experience. Rather it illustrates Plato’s demand for the insertion of the How Many (o{posa) between the One and the Many, while we leave the multitude of particulars to “go to infinity”.15 But that is to throw up the game, for the crux of the problem is the relation, not of genus to species, but of both genus and species to individuals.

Plato and Aristotle were, both of them, vividly aware of the problem. The Ideal Theory of Plato, with its difficult appendix concerning a realm that lies between Being and Not-Being—the realm of phenomena—is an explicit recognition of it. Aristotle’s difficulties with Matter—the substrate of the correlate of definitions—are in like manner a recognition of it. But the scholastic philosophers tended to forget it, and to pass, by an unrecognised and uncriticised form of the Ontological Argument, from the completeness of the intellectual system to an affirmation of its existential reality. Thus when modern science began its enquiry into efficient as distinct from formal causation, as the way to an explanation of the existence of actual things, it was following a method so diverse from that of the accepted Logic that Science and Logic tended to part company, to the serious injury of both. And the only method of reconciliation is by the discovery of some form of Universal which “can be shown to include the residual element”—the individuality of the extant thing—“within the scope of its determination”.16 Bosanquet, who treated Logic as the “Morphology of Knowledge”, was conscious of the problem and attempted to solve it by means of reflections drawn from the “comparative” sciences; but the attempt is only successful in relation to the field of those sciences. Here I quote Mr. Foster’s comment in full:

“Scholasticism had declared that the whole being of a substance was determined by its character (substantial form). The new philosophy based on Galilean physics had retorted that everything was determined to be what it was by its causal relation to the infinite system of everything else, and that its character or form was wholly inoperative. Bosanquet, basing himself upon the ‘comparative sciences’, cries in reaction against the physicists, ‘Form is operative’. He forgets that the physical sciences continue to exist none the less because the comparative sciences have arisen by their side, and that his results can claim an application at least no more extended than the sphere of observation from which they were derived.
“The understanding of Bosanquet’s Logic becomes suddenly illuminated by the recognition that it is derived almost exclusively from reflection upon the ‘comparative’ sciences (Botany, Zoology, Anthropology) and its conclusions then applied uncritically to the whole of knowledge. The comparative sciences are, roughly, the sciences of Life; and to them Bosanquet’s conception of ‘individual system’ is, as we shall see, genuinely adequate. But below them exist the sciences of matter, which work with the conception of causal law; and above them exist the sciences of spirit (the historical sciences) which work with the notion of the individual. Since system is intermediate between (causal) law and individual, it was inevitable that an attempt to extend this conception beyond its proper sphere into the spheres both above and below it should lead to that confusion of thought and terminology in which Bosanquet’s doctrine is in fact obscured.”17

But this is not the end of the trouble, nor the worst of it. For, as Mr. Foster goes on to show, Bosanquet’s doctrine requires, not only that “there must be a systematic articulation in the world of universals”, but that “this articulated system must have an historical existence in space and time”; and the arguments (so far as he offers any) by which Bosanquet accomplishes the transition “are in essence the ontological argument from the organised complexity of a system to the necessity of its real existence”.18 And this is a mere postulate of which the legitimacy is by no means apparent.

Logicians had continued to insist upon the doctrine of the “determining activity of the generic concept” in protest against “the opposite doctrine that physical causation is the only active determinant”.19 And philosophers who based themselves primarily on the scientific investigation of phenomena in the causal series retaliated by adopting a short way with Logic. It was perhaps impossible to do justice to both claims until the idea of Evolution had taken possession of men’s minds; but when once that had happened, it was possible to put in place of “the activity of the universal in determining its own specific determinations”, “the development of the species through actual generations. The race which develops is the concrete universal which needs no ontological argument to add concreteness to it.”20

