This conclusion is quite correct: if there is only composite and we imagine everything composite removed, then nothing remains; we may grant this, but this tautological superfluity might be spared, and the proof might begin with what follows, namely : —
“Either all composition cannot be mentally removed, or, it being removed, something subsisting without composition — namely, the simple — must remain.”
“But in the first case again the composite would not consist of substances (for with these compositeness is only a contingent relation of the substances,! and they must be capable of existing as persistent beings without it). — But this case is contradictory to the hypothesis, and thereiore only the second remains: namely, that all composite substance consists of simple parts.”
The reason which is the main point, compared with which all that has been said is quite superfluous, is relegated to a parenthesis. The dilemma is this : — That which persists is either the composite or else the simple. If the first (the composite) were that which persists, then that which persists is not the substances, for in these compositeness is a contingent relation; but the substances are that which persists, therefore that which persists is the simple.
It is evident that the apagogic detour might be avoided, and
' Here redundance of language it added to redundance in proof, — “for with these (to wit, the substances) compositeness is only a contingent relation of Its
that to the Thesis (“Composite substance consists of simple parts”) this reason might immediately be added as proof, namely that compositeness is merely a contingent relation of the substances, external to and not concerned with them. — If compositeness is in truth contingent, then the essence is of course the simple. But this contingency, which is the sole point at issue, is not proved but simply assumed (in parenthesis) as though it were obvious or beside the point. It is, of course, self-evident that compositeness is the determination of con¬ tingency and externality; but, if we were only about to deal with a contingent juxtaposition instead of continuity, it was not worth while to set up an antinomy about it, or, rather, it was impossible ; in that case it is mere tautology (as was said) to assert that the parts are simple.
Thus in the apagogic detour we see that the very assertion which is to be its result already occurs. Thus the proof might more briefly be put as follows : —
Let it be assumed that substances do not consist of simple parts, but are composite. Now all composition can be thought away (since it is a merely contingent relation) ; it being removed, therefore, no substances remain unless they consist of simple parts. But we must have substances, since we assumed their existence ; everything must not vanish, something must remain ; for we have presupposed that some such persistent entity (which we called substance) exists. Therefore this something must be simple.
To complete matters, we must consider the conclusion, which runs as follows : —
“Hence it follows immediately that all the things in the world are simple essences, that compositeness is a condition merely external to them, and that reason must consider the elementary substances to be simple essences.”
Here the external, that is, the contingent, nature of com¬ positeness is cited as a consequence, after having already been introduced parenthetically and used in the proof.
Kant protests much that he is not looking for sophisms in the conflicting propositions of the antinomy, in order to effect (as the phrase is) an advocate’s proof. Indeed, this proof is not really to be accused of sophistry so much as of a forced and useless tortuosity, which serves only to produce the outer form
of a proof, not allowing it to be perfectly transparent that that which should stand out as conclusion is the (parenthetic) hinge of the proof, and that there is no proof but only an assumption.
“No composite thing in the world consists of simple parts, and nothing simple anywhere exists in it.”
The proof again has the apagogic turn, and in another manner is as faulty as the previous.
“Assume,” it runs, “that a composite thing as substance consists of simple parts. Now all external relation (and therefore also the composition of substances) is possible only in space, and therefore the space which it occupies must consist of as many parts as those of which the composite thing consists. Now space does not consist of simple parts, but of spaces. Therefore each part of the composite thing must occupy one space.”
“But the absolutely first parts of every composite thing are simple.”
“Now every reality which occupies a space comprehends a manifold of mutual externalities ; it is therefore composed (of substances), and thus the simple would be a substantial com¬ posite, which is self-contradictory.”
This proof may be called a perfect nest (to use an expression which is elsewhere employed by Kant) of faulty procedure.
First, the apagogic turn is a baseless fabric. The assumption that whatever is substantial is spatial, while space does not consist of simple parts, is a direct assertion which is made the immediate foundation of that which is to be proved ; after this there is no need for further proof.
Next, this apagogic proof begins with the proposition “that all composition of substances is an external relation,” but, strangely enough, immediately forgets it. For the further con¬ clusion is made that composition can take place only in space, that space does not consist of simple parts, and that therefore the reality occupying a space is composite. If once composite¬ ness has been taken as an external relation, then spatiality itself (since composition is supposed to be possible in it alone) is for that very reason a relation external to the substances, which does sot concern them nor touch their nature any more than
does all the rest that can be inferred from the determination of spatiality. For this very reason the substances should not be placed in space.
Further, it is assumed that the space into which the sub¬ stances are here transplanted does not consist of simple parts ; since it is an intuition (that is, according to Kant’s determina¬ tion, a sensuous representation) which can be given only by one single object, and is not a so-called discursive concept. — This distinction of Kant’s between intuition and concept has, of course, led to the abuse of the former, and, in order to save the labour of achieving the latter, its value and sphere have been extended to all cognition. What is relevant here is this, that space, like intuition itself, must be conceived if we would have any concept at all. And thus the question would arise whether space, though simple continuity for intuition, might not conceptually have to be taken as composed of simple parts, since otherwise space would be affected by that same antinomy into which substance only was transplanted. Indeed, if the antinomy is taken in the abstract, it extends to all Quantity (as was said above) and therefore also to Space and Time.
