CHAPTER IV¶
ON THE PROBLEM OF EXACTITUDE — PART II — ON EXACT FIGURES AND RELATIONS
Preliminary Remarks.¶
In the preceding chapters we dealt with the two principal problems, of necessity and of exactitude. Of this problem of exactitude, only the first part was considered by us; namely that part which, as was said there (Chapter III), does not require the consideration of extension, such as the “principle of individuation” in mathematical entities. The second part of this problem, whose solution depends upon this principle, now comes to be considered.
Let us briefly recall why the problem of geometric exactitude arises; let it be permitted to repeat here what was said earlier (Chapter III, § 1, pp. 65–69 ff.):
“What geometry asserts concerning its objects: points, lines, surfaces, concerning their mutual relations, concerning the shapes of figures, e.g., concerning the straightness of a line, concerning the equality or proportions of figures: all these are considered as absolutely exact. Sensitive data, however, from which classical geometry derives its origin, do not possess this exactitude. Neither sense nor imagination can perceive points and lines, but only bodies extended according to three (or at least according to two) dimensions. They cannot distinguish lines, slightly curved, from straight ones; consequently, they can never decide that a given line is straight. Equality between two lines or surfaces they can perceive approximately, but not exactly; the same must be said of proportions. Things that are very distant cannot be perceived, and yet geometry deals with extended entities that extend into infinity.
If, therefore, geometry in its origin depends upon sensitive data, how can this transition to perfect exactitude be explained; or is it perhaps merely an illusion?”
Regarding the existence of objects that correspond to the exact notions of point, line, and surface, in the third chapter we examined the question and vindicated the classical notions; we elucidated the transition from sensitive data to perfectly exact intellective notions. We can therefore safely use these notions in what follows; we know how the intellect, in the abstraction of these exact notions from the inexact phantasm, can transcend sensitive data; we know that this operation is critically justified. It remains for us to examine other questions. Now, as before, we look more to noetic questions that pertain to the theory of knowledge than to those that must be examined to expound fully the foundations of geometry; for purely noetic questions are most worthy of being considered in themselves. Therefore, this inquiry too intends to be purely philosophical, noetic, not mathematical.
We must diligently examine the processes that are completed in the human mind while it intends to constitute elementary geometry from its first principles. There may lie hidden there operations of the intellect, operations that are simultaneously abstractive—for they have their origin from the phantasm—and intuitive; for by a simple intuition of the mind we can intuit and affirm both the necessity of certain connections and their exactitude. We say these operations “lie hidden” there; not as though they were unconscious, on the contrary they are cognitive and intuitive; not as though they were without influence upon subsequent judgments, indeed they determine these judgments; but because their objects are not explicitly affirmed, much less enunciated by means of propositions. We are dealing here with virtual judgments, which we already encountered earlier (Chapter II, § 2, no. 6; § 3, no. 3; Chapter III, § 2, no. 5 A); there it was also said that at times—or often—it is difficult to accurately detect, explicitly affirm, and enunciate these virtual judgments. The necessary connections that in these judgments, if explicitly formed, are affirmed, are certainly known intuitively in virtual judgments, and in the mental process these intuitions exert an influence. Thus (loc. cit.) we found that two intuitive cognitions regarding the “proper attributes” (passiones propriae) of multitudes always precede the simple cognition and affirmation of what results from the addition of two numbers. Such prior virtual judgments are also found in examining the problem of geometric exactitude [1]. And in order to understand well the character and mode of cognition of the human mind in general, careful attention must be paid to what we found earlier: The function of these prior judgments is not like the office of a major premise, i.e., of a universal principle in a syllogism; for there it was demonstrated that the movement of which virtual judgments constitute a part is not syllogistic; it is not the deduction of a conclusion from those judgments; the movement that leads to the universal affirmation of these judgments is a movement precisely the inverse of the syllogism. These things must be attended to if we wish to understand accurately the mode of operation of the human mind. For such virtual judgments are accustomed to precede, as intellective intuitions of necessary relations, those judgments that affirm other relations subsequent to those prior ones, namely, those judgments that are customarily enunciated as classical axioms. The office of the philosopher is, as was already said before (Chapter I, § 1), to examine those operations of the human mind from which the cognition of classical axioms flows forth. This must not only be attended to in order that the foundations of geometry may be examined, but must also be accurately considered in examining the noetic nature of the human mind itself in general. St. Thomas says (Q. D. de Ver. q. 10 a. 6 « sed contra » 2):
“All our cognition originally consists in the knowledge of the first indemonstrable principles. The cognition of these in us arises from sense, as is clear at the end of the Posterior Analytics. Therefore our science arises from sense.”
