Kant's Theory of Knowledge Part 17

"The fact is that it is not images of objects, but schemata, which lie at the foundation of our pure sensuous conceptions. No image could ever be adequate to our conception of a triangle in general. For it would not attain the generality of the conception which makes it valid for all triangles, whether right-angled, acute-angled, &c., but would always be limited to one part only of this sphere. The schema of the triangle can exist nowhere else than in thought, and signifies a rule of the synthesis of the imagination in regard to pure figures in s.p.a.ce. An object of experience or an image of it always falls short of the empirical conception to a far greater degree than does the schema; the empirical conception always relates immediately to the schema of the imagination as a rule for the determination of our perception in conformity with a certain general conception. The conception of 'dog'

signifies a rule according to which my imagination can draw the general outline of the figure of a four-footed animal, without being limited to any particular single form which experience presents to me, or indeed to any possible image that I can represent to myself _in concreto_. This schematism of our understanding in regard to phenomena and their mere form is an art hidden in the depths of the human soul, whose true modes of action we are not likely ever to discover from Nature and unveil. Thus much only can we say: the _image_ is a product of the empirical faculty of the productive imagination, while the _schema_ of sensuous conceptions (such as of figures in s.p.a.ce) is a product and, as it were, a monogram of the pure _a priori_ imagination, through which, and according to which, images first become possible, though the images must be connected with the conception only by means of the schema which they express, and are in themselves not fully adequate to it. On the other hand, the schema of a pure conception of the understanding is something which cannot be brought to an image; on the contrary, it is only the pure synthesis in accordance with a rule of unity according to conceptions in general, a rule of unity which the category expresses, and it is a transcendental product of the imagination which concerns the determination of the inner sense in general according to conditions of its form (time) with reference to all representations, so far as these are to be connected _a priori_ in one conception according to the unity of apperception."[6]

[6] B. 179-81, M. 109-10.

Now, in order to determine whether schemata can const.i.tute the desired link between the pure conceptions or categories and the manifold of sense, it is necessary to follow closely this account of a schema.

Kant unquestionably in this pa.s.sage treats as a mental image related to a conception what really is, and what on his own theory ought to have been, an individual object related to a conception, i. e. an instance of it. In other words, he takes a mental image of an individual for the individual itself.[7] On the one hand, he treats a schema of a conception throughout as the thought of a procedure of the imagination to present to the conception its _image_, and he opposes schemata not to objects but to _images_; on the other hand, his problem concerns subsumption under a conception, and what is subsumed must be an instance of the conception, i. e. an individual object of the kind in question.[8] Again, in a.s.serting that if I place five points one after another, . . . . . this is an image of the number five, he is actually saying that an individual group of five points is an image of a group of five in general.[9] Further, if the process of schematizing is to enter--as it must--into knowledge of the phenomenal world, what Kant here speaks of as the images related to a conception must be taken to be individual instances of the conception, whatever his language may be. For, in order to enter into knowledge, the process referred to must be that by which _objects of experience_ are constructed. Hence the pa.s.sage should be interpreted as if throughout there had been written for 'image' 'individual instance' or more simply 'instance'. Again, the process of schematizing, although _introduced_ simply as a process by which an individual is to be subsumed indirectly under a conception, is a.s.sumed in the pa.s.sage quoted to be a process of _synthesis_. Hence we may say that the process of schematizing is a process by which we combine the manifold of perception into an individual whole in accordance with a conception, and that the schema of a conception is the thought of the rule of procedure on our part by which we combine the manifold in accordance with the conception, and so bring the manifold under the conception. Thus the schema of the conception of 100 is the thought of a process of synthesis by which we combine say 10 groups of 10 units into 100, and the schematizing of the conception of 100 is the process by which we do so. Here it is essential to notice three points. In the first place, the schema is a conception which relates not to the reality apprehended but to us. It is the thought of a rule of procedure on our part by which an instance of a conception is constructed, and not the thought of a characteristic of the reality constructed. For instance, the thought of a rule by which we can combine points to make 100 is a thought which concerns us and not the points; it is only the conception corresponding to this schema, viz.

the thought of 100, which concerns the points. In the second place, although the thought of time is involved in the schema, the succession in question lies not in the object, but in our act of construction or apprehension. In the third place, the schema presupposes the corresponding conception and the process of schematizing directly brings the manifold of perception under the conception. Thus the thought of combining 10 groups of 10 units to make 100 presupposes the thought of 100, and the process of combination brings the units under the conception of 100.

[7] Cf. pp. 240-1. The mistake is, of course, facilitated by the fact that 'objects in nature', being for Kant only 'appearances', resemble mental images more closely than they do as usually conceived.