Now in the proof it is assumed that space does not consist of simple parts ; and this should have been sufficient reason for not transplanting the simple into this element, which is inade¬ quate to the determination of the simple. — Here also the con¬ tinuity of space comes into collision with compositeness ; these two are confused, and the former is inserted in the place of the latter (which results in a Quaternio Urminorum in the con¬ clusion). Kant expressly determines space as being one, its parts being based only on limitations, so that they do not precede space, sole and all-comprehending, as its component parts from which it could be put together. ( Critique of Pure Reason, Second Ed., p. 39.) Here continuity is predicated of space, very justly and definitely, in opposition to its compositeness out of parts. In the proof, on the other hand, the transplanting of the sub¬ stances into space is to involve “a manifold of mutual exter¬ nalities,” which, further, is “therefore composite.” Yet we have quoted what shows that the manner in which a manifold exists in space is expressly designed to exclude compositeness and parts which precede its unity.
In the Note to the proof of the Antithesis, the other Junda-
mental notion of the critical philosophy is further expressly adduced, namely, that we have a concept of bodies only as phenomena, which, as such, necessarily presuppose space as the condition of the possibility of all external manifestation. If by substances mere bodies are meant, such as are seen, felt, tasted, and so forth, then there is really no question of their concep¬ tual meaning : sense-perception only is in question. The proof of the Antithesis might then be put in brief thus : — All visual, sensory, and other experience shows us only what is composite; the best microscopes and the keenest knives have never yet allowed us to hit on anything simple. Therefore Reason, too, should not want to hit on anything simple.
And thus when now we regard more closely this opposition of Thesis and Antithesis, liberating their proofs of all idle superfluity and tortuousness, we find that the proof of the Antithesis, by transplanting the substances into space, contains a dogmatic assumption of Continuity, and that the proof of the Thesis, by assuming Compositeness as the class of relation sub¬ sisted between the substances, contains the dogmatic assump¬ tion of the contingency of this relation — that is, the assumption that the substances are absolute Ones. Thus the whole antinomy reduces itself to the separation and direct assertion of the two moments of Quantity as being absolutely separate. Looked at from the point of view of mere discreteness, substance, matter, space and time, and so on, are absolutely divided, and the One is their principle. From the point of view of continuity, this One is merely suspended: division remains divisibility, the possibility of dividing remains as possibility, without ever actually reaching the atom. Now even if we do not move beyond the determination implied in what has been said about these contradictions, still continuity itself contains the moment of the atom, since continuity exists simply as the possibility of division ; just as accomplished division, or discreteness, cancels all distinction between the Ones — for each simple One is what every other is, — and for that very reason contains their equality and therefore their continuity. Each of the two opposed sides contains its other in itself, and neither can be thought of with¬ out the other; and thus it follows that, taken alone, neither determination has truth, but only their unity. This is the true dialectic consideration of them, and the true result.
The dialectic examples of the old Eleatic school, especially those which deal with motion, are incomparably deeper and richer in meaning than the antinomy of Kant which we have considered. They, too, are based on the concept of Quantity, and in it have their solution. To consider them here would lead too far ; they concern the concepts of Space and Time, and can be dealt with in connexion with these subjects and with the history of philosophy. — They do the highest honour to the reason of their inventors; their result is the pure Being of Parmenides, since they demonstrate the solution into itself of all determinate being : in themselves, therefore, they are the Flux of Herakleitos. They therefore deserve a fuller considera¬ tion than the ordinary explanation that they are just sophisms ; — an assertion which clings to empirical perception, following that method of Diogenes (so convincing to common sense), who, when a dialectician demonstrated the contradiction con¬ tained in motion, is said to have put no further strain on his reason, but, by mutely walking up and down, to have referred to the evidence of the eyes; — an assertion and a refutation which of course it is easier to make than to enter upon thought, to seize the confusions into which thought, quite unforced, leads when it formulates itself in ordinary consciousness, and to solve them by means of thought.
The solution of these dialectic formations which Aristotle effects is highly to be praised : it is implied in his truly specu¬ lative concepts of Space, Time, and Motion. Infinite divisibility (which, being imagined as actually completed, is equivalent to infinite division, or the atoms), as being the principle on which are based the most famous of these proofs, is by him opposed to Continuity (of Time as well as of Space), in such a manner that infinite — that is, abstract — plurality is contained in continuity only in itself, or potentially. The actual, as opposed to abstract plurality (or to abstract continuity), is their concrete form, it is* Time and Space itself; and these in turn are opposed to Motion and Matter. The abstract exists only in itself, or potentially ; it exists only as a moment of the Real. Bayle in his Dictionnaire (Article, Zenon) considers Aristotle’s solution of Zeno’s dialectic “ pitoyable ” : he does not understand the meaning of the potentially infinite divisibility of matter. He replies that if matter is infinitely divisible, it actually contains
an infinite number of parts", so that this is not an infinite en puissance but an infinite really and actually existing. — No: divisibility itself is only a potentiality, not an existence of parts ; and in any case, in Continuity, plurality is posited only as a moment already transcended. — Keen understanding — though in this, too, probably Aristotle has not been excelled — is not sufficient to seize and judge his speculative concepts; nor is the clumsy method of sensuous representation which we cited sufficient to refute such arguments as Zeno used. Such an understanding makes the mistake of taking these objects of thought or abstraction, such as an infinite number of parts, for Something, for true and actual ; and this sensuous conscious¬ ness will never let itself be raised from the sphere of the empirical to that of Thought.
Kant’s solution of the antinomy, too, consists only in demand¬ ing that Reason must not rise above sense-perception, but must take the phenomenon as it is. This solution leaves on one side the content of the antinomy ; it does not reach the nature of the concept of its determinations, each of which, isolated by itself, is null, being no more than a transition to its Other, while Quantity is their unity and therefore their truth.
I. Quantity contains the two moments of Continuity and Discreteness. Each is its determination, and it must therefore be posited in each. — Primarily it is their immediate unity, that is, primarily it is posited only in one of its determinations, namely, continuity ; and thus is Continuous Magnitude.
