I SHALL first[143] give a restatement, partly historical and partly explanatory, of Kant’s main argument as contained in the enlarged Introduction of the second edition.

There were two stages in the process by which Kant came to full realisation of the Critical problem. There is first the problem as formulated in his letter of 1772 to Herz: how the a priori can yield knowledge of the independently real.[144] This, as he there states it, is an essentially metaphysical problem. It is the problem of the possibility of transcendent metaphysics. He became aware of it when reflecting upon the function which he had ascribed to intellect in the Dissertation. Then, secondly, this problem was immeasurably deepened, and at the same time the proper line for its treatment was discovered, through the renewed influence which Hume at some date subsequent to February 1772 exercised upon Kant’s thought.[145] Hume awakened Kant to what may be called the immanent problem involved in the very conception of a priori knowledge as such. The primary problem to be solved is not how we advance by means of a priori ideas to the independently real, but how we are able to advance beyond a subject term to a predicate which it does not appear to contain. The problem is indeed capable of solution, just because it takes this logical form. Here as elsewhere, ontological questions are viewed by Kant as soluble only to the extent to which they can be restated in logical terms. Now also the enquiry becomes twofold: how and in what degree are a priori synthetic judgments possible, first in their employment within the empirical sphere (the problem of immanent metaphysics) and secondly in their application to things in themselves (the problem of transcendent metaphysics). The outcome of the Critical enquiry is to establish the legitimacy of immanent metaphysics and the impossibility of all transcendent speculation.

The argument of Kant’s Introduction follows the above sequence. It starts by defining the problem of metaphysical knowledge a priori, and through it leads up to the logical problem of the a priori synthetic judgment. In respect of time all knowledge begins with experience. But it does not therefore follow that it all arises from experience. Our experience may be a compound of that which we receive through impressions, and of that which pure reason supplies from itself.[146] The question as to whether or not any such a priori actually exists, is one that can be answered only after further enquiry. The two inseparable criteria of the a priori are necessity and universality. That neither can be imparted to a proposition by experience was Kant’s confirmed and unquestioned belief. He inherited this view both from Leibniz and from Hume. It is one of the presuppositions of his argument. Experience can reveal only co-existence or sequence. It enables us only to assert that so far as we have hitherto observed, there is no exception to this or that rule. A generalisation, based on observation, can never possess a wider universality than the limited experience for which it stands. If, therefore, necessary and universal judgments can anywhere be found in our knowledge, the existence of an a priori that originates independently of experience is ipso facto demonstrated.[147]

The contrast between empirical and a priori judgments, as formulated from the dogmatic standpoint, is the most significant and striking fact in the whole range of human knowledge. A priori judgments claim absolute necessity. They allow of no possible exception. They are valid not only for us, but also for all conceivable beings, however different the specific conditions of their existence, whether they live on the planet Mars or in some infinitely remote region of stellar space, and no matter how diversely their bodily senses may be organised. Through these judgments a creature five feet high, and correspondingly limited by temporal conditions, legislates for all existence and for all time. Empirical judgments, on the other hand, possess only a hypothetical certainty. We recognise that they may be overturned through some addition to our present experience, and that they may not hold for beings on other planets or for beings with senses differently constituted. Whereas the opposite of a rational judgment is not even conceivable, the opposite of an empirical judgment is always possible. The one depends upon the inherent and inalienable nature of our thinking; the other is bound up with the contingent material of sense. The one claims absolute or metaphysical truth: the other is a merely tentative rÉsumÉ of a limited experience.

The possibility of such a priori judgments had hitherto been questioned only by those who sought to deny to them all possible objective validity. Kant, as a rationalist, has no doubt as to their actual existence. In the Introduction to the second edition he bluntly asserts their de facto existence, citing as instances the propositions of mathematics and the fundamental principles of physical science. Their possibility can be accounted for through the assumption of a priori forms and principles.[148] But with equal emphasis he questions the validity of their metaphysical employment. For that is an entirely different matter. We then completely transcend the world of the senses and pass into a sphere where experience can neither guide nor correct us. In this sphere the a priori is illegitimately taken as being at once the source of our professed knowledge and also the sole criterion of its own claims.