It is certainly true that the “race which develops” is different from the Universal as commonly understood, for it is itself, from one point of view, a Particular; but from another point of view it is not a Particular, because it is certainly not apprehensible by sense. The fact is that here the sharp division between thought and sense is already broken down; but the generic character of the race does not completely pervade and interpenetrate its members; each has many characteristics which cannot be regarded as determined by it. It is only when we come to “the historical individual” that we return to that synthesis of sense and thought with which all intelligent experience begins. For this “historical individual”—by which is meant not a person but such an entity as the British Empire or such an occurrence as the Renaissance or the Reformation—is a Universal which profoundly penetrates its constituent elements and points to that ideal wherein the distinction between accidental and essential qualities is eliminated. It is true that Bosanquet glanced at the “historical individual” as a possible solution of his problem, and explicitly rejected it.21 His reason for doing so is to be found in his dependence upon the “comparative sciences”.22

All modern thought and science is historical in method. Whatever is studied is considered not only as it is now observed to be, but in the light of the process by which it has come to be. Natural History, until recently, was the classification of existing species; now it is quite equally concerned with the origin of species. Geology is not only a study of the crust of the earth but an examination of the question how that crust has been formed. Astronomy aspires to give us a history of the heavens. Similarly in the Humanities we no longer treat the utterances of prophets or philosophers as oracles to be accepted or rejected, but seek to understand them in relation to their historical context, reconstructed as fully as possible, and to evaluate them in the light of that understanding. This use of the historical method is the main distinguishing characteristic of our own modern thought as compared with the thought of all former ages; and it coheres closely with the notion of Evolution as a general term for the process whereby things not only are but come to be, and indeed have their being in the process of coming to be.

Now of such a process there cannot be exact analytical knowledge of the mathematical type. The Greeks sharply distinguished the realm of Being from the realm of Becoming. Knowledge, and accordingly Logic, were concerned with the former; the latter was the sphere of Opinion or Belief, and of Art or Skill, and the study of the process by which this was reached was, as it still is, Psychology. Knowledge was concerned with the unchanging Forms, the Ideas, the Kinds of Being (gevnh tw¤n o[ntwn). The particulars of our actual experience might “go to infinity”; science was not concerned with them. This inevitably, though undesignedly, involved the elaboration of an a priori science, which was imposed dogmatically on a more or less recalcitrant experience. The heart of this method was the conception of static Real Kinds, of which the Form or Essence could be defined, so that from the definition valid inferences could be drawn by deduction and the volume of ascertained truth thus be increased. This coheres closely with such a conception of the act of Creation as is formed by an acceptance of the first chapter of Genesis as literal historical truth. Perhaps that is part of the reason why Darwin’s success in making the popular mind aware of the implications of Evolution provoked so great an uproar; it not only set aside the literal and (at that time, though not always) traditional interpretation of a passage in the Bible, but it assaulted the doctrine of Real Kinds which was the central position of the traditional Aristotelian logic in its conflict with Empiricism.

The philosophers of the post-Reformation period started from the science, specially the physical science, which was entering then on its independent career.23 Two novelties in their equipment and method call for special notice. One is that Mathematics was regarded as supplying an account of the physical world. To Plato its great value was that it effected a transition away from the physical world,24 which by its help we might learn to leave behind; it was the science of extended form as apprehended by the pure intellect. But for the Cartesians Mathematics is a study of the extended world. An independent science of Arithmetic, and (still more influential) of Algebra, was growing up side by side with Geometry, and this made possible the universal application of numeration as distinct from measurement throughout the natural world, whereby modern Mathematics obtained its distinguishing characteristic of being applicable to nature; and besides this, Geometry itself was treated as thus applicable. Einstein is reported as having said,25 that the great distinction between Euclidean and Cartesian Geometry was that the Greeks lacked the conception of space as a single continuum. From this it would follow that the Greeks studied geometrical figures as species of an Idea, conceiving them as related to one another not physically at all, but only intelligibly or logically. But if every geometrical figure is a part of one single space, it follows that geometrical knowledge is knowledge of that single spatial system which comprises (or even which is) the physical universe.

