phenomena, the category of substance and its accidents is not suitable for the formation of a transcendental idea; that is to say, reason has no ground, in regard to it, to proceed regressively with conditions.
For accidents (in so far as they inhere in a substance) are co-ordinated with each other, and do not constitute a series. And, in relation to substance, they are not properly subordinated to it, but are the mode of existence of the substance itself. The conception of the substantial might nevertheless seem to be an idea of the transcendental reason. But, as this signifies nothing more than the conception of an object in general, which subsists in so far as we cogitate in it merely a transcendental subject without any predicates; and as the question here is of an unconditioned in the series of phenomena—it is clear that the substantial can form no member thereof.
The same holds good of substances in community, which are mere aggregates and do not form a series. For they are not subordinated to each other as conditions of the possibility of each other; which, however, may be affirmed of spaces, the limits of which are never determined in themselves, but always by some other space. It is, therefore, only in the category of causality that we can find a series of causes to a given effect, and in which we ascend from the latter, as the conditioned, to the former as the conditions, and thus answer the question of reason.
Fourthly, the conceptions of the possible, the actual, and the necessary do not conduct us to any series—excepting only in so far as the contingent in existence must always be regarded as conditioned, and as indicating, according to a law of the understanding, a condition, under which it is necessary to rise to a higher, till in the totality of the series, reason arrives at unconditioned necessity.
There are, accordingly, only four cosmological ideas, corresponding with the four titles of the categories. For we can select only such as necessarily furnish us with a series in the synthesis of the manifold.
1 The absolute Completeness of the COMPOSITION of the given totality of all phenomena.
2 The absolute Completeness of the DIVISION of given totality in a phenomenon.
3 The absolute Completeness of the ORIGINATION of a phenomenon.
4 The absolute Completeness of the DEPENDENCE of the EXISTENCE of what is changeable in a phenomenon.
We must here remark, in the first place, that the idea of absolute totality relates to nothing but the exposition of phenomena, and therefore not to the pure conception of a totality of things. Phenomena are here, therefore, regarded as given, and reason requires the absolute completeness of the conditions of their possibility, in so far as these conditions constitute a series—consequently an absolutely (that is, in every respect) complete synthesis, whereby a phenomenon can be explained according to the laws of the understanding.
Secondly, it is properly the unconditioned alone that reason seeks in this serially and regressively conducted synthesis of conditions. It wishes, to speak in another way, to attain to completeness in the series of premisses, so as to render it unnecessary to presuppose others. This unconditioned is always contained in the absolute totality of the series, when we endeavour to form a representation of it in thought. But this absolutely complete synthesis is itself but an idea; for it is impossible, at least before hand, to know whether any such synthesis is possible in the case of phenomena. When we represent all existence in thought by means of pure conceptions of the understanding, without any conditions of sensuous intuition, we may say with justice that for a given conditioned the whole series of conditions subordinated to each other is also given; for the former is only given through the latter. But we find in the case of phenomena a particular limitation of the mode in which conditions are given, that is, through the successive synthesis of the manifold of intuition, which must be complete in the regress. Now whether this completeness is sensuously possible, is a problem. But the idea of it lies in the reason—be it possible or impossible to connect with the idea adequate empirical conceptions. Therefore, as in the absolute totality of the regressive synthesis of the manifold in a phenomenon (following the guidance of the categories, which represent it as a series of conditions to a given conditioned) the unconditioned is necessarily contained—it being still left unascertained whether and how this totality exists; reason sets out from the idea of totality, although its proper and final aim is the unconditioned—of the whole series, or of a part thereof.
This unconditioned may be cogitated—either as existing only in the entire series, all the members of which therefore would be without exception conditioned and only the totality absolutely unconditioned—and in this case the regressus is called infinite; or the absolutely unconditioned is only a part of the series, to which the other members are subordinated, but which Is not itself submitted to any other condition.[48] In the former case the series is a parte priori unlimited (without beginning), that is, infinite, and nevertheless completely given. But the regress in it is never completed, and can only be called potentially infinite. In the second case there exists a first in the series. This first is called, in relation to past time, the beginning of the world; in relation to space, the limit of the world; in relation to the parts of a given limited whole, the simple; in relation to causes, absolute spontaneity (liberty); and in relation to the existence of changeable things, absolute physical necessity.