[8] Cf. B. 176, M. 107. That individuals are really referred to is also implied in the a.s.sertion that 'the synthesis of imagination has for its aim no single _perception_, but merely unity in the determination of sensibility'. (The italics are mine.)

[9] Two sentences treat individual objects and images as if they might be mentioned indifferently. "An object of experience or an image of it always falls short of the empirical conception to a far greater degree than does the schema." "The conception of a 'dog' signifies a rule according to which my imagination can draw the general outline of the figure of a four-footed animal without being limited to any single particular form which experience presents to me, or indeed to any possible image that I can represent to myself _in concreto_."

If, however, we go on to ask what is required of schemata and of the process of schematizing, if they are to enable the manifold to be subsumed under the categories, we see that each of these three characteristics makes it impossible for them to fulfil this purpose.

For firstly, an individual manifold A has to be brought under a category B. Since _ex hypothesi_ this cannot be effected directly, there is needed a mediating conception C. C, therefore, it would seem, must be at once a species of B and a conception of which A is an instance. In any case C must be a conception relating to the reality to be known, and not to any process of knowing on our part, and, again, it must be more concrete than B. This is borne out by the list of the schemata of the categories. But, although a schema may be said to be more concrete than the corresponding conception, in that it presupposes the conception, it neither is nor involves a more concrete conception of an _object_ and in fact, as has been pointed out, relates not to the reality to be known but to the process on our part by which we construct or apprehend it.[10] In the second place, the time in respect of which the category B has to be made more concrete must relate to the object, and not to the successive process by which we apprehend it, whereas the time involved in a schema concerns the latter and not the former. In the third place, from the point of view of the categories, the process of schematizing should be a process whereby we combine the manifold into a whole A in accordance with the conception C, and thereby render _possible_ the subsumption of A under the category B. If it be a process which actually subsumes the manifold under B, it will _actually_ perform that, the very impossibility of which has made it necessary to postulate such a process at all. For, according to Kant, it is just the fact that the manifold cannot be subsumed directly under the categories that renders schematism necessary. Yet, on Kant's general account of a schema, the schematizing must actually bring a manifold under the corresponding conception. If we present to ourselves an individual triangle by successively joining three lines according to the conception of a triangle, i. e. so that they enclose a s.p.a.ce, we are directly bringing the manifold, i. e. the lines, under the conception of a triangle.

Again, if we present to ourselves an instance of a group of 100 by combining 10 groups of 10 units of any kind, we are directly bringing the units under the conception of 100. If this consideration be applied to the schematism of a category, we see that the process said to be necessary because a certain other process is impossible is the very process said to be impossible.

[10] It may be objected that, from Kant's point of view, the thought of a rule of construction, and the thought of the principle of the whole to be constructed, are the same thing from different points of view. But if this be insisted on, the schema and its corresponding conception become the same thing regarded from different points of view; consequently the schema will not be a more concrete conception of an object than the corresponding conception, but it will be the conception itself.

If, therefore, Kant succeeds in finding schemata of the categories in detail in the sense in which they are required for the solution of his problem, i. e. in the sense of more concrete conceptions involving the thought of time and relating to objects, we should expect either that he ignores his general account of a schema, or that if he appeals to it, the appeal is irrelevant. This we find to be the case. His account of the first two transcendental schemata makes a wholly irrelevant appeal to the temporal process of synthesis on our part, while his account of the remaining schemata makes no attempt to appeal to it at all.

"The pure _schema_ of _quant.i.ty_, as a conception of the understanding, is _number_, a representation which comprises the successive addition of one to one (h.o.m.ogeneous elements). Accordingly, number is nothing else than the unity of the synthesis of the manifold of a h.o.m.ogeneous perception in general, in that I generate time itself in the apprehension of the perception."[11]

[11] B. 182, M. 110.

It is clear that this pa.s.sage, whatever its precise interpretation may be,[12] involves a confusion between the thought of counting and that of number. The thought of number relates to objects of apprehension and does not involve the thought of time. The thought of counting, which presupposes the thought of number, relates to our apprehension of objects and involves the thought of time; it is the thought of a successive process on our part by which we count the number of units contained in what we already know to consist of units.[13] Now we must a.s.sume that the schema of quant.i.ty is really what Kant says it is, viz. number, or to express it more accurately, the thought of number, and not the thought of counting, with which he wrongly identifies it.