The Cartesians discarded Logic. They did not need it as a criterion of truth, because the guarantee of truth was found in the perception of clear and distinct ideas; the power by which the mind could judge the principles of Logic to be valid enabled it also to grasp the truth of propositions or arguments without reference to those principles. As a normative science telling us how to think correctly it was otiose. As a basis for physical science it was even misleading; for it led men to seek for the explanation of things in the self-differentiation of a generic concept and not in the actual process of efficient causation. Some philosophers, indeed, among whom Malebranche is the most conspicuous, retained the Intelligible Forms of Greek and Scholastic Logic as objects of the Divine Thought informing the Divine Purpose, which was itself the sole cause of actual occurrences. But this, though sound in itself as I believe, is to abandon the notion of the Form as itself determining events. Formal causation, as such, had disappeared. What Malebranche represents is an attempt to salvage Final Causation and what had been serviceable in Formal Causation from the floodtide of Cartesian Rationalism and pure Empiricism, which were alike in treating Efficient Causation as alone truly causative and were thus sweeping human thought into the abyss of mechanical determinism.

The Cartesians were right to repudiate Scholastic Logic as a discipline of universal applicability. They were not right in discarding the requirement of a discipline which should result in accurate and valid thought. They trusted to a “natural light”—the philosophic counterpart of the “inner light” of the Quakers; they spoke of Reason as “the eye” of the mind. We must accept their ultimate appeal on each occasion to the apprehensions of the individual mind. But this need not involve acquiescence in the notion that all men have, by natural endowment, a faculty of “seeing” that a proposition is self-evidently true or that an action is self-evidently right. Such intuitions are only reliable when they are the apprehensions of trained minds. The Middle Ages had supplied the apprenticeship for that Conscience and Reason to which Luther and Descartes and Locke appealed. Such insight as is a guarantee of truth only comes at the end of a long process of training alike in the individual and in the race. But if we have understood this, we must realise that the end of our discipline is the escape from our temporal and personal contingency, not into a timeless realm of static Truth, Beauty and Goodness, but into the full historic process wherein both we and those sublimities have actual being.

Now the discipline which is aiming at this result must be itself concerned with process, whereas the traditional Logic was concerned with the unchanging Forms and Kinds. Kant perceived that the mental act which is the ground of the validity of a conclusion is not immediate (as the Cartesian language implied) but conditioned—it is true that he says “transcendentally conditioned”, and the value of his argument is diminished by the presence of that perplexing adverb. The conditions in question are discovered by critical reflection, which resolves the act of knowing into its constituent elements. Unfortunately by the introduction of the term “transcendental” Kant lifts the process of conditioning or mediation out of the time process, and though the object of “Transcendental Logic” is a “process” it is declared to be a “timeless process”. Yet still he does recognise, in opposition to traditional Logic, that the study of a process may still be a study of the grounds of validity—not only a study of how men come to hold conclusions, which is Psychology, but a study of their justification in holding conclusions, which is Logic. Hegel seized upon this and developed it, so that for him Logic is the science of the Dialectical Process; but he still regards this as timeless, and so makes it after all only a variant of the self-differentiation of the Intelligible Form.

But what is there, in fact, of which Logic so conceived can be the study? As we descend through the scale of categories from the more concrete—such as Life or Personality—to the more abstract, we arrive at Mathematics, the science of Quantity, which “tells us something about everything, but very little about anything”. Logic, if it is to be the universal science of the validity of thought, must be more abstract still. Can it be occupied with Being as such? But that, as Hegel showed, is indistinguishable from Not-Being. To Be is to be Something. The study of existence apart from all study of what exists is the study of sheer vacuity. Or perhaps Logic is the study of thought, not as an actual psychological process but as the pure activity of Mind acting according to its true nature. But then we have no data. Such thought may exist in God; in us it certainly does not, and so far as we have need of Logic as a discipline, we need it to aid our estimate of our approximations to valid thought or true knowledge, in an experience where these exist side by side with, and at first undistinguishable from, prejudices, casual opinions, and products of mere self-assertiveness.