Or again, continuity is one of the moments of Quantity which requires the other moment, discreteness, to complete it. But still Quantity is concrete unity only in so far as it is the unity of distinct moments. The moments therefore must be taken as distinct, but must not be resolved back into Attraction and Repulsion ; they must be taken in their truth, as each forming the whole in its unity with the other. Continuity is only coherent and homogeneous unity as unity of discrete elements.
and, posited thus, it is no longer mere moment but complete Quantity: this is Continuous Magnitude.
2. Immediate quantity is Continuous Magnitude. But in fact Quantity is not immediate; immediacy is a determinate¬ ness, and Quantity itself is this immediacy transcended. It must therefore be posited in the determinateness immanent in it, and this is the One. Quantity is Discrete Magnitude.
Discreteness, like continuity, is a moment of Quantity, but is also the whole of Quantity just because it is a moment of this whole, and thus, as distinct from it, does not relinquish its unity with the other moment. — Quantity is externality in itself, and Continuous Magnitude is this externality as pro¬ pagating itself without negation, as a context which remains at one in itself. Discrete Magnitude is this externality as non- continuous or interrupted. But though we thus have a multi¬ tude of Ones, this is not the multitude of atoms and the void, or Repulsion in general, meeting us once again: Discrete Magnitude is Quantity; and for that very reason their dis¬ creteness is continuous. The continuity in discreteness consists in the fact that the Ones are equal to one another, or have the same unity. Discrete Magnitude, then, is the externality of much One posited as the same, and not of the many Ones in general ; it is posited as the Many of one unity.
Ordinarily, when an image is formed of Continuous and of Discrete Magnitude, it is overlooked that each of these mag¬ nitudes has both the moments, continuity and discreteness, and that their difference is only constituted by which of the two moments is posited determinateness, and which that which only is in itself. Space, Time, Matter, and so on, are fixed mag¬ nitudes because each is a repulsion from itself, where there is a stream of extrogression without any transition or relation to what is qualitatively Other. They have the absolute poten¬ tiality of the One being posited relatively to them, and not merely as the empty potentiality of a mere otherness (as we say that a tree might occupy the space occupied by a stone) : they contain in themselves the principle of the One, it is one of the determinations which constitute them.
Conversely, continuity must not be overlooked in Discrete Magnitude: this moment, as has been shown, is the One as unity.
Continuous and Discrete Magnitude may be considered as species of Quantity, but only in so far as magnitude is posited not under any external determinateness, but solely under the determinatenesses of its own moments ; the ordinary transition from genus to species allows external determinations to be applied to these according to some external principle of classi¬ fication. And so far Continuous and Discrete Magnitude are not yet Quanta: they are Quantity itself in each one of its two forms. If they are called magnitude, this is because they have this in common with Quantum, to be a determinateness applied to Quantity.
Discrete Magnitude, first, has the One for principle ; secondly, it is a plurality of Ones ; thirdly, it is, essentially, static ; it is One, but One transcended, or as unity; it is self-continuation as such in the discreteness of the Ones. It is therefore posited as one magnitude, and its determinateness is the One, which, in this positedness and determinate being is exclusive One, or Limit, applied to unity. Discrete Magnitude as such is not to be limited immediately; but, as distinct from Continuous Magnitude, it exists as a Determinate Being and a Something whose determinateness is One, and, as being in a Determinate Being, is also first negation and Limit.
This Limit is referred to unity and applied to it as negation, and, besides this, as One is also referred to itself; it is thus including and comprehending Limit. Here the Limit is not distinguished from the Something of its Determinate Being, but, as One, is immediately this negative point itself. But the Being which here is limited exists essentially as continuity, by virtue of which it passes beyond the Limit and beyond this One, and is indifferent to them. Real discrete Quantity is therefore one Quantity, or Quantum, — Quantity as Determinate Being and Something.
Since the One, which is Limit, comprehends in itself the many Ones of discrete Quantity, it in so doing also posits them as transcended in itself. It is Limit as applied to continuity in general as such, and hence the distinction between Continuous and Discrete Magnitude is here immaterial ; or, rather, it is Limit to the continuity of the one as much as of the other: both pass over into the state of Quantum.
Quantum — which, first, means Quantity having any deter¬ minateness or limit at all — is, in its complete determinateness, Number. It is divided into
secondly [a) Extensive Quantum, where the Limit is the Barrier of a plurality there existing; ( b ) Intensive Quantum or Degree, the plurality having passed over into bcing-for-self. This Barrier being for itself, and therefore as Limit indifferent, is equally immediately outside itself and has its determinate¬ ness in an Other. Thus Quantum is a posited contradiction, it is simply determined for itself and also has its determinate¬ ness outside itself and thus refers beyond itself; it therefore
thirdly passes over, as that which is itself posited as external to itself, into Quantitative Infinity.
Quantity is Quantum ; it has a Limit, whether it be taken as continuous or as discrete magnitude. The distinction between these two species is here of no importance.
As transcended Being-for-Self, Quantity in itself already is indifferent to its Limit. For that very reason Limit (or the fact that it is a Quantum) is not indifferent to it ; for it contains the One, or absolute determinateness, in itself as its own moment, which thus posited in its continuity or oneness is its Limit, but yet remains that One which it has become.
This One is thus the principle of Quantum; but it is the One of Quantity. Hence, first, it is continuous, or Unity; secondly, it is discrete, plurality of Ones which is in itself (as in continuous magnitude) or posited (as in discrete mag¬ nitude) : these Ones are equal to one another, they have this continuity, this same unity. Thirdly, this One is also negation of the many Ones as simple Limit, an exclusion of otherness.
or a determination of itself in opposition to other Quanta. One is thus Limit (a) referring to itself, (j8) inclusive, and (y) ex¬ clusive of Other.