This is the problem, semi-Critical, semi-dogmatic, which is formulated in the letter of 1772 to Herz.[149] What right have we to regard ideas, which as a priori originate from within, as being valid of things in themselves? In so doing we are assuming a pre-established harmony between our human faculties and the ultimately real; and that is an assumption which by its very nature is incapable of demonstration. The proofs offered by Malebranche and by Leibniz are themselves speculative, and consequently presuppose the conclusion which they profess to establish.[150] As above stated, Kant obtained his answer to this problem by way of the logical enquiry into the nature and conditions of a priori judgment.

One of the chief causes, Kant declares, why hitherto metaphysical speculation has passed unchallenged among those who practise it, is the confusion of two very different kinds of judgment, the analytic and the synthetic. Much the greater portion of what reason finds to do consists in the analysis of our concepts of objects.

The concepts which are analytically treated may be either empirical or a priori. When they are empirical, the judgments which they involve can have no wider application than the experience to which they give expression; and in any case can only reveal what has all along been thought, though confusedly, in the term which serves as subject of the proposition. They can never reveal anything different in kind from the contents actually experienced. This limitation, to which the analysis of empirical concepts is subject, was admitted by both empiricists and rationalists. The latter sought, however, to escape its consequences by basing their metaphysics upon concepts which are purely a priori, and which by their a priori content may carry us beyond the experienced. But here also Kant asserts a non possibile. A priori concepts, he seeks to show, are in all cases purely logical functions without content, and accordingly are as little capable as are empirical concepts of carrying us over to the supersensible. This is an objection which holds quite independently of that already noted, namely, that their objective validity would involve a pre-established harmony.

What, then, is the nature and what are the generating conditions of synthetic judgments that are also a priori? In all judgments there is a relation between subject and predicate, and that can be of two kinds. Either the predicate B belongs to the subject A, or B lies outside the sphere of the concept A though somehow connected with it. In the former case the judgment is analytic; in the latter it is synthetic. The one simply unfolds what has all along been conceived in the subject concept; the other ascribes to the concept of the subject a predicate which cannot be found in it by any process of analysis. Thus the judgment ‘all bodies are extended’ is analytic. The concept of body already contains that of extension, and is impossible save through it. On the other hand, the judgment ‘all bodies are heavy’ is synthetic. For not body as such, but only bodies which are in interaction with other bodies, are found to develop this property. Bodies can very well be conceived as not influencing one another in any such manner.

There is no difficulty in accounting for analytic judgments. They can all be justified by the principle of contradiction. Being analytic, they can be established a priori. Nor, Kant here claims, is there any difficulty in regard to synthetic judgments that are empirical. Though the predicate is not contained in the subject concept, they belong to each other (though accidentally) as parts of a given empirical whole. Experience is the x which lies beyond the concept A, and on which rests the possibility of the synthesis of B with A. In regard, however, to synthetic judgments which are likewise a priori, the matter is very different. Hitherto, both by the sensationalists and by the rationalists, all synthetic judgments have been regarded as empirical, and all a priori judgments as analytic. The only difference between the opposed schools lies in the relative value which they ascribe to the two types of judgment. For Hume the only really fruitful judgments are the synthetic judgments a posteriori; analytic judgments are of quite secondary value; they can never extend our knowledge, but only clarify its existing content. For Leibniz, on the other hand, true knowledge consists only in the analysis of our a priori concepts, which he regards as possessing an intrinsic and fruitful content; synthetic judgments are always empirical, and as such are purely contingent.[152]