The fact is that the Logic of Inference as traditionally conceived has no object or sphere of its own at all. There is a certain application of mathematical principles which gives rise to Formal Logic, and this has complete validity in relation to certain types of argument, and a certain very real value for all types as supplying a norm of cogency, but provides no criterion of the subject-matter to which it is properly applicable. We now recognise that the understanding of all other existing things is to be reached in part by study of the historical process which has led to their being what they are; is it not probable that in like manner the understanding and consequent evaluation of thought is to be reached by a study of the history of thought? At any rate that is the study which is chiefly offered in our Universities as the required discipline of the mind. No one now teaches Philosophy otherwise than by teaching the History of Philosophy; and though this would not be everywhere avowed as a principle in relation to Logic, yet in practice that subject too is largely taught by reference to the theories of Logic set forth by various thinkers in different ages, and to the process by which one of these gave rise to another either through the interplay among themselves of the notions expounded, or through the reaction upon these of processes in vogue in various branches of scientific enquiry. The conclusion we reach is this: that the discipline required to perform the function traditionally ascribed to Logic is the History of Thought, and especially the History of Philosophy.

Especially—but not only: for we have returned to the doctrine of Aristotle at the close of the Posterior Analytics. Understanding, Appreciation, Knowledge come by the process of Sensation, Memory, Training by Experience, Adduction of Relevant Instances, Rational Intuition. The last is the “natural light”, the “eye of the soul” relied on by the Cartesians. But it only deserves such reliance when the stages of the process have been thoroughly traversed; and of these the most determinant is Training by Experience. For thinking, as will later appear more clearly, is grounded in, and is an extension of, the adjustment of organism to environment or vice versa. Therefore the experience must be relevant, or else the aptitude developed by it will not be relevant. Thus, for example, experience of engineering does not train a man to appreciate Flemish painting. For this reason there is a different discipline for the mind in every department of enquiry, and only the mind trained in relation to any department is capable of secure judgements with regard to it. Lawyers are as a rule not good judges of historical evidence; mathematicians are not qualified as such to pronounce upon questions calling for spiritual perception. There is a different discipline for every department. The traditional Logic has a certain value as a standard of reference and norm of procedure. It is, as has been said, a special application of the principle of Mathematics, and has that degree of universal relevance which is involved in the truth that “Mathematics tells us something about everything”, but only that degree which is compatible with the further truth that “Mathematics tells us very little about anything”. It is all extremely useful preparatory discipline, but it is not a universal guide to valid thinking.

It is important to determine our use of terms. We might decide, as the upshot of this discussion, to confine the term Logic to what is recognised to be a very subsidiary discipline; or we might keep it with a modified significance to represent the discipline that results in accurate and adequate thinking. What always leads to dangerous confusion is to use it with an undefined significance, as is commonly done by persons who say that Life is wider or richer than Logic. If they mean that Life is too rich to be articulated in the forms of Deductive or Subsumptive Logic, that is true and may be important. But there is usually in the minds of those who use such expressions a sense that Logic covers the field of intelligence, and their meaning is that we often get on better if we stop trying to think accurately. Well, we may; but if so that is because we think badly, not because good thinking can ever be misleading. And there is a wider use of the term in vogue, as when we speak of “the Logic of facts” or “the Logic of the situation”. It would be well to adopt such a use of terms that these phrases would not be metaphors, but would point to the only Logic that has any bearing on the occasion. With such a use of the term Bosanquet says that “freedom is the logic of individuality”.26 And he adopts as a synonym of Logic “the spirit of totality”.27 With the same conception in mind he declares that “love is the mainspring of Logic”,28 for “by Logic we understand, with Plato and Hegel, the supreme Law or nature of experience, the impulse towards unity and coherence (the positive spirit of non-contradiction) by which every fragment yearns towards the whole to which it belongs”.29 “All logical activity is a world of content reshaping itself by its own spirit and laws in presence of new suggestions; a syllogism is in principle nothing less, and a Parthenon or Paradise Lost is in principle nothing more.”