Quantum completely posited in these determinations is Number. Complete positedness is implied by the existence of the Limit as plurality and its consequent distinctness from unity Hence Number appears as discrete magnitude, but in unity it also has continuity. Hence too it is Quantum in com¬ plete .determinateness : for in it Limit is determinate plurality, having for principle One, the absolutely determinate. Con¬ tinuity, where One is only in itself or as transcended (posited as unity) — is the form of indeterminateness.
Quantum, merely as such, is limited generally; its limit is its abstract and simple determinateness. But since it is Number, the Limit is posited as being manifold within itself. It contains the many Ones which constitute its Determinate Being, but not in a merely indeterminate manner: the determinateness of the Limit falls within it. The Limit excludes other Deter¬ minate Being, that is, other Manies : the Ones included in it are a determinate heap, an Amount. This is discreteness as it is in Number, and hence its other is Unity, the continuity of this. Amount and Unit constitute the moments of Number.
With regard to Amount, it must further be considered in what manner the many Ones of which it consists are in the Limit; it is correct to say of Amount that it consists of the Many, for in it the Ones do not exist as transcended, but in it they are , posited only with the excluding Limit, to which they are indifferent. The Limit is not, however, indifferent to them. The relation of Determinate Bang to Limit (as was seen when that category was treated) was such that Determinate Being, as the affirmative, remained on this side of its Limit, while the Limit (or negation) remained outside, at the edge; similarly, with the many Ones, the abrupt exclusion of other Ones appears as a determination falling without the included Ones. But it was seen above that Determinate Being is pene¬ trated by Limit and is coextensive with it, and that thus some¬ thing is inherently limited, that is, finite. — Thus Number — which is Quantitative— is imagined in such a way that with one hundred, for instance, the hundredth One alone shall limit the Many in such a manner as to make them one hundred-
This is correct ; yet among the hundred Ones none is preferred individually, since all are equal : each equally is the hundredth ; they all equally belong to the Limit which makes the number one hundred, and, for the determinateness of the number, not one can be spared. Thus the others do not constitute relatively to the hundredth One a Determinate Being external to the limit, or distinct from it within it. Amount then is not a plurality as opposed to the inclusive and limiting One, but rather itself constitutes the limitation which is a determinate Quantum : the Many constitute one number, One two, One ten, One hundred, and so forth.
Now the limiting One is determinateness against Other, the distinguishing of one number from another. But this dis¬ tinguishing does not become qualitative determinateness; it remains quantitative and appertains only to external com¬ paring reflection: Number, as One, has returned upon itself, and at this resting-place it remains indifferent to Others. This indifference of Number to Others is its essential determination : by virtue of it, Number is determinate-in-itself, but also in itself external. — It is thus a numerical One, as absolutely determinate ; it has the form of simple immediacy, and there¬ fore relation to Other is quite external to it. And as being One which is Number, it further has determinateness (in so far as it is relation to Other) as its moments in itself, in its distinction between Unit and Amount ; and Amount itself is plurality of Ones, that is, in itself it is this absolute externality. — This contradiction of Number, or of Quantum in general, in itself, is the quality of Quantum : this contradiction develops itself in the further determinations of the latter.
Spatial and numerical magnitude are generally considered as two species, each being a determinate magnitude as much as the other; and it is thought that, the distinction being only in the different determinations of continuity and discreteness, they rank equally as Quantum. Geometry, in spatial magni¬ tude, generally has continuous magnitude for object; and arithmetic, in numerical, has discrete magnitude for object. But, the objects being thus different, they are not limited or
determined in the same manner or with the same completeness. Spatial magnitude is merely limited in the abstract ; if it must be considered as a definitely determinate Quantum, Number is required. Geometry as such does not measure spatial figures — it is not the art of measuring: it only compares. In its definitions too the determinations partly are derived from equality of sides and angles, or from equidistance. Thus the circle, since it is based solely upon the equidistance of all possible points in it from a centre, can be determined with¬ out Number. These determinations, based on equality and inequality, are true geometry. But they are not sufficient, and others, such as triangle or quadrilateral, require Number, which in its principle, the One, contains determinedness-for- self, not determinedness by the aid of an Other, hence not by comparison. It is true that, in the point, spatial magnitude has a determinateness corresponding to the One; but the point, in so far as it passes out of itself, becomes an Other, becomes the line ; and, being essentially the One of Space, it becomes, when related, a continuity, in which its puncticity (or determinedness-for-self, or the One) is transcended. In so far as determinedness-for-self is to preserve itself in self-exter¬ nality, the line must be imagined as a multitude of Ones, and the Limit must contain the determination of many Ones ; that is, the magnitude of the line — like other spatial determinations — must be taken as Number.
Arithmetic contemplates Number and its figures ; or, rather, it operates with them and does not contemplate. For Number is indifferent determinateness and inert; it must be actuated and brought into relation from without. The different methods of relation are the species of calculation. In arithmetic they are enumerated in series, and it is evident that they are mutually dependent. But arithmetic does not give prominence to the thread which guides their progress. However, the systematic arrangement which the exposition of these elements in the text-books justly claims, results easily from the conceptual determination of Number itself. We will here briefly notice these cardinal determinations.
The principle of Number is the One, and for this reason it is a purely analytic figure ; its elements are connected externally, not internally. It is thus only an external creation ; for which
reason, too, all arithmetical operations are the production of numbers, — numbering, or, more closely, adding together. Variations in this external production, which ever proceeds in the same manner, can depend only on relative diversity between the numbers which are to be added together; such diversity itself must come from another source, and from external determination.