[48] The absolute totality of the series of conditions to a given conditioned is always unconditioned; because beyond it there exist no other conditions, on which it might depend. But the absolute totality of such a series is only an idea, or rather a problematical conception, the possibility of which must be investigated—particularly in relation to the mode in which the unconditioned, as the transcendental idea which is the real subject of inquiry, may be contained therein.
We possess two expressions, world and nature, which are generally interchanged. The first denotes the mathematical total of all phenomena and the totality of their synthesis—in its progress by means of composition, as well as by division. And the world is termed nature,[49] when it is regarded as a dynamical whole—when our attention is not directed to the aggregation in space and time, for the purpose of cogitating it as a quantity, but to the unity in the existence of phenomena. In this case the condition of that which happens is called a cause; the unconditioned causality of the cause in a phenomenon is termed liberty; the conditioned cause is called in a more limited sense a natural cause. The conditioned in existence is termed contingent, and the unconditioned necessary. The unconditioned necessity of phenomena may be called natural necessity.
[49] Nature, understood adjective (formaliter), signifies the complex of the determinations of a thing, connected according to an internal principle of causality. On the other hand, we understand by nature, substantive (materialiter), the sum total of phenomena, in so far as they, by virtue of an internal principle of causality, are connected with each other throughout. In the former sense we speak of the nature of liquid matter, of fire, etc., and employ the word only adjective; while, if speaking of the objects of nature, we have in our minds the idea of a subsisting whole.
The ideas which we are at present engaged in discussing I have called cosmological ideas; partly because by the term world is understood the entire content of all phenomena, and our ideas are directed solely to the unconditioned among phenomena; partly also, because world, in the transcendental sense, signifies the absolute totality of the content of existing things, and we are directing our attention only to the completeness of the synthesis—although, properly, only in regression.
In regard to the fact that these ideas are all transcendent, and, although they do not transcend phenomena as regards their mode, but are concerned solely with the world of sense (and not with noumena), nevertheless carry their synthesis to a degree far above all possible experience—it still seems to me that we can, with perfect propriety, designate them cosmical conceptions. As regards the distinction between the mathematically and the dynamically unconditioned which is the aim of the regression of the synthesis, I should call the two former, in a more limited signification, cosmical conceptions, the remaining two transcendent physical conceptions. This distinction does not at present seem to be of particular importance, but we shall afterwards find it to be of some value.
Section II. Antithetic of Pure Reason Thetic is the term applied to every collection of dogmatical propositions. By antithetic I do not understand dogmatical assertions of the opposite, but the self-contradiction of seemingly dogmatical cognitions (thesis cum antithesis), in none of which we can discover any decided superiority. Antithetic is not, therefore, occupied with one-sided statements, but is engaged in considering the contradictory nature of the general cognitions of reason and its causes.
Transcendental antithetic is an investigation into the antinomy of pure reason, its causes and result. If we employ our reason not merely in the application of the principles of the understanding to objects of experience, but venture with it beyond these boundaries, there arise certain sophistical propositions or theorems. These assertions have the following peculiarities: They can find neither confirmation nor confutation in experience; and each is in itself not only self-consistent, but possesses conditions of its necessity in the very nature of reason—only that, unluckily, there exist just as valid and necessary grounds for maintaining the contrary proposition.
The questions which naturally arise in the consideration of this dialectic of pure reason, are therefore: 1st. In what propositions is pure reason unavoidably subject to an antinomy? 2nd. What are the causes of this antinomy? 3rd. Whether and in what way can reason free itself from this self-contradiction?
A dialectical proposition or theorem of pure reason must, according to what has been said, be distinguishable from all sophistical propositions, by the fact that it is not an answer to an arbitrary question, which may be raised at the mere pleasure of any person, but to one which human reason must necessarily encounter in its progress.
In the second place, a dialectical proposition, with its opposite, does not carry the appearance of a merely artificial illusion, which disappears as soon as it is investigated, but a natural and unavoidable illusion, which, even when we are no longer deceived by it, continues to mock us and, although rendered harmless, can never be completely removed.
This dialectical doctrine will not relate to the unity of understanding in empirical conceptions, but to the unity of reason in pure ideas. The conditions of this doctrine are—inasmuch as it must, as a synthesis according to rules, be conformable to the understanding, and at the same time as the absolute unity of the synthesis, to the reason—that, if it is adequate to the unity of reason, it is too great for the understanding, if according with the understanding, it is too small for the reason. Hence arises a mutual opposition, which cannot be avoided, do what we will.