This use of the word is fully justified, and it seems better to retain the word with this use than to abandon it. For this self-shaping of our experience is apprehensible by mind and follows principles which mind recognises as its own, while mind fulfils its functions precisely in that apprehension and that recognition.

  • 1. Op. cit. pp. 23, 24.
  • 2. Discourse on Method (Veitch’s translation), p. 25.
  • 3. Professor Boyce Gibson claims that Descartes was in fact a truly religious man; that may be true, but it does not make his philosophy religiously tolerable. It is urged that his philosophy also is religious, because God is the pivot of it, and apart from the reality of God it would fall to pieces. That also is true; yet it is still not a religious philosophy, for it sets no value on God in Himself, but only as the lynch-pin of its own mechanism. It does not interpret the world in the light of knowledge of God, but makes use of God to vindicate its own interpretation of the world, and constructs its concept of Him with that in view. He is to be used for our purpose, not we for His. This is the essential principle of magic, which is thus found as a canker at the heart of Rationalism.
  • 4. Discourse on Method (Veitch’s translation), p. 19.
  • 5. “The deliverances of clear and distinct consciousness require criticism by reference to elements in experience which are neither clear nor distinct.”—Whitehead, Adventures of Ideas, p. 348.
  • 6. Cf. Process and Reality, pp. 129–133, and many other passages.
  • 7. Plato, Republic, 477 A.
  • 8. Bosanquet, Logic, vol. ii. p. 4.
  • 9. As a matter of fact I think Plato was predisposed to believe in the reality of his Ideas by an experience which he interpreted as an actual apprehension of the Idea of Beauty: see my article on “Plato’s Vision of the Ideas” in Mind, N.S. 68. But this is at most the psychological predisposition, not the philosophical ground, of the belief.
  • 10. Posterior Analytics, ad finem.
  • 11. Metaphysics, 1087 a 10–25; cf. De Anima, 417 a 21–29.
  • 12. Ethics, Part II, Props. XL.–XLII.; Part V. Props. XXV.–XXVII.
  • 13. Metaphysics, 1074 b 32–35.
  • 14. Dr. Schiller had opened up the subject in Formal Logic: see pp. 55–57 and 333. But he did not develop the argument with the thoroughness which is found in Mr. Foster’s treatment of it.
  • 15. Philebus, 16 c.
  • 16. Foster, Mind, N.S. 157, p. 7.
  • 17. L.c. pp. 13, 14. Mr. Foster, in a footnote, supplies the qualification of his remarks about the sciences of matter which recent physical theories may be held to require.
  • 18. L.c. p. 15, with quotations from Bosanquet’s Logic there given: Logic, i. pp. 225 and 230. The difference between development and mere change consists, I take it, in the fact that development is change either in the direction of a fixed form in attaining to which the development culminates and ceases, or in the ever fuller articulation or application of a principle which is dominant throughout the process.
  • 19. L.c. p. 16.
  • 20. L.c.. p. 17.
  • 21. Cf. The Principle of Individuality and Value, pp. 77–81.
  • 22. Cf. p. 99 supra.
  • 23. Throughout this section I am indebted to suggestions made to me in private correspondence by Mr. Michael Foster. But he is not responsible for the use I make of his suggestions.
  • 24. Republic, 521 D.
  • 25. In a lecture delivered at Oxford.
  • 26. Principle of Individuality and Value, p. 80.
  • 27. Ibid. p. 23.
  • 28. Ibid. p. 341.
  • 29. Ibid. p. 340. Cf. also the eloquent exposition of the Logic in Great Art, pp. 332, 333, of which the closing sentence is quoted above.
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