Thus for pre-Kantian philosophy analytic is interchangeable with a priori, and synthetic with a posteriori. Kant’s Critical problem arose from the startling discovery that the a priori and the synthetic do not exclude one another. A judgment may be synthetic and yet also a priori. He appears to have made this discovery under the influence of Hume, through study of the general principle of causality—every event must have a cause.[153] In that judgment there seems to be no connection of any kind discoverable between the subject (the conception of an event as something happening in time) and the predicate (the conception of another event preceding it as an originating cause); and yet we not merely ascribe the one to the other but assert that they are necessarily connected. We can conceive an event as sequent upon a preceding empty time; none the less, in physical enquiry, the causal principle is accepted as an established truth. Here, then, is a new and altogether unique type of judgment, of thoroughly paradoxical nature. So entirely is it without apparent basis, that Hume, who first deciphered its strange character, felt constrained to ascribe our belief in it to an unreasoning and merely instinctive, ‘natural’ habit or custom.

Kant found, however, that the paradoxical characteristics of the causal principle also belong to mathematical and physical judgments. This fact makes it impossible to accept Hume’s sceptical conclusion. If even the assertion 7 + 5 = 12 is both synthetic and a priori, it is obviously impossible to question the validity of judgments that possess these characteristics. But they do not for that reason any the less urgently press for explanation. Such an enquiry might not, indeed, be necessary were we concerned only with scientific knowledge. For the natural sciences justify themselves by their practical successes and by their steady unbroken development. But metaphysical judgments are also of this type; and until the conditions which make a priori synthetic judgment possible have been discovered, the question as to the legitimacy of metaphysical speculation cannot be decided. Such judgments are plainly mysterious, and urgently call for further enquiry.

The problem to be solved concerns the ground of our ascription to the subject concept, as necessarily belonging to it, a predicate which seems to have no discoverable relation to it. What is the unknown x on which the understanding rests in asserting the connection? It cannot be repeated experience; for the judgments in question claim necessity. Nor can such judgments be proved by means of a logical test, such as the inconceivability of the opposite. The absence of all apparent connection between subject and predicate removes that possibility. These, however, are the only two methods of proof hitherto recognised in science and philosophy. The problem demands for its solution nothing less than the discovery and formulation of an entirely novel method of proof.

The three main classes of a priori synthetic judgments are, Kant proceeds, the mathematical, the physical, and the metaphysical. The synthetic character of mathematical judgments has hitherto escaped observation owing to their being proved (as is required of all apodictic certainty) according to the principle of contradiction. It is therefrom inferred that they rest on the authority of that principle, and are therefore analytic. That, however, is an illegitimate inference; for though the truth of a synthetic proposition can be thus demonstrated, that can only be if another synthetic principle is first presupposed. It can never be proved that its truth, as a separate judgment, is demanded by the principle of contradiction. That 7 + 5 must equal 12 does not follow analytically from the conception of the sum of seven and five. This conception contains nothing beyond the union of both numbers into one; it does not tell us what is the single number that combines both. That five should be added to seven is no doubt implied in the conception, but not that the sum should be twelve. To discover that, we must, Kant maintains, go beyond the concepts and appeal to intuition. This is more easily recognised when we take large numbers. We then clearly perceive that, turn and twist our concepts as we may, we can never, by means of mere analysis of them, and without the help of intuition, arrive at the sum that is wanted. The fundamental propositions of geometry, the so-called axioms, are similarly synthetic, e.g. that the straight line between two points is the shortest. The concept ‘straight’ only defines direction; it says nothing as to quantity.

As an instance of a synthetic a priori judgment in physical science Kant cites the principle: the quantity of matter remains constant throughout all changes. In the conception of matter we do not conceive its permanency, but only its presence in the space which it fills. The opposite of the principle is thoroughly conceivable.

Metaphysics is meant to contain a priori knowledge. For it seeks to determine that of which we can have no experience, as e.g. that the world must have a first beginning. And if, as will be proved, our a priori concepts have no content, which through analysis might yield such judgments, these judgments also must be synthetic.