For his main problem is to find conceptions which at once are more concrete than the categories and, at the same time, like the categories, relate to objects, and the thought of counting, though more concrete than that of number, does not relate to objects. Three consequences follow. In the first place, although the schema of quant.i.ty, i. e. the thought of number, is more concrete than the thought of quant.i.ty,[14] it is not, as it should be, more concrete in respect of time; for the thought of number does not include the thought of time. Secondly, the thought of time is only introduced into the schema of quant.i.ty irrelevantly by reference to the temporal process of _counting_, by which we come to apprehend the number of a given group of units. Thirdly, the schema of quant.i.ty is only in appearance connected with the nature of a schema in general, as Kant describes it, by a false identification of the thought of number with the thought of the process on our part by which we count groups of units, i. e. numbers.

[12] The drift of the pa.s.sage would seem to be this: 'If we are to present to ourselves an instance of a quant.i.ty, we must successively combine similar units until they form a quant.i.ty. This process involves the thought of a successive process by which we add units according to the conception of a quant.i.ty. This thought is the thought of number, and since by it we present to ourselves an instance of a quant.i.ty, it is the schema of quant.i.ty.' But if this be its drift, considerations of sense demand that it should be rewritten, at least to the following extent: 'If we are to present to ourselves an instance of a _particular_ quant.i.ty [which will really be a particular number, for it must be regarded as discrete, (cf. B. 212, M. 128 fin., 129 init.)] e. g. three, we must successively combine units until they form _that_ quant.i.ty. This process involves the thought of a successive process, by which we add units according to the conception of _that_ quant.i.ty. This thought is the thought of a particular number, and since by it we present to ourselves an instance of _that_ quant.i.ty, this thought is the schema of _that_ quant.i.ty.' If this rewriting be admitted to be necessary, it must be allowed that Kant has confused (_a_) the thoughts of particular quant.i.ties and of particular numbers with those of quant.i.ty and of number in general respectively, (_b_) the thought of a particular quant.i.ty with that of a particular number (for the process referred to presupposes that the particular quant.i.ty taken is known to consist of a number of equal units) and (_c_) the thought of counting with that of number.

[13] This statement is, of course, not meant as a definition of counting, but as a means of bringing out the distinction between a process of counting and a number.

[14] For the thought of a number is the thought of a quant.i.ty of a special kind, viz. of a quant.i.ty made up of a number of similar units without remainder.

The account of the schema of reality, the second category, runs as follows: "Reality is in the pure conception of the understanding that which corresponds to a sensation in general, that therefore of which the conception in itself indicates a being (in time), while negation is that of which the conception indicates a not being (in time). Their opposition, therefore, arises in the distinction between one and the same time as filled or empty. Since time is only the form of perception, consequently of objects as phenomena, that which in objects corresponds to sensation is the transcendental matter of all objects as things in themselves (thinghood, reality).[15] Now every sensation has a degree or magnitude by which it can fill the same time, i. e. the internal sense, in respect of the same representation of an object, more or less, until it vanishes into nothing ( = 0 = _negatio_). There is, therefore, a relation and connexion between reality and negation, or rather a transition from the former to the latter, which makes every reality representable as a _quantum_; and the schema of a reality, as the quant.i.ty of something so far as it fills time, is just this continuous and uniform generation of the reality in time, as we descend in time from the sensation which has a certain degree, down to the vanishing thereof, or gradually ascend from negation to the magnitude thereof."[16]

[15] It is difficult to see how Kant could meet the criticism that here, contrary to his intention, he is treating physical objects as things in themselves. Cf. p. 265.

[16] B. 182-3, M. 110-11.

This pa.s.sage, if it be taken in connexion with the account of the antic.i.p.ations of perception,[17] seems to have the following meaning: 'In thinking of something as a reality, we think of it as that which corresponds to, i. e. produces, a sensation, and therefore as something which, like the sensation, is in time; and just as every sensation, which, as such, occupies time, has a certain degree of intensity, so has the reality which produces it. Now to produce for ourselves an instance of a reality in this sense, we must add units of reality till a reality of the required degree is produced, and the thought of this method on our part of constructing an individual reality is the schema of reality.' But if this represents Kant's meaning, the schema of reality relates only to our process of apprehension, and therefore is not a conception which relates to objects and is more concrete than the corresponding category in respect of time. Moreover, it is matter for surprise that in the case of this category Kant should have thought schematism necessary, for time is actually included in his own statement of the category.

[17] B. 207-18, M. 125-32.

The account of the schemata of the remaining categories need not be considered. It merely _a.s.serts_ that certain conceptions relating to objects and involving the thought of time are the schemata corresponding to the remaining categories, without any attempt to connect them with the nature of a schema. Thus, the schema of substance is a.s.serted to be the _permanence_ of the real _in time_, that of cause the _succession_ of the manifold, in so far as that succession is subjected to a rule, that of interaction the _coexistence_ of the determinations or accidents of one substance with those of another according to a universal rule.[18] Again, the schemata of possibility, of actuality and of necessity are said to be respectively the accordance of the synthesis of representations with the conditions of time in general, existence in a determined time, and existence of an object in all time.