The qualitative distinction which constitutes the deter¬ minateness of Number is, as we have seen, that which subsists between Unit and Amount; hence all conceptual determinate¬ ness which can arise in the species of calculation reduces itself to this. But the distinction proper to numbers as Quanta is external identity and external difference, or equality and inequality ; and these are moments of reflection, and must be dealt with when we deal with the determinations of Essence (under the heading of Difference).
And further we must premise that numbers can in general be produced in two ways — by joining, or by disjoining what was already joined. Each takes place in a manner of numbering determined in the same way ; and thus to the joining of numbers there corresponds what may be called a positive kind of arith¬ metic ; and to disjoining, a negative ; the determination of each species of calculation itself is independent of this contrast.
i. We proceed to indicate the species. The first creation of Number is the joining together of Manies as such, that is, such as are posited each as One only, — Numbering. The Ones are external to one another, and are thus imagined under a sense- image: the operation which creates the number becomes a process of counting fingers, dots, and so forth. We can only point to an example of four, or of five, and so on. And the point where the process of joining is broken is (since the Limit is external) contingent and arbitrary. — The distinction between Amount and Unit which arises in the process of the species of calculation, becomes the basis of a system of numbers, such as the dyadic, decadic, and so on; in general such system tests on the choice of the amount which is consistently to be taken as unit.
The numbers produced by numbering are once more num¬ bered: posited thus immediately they are determined as mutually quite unrelated, and indifferent to equality and
inequality; their relative magnitude is contingent; they are hence altogether unequal. This is Addition. We discover that 7 and 5 make twelve by adding (by counting on our fingers or otherwise) 5 more Ones to the 7; the result is afterwards learnt by heart : but there is nothing internal here. We similarly learn that 7x5 = 35 by counting on the fingers, and so forth, one seven being added to another, this process being gone through five times, and the result again being learnt by heart. The trouble of numbering, the discovering of sums or products, is abolished by the finished “one and one” or multiplication- tables, which have only to be learnt by heart.
In the Introduction to the Critique of Pure Reason, V, Kant has considered the proposition that 7+5 =*= 12 as a synthetic proposition. “At first,” he says, “one should think [of course !] that it is merely an analytical proposition, resulting from the concept of a sum of seven and five by the Law of Contradiction.” The concept of a sum is just this abstract determination that these two numbers ought to be joined together, and that too (since they are numbers) in an external and conceptless manner, — that we are to count on from seven until the Ones which are to be added (their amount being determined as five) are exhausted ; the result bears the name independently known as twelve. “But,” so Kant continues, “if we look more closely, we find that the concept of the sum of seven and five contains no more than the joining of these two numbers into one ; it is not thought at all what this single number is which compre¬ hends these two;” “ — however much I may analyse my con¬ cept of such a possible sum, I shall never find twelve therein.” The transition from the problem to the result has indeed nothing to do with the thinking of the sum and the analysis of the concept; and he adds, “We must pass beyond these concepts and take to intuition, to five fingers and so on, adding the five ones given by intuition to the concept of seven.” Of course five is given intuitively, that is, it is an entirely external agglomeration of the thought of one, arbitrarily repeated-; but equally seven itself is no concept ; there are no concepts beyond which a transition here is made. The sum of 5 and 7 is the conceptless connexion of these two numbers, and the concept¬ less process of numbering from seven on, until five are ex¬ hausted, can literally be called a synthesis, which name can
also be given to the process of numbering from one onwards ; — but this synthesis is wholly analytical in nature, for the con¬ nexion is quite artificial and nothing is introduced that is not at hand externally. The postulate of adding 5 to 7 is to the postulate of numbering at all as is the postulate of producing a straight line to that of drawing a straight line.
The expression “synthesis” is void, and equally void is the determination that it takes place a priori. Counting is not, of course, a determination of sensation (which, according to Kant’s determination of intuition, is all that remains to be predicated of whatever is a posteriori ), and it is an occupation on the basis of abstract intuition: that is, it is determined through the category of the One, and abstraction is here made from all other determinations of sensation, as well as from concepts. Altogether, the a priori is quite vague; and deter¬ mination of feeling, in the shape of impulse, tendency, and so forth, contains the moment of apriority as much as space and time are determined as existing, and the temporal and the spatial are determined, a posteriori.
We may add in this connexion that Kant’s assertion of the synthetic nature of the foundations of pure geometry equally lacks solidity. He asserts that several are really analytic, but cites in favour of this idea only the proposition that the straight line is the shortest between any two points. “My concept of the straight does not contain anything about magnitude, but only a Quality ; thus the concept of shortest is a pure addition, which can be extracted by no analysis from the concept of the straight line: we must have recourse to intuition, and then alone synthesis is possible.” — But here again we are not dealing with a concept of the straight in general, but with that of a straight line, and this is already spatial and intuited. The determination (or, if you will, the concept) of the straight line is surely none other than this, that it is the absolutely simple line, that is, that, in passing beyond itself (the so-called move¬ ment of the point) it is simply self-related ; in its extension no sort of differentiation of the determination, or relation to any other point or line without it, is posited;— -it is just simple direction in itself. This simplicity is indeed its Quality; and, if it should seem hard to define the straight line analytically, this is(so only because of this determination of simplicity or
self-relation, and solely because, when determination is thought of, a multiplicity, or determining through others, is primarily imagined. In itself, however, it is not difficult to seize on this determination of the simplicity in itself of extension, and of its indeterminedness by Other; — Euclid’s definition contains nothing else than this simplicity. — Now the transition of this Quality to quantitative determination (namely, that of being the shortest), which is supposed to constitute the synthetic element, is purely analytical. Being spatial, the line is Quantity in general; the most simple thing, if predicated of Quantum, is “least”; and this, predicated of a line, is “shortest.” Geometry can take these determinations as corollary to the definition ; but Archimedes in his books on Sphere and Cylinder (translation Hauber, p. 4) did more suitably in making this determination of the straight line fundamental : and therein he was right, as was Euclid when he placed the determination relating to parallel lines among the fundamentals ; for, in order to become a definition, the development of this determina¬ tion too would have required qualitative determinations not properly spatial but more abstract, namely (as simplicity above, so here) similarity of direction and the like. The ancients gave a plastic character even to their sciences; they kept their exposition strictly within the peculiarities of their material, and therefore excluded what would have been heterogeneous.