Here, then, we find the essential problem of pure reason. Expressed in a single formula, it runs: How are synthetic a priori judgments possible? To ask this question is to enquire, first, how pure mathematics is possible; secondly, how pure natural science is possible; and thirdly, how metaphysics is possible. That philosophy has hitherto remained in so vacillating a state of ignorance and contradiction is entirely due to the neglect of this problem of a priori synthesis. “Its solution is the question of life and death to metaphysics.” Hume came nearest to realising the problem, but he discovered it in too narrow a form to appreciate its full significance and its revolutionary consequences.

These statements are decidedly ambiguous, owing to Kant’s failure to distinguish in any uniform and definite manner between immanent and transcendent metaphysics.[155] The term metaphysics is used to cover both. Sometimes it signifies the one, sometimes the other; while in still other passages its meaning is neutral. But if we draw the distinction, Kant’s answer is that a genuine and valid immanent metaphysics is for the first time rendered possible by his Critique; its positive content is expounded in the Analytic. Transcendent metaphysics, on the other hand, is criticised in the Dialectic; it is never possible. The existing speculative sciences transgress the limits of experience and yield only a pretence of knowledge. This determination of the limits of our possible a priori knowledge is the second great achievement of the Critique. Thus the Critique serves a twofold purpose. It establishes a new a priori system of metaphysics, and also determines on principles equally a priori the ultimate limits beyond which metaphysics can never advance. The two results, positive and negative, are inseparable and complementary. Neither should be emphasised to the neglect of the other.

Comment on the Argument of Kant’s Introduction

This Introduction, though a document of great historical importance as being the first definite formulation of the generating problem of Kant’s new philosophy, is extremely unsatisfactory as a statement of Critical teaching. The argument is developed in terms of distinctions which are borrowed from the traditional logic, and which are not in accordance with the transcendental principles that Kant is professing to establish. This is, indeed, a criticism which may be passed upon the Critique as a whole. Though Kant was conscious of opening a new era in the history of philosophy, and compares his task with that of Thales, Copernicus, Bacon and Galileo, it may still be said that he never fully appreciated the greatness of his own achievement. He invariably assumes that the revolutionary consequences of his teaching will not extend to the sphere of pure logic. They concern, as he believed, only our metaphysical theories regarding the nature of reality and the determining conditions of our human experience. As formal logic prescribes the axiomatic principles according to which all thinking must proceed, its validity is not affected by the other philosophical disciplines, and is superior to the considerations that determine their truth or falsity. Its distinctions may be securely relied upon in the pioneer labours of Critical investigation. This was, of course, a very natural assumption for Kant to make; and many present-day thinkers will maintain that it is entirely justified. Should that be our attitude, we may approve of Kant’s general method of procedure, but shall be compelled to dissent from much in his argument and from many of his chief conclusions. If, on the other hand, we regard formal logic as in any degree adequate only as a theory of the thought processes involved in the formation and application of the generic or class concept,[156] we shall be prepared to find that the equating of this highly specialised logic with logic in general has resulted in the adoption of distinctions which may be fairly adequate for the purposes in view of which they have been formulated, but which must break down when tested over a wider field. So far from condemning Kant for departing in his later teaching from these hard and fast distinctions, we shall welcome every sign of his increasing independence.

Kant was not, of course, so blind to the real bearing of his principles as to fail to recognise that they have logical implications.[157] He speaks of the new metaphysics which he has created as being a transcendental logic. It is very clear, however, that even while so doing he does not regard it as in any way alternative to the older logic, but as moving upon a different plane, and as yielding results which in no way conflict with anything that formal logic may teach. Indeed Kant ascribes to the traditional logic an almost sacrosanct validity. Both the general framework of the Critique and the arrangement of the minor subdivisions are derived from it. It is supposed to afford an adequate account of discursive thinking, and such supplement as it may receive is regarded as simply an extension of its carefully delimited field. There are two logics, that of discursive or analytic reasoning, and that of synthetic interpretation. The one is formal; the other is transcendental. The one was created by Aristotle, complete at a stroke; Kant professes to have formulated the other in an equally complete and final manner.