We are dealing with all our cognition considered in its origin, “originally”; the principles, however, in which this origin is said to consist, are indemonstrable principles that regard the “proper object” of the human mind, i.e., the form that is in matter; and the first principles of this are above all geometric principles, for these are first and best known to us. In order that we may have clear insight into the mode of operation of our mind, we must institute a “resolution” (resolutio) to first principles: and therefore (Q. D. de Ver. q. 12 a. 3 ad 2):
“Because the first principle of our cognition is sense, it is necessary to resolve in some way to sense all things concerning which we judge.”
St. Thomas says that “all things concerning which we judge” must be resolved to sense; but he adds “in some way” (quodammodo); in order that this “mode” may become known to us, the best, if not the sole, object of examination will be: the consideration of those things from which the intuition of the first principles of geometry arises. This intuition transcends the sensitive data from which it arises in two respects: 1° relative to necessity (and consequently universality); 2° relative to exactitude.
In this second part of our inquiry into the problem of exactitude, cases therefore occur that deal with the perfect equality (congruence) of two or more figures, with the existence of perfect lines, i.e., of a perfectly straight line, of a circle whose radii are perfectly equal, with the existence of a perfect plane, with the perfect equality of two proportions, with the perfect concurrence of three straight lines in a single point, etc. Judgments concerning such cases certainly cannot be directly formed through abstraction from the inspection of a phantasm or sensitive datum; in all these cases, figures can be given that are in reality inexact, but which approach so close to exactitude that the imperfection cannot be apprehended by the senses. Geometry, nevertheless, deals with certitude concerning perfectly exact relations of figures. We must inquire whether a principle is given from which this certitude legitimately arises. We shall see that this principle is given, and that it is the cognition of extension as it is the principle of individuation for mathematical entities. Moreover, it will be clear that this principle is indeed everywhere employed as a virtual judgment.
§ 1. ON THE PRINCIPLE OF INDIVIDUATION IN GEOMETRIC ENTITIES.¶
We are not dealing here with the metaphysical principle of individuation and with the celebrated metaphysical question that is raised relative to this principle; we are dealing only with geometric entities, as they are geometric; but what we shall say without doubt coheres with the metaphysical question.
Here, therefore, mention is made only of geometric entities, i.e., of figures that are the boundaries (limites) of an extended entity, of lines and surfaces, and of the parts of extension that can be circumscribed by these boundaries; figures can differ and agree both in quantity and in shape (forma).
Can multiple figures be given that agree exactly both in quantity and in shape, and that therefore differ by number alone? Geometry answers affirmatively, considering what is called the congruence of figures and establishing theses that regard that congruence.
What is the foundation of this affirmation? It cannot be sensitive experience. This experience certainly induces the human mind to conceive the idea of congruent figures. Two circles, two triangles, or any other figures whatever can be observed, between which the senses can perceive no difference relative to quantity and shape. But since we know that the senses cannot arrive at perfect exactitude, it is not on that account permissible to affirm that these two figures are exactly congruent; indeed, it is not on that account permissible to say that two such figures can be given, i.e., that they exist mathematically.
But our intellect in this case too can transcend sense and affirm the possibility, the mathematical existence, of congruent figures; and indeed it can do this in the same way in which it was done in the preceding case, namely in the case of indivisibles, which cannot be perceived by sense: by understanding the nature of extension, as it is extended, and the nature of a certain proper attribute (proprium) that flows from it. For considering any figure whatever, the boundary of extension, our mind understands that this bounded extension, by its nature, can be a part of a greater extension, in such a way indeed that other parts of this extension, as they are extended, are among themselves and with the first part of perfectly the same nature; the parts of extension, as it is extended, are perfectly homogeneous, uniform (i.e., unformed, not determined through a form) with the whole and with each of the other parts. Whence it immediately follows: if any figure (determination) is possible in one part, in any other part whatever a figure specifically and perfectly the same will be possible; and in such a way indeed that its site (situs), and its site alone, differs in a continuous manner from the site of the prior figure. Or thus: alongside every figure another figure is possible that differs by site alone, and that “alongside” (iuxta) is to be understood in such a way that it can be continuously varied.