The concept advanced by Kant in the synthetic judgments a priori — the concept of differentiation with inseparability, and of identity which in itself is unseparated differentiation — belongs to the great and immortal part of his philosophy. In intuition this concept — since it is the concept itself, and everything in itself is concept — likewise is certainly present; but the deter¬ minations which have been taken in those examples do not represent it; Number, and numbering, rather is an identity and the production of an identity which is merely external, or merely a superficial synthesis ; a unity of Ones, such as are posited as not identical with one another, but rather as mutually separate and external. With the straight line, the determination of being the shortest line between two points is based rather on the moment of abstract identity, without differentiation in itself.
From this digression I revert to addition. The corresponding
negative arithmetical process, subtraction, likewise is the purely analytical severing of numbers, which, as in addition, are determined merely as mutually altogether unequal.
2. The next determination is the equality of the numbers which are to be numbered. This equality makes them into a unity; and thus there is applied to Number the distinction between Unit and Amount. Multiplication is the problem of adding an amount of units of which each is an amount. It is here indifferent which of the two numbers is taken as unit and which as amount, — whether we say 4 times 3 (where 4 is the amount and 3 the unit) or, conversely, 3 times 4. — We have already indicated above that the original discovery of the product is effected by simple numbering, by counting on the fingers and so forth; the product, later, can be indicated immediately because there is a collection of these products (the multiplication table) which is known by heart.
Division is the negative arithmetical process with the same determination of the distinction. It is equally indifferent which of the two factors, divisor or quotient, is determined as unit or as amount. The divisor is determined as unit and the quotient as amount when the problem in division is ex¬ pressed in the words that it is desired to see how many times (amount) one number (unit) is contained in a given number; conversely the divisor is taken as amount and the quotient as unit, when it is said that a number is to be divided into a given amount of equal parts, and that the magnitude of such a part (the unit) is to be found.
3. The two numbers which are determined as being related to one another as unit and amount, are still immediately opposed as numbers, and therefore are altogether unequal. The equality which is attained is next that of unit and of amount themselves: thus the progress towards the equality of the determinations immanent in the determination of Number, is completed. According to this complete equality, counting is the raising to a certain power (the negative arithmetical process being to take the root) — and, first, the squaring of a number; — numbering is here completely determinate in itself, when (1) the many numbers which are to be added are the same, and (2) their plurality, or amount, is the same as the number which is posted so many times, that is, which is unit. There are no
further determinations in the concept of Number which could offer a differentiation; nor can the differentiation immanent in Number be made more homogeneous. Where a number is raised to a higher power than the square, there is a formal continuation ; partly (with the even exponents) there is a mere repetition of squaring; partly (with the odd powers) inequality returns; for, although the new factor is formally equal (for example, in a first instance with the cube) both to amount and to unit, yet it stands, as unit, in a relation of inequality to the amount (the square, 3 against 3 times 3) ; still more in the cube of four, where the amount, 3, according to which the number which is unit should be multiplied by itself, is actually different from the latter. — Where these determinations — amount and unit — are present, they constitute in themselves the essen¬ tial differentiation of the concept: in order that that which has gone outside itself may completely return upon itself, these two must be equalized. The above exposition contains the reason why partly the solution of higher equations must consist in the reduction to quadratic equations, and partly why the equations of odd exponents can only be formally determined, and, just when the roots are rational, they cannot be found otherwise than by an imaginary expression, that is, by the opposite of that which the roots are and express. — From what has been said it appears that the arithmetical square alone contains absolute self-determinedness in itself; hence the equations with further formal powers must be reduced to it; just as the right-angled triangle in geometry con¬ tains that absolute self-determinedness in itself which is set forth in the theorem of Pythagoras, and so all other geo¬ metrical figures must be reduced to it for their complete determination.
A method proceeding under the guidance of a judgment logically constituted treats of powers before it treats of pro¬ portions; it is true that the latter are connected with the difference between unit and amount, which constitutes the determination of the second arithmetical process; but they step beyond the One of immediate Quantum, in which unit and amount are only moments ; and the further determination according to it still remains external to it. In Ratio, Number no longer is immediate Quantum : there it has its determinate-
ness, as mediation. Qualitative Relation will be considered in what follows.
We may say of this exposition of the progressive determina¬ tion of the arithmetical processes, that it is no philosophy of them nor demonstration of their inner meaning, since in fact it is not an immanent development of the concept. But philo¬ sophy must know how to distinguish what is in its nature a material external to itself, so that here the progress of the concept can take place only in an external manner, while its moments can have only the peculiar form of their externality, such as oddness and evenness in this example. It is an essential condition, if one would philosophize about real objects, that the spheres be distinguished in which a definite form of the concept belongs, or, in other words, is present as existence: else what is external and contingent will be disturbed in its peculiarity by ideas ; and also these ideas, through the inade¬ quacy of the material, will be distorted and formalized. Now this externality in which the moments of the concept appear in this external material — Number — is here the adequate form ; they represent the object in its understanding, and further contain no speculative demands and thus appear easy: they therefore deserve to be employed in text-books of the elements.