This latter claim, which is expressed in the most unqualified terms in the Prefaces to the first and second editions, is somewhat startling to a modern reader, and would seem to imply the adoption of an ultra-rationalistic attitude, closely akin to that of Wolff.

These sanguine expectations—by no means supported by the after-history of Kant’s system—are not really due to Kant’s immodest over-estimate of the importance of his work. They would rather seem to be traceable, on the one hand to his continuing acceptance of rationalistic assumptions proper only to the philosophy which he is displacing, and on the other to his failure to appreciate the full extent of the revolutionary consequences which his teaching was destined to produce in the then existing philosophical disciplines. Kant, like all the greatest reformers, left his work in the making. Both his results and his methods call for modification and extension in the light of the insight which they have themselves rendered possible. Indeed, Kant was himself constantly occupied in criticising and correcting his own acquired views; and this is nowhere more evident than in the contrast between the teaching of this Introduction and that of the central portions of the Analytic. But even the later expressions of his maturer views reveal the persisting conflict. They betray the need for further reconstruction, even in the very act of disavowing it. Not an additional logic, but the demonstration of the imperative need for a complete revisal of the whole body of logical science, is the first, and in many respects the chief, outcome of his Critical enquiries.

The broader bearings of the situation may perhaps be indicated as follows. If our account of Kant’s awakening from his dogmatic slumber[160] be correct, it consisted in his recognition that self-evidence will not suffice to guarantee any general principle. The fundamental principles of our experience are synthetic. That is to say, their opposite is in all cases conceivable. Combining this conclusion with his previous conviction that they can never be proved by induction from observed facts, he was faced with the task of establishing rationalism upon a new and altogether novel basis. If neither empirical facts nor intuitive self-evidence may be appealed to, in what manner can proof proceed? And how can we make even a beginning of demonstration, if our very principles have themselves to be established? Principles are never self-evident, and yet principles are indispensable. Such was Kant’s unwavering conviction as regards the fundamental postulates alike of knowledge and of conduct.

This is only another way of stating that Kant is the real founder of the Coherence theory of truth.[161] He never himself employs the term Coherence, and he constantly adopts positions which are more in harmony with a Correspondence view of the nature and conditions of knowledge. But all that is most vital in his teaching, and has proved really fruitful in its after-history, would seem to be in line with the positions which have since been more explicitly developed by such writers as Lotze, Sigwart, Green, Bradley, Bosanquet, Jones and Dewey, and which in their tenets all derive from Hegel’s restatement of Kant’s logical doctrines. From this point of view principles and facts mutually establish one another, the former proving themselves by their capacity to account for the relevant phenomena, and the latter distinguishing themselves from irrelevant accompaniments by their conformity to the principles which make insight possible. In other words, all proof conforms in general type to the hypothetical method of the natural sciences. Kant’s so-called transcendental method, the method by which he establishes the validity of the categories, is itself, as we have already observed,[162] of this character. Secondly, the distinction between the empirical and the a priori must not be taken (as Kant himself takes it in his earlier, and occasionally even in his later utterances) as marking a distinction between two kinds of knowledge. They are elements inseparably involved in all knowledge. And lastly, the contrast between analysis and synthesis becomes a difference not of kind but of degree. Nothing can exist or be conceived save as fitted into a system which gives it meaning and decides as to its truth. In the degree to which it can be studied in relative independence of the supporting system analysis will suffice; in the degree to which it refers us to this system it calls for synthetic interpretation. But ultimately the needs of adequate understanding must constrain us to the employment of both methods of enquiry. Nothing can be known save in terms of the wider whole to which it belongs.

There is, however, one important respect in which Kant diverges in very radical fashion from the position of Hegel. The final whole to which all things must be referred is represented to us only through an “Idea,” for which no corresponding reality can ever be found. The system which decides what is to be regarded as empirically real is the mechanical system of natural science. We have no sufficient theoretical criterion of absolute reality.