[18] The italics are mine.

The main confusion pervading the chapter is of course that between temporal relations which concern the process of apprehension and temporal relations which concern the realities apprehended. Kant is continually referring to the former as if they were the latter. The cause of this confusion lies in Kant's reduction of physical realities to representations. Since, according to him, these realities are only our representations, all temporal relations are really relations of our representations, and these relations have to be treated at one time as relations of our apprehensions, and at another as relations of the realities apprehended, as the context requires.

CHAPTER XI

THE MATHEMATICAL PRINCIPLES

As has been pointed out,[1] the aim of the second part of the _a.n.a.lytic of Principles_ is to determine the _a priori_ principles involved in the use of the categories under the necessary sensuous conditions. These principles Kant divides into four cla.s.ses, corresponding to the four groups of categories, and he calls them respectively 'axioms of perception', 'antic.i.p.ations of sense-perception', 'a.n.a.logies of experience', and 'postulates of empirical thought'. The first two and the last two cla.s.ses are grouped together as 'mathematical' and 'dynamical' respectively, on the ground that the former group concerns the perception of objects, i. e. their nature apprehended in perception, while the latter group concerns their existence, and that consequently, since a.s.sertions concerning the existence of objects presuppose the realization of empirical conditions which a.s.sertions concerning their nature do not, only the former possesses an absolute necessity and an immediate evidence such as is found in mathematics.[2]

[1] p. 246.

[2] The a.s.sertion that all perceptions (i. e. all objects of perception) are extensive quant.i.ties relates, according to Kant, to the nature of objects, while the a.s.sertion that an event must have a necessary antecedent affirms that such an antecedent must exist, but gives no clue to its specific nature. Compare "But the existence of phenomena cannot be known _a priori_, and although we could be led in this way to infer the fact of some existence, we should not know this existence determinately, i. e. we could not antic.i.p.ate the respect in which the empirical perception of it differed from that of other existences". (B. 221, M. 134). Kant seems to think that the fact that the dynamical principles relate to the existence of objects is a sufficient justification of their name.

It needs but little reflection to see that the distinctions which Kant draws between the mathematical and the dynamical principles must break down.

These two groups of principles are not, as their names might suggest, principles within mathematics and physics, but presuppositions of mathematics and physics respectively. Kant also claims appropriateness for the special terms used of each minor group to indicate the kind of principles in question, viz. 'axioms', 'antic.i.p.ations', 'a.n.a.logies', 'postulates'. But it may be noted as an indication of the artificiality of the scheme that each of the first two groups contains only one principle, although Kant refers to them in the plural as axioms and antic.i.p.ations respectively, and although the existence of three categories corresponding to each group would suggest the existence of three principles.

The axiom of perception is that 'All perceptions are extensive quant.i.ties'. The proof of it runs thus:

"An extensive quant.i.ty I call that in which the representation of the parts renders possible the representation of the whole (and therefore necessarily precedes it). I cannot represent to myself any line, however small it may be, without drawing it in thought, that is, without generating from a point all its parts one after another, and thereby first drawing this perception. Precisely the same is the case with every, even the smallest, time.... Since the pure perception in all phenomena is either time or s.p.a.ce, every phenomenon as a perception is an extensive quant.i.ty, because it can be known in apprehension only by a successive synthesis (of part with part). All phenomena, therefore, are already perceived as aggregates (groups of previously given parts), which is not the case with quant.i.ties of every kind, but only with those which are represented and apprehended by us as _extensive_."[3]

[3] B. 203-4, M. 123.

Kant opposes an extensive quant.i.ty to an intensive quant.i.ty or a quant.i.ty which has a degree. "That quant.i.ty which is apprehended only as unity and in which plurality can be represented only by approximation to negation = 0, I call _intensive quant.i.ty_."[4] The aspect of this ultimate distinction which underlies Kant's mode of stating it is that only an extensive quant.i.ty is a whole, i. e.

something made up of parts. Thus a mile can be said to be made up of two half-miles, but a velocity of one foot per second, though comparable with a velocity of half a foot per second, cannot be said to be made up of two such velocities; it is essentially one and indivisible. Hence, from Kant's point of view, it follows that it is only an extensive magnitude which can, and indeed must, be apprehended through a successive synthesis of the parts. The proof of the axiom seems to be simply this: 'All phenomena as objects of perception are subject to the forms of perception, s.p.a.ce and time. s.p.a.ce and time are [h.o.m.ogeneous manifolds, and therefore] extensive quant.i.ties, only to be apprehended by a successive synthesis of the parts. Hence phenomena, or objects of experience, must also be extensive quant.i.ties, to be similarly apprehended.' And Kant goes on to add that it is for this reason that geometry and pure mathematics generally apply to objects of experience.