It is well known that Pythagoras represented intellectual relations (or philosophemata) by numbers; in more modem times too numbers and the forms of their relations (such as powers and so on) have been used in philosophy, in order to regulate thoughts or to express them. — From an educational point of view Number has been considered the most suitable object of inner intuition, and the arithmetical manipulation of its relations has been considered that activity of the mind where it manifests its most peculiar relations and altogether the fundamental relations of Essence. — The concept of Number, as it has here yielded itself, shows how far Number may claim this high worth.
We have seen in Number the absolute determinateness of Quantity, and in its element differentiation which has become indifferent; — determinateness in itself, which also is posited
as wholly external. Arithmetic is an analytic science, because all connexions and distinctions which occur in its object are not inherent in it, but are applied to it wholly from without. It has no concrete object containing internal relations which, at first hidden from knowledge and not given in the immediate idea of the object, have to be elaborated by the efforts of cognition. Not only does it not contain the concept and there¬ with the task for conceptual thought: it even is its opposite. The connected terms are indifferent to the connexion, which lacks necessity: the activity of thought, therefore, is here one which is the extremest self-renunciation, an activity which forces it to move in thoughtlessness and to connect terms which admit of no necessity. The object is the abstract thought of externality itself.
Number thus being the very thought of externality, is also the abstraction from the manifold of the senses ; it has retained nothing of the sensuous except the abstract determination of externality itself; and the latter therefore approaches nearest to it in thought ; it is the pure thought of the self-renunciation of thought.
Mind, which rises above the world of the senses and under¬ stands its own essence, when it seeks an element for its pure imagination, for the expression of its essence, may therefore strike upon that inner and abstract externality, Number, before it seizes thought itself as this element and achieves the purely spiritual expression fit to represent it. Hence we see, early in the history of Science, Number being used to express philoso- phemata. It constitutes the last step of that incomplete method which apprehends the universal as affected by the sensuous. The ancients were clearly conscious that Number is midway between the sensuous and thought. Aristotle ( Metaphys . I. 5) quotes Plato as saying that the mathematical determinations of things stand apart from and between the sensuous and the Ideas, distinguished from the former because they are invisible (eternal) and unmoved, and from the latter because they are a manifold and have similarity, while the Idea is simply self¬ identical and one within itself. — A more detailed and profound consideration of this matter, by Moderatus1 of Cadiz, is quoted in Malcki Vita Pythagorae ed. Ritterhus, pp. 30 sq. He thinks that 1 A Neopythagorean who lived in the time of Nero.
the Pythagoreans hit upon numbers because they were not yet able clearly to seize in Reason the fundamental ideas and fast principles, these principles being hard to think and hard to express. Numbers serve the teacher well as designations; and the Pythagoreans imitated, herein as elsewhere, the geometers, who cannot express the corporeal in thought, and therefore use figures, saying “this is a triangle” when, however, they do not mean that this optically visible figure is to be taken for a triangle, but that only the concept is to be represented. And thus the Pythagoreans used the expression “One” for the concept of unity, of identity and equality, the basis of agree¬ ment, of connexion, of the preservation of all, of the self¬ identical and so forth. — It is superfluous to observe that the Pythagoreans passed beyond the expression of Number to the expression of Thought, to the express categories of equal and unequal, to limit and infinity; indeed, at the place where these numerical expressions are considered (ibid., note to p. 31, 1. s., from a Life of Pythagoras in Photius, p. 722), it is mentioned that the Pythagoreans distinguished between the Monas and the One, taking the Monas as concept and the One as number, and, similarly, the Two for the arithmetical term, but the Dyas (for this seems the correct reading) for the concept of the indeterminate. — Thus these ancient writers correctly perceived the inadequacy of these numerical forms as determinations of thought, and equally rightly they demanded, in place of this first substitute for thought, its characteristic expression; how much further had they progressed in thought than those who in our day, when some put in the place of determinations of thought numbers and determinations of numbers (like powers), next the infinitely great and the infinitely small, one divided by infinity, and other such determinations, which often are a perverted mathematical formalism, take the return to this impotent childishness for something praiseworthy and even for something thorough and profound.
We quoted above the expression that number stands between the sensuous and thought, since it shares with the former the quality of being in itself the Many, or separate existence : it is here to be noted that this Many itself, the sensuous which is taken up into thought, is that category of the self-external which is proper to it. Further, concrete, veritable thoughts,
which of all things are the most quick and flexible, conceived only where there is relation, when translated into this element of being beyond self become dead and rigid determinations. The richer in determinateness, and hence in relation, thoughts become, the more confused and the more arbitrary and sense¬ less becomes their representation in such forms as numbers. The One, the Two, the Three, and the Four, Henas or Monas, Dyas, Trias, Tetraktys, still approximate to the simple and abstract concepts ; but when numbers are required to pass over into concrete relations it is a vain attempt which would keep them close to the concept.
But the hardest thing is asked of thought, when the deter¬ minations of thought by One, Two, Three, Four are desig¬ nated as that movement of the concept through which alone it is concept. It moves in the element of its opposite, which is unrelatedness ; its work is the work of derangement. It is a hard suggestion to conceive that One is Three (for instance) or Three One, because the One is the unrelated and, therefore, does not show in itself the determination by virtue of which it passes over into its opposite, since its very being is the absolute exclusion and negation of such a relation. Conversely, Understanding makes use of this against speculative truth (for instance against that which is laid down in the doctrine called that of the Trinity), and counts those determinations of it which constitute One Unity, in order to demonstrate it to be a clear contradiction; — that is, itself contradictorily makes unrelated that which essentially is relation. It was not expected, when the name of Trinity was coined, that Understanding would consider One and Number as the essential determinateness of the content. This name expresses contempt for Understanding, which has nevertheless confirmed itself in its vanity in clinging to One and Number as such, and has set it up against Reason.