These somewhat general considerations may be made more definite if we now endeavour to determine in what specific respects the distinctions employed in the Introduction fail to harmonise with the central doctrines of the Analytic.

In the first place, Kant states his problem in reference only to the attributive judgment. The other types of relational judgment are entirely ignored. For even when he cites judgments of other relational types, such as the propositions of arithmetic and geometry, or that which gives expression to the causal axiom, he interprets them on the lines of the traditional theory of the categorical proposition. As we shall find,[163] it is with the relational categories, and consequently with the various types of relational judgment to which they give rise, that the Critique is alone directly concerned. Even the attributive judgment is found on examination to be of this nature. What it expresses is not the inclusion of an attribute within a given group of attributes, but the organisation of a complex manifold in terms of the dual category of substance and attribute.

Secondly, this exclusively attributive interpretation of the judgment leads Kant to draw, in his Introduction, a hard and fast distinction between the analytic and the synthetic proposition—a distinction which, when stated in such extreme fashion, obscures the real implications of the argument of the Analytic. For Kant here propounds[164] as an exhaustive division the two alternatives: (a) inclusion of the predicate concept within the subject concept, and (b) the falling of the predicate concept entirely outside it. He adds, indeed, that in the latter case the two concepts may still be in some way connected with one another; but this is a concession of which he takes no account in his subsequent argument. He leaves unconsidered the third possibility, that every judgment is both analytic and synthetic. If concepts are not independent entities,[165] as Kant, in agreement with Leibniz, still continues to maintain, but can function only as members of an articulated system, concepts will be distinguishable from one another, and yet will none the less involve one another. In so far as the distinguishable elements in a judgment are directly related, the judgment may seem purely analytic; in so far as they are related only in an indirect manner through a number of intermediaries, they may seem to be purely synthetic. But in every case there is an internal articulation which is describable as synthesis, and an underlying unity that in subordinating all differences realises more adequately than any mere identity the demand for connection between subject and predicate. In other words, all judgments will, on this view, be of the relational type. Even the attributive judgment, as above noted, is no mere assertion of identity. It is always expressed in terms of the dual category of substance and attribute, connecting by a relation contents that as contents may be extremely diverse.

This would seem to be the view to which Kant’s Critical teaching, when consistently developed, is bound to lead. For in insisting that the synthetic character of a judgment need not render it invalid, and that all the fundamental principles and most of the derivative judgments of the positive sciences are of this nature, Kant is really maintaining that the justification of a judgment is always to be looked for beyond its own boundaries in some implied context of coherent experience. But though the value of his argument lies in clear-sighted recognition of the synthetic factor in all genuine knowledge, its cogency is greatly obscured by his continued acceptance of the possibility of judgments that are purely analytic. Thus there is little difficulty in detecting the synthetic character of the proposition: all bodies are heavy. Yet the reader has first been required to admit the analytic character of the proposition: all bodies are extended. The two propositions are really identical in logical character. Neither can be recognised as true save in terms of a comprehensive theory of physical existence. If matter must exist in a state of distribution in order that its parts may acquire through mutual attraction the property of weight, the size of a body, or even its possessing any extension whatsoever, may similarly depend upon specific conditions such as may conceivably not be universally realised. We find the same difficulty when we are called upon to decide whether the judgment 7 + 5 = 12 is analytic or purely synthetic. Kant speaks as if the concepts of 7, 5, and 12 were independent entities, each with its own quite separate connotation. But obviously they can only be formed in the light of the various connected concepts which go to constitute our system of numeration. The proposition has meaning only when interpreted in the light of this conceptual system. It is not, indeed, a self-evident identical proposition; but neither is the connection asserted so entirely synthetic that intuition will alone account for its possibility. That, however, brings us to the third main defect in Kant’s argument.