These sophistical assertions of dialectic open, as it were, a battle-field, where that side obtains the victory which has been permitted to make the attack, and he is compelled to yield who has been unfortunately obliged to stand on the defensive. And hence, champions of ability, whether on the right or on the wrong side, are certain to carry away the crown of victory, if they only take care to have the right to make the last attack, and are not obliged to sustain another onset from their opponent. We can easily believe that this arena has been often trampled by the feet of combatants, that many victories have been obtained on both sides, but that the last victory, decisive of the affair between the contending parties, was won by him who fought for the right, only if his adversary was forbidden to continue the tourney.
As impartial umpires, we must lay aside entirely the consideration whether the combatants are fighting for the right or for the wrong side, for the true or for the false, and allow the combat to be first decided. Perhaps, after they have wearied more than injured each other, they will discover the nothingness of their cause of quarrel and part good friends.
This method of watching, or rather of originating, a conflict of assertions, not for the purpose of finally deciding in favour of either side, but to discover whether the object of the struggle is not a mere illusion, which each strives in vain to reach, but which would be no gain even when reached—this procedure, I say, may be termed the sceptical method. It is thoroughly distinct from scepticism—the principle of a technical and scientific ignorance, which undermines the foundations of all knowledge, in order, if possible, to destroy our belief and confidence therein. For the sceptical method aims at certainty, by endeavouring to discover in a conflict of this kind, conducted honestly and intelligently on both sides, the point of misunderstanding; just as wise legislators derive, from the embarrassment of judges in lawsuits, information in regard to the defective and ill-defined parts of their statutes. The antinomy which reveals itself in the application of laws, is for our limited wisdom the best criterion of legislation. For the attention of reason, which in abstract speculation does not easily become conscious of its errors, is thus roused to the momenta in the determination of its principles.
But this sceptical method is essentially peculiar to transcendental philosophy, and can perhaps be dispensed with in every other field of investigation. In mathematics its use would be absurd; because in it no false assertions can long remain hidden, inasmuch as its demonstrations must always proceed under the guidance of pure intuition, and by means of an always evident synthesis. In experimental philosophy, doubt and delay may be very useful; but no misunderstanding is possible, which cannot be easily removed; and in experience means of solving the difficulty and putting an end to the dissension must at last be found, whether sooner or later. Moral philosophy can always exhibit its principles, with their practical consequences, in concreto—at least in possible experiences, and thus escape the mistakes and ambiguities of abstraction. But transcendental propositions, which lay claim to insight beyond the region of possible experience, cannot, on the one hand, exhibit their abstract synthesis in any à priori intuition, nor, on the other, expose a lurking error by the help of experience.
Transcendental reason, therefore, presents us with no other criterion than that of an attempt to reconcile such assertions, and for this purpose to permit a free and unrestrained conflict between them. And this we now proceed to arrange.[50] [50] The antinomies stand in the order of the four transcendental ideas above detailed.
FIRST CONFLICT OF THE TRANSCENDENTAL IDEAS. THESIS.
The world has a beginning in time, and is also limited in regard to space.
PROOF.
Granted that the world has no beginning in time; up to every given moment of time, an eternity must have elapsed, and therewith passed away an infinite series of successive conditions or states of things in the world. Now the infinity of a series consists in the fact that it never can be completed by means of a successive synthesis. It follows that an infinite series already elapsed is impossible and that, consequently, a beginning of the world is a necessary condition of its existence. And this was the first thing to be proved.
As regards the second, let us take the opposite for granted. In this case, the world must be an infinite given total of coexistent things.
Now we cannot cogitate the dimensions of a quantity, which is not given within certain limits of an intuition,[51] in any other way than by means of the synthesis of its parts, and the total of such a quantity only by means of a completed synthesis, or the repeated addition of unity to itself. Accordingly, to cogitate the world, which fills all spaces, as a whole, the successive synthesis of the parts of an infinite world must be looked upon as completed, that is to say, an infinite time must be regarded as having elapsed in the enumeration of all co-existing things; which is impossible. For this reason an infinite aggregate of actual things cannot be considered as a given whole, consequently, not as a contemporaneously given whole. The world is consequently, as regards extension in space, not infinite, but enclosed in limits. And this was the second thing to be proved.
[51] We may consider an undetermined quantity as a whole, when it is enclosed within limits, although we cannot construct or ascertain its totality by measurement, that is, by the successive synthesis of its parts. For its limits of themselves determine its completeness as a whole.