Extension as such is therefore truly “intelligible matter” (materia intelligibilis) for geometric entities; out of this matter, figures specifically the same can be actuated in diverse parts. And thus extension contains the twofold element required for a principle of individuation. First, because it implies the distinction of parts of perfectly the same nature, by that very fact it brings with it the possibility of the multiplication of geometric figures in the same lowest species (species specialissima), with respect to their quantity and with respect to their shape. This is the first element of the principle of individuation. Second, it likewise implies the other element, namely that in which those forms differ among themselves. For it necessarily brings with it a possible diversity of site (situs). These figures, specifically the same, therefore differ by site alone, by position alone in extension, by number alone. All these things are clear and are immediately understood to flow from the understood nature of extension as it is extended. They are understood by a virtual judgment, even if they are never enunciated; and this judgment is always associated with our concept of extension as such. Extension so conceived we shall indicate in what follows (as is customary) by the word “space”; we certainly do not intend to speak of imaginary space, the receptacle of bodies, which is non-being (non-ens); we intend to speak of a real extended entity, but as it is extended only; and in this sense it can be applied to imaginary space, which is conceived as extended after the manner of a real entity; this is a receptacle because it is extended.
In passing we note: from this consideration it follows that extension as such is without doubt some principle of individuation (for geometric things); it does not follow that it is the sole principle of individuation; hence the metaphysical question is not yet resolved from these considerations.
At first sight a certain difficulty could be raised, e.g., from considering an ellipsoidal figure. On such a surface a figure traced in one part cannot be traced in any other part whatever, because the “curvature” of this surface is not everywhere the same. We do not think this difficulty ought to detain us long. This surface itself is already not an extension determined by no form; it is not pure extension. Moreover, the extension thus bounded through this surface is, or can be, a part of a greater extension; and in any part of this, a figure can be repeated that is congruent to the first.
Congruent figures differ by site alone. We observe that the slightest difference in site already brings with it a numerically different figure. Let us take a simple example in which all that we were saying is immediately understood from the inspection of the phantasm. We have a certain straight line, let us say of one meter. From this extension there can be cut off—in this intelligible matter there can be actuated—figures that are specifically determined thus: they are the half part of the given line, i.e., lines of half a meter. Infinitely many lines of this species can be actuated from the given line through division; not, of course, simultaneously. Simultaneously there can be actuated, through division at the accurately middle point, the right half part and the left half part. But these are not the only ones. For a line that begins from a point at a distance of 1/3 from the beginning of the line, and ends at a point situated at a distance of 5/6, is a half part of the line; it is an individual of this species. And thus it is clear, on account of the divisibility into infinity of the line, that infinitely many half-lines can be actuated; not, of course, simultaneously; each is of the same species, they differ among themselves purely numerically, and this difference depends upon the sole diversity of site in which they arise; and the slightest diversity of site is the cause of numerical difference [2].
In this entire exposition special attention must be paid to this: the understanding of extension, as it is the principle of individuation, does not arise from the fact that we perceive by sense sensibly congruent figures; such a perception can certainly be the occasion from which we conceive the notion of congruence; but that exact congruence is possible cannot be justified from that perception alone; the transition from this perception to the affirmation of congruence would be illegitimate; indeed, the transition to the affirmation of congruence realized in this case remains illegitimate from that perception alone. In order that the transition to the affirmation of the possibility of congruence be legitimate, an intellective intuition, flowing from the understanding of the nature of extension, is required and suffices, which recognizes extension as the principle of individuation in geometric entities. And this intuition was without any doubt naturally present in the human mind while it was constructing classical geometry, just as there was present the intuition of points, lines, and surfaces through which it already transcended sensitive cognition; they were present as virtual judgments.
From what has been said so far, however, we cannot yet know which exact figures are possible, whether perfectly straight lines, perfect circles, etc., exist mathematically. What is established so far is only this: if they are possible in one part of extension, they are possible in another as well. Nevertheless, this already suffices for us to deduce diverse geometric theses relative to exact congruence, as we shall immediately show by an example.
First, however, another matter must be considered. Geometers often use the “motion” of figures in order to prove such theses; Euclid already did this in the first thesis that he demonstrates concerning this matter. Because difficulties are raised against this mode of procedure, and in general against the use of the notion of “motion” in geometry, this notion, which intimately coheres with the nature of extension as it is the principle of individuation, must first be elucidated.
§ 2. ON MOTION IN GEOMETRIC ENTITIES.¶
Euclid employs the motion of figures in the first theorem that he demonstrates concerning the congruence of figures (Book I, Theorem 4; ed. Heiberg, pp. 16–19); so at least it is accustomed to be understood by interpreters, although, as we shall see, another interpretation is possible. There he proves the congruence of two triangles ABC and DEZ that have two sides equal (AB = DE and AC = DZ) and the angles between them, A and D, also equal; the possibility of these data already depends upon the principle of individuation. In order to demonstrate the thesis, he transfers, they say, the first triangle and applies it to the other: A to D, AB to DE; by that very fact, on account of the data, B will fall upon E, AC will be applied to DZ, and the point C to Z. Because between E and Z there exists only a single straight line (for “two straight lines cannot enclose a space”), BC will fall upon EZ; and hence the whole triangles themselves coincide.