When Kant states[166] that in synthetic judgments we require, besides the concept of the subject, something else on which the understanding can rely in knowing that a predicate, not contained in the concept, nevertheless belongs to it, he entitles this something x. In the case of empirical judgments, this x is brute experience. Such judgments, Kant implies, are merely empirical. No element of necessity is involved, not even in an indirect manner; in reference to empirical judgments there is no problem of a priori synthesis. Now in formulating the issue in this way, Kant is obscuring the essential purpose of his whole enquiry. He may, without essential detriment to his central position, still continue to preserve a hard-and-fast distinction between analytic and synthetic judgments. In so doing he is only failing to perceive the ultimate consequences of his final results. But in viewing empirical judgments as lacking in every element of necessity, he is destroying the very ground upon which he professes to base the a priori validity of general principles. All judgments involve relational factors of an a priori character. The appeal to experience is the appeal to an implied system of nature. Only when fitted into the context yielded by such a system can an empirical proposition have meaning, and only in the light of such a presupposed system can its truth be determined. It can be true at all, only if it can be regarded as necessarily holding, under the same conditions, for all minds constituted like our own. Assertion of a contingent relation—as in the proposition: this horse is white—is not equivalent to contingency of assertion. Colour is a variable quality of the genus horse, but in the individual horse is necessarily determined in some particular mode. If a horse is naturally white, it is necessarily white. Though, therefore, in the above proposition, necessity receives no explicit verbal expression, it is none the less implied.

In other words, the distinction between the empirical and the a priori is not, as Kant inconsistently assumes in this Introduction, a distinction between two kinds of synthesis or judgment, but between two elements inseparably involved in every judgment. Experience is transcendentally conditioned. Judgment is in all cases the expression of a relation which implies an organised system of supporting propositions; and for the articulation of this system a priori factors are indispensably necessary.

But the most flagrant example of Kant’s failure to live up to his own Critical principles is to be found in his doctrine of pure intuition. It represents a position which he adopted in the pre-Critical period. It is prefigured in Ueber die Deutlichkeit der GrundsÄtze (1764),[167] and in Von dem ersten Grunde des Unterschiedes der Gegenden im Raume (1768),[168] and is definitely expounded in the Dissertation (1770).[169] That Kant continued to hold this doctrine, and that he himself regarded it as an integral part of his system, does not, of course, suffice to render it genuinely Critical. As a matter of fact, it is really as completely inconsistent with his Critical standpoint as is the view of the empirical proposition which we have just been considering. An appeal to our fingers or to points[170] is as little capable, in and by itself, of justifying any a priori judgment as are the sense-contents of grounding an empirical judgment. Even when Kant is allowed the benefit of his own more careful statements,[171] and is taken as asserting that arithmetical propositions are based on a pure a priori intuition which can find only approximate expression in sensuous terms, his statements run counter to the main tendencies of his Critical teaching, as well as to the recognised methods of the mathematical sciences. Intuition may, as PoincarÉ and others have maintained, be an indispensable element in all mathematical concepts; it cannot afford proof of any general theorem. The conceptual system which directs our methods of decimal counting is what gives meaning to the judgment 7 + 5 = 12; it is also what determines that judgment as true. The appeal to intuition in numerical judgments must be regarded only as a means of imaginatively realising in a concrete form the abstract relations of some such governing system, or else as a means of detecting relations not previously known. The last thing in the world which such a method can yield is universal demonstration. This is equally evident in regard to geometrical propositions. That a straight line is the shortest distance between two points, cannot be proved by any mere appeal to intuition. The judgment will hold if it can be assumed that space is Euclidean in character; and to justify that assumption it must be shown that Euclidean concepts are adequate to the interpretation of our intuitional data. Should space possess a curvature, the above proposition might cease to be universally valid. Space is not a simple, unanalysable datum. Though intuitionally apprehended, it demands for its precise determination the whole body of geometrical science.[172]