ANTITHESIS.
The world has no beginning, and no limits in space, but is, in relation both to time and space, infinite.
PROOF.
For let it be granted that it has a beginning. A beginning is an existence which is preceded by a time in which the thing does not exist. On the above supposition, it follows that there must have been a time in which the world did not exist, that is, a void time. But in a void time the origination of a thing is impossible; because no part of any such time contains a distinctive condition of being, in preference to that of non-being (whether the supposed thing originate of itself, or by means of some other cause). Consequently, many series of things may have a beginning in the world, but the world itself cannot have a beginning, and is, therefore, in relation to past time, infinite.
As regards the second statement, let us first take the opposite for granted—that the world is finite and limited in space; it follows that it must exist in a void space, which is not limited. We should therefore meet not only with a relation of things in space, but also a relation of things to space. Now, as the world is an absolute whole, out of and beyond which no object of intuition, and consequently no correlate to which can be discovered, this relation of the world to a void space is merely a relation to no object. But such a relation, and consequently the limitation of the world by void space, is nothing.
Consequently, the world, as regards space, is not limited, that is, it is infinite in regard to extension.[52] [52] Space is merely the form of external intuition (formal intuition), and not a real object which can be externally perceived.
Space, prior to all things which determine it (fill or limit it), or, rather, which present an empirical intuition conformable to it, is, under the title of absolute space, nothing but the mere possibility of external phenomena, in so far as they either exist in themselves, or can annex themselves to given intuitions. Empirical intuition is therefore not a composition of phenomena and space (of perception and empty intuition). The one is not the correlate of the other in a synthesis, but they are vitally connected in the same empirical intuition, as matter and form. If we wish to set one of these two apart from the other—space from phenomena—there arise all sorts of empty determinations of external intuition, which are very far from being possible perceptions. For example, motion or rest of the world in an infinite empty space, or a determination of the mutual relation of both, cannot possibly be perceived, and is therefore merely the predicate of a notional entity.
OBSERVATIONS ON THE FIRST ANTINOMY. ON THE THESIS.
In bringing forward these conflicting arguments, I have not been on the search for sophisms, for the purpose of availing myself of special pleading, which takes advantage of the carelessness of the opposite party, appeals to a misunderstood statute, and erects its unrighteous claims upon an unfair interpretation. Both proofs originate fairly from the nature of the case, and the advantage presented by the mistakes of the dogmatists of both parties has been completely set aside.
The thesis might also have been unfairly demonstrated, by the introduction of an erroneous conception of the infinity of a given quantity. A quantity is infinite, if a greater than itself cannot possibly exist. The quantity is measured by the number of given units—which are taken as a standard—contained in it. Now no number can be the greatest, because one or more units can always be added. It follows that an infinite given quantity, consequently an infinite world (both as regards time and extension) is impossible. It is, therefore, limited in both respects. In this manner I might have conducted my proof; but the conception given in it does not agree with the true conception of an infinite whole. In this there is no representation of its quantity, it is not said how large it is; consequently its conception is not the conception of a maximum. We cogitate in it merely its relation to an arbitrarily assumed unit, in relation to which it is greater than any number. Now, just as the unit which is taken is greater or smaller, the infinite will be greater or smaller; but the infinity, which consists merely in the relation to this given unit, must remain always the same, although the absolute quantity of the whole is not thereby cognized.
The true (transcendental) conception of infinity is: that the successive synthesis of unity in the measurement of a given quantum can never be completed.[53] Hence it follows, without possibility of mistake, that an eternity of actual successive states up to a given (the present) moment cannot have elapsed, and that the world must therefore have a beginning.
[53] The quantum in this sense contains a congeries of given units, which is greater than any number—and this is the mathematical conception of the infinite.
In regard to the second part of the thesis, the difficulty as to an infinite and yet elapsed series disappears; for the manifold of a world infinite in extension is contemporaneously given. But, in order to cogitate the total of this manifold, as we cannot have the aid of limits constituting by themselves this total in intuition, we are obliged to give some account of our conception, which in this case cannot proceed from the whole to the determined quantity of the parts, but must demonstrate the possibility of a whole by means of a successive synthesis of the parts. But as this synthesis must constitute a series that cannot be completed, it is impossible for us to cogitate prior to it, and consequently not by means of it, a totality. For the conception of totality itself is in the present case the representation of a completed synthesis of the parts; and this completion, and consequently its conception, is impossible.