[4] B. 210, M. 127.

We need only draw attention to three points. Firstly, no justification is given of the term 'axiom'. Secondly, the argument does not really appeal to the doctrine of the categories, but only to the character of s.p.a.ce and time as forms of perception. Thirdly, it need not appeal to s.p.a.ce and time as forms of perception in the proper sense of ways in which we apprehend objects, but only in the sense of ways in which objects are related[5]; in other words, it need not appeal to Kant's theory of knowledge. The conclusion follows simply from the nature of objects as spatially and temporally related, whether they are phenomena or not. It may be objected that Kant's thesis is that _all_ objects of perception are extensive quant.i.ties, and that unless s.p.a.ce and time are allowed to be ways in which _we must perceive_ objects, we cannot say that all objects will be spatially and temporally related, and so extensive quant.i.ties. But to this it may be replied that it is only true that all objects of perception are extensive quant.i.ties if the term 'object of perception' be restricted to parts of the physical world, i. e. to just those realities which Kant is thinking of as spatially and temporally related,[6] and that this restriction is not justified, since a sensation or a pain which has only intensive quant.i.ty is just as much ent.i.tled to be called an object of perception.

[5] Cf. pp. 37-9.

[6] The context shows that Kant is thinking only of such temporal relations as belong to the physical world, and not of those which belong to us as apprehending it. Cf. p. 139.

The antic.i.p.ation of sense-perception consists in the principle that 'In all phenomena, the real, which is an object of sensation, has intensive magnitude, i. e. a degree'. The proof is stated thus:

"Apprehension merely by means of sensation fills only one moment (that is, if I do not take into consideration the succession of many sensations). Sensation, therefore, as that in the phenomenon the apprehension of which is not a successive synthesis advancing from parts to a complete representation, has no extensive quant.i.ty; the lack of sensation in one and the same moment would represent it as empty, consequently = 0. Now that which in the empirical perception corresponds to sensation is reality (_realitas phaenomenon_); that which corresponds to the lack of it is negation = 0. But every sensation is capable of a diminution, so that it can decrease and thus gradually vanish. Therefore, between reality in the phenomenon and negation there exists a continuous connexion of many possible intermediate sensations, the difference of which from each other is always smaller than that between the given sensation and zero, or complete negation. That is to say, the real in the phenomenon has always a quant.i.ty, which, however, is not found in apprehension, since apprehension takes place by means of mere sensation in one moment and not by a successive synthesis of many sensations, and therefore does not proceed from parts to the whole. Consequently, it has a quant.i.ty, but not an extensive quant.i.ty."

"Now that quant.i.ty which is apprehended only as unity, and in which plurality can be represented only by approximation to negation = 0, I call an _intensive quant.i.ty_. Every reality, therefore, in a phenomenon has intensive quant.i.ty, that is, a degree."[7]

[7] B. 209-10, M. 127.

In other words, 'We can lay down _a priori_ that all sensations have a certain degree of intensity, and that between a sensation of a given intensity and the total absence of sensation there is possible an infinite number of sensations varying in intensity from nothing to that degree of intensity. Therefore the real, which corresponds to sensation, can also be said _a priori_ to admit of an infinite variety of degree.'

Though the principle established is of little intrinsic importance, the account of it is noticeable for two reasons. In the first place, although Kant clearly means by the 'real corresponding to sensation' a body in s.p.a.ce, and regards it as a phenomenon, it is impossible to see how he can avoid the charge that he in fact treats it as a thing in itself.[8] For the correspondence must consist in the fact that the real causes or excites sensation in us, and therefore the real, i. e.

a body in s.p.a.ce, is implied to be a thing in itself. In fact, Kant himself speaks of considering the real in the phenomenon as the cause of sensation,[9] and, in a pa.s.sage added in the second edition, after proving that sensation must have an intensive quant.i.ty, he says that, corresponding to the intensive quant.i.ty of sensation, an intensive quant.i.ty, i. e. _a degree of influence on sense_, must be attributed to all objects of sense-perception.[10] The difficulty of consistently maintaining that the real, which corresponds to sensation, is a phenomenon is, of course, due to the impossibility of distinguishing between reality and appearance within phenomena.[11]