The comparative simplicity of Kant’s intuitional theory of mathematical science, supported as it is by the seemingly fundamental distinction between abstract concepts of reflective thinking and the construction of concepts[173] in geometry and arithmetic, has made it intelligible even to those to whom the very complicated argument of the Analytic makes no appeal. It would also seem to be inseparably bound up with what from the popular point of view is the most striking of all Kant’s theoretical doctrines, namely, his view that space and time are given subjective forms, and that the assertion of their independent reality must result in those contradictions to which Kant has given the title antinomy. For these reasons his intuitional theory of mathematical science has received attention out of all proportion to its importance. Its pre-Critical character has been more or less overlooked, and instead of being interpreted in the light of Critical principles, it has been allowed to obscure the sounder teaching of the Analytic. In this matter Schopenhauer is a chief culprit. He not only takes the views of mathematical science expounded in the Introduction and Aesthetic as being in line with Kant’s main teaching, but expounds them in an even more unqualified fashion than does Kant himself.

There are thus four main defects in the argument of this Introduction, regarded as representative of Critical teaching. (1) Its problems are formulated exclusively in terms of the attributive judgment; the other forms of relational judgment are ignored. (2) It maintains that judgments are either merely analytic or completely synthetic. (3) It proceeds in terms of a further division of judgments into those that are purely empirical and those that are a priori. (4) It seems to assert that the justification for mathematical judgments is intuitional. All these four positions are in some degree retained throughout the Critique, but not in the unqualified manner of this Introduction. In the Analytic, judgment in all its possible forms is shown to be a synthetic combination of a given manifold in terms of relational categories. This leads to a fourfold conclusion. In the first place, judgment must be regarded as essentially relational. Secondly, the a priori and the empirical must not be taken as two separate kinds of knowledge, but as two elements involved in all knowledge. Thirdly, analysis and synthesis must not be viewed as co-ordinate processes; synthesis is the more fundamental; it conditions all analysis. And lastly, it must be recognised that nothing is merely given; intuitional experience, whether sensuous or a priori, is conditioned by processes of conceptual interpretation. Though the consequences which follow from these conclusions, if fully developed, would carry us far beyond any point which Kant himself reached in the progressive maturing of his views, the next immediate steps would still be on the strict lines of the Critical principles, and would involve the sacrifice only of such pre-Critical doctrines as that of the intuitive character of mathematical proof. Such correction of Kant’s earlier positions is the necessary complement of his own final discovery that sense-intuition is incapable of grounding even the so-called empirical judgment.

The Introduction to the first edition bears all the signs of having been written previous to the central portions of the Analytic.[174] That it was not, however, written prior to the Aesthetic seems probable. The opening sections of the Aesthetic represent what is virtually an independent introduction which takes no account of the preceding argument, and which redefines terms and distinctions that have already been dwelt upon. The extensive additions which Kant made in recasting the Introduction for the second edition are in many respects a great improvement. In the first edition Kant had not, except when speaking of the possibility of constructing the concepts of mathematical science, referred to the synthetic character of mathematical judgments. This is now dwelt upon in adequate detail. Kant’s reason for not making the revision more radical was doubtless his unwillingness to undertake the still more extensive alterations which this would have involved. Had he expanded the opening statement of the second edition Introduction, that even our empirical knowledge is a compound of the sensuous and the a priori, an entirely new Introduction would have become necessary. The additions made are therefore only such as will not markedly conflict with the main tenor of the argument of the first edition.

How Are Synthetic a priori Judgments Possible?

Treatment of detailed points will be simplified if we now consider in systematic fashion the many difficulties that present themselves in connection with Kant’s mode of formulating his central problem: How are synthetic a priori judgments possible? This formula is less definite and precise than would at first sight appear. The central phrase ‘synthetic a priori’ is sufficiently exact (the meaning to be attached to the a priori has already been considered[175]), but ambiguities of the most various kinds lurk in the seemingly innocent and simple terms with which the formula begins and ends:

A. ‘How’ has two very different meanings:

(a) How possible = in what manner possible = wie.

(b) How possible = in how far possible, i.e. whether possible = ob.