ON THE ANTITHESIS.
The proof in favour of the infinity of the cosmical succession and the cosmical content is based upon the consideration that, in the opposite case, a void time and a void space must constitute the limits of the world. Now I am not unaware, that there are some ways of escaping this conclusion. It may, for example, be alleged, that a limit to the world, as regards both space and time, is quite possible, without at the same time holding the existence of an absolute time before the beginning of the world, or an absolute space extending beyond the actual world—which is impossible. I am quite well satisfied with the latter part of this opinion of the philosophers of the Leibnitzian school. Space is merely the form of external intuition, but not a real object which can itself be externally intuited; it is not a correlate of phenomena, it is the form of phenomena itself. Space, therefore, cannot be regarded as absolutely and in itself something determinative of the existence of things, because it is not itself an object, but only the form of possible objects. Consequently, things, as phenomena, determine space; that is to say, they render it possible that, of all the possible predicates of space (size and relation), certain may belong to reality.
But we cannot affirm the converse, that space, as something self-subsistent, can determine real things in regard to size or shape, for it is in itself not a real thing. Space (filled or void)[54] may therefore be limited by phenomena, but phenomena cannot be limited by an empty space without them. This is true of time also. All this being granted, it is nevertheless indisputable, that we must assume these two nonentities, void space without and void time before the world, if we assume the existence of cosmical limits, relatively to space or time.
[54] It is evident that what is meant here is, that empty space, in so far as it is limited by phenomena—space, that is, within the world—does not at least contradict transcendental principles, and may therefore, as regards them, be admitted, although its possibility cannot on that account be affirmed.
For, as regards the subterfuge adopted by those who endeavour to evade the consequence—that, if the world is limited as to space and time, the infinite void must determine the existence of actual things in regard to their dimensions—it arises solely from the fact that instead of a sensuous world, an intelligible world—of which nothing is known—is cogitated; instead of a real beginning (an existence, which is preceded by a period in which nothing exists), an existence which presupposes no other condition than that of time; and, instead of limits of extension, boundaries of the universe. But the question relates to the mundus phaenomenon, and its quantity; and in this case we cannot make abstraction of the conditions of sensibility, without doing away with the essential reality of this world itself. The world of sense, if it is limited, must necessarily lie in the infinite void. If this, and with it space as the à priori condition of the possibility of phenomena, is left out of view, the whole world of sense disappears. In our problem is this alone considered as given. The mundus intelligibilis is nothing but the general conception of a world, in which abstraction has been made of all conditions of intuition, and in relation to which no synthetical proposition—either affirmative or negative—is possible.
SECOND CONFLICT OF TRANSCENDENTAL IDEAS. THESIS.
Every composite substance in the world consists of simple parts; and there exists nothing that is not either itself simple, or composed of simple parts.
PROOF.
For, grant that composite substances do not consist of simple parts; in this case, if all combination or composition were annihilated in thought, no composite part, and (as, by the supposition, there do not exist simple parts) no simple part would exist. Consequently, no substance; consequently, nothing would exist. Either, then, it is impossible to annihilate composition in thought; or, after such annihilation, there must remain something that subsists without composition, that is, something that is simple. But in the former case the composite could not itself consist of substances, because with substances composition is merely a contingent relation, apart from which they must still exist as self-subsistent beings. Now, as this case contradicts the supposition, the second must contain the truth—that the substantial composite in the world consists of simple parts.
It follows, as an immediate inference, that the things in the world are all, without exception, simple beings—that composition is merely an external condition pertaining to them—and that, although we never can separate and isolate the elementary substances from the state of composition, reason must cogitate these as the primary subjects of all composition, and consequently, as prior thereto—and as simple substances.
ANTITHESIS.
No composite thing in the world consists of simple parts; and there does not exist in the world any simple substance.
PROOF.
Let it be supposed that a composite thing (as substance) consists of simple parts. Inasmuch as all external relation, consequently all composition of substances, is possible only in space; the space, occupied by that which is composite, must consist of the same number of parts as is contained in the composite. But space does not consist of simple parts, but of spaces. Therefore, every part of the composite must occupy a space. But the absolutely primary parts of what is composite are simple. It follows that what is simple occupies a space.