It is, in a manner, harmless, when numbers or geometrical figures are taken as mere symbols, as has often been done with the circle, the triangle, and so on, the circle for instance standing for eternity and the triangle for the Trinity; but it is foolish to think that more is thus expressed than can be comprehended or expressed by thought. These symbols, like others created by fancy in the mythologies of peoples and in poetry generally, — compared with which the bare geometrical
figures are in any case meagre, — are supposed to contain a profound wisdom and a profound meaning ; if it is so, then the sole purpose of thought is, to extract the wisdom which lies only here, and not merely in symbols but also in Nature and in Spirit. In symbols truth is darkened and veiled by the sensuous element; it is wholly revealed to consciousness only in the form of thought : Meaning is only the thought itself.
But when mathematical categories are used to determine something bearing upon the method or content of philosophic science, such a procedure proves its preposterous nature chiefly herein, that, in so far as mathematical formulae mean thoughts and conceptual distinctions, such meaning must first report, determine, and justify itself in philosophy. In its concrete sciences, philosophy takes the logical element from logic and not from mathematics ; it must be a mere refuge of philosophic impotence when it flies to the formations which logic takes in other sciences, of which many are only dim presentiments and others stunted forms of it, in order to get logic for philosophy. The mere employment of such borrowed forms is in any case an external attitude : a knowledge of their worth and of their meaning should precede their use ; but such knowledge results only from thoughtful contemplation, and not from the authority which mathematics gives them. Logic itself is such a conscious¬ ness; and such consciousness strips them of their particular form, making it superfluous and useless, and, while correcting them, alone gives them justification, meaning, and worth.
The value of the use of Number and of Arithmetic, in so far as it is supposed to be a main basis of education, is evident from the above. Number is a non-sensuous object, and an occupation with it and its combinations a non-sensuous busi¬ ness : thus the mind is urged to reflect in itself and to do inner and abstract work ; which is of great though one-sided impor¬ tance. For, on the other hand. Number being based only on external and non-conceptual differentiation, this occupation becomes thoughtless and mechanical. The effort chiefly consists in seizing units void of concept and combining them without the use of concept. The content is the empty One : that rich content of moral and spiritual life, and its individual growth, on which, as its noblest nourishment, education should rear the young mind, would here be ousted by the empty One; and
when such exercises form the main matter and the main occupation, the effect can be no other than to hollow and blunt the spirit in form and concept. Arithmetic being so extremely external and therefore mechanical a matter, it has been possible to construct machines which execute the arith¬ metical operations in the most perfect manner. If this one circumstance were known about the nature of arithmetic, it alone would contain a judgment about the value of the notion which would make arithmetic the chief instrument for educating the mind by putting it on the rack which would perfect it into a machine.
1. We saw that Quantum has its determinateness as limit in Amount. It is a something discrete and manifold in itself, and has no Being distinct from its limit and having this outside itself. Quantum and its limit (and the limit in itself is a mani¬ fold) together thus are Extensive Magnitude.
Extensive Magnitude must be distinguished from continuous magnitude: the former is opposed to intensive and not to dis¬ crete magnitude. Of extensive and intensive magnitude each is a determinateness of the quantitative limit itself, while Quantum is identical with its limit; continuous and discrete magnitude, on the other hand, are determinations of magnitude in itself, that is, of Quantity as such, in so far as in Quantum abstraction is made from limit. — The moment of continuity is attached to extensive magnitude and to its limit, for its Many is simply continuous ; the limit as negation thus appears in this levelling of the Many, as limitation of unity. Continuous mag¬ nitude is self-repeating Quantity without regard to any limit, and, in so far as it is imagined as having a limit, this is limitation in general, without discreteness being posited in it. Quantum taken merely as continuous magnitude is not yet truly deter¬ minate for itself, for it lacks Number and the One which implies that it is determinate for itself. And similarly discrete magnitude is, immediately, only differentiated Many in general,
which, in so far as it had a limit as such, would be just a multitude, that is, a something limited but limited indeter¬ minately; in order to make it into a determinate Quantum, the Many must be subsumed into One, by which process they are posited as identical with the limit. Continuous and discrete magnitude, taken as Quantum generally, have each posited in it only one of the sides by which it is completely determined and made into Number. The latter is immediate Extensive Quantum, — simple determinateness, existing essentially as amount , though only as an amount of one and the same unit, being distinguished from Number only in this, that in the latter determinateness is expressly posited as plurality.
2. However, when the magnitude of something is determined by number, no contrast with another magnitude is needed: the determinateness of this magnitude does not depend upon itself and another magnitude, for the determinateness of mag¬ nitude in general is self-determinate, indifferent and simply self-related Limit; which, in Number, is posited as enclosed in the self-existing One, externality and relation to Other being within itself. This Many of the Limit, further, (like the Many in general) is not differentiated within itself, but continuous; each of the Many is what every other is; its determinateness as such does not consist in its quality of mutual externality of parts, or discreteness. Thus this Many for itself collapses into its continuity and becomes simple unity. — Amount is only a moment of Number, but does not constitute the determinate¬ ness of Number as a multitude of numerical Ones : the Ones, as indifferent and self-external, are transcended when Number has returned upon itself. Externality, which constituted the Ones of plurality, vanishes in the One as self-relation of Number.
Quantum, as extensive, had its existing determinateness as self-external Amount: its limit thus passes over into simple determinateness. In this simple determination of the limit. Quantum is intensive magnitude; and the limit or deter¬ minateness which is identical with Quantum is now also posited as simple — it is Degree.
Degree thus is determinate magnitude or Quantum, but it is not also a multitude or multiplicity within itself; it is only a “more”; and the quality of “more” is the manv taken