Now, as everything real that occupies a space, contains a manifold the parts of which are external to each other, and is consequently composite—and a real composite, not of accidents (for these cannot exist external to each other apart from substance), but of substances—it follows that the simple must be a substantial composite, which is self-contradictory.
The second proposition of the antithesis—that there exists in the world nothing that is simple—is here equivalent to the following: The existence of the absolutely simple cannot be demonstrated from any experience or perception either external or internal; and the absolutely simple is a mere idea, the objective reality of which cannot be demonstrated in any possible experience; it is consequently, in the exposition of phenomena, without application and object. For, let us take for granted that an object may be found in experience for this transcendental idea; the empirical intuition of such an object must then be recognized to contain absolutely no manifold with its parts external to each other, and connected into unity. Now, as we cannot reason from the non-consciousness of such a manifold to the impossibility of its existence in the intuition of an object, and as the proof of this impossibility is necessary for the establishment and proof of absolute simplicity; it follows that this simplicity cannot be inferred from any perception whatever. As, therefore, an absolutely simple object cannot be given in any experience, and the world of sense must be considered as the sum total of all possible experiences: nothing simple exists in the world.
This second proposition in the antithesis has a more extended aim than the first. The first merely banishes the simple from the intuition of the composite; while the second drives it entirely out of nature. Hence we were unable to demonstrate it from the conception of a given object of external intuition (of the composite), but we were obliged to prove it from the relation of a given object to a possible experience in general.
OBSERVATIONS ON THE SECOND ANTINOMY. THESIS.
When I speak of a whole, which necessarily consists of simple parts, I understand thereby only a substantial whole, as the true composite; that is to say, I understand that contingent unity of the manifold which is given as perfectly isolated (at least in thought), placed in reciprocal connection, and thus constituted a unity. Space ought not to be called a compositum but a totum, for its parts are possible in the whole, and not the whole by means of the parts. It might perhaps be called a compositum ideale, but not a compositum reale. But this is of no importance. As space is not a composite of substances (and not even of real accidents), if I abstract all composition therein—nothing, not even a point, remains; for a point is possible only as the limit of a space—consequently of a composite. Space and time, therefore, do not consist of simple parts. That which belongs only to the condition or state of a substance, even although it possesses a quantity (motion or change, for example), likewise does not consist of simple parts. That is to say, a certain degree of change does not originate from the addition of many simple changes. Our inference of the simple from the composite is valid only of self-subsisting things. But the accidents of a state are not self-subsistent. The proof, then, for the necessity of the simple, as the component part of all that is substantial and composite, may prove a failure, and the whole case of this thesis be lost, if we carry the proposition too far, and wish to make it valid of everything that is composite without distinction—as indeed has really now and then happened. Besides, I am here speaking only of the simple, in so far as it is necessarily given in the composite—the latter being capable of solution into the former as its component parts. The proper signification of the word monas (as employed by Leibnitz) ought to relate to the simple, given immediately as simple substance (for example, in consciousness), and not as an element of the composite. As an element, the term atomus would be more appropriate. And as I wish to prove the existence of simple substances, only in relation to, and as the elements of, the composite, I might term the antithesis of the second Antinomy, transcendental Atomistic. But as this word has long been employed to designate a particular theory of corporeal phenomena (moleculae), and thus presupposes a basis of empirical conceptions, I prefer calling it the dialectical principle of Monadology.
ANTITHESIS.
Against the assertion of the infinite subdivisibility of matter whose ground of proof is purely mathematical, objections have been alleged by the Monadists. These objections lay themselves open, at first sight, to suspicion, from the fact that they do not recognize the clearest mathematical proofs as propositions relating to the constitution of space, in so far as it is really the formal condition of the possibility of all matter, but regard them merely as inferences from abstract but arbitrary conceptions, which cannot have any application to real things. Just as if it were possible to imagine another mode of intuition than that given in the primitive intuition of space; and just as if its à priori determinations did not apply to everything, the existence of which is possible, from the fact alone of its filling space. If we listen to them, we shall find ourselves required to cogitate, in addition to the mathematical point, which is simple—not, however, a part, but a mere limit of space—physical points, which are indeed likewise simple, but possess the peculiar property, as parts of space, of filling it merely by their aggregation. I shall not repeat here the common and clear refutations of this absurdity, which are to be found everywhere in numbers: every one knows that it is impossible to undermine the evidence of mathematics by mere discursive conceptions; I shall only remark that, if in this case philosophy