Euclid therefore in his argument applies the translation, and therefore the motion, of figures; which he does very rarely (besides this passage, I, 8 and III, 24 are cited); certain moderns, however, vehemently carp at this method. We shall immediately see that this method can be avoided in such a way that the structure of the argument is scarcely changed; nevertheless, its legitimacy must be entirely sustained. No one carps at Euclid more vehemently than B. Russell [3]:
Our experience, however, seems to testify that the argument of Euclid is no less convincing to intelligent children than any other arguments whatever, in this and in other matters. But let us hear the first reason that Russell brings forward. He says:
“In the first place, to speak of motion implies that our triangles are not spatial but material. For a point of space is a position, and can no more change its position than the leopard can change its spots … motion, in the ordinary sense, is only possible to matter, not to space” [5].
The other reasons especially regard the notion of a “rigid body,” i.e., of a figure invariable in translation, and will immediately be considered by us, also according to other modern authors. This first reason is the most fundamental, because it regards the very notion of “motion in geometric entities”; and therefore it must be examined more closely.
Russell says, therefore, that motion pertains to bodies, not to space, which is considered by geometry, whose points are positions that themselves cannot be moved. But Russell errs; geometry is certainly the science of extension as it is extended (not, e.g., as it is colored); this they often express by saying: geometry is the science of “space”; which can be admitted if the word “space” here signifies: extension as such; but it is false if the word is taken for “imaginary space,” which is a fictitious receptacle of bodies. Geometry certainly holds validly of that imaginary entity, but only because this is conceived as extended; geometry, however, also holds validly of a real body as it is extended. A figure realized in such a body can be moved, just as the body itself, because extended, is mobile according to place.
These things must be strongly urged, because the possibility of motion is a “proper attribute” (proprium), in the strict sense, of extension as it is extended; motion would not be possible in a purely spiritual world; the possibility of motion is immediately understood to flow from extension as it is extended. And indeed extension is required under a twofold respect. First, that which is moved must be extended; at least that which is moved per se; things that are joined with an extended entity can be moved per accidens, according to the excellent theory of Aristotle. Next: in order that motion be possible, another extension is required (namely, place) relative to which the former is moved. If these two are really present, then and then only is the possibility of motion present. And because these things are understood to flow from extension itself as such, extension, the object of geometry, is not fully understood unless the possibility of motion is understood, even of a figure, which is the boundary of a part of extension. The cognition of this proper attribute is again naturally present in the human mind, as a virtual judgment that is always associated with the notion of extension.
But the notion of motion does not under every respect involved in it pertain to geometry; in part it is the object of metaphysics and physics; and therefore diverse respects must be distinguished.
Let us distinguish beforehand motion itself from that which results from motion: the new position of the figure. And it must be observed that the argument of Euclid considers only the new position resulting from motion, not the motion itself in itself; and this would already suffice to justify the argument.
More, however, must be said: even motion itself, according to certain of its respects, pertains to the field of pure geometry; and here we must carefully distinguish.
In motion, its velocity and its cause can be attended to. The former does not pertain to geometry, insofar as this velocity is a certain intensity. This intensity certainly, like every intensity (cf. the author’s Cosmologia, ch. "De mensura qualitatum," 4th ed., pp. 184–203, 518–527; Filosofia della natura inorganica, pp. 176–186), can be “measured,” and indeed through purely quantitative effects; for it is measured through the length of the line that the mobile traverses in a certain time (in a unit of time). But the numbers resulting from that measure are not purely cardinal numbers; the quantity that is thus measured is not predicamental quantity, but “virtual quantity” (quantitas virtutis), intensity. And so considered, motion does not pertain to geometry, but to physics or natural philosophy. Immediately, however, in velocity itself we shall detect one aspect that is geometric; which is not to be wondered at, for it is an intensity in a geometric thing.
Next: because motion is a change, it requires an efficient cause; under this respect too it is a notion alien to geometry, and pertains to metaphysics and physics. Whence this aspect is completely neglected in geometry.
There are, however, two aspects in motion itself according to which it pertains to geometry, in such a way that this science of extended being would be imperfect if they were omitted. For we are dealing with two “proper attributes” (propria) of extension as such, namely with the prerequisite conditions that motion may be possible. The first was already partly described above: in order that motion be possible, it is required that both the mobile and that according to which it is moved, I mean place, be extended; from extension as such, therefore, flows the possibility of motion. But now something must be added: for it is clear, in order that a figure, or a figured body, may be moved without any intrinsic change, that the extension alongside which, or rather in which, it is moved—whether this be considered purely geometrically (as a system of coordinates) or be a real place, such as the ether—must in each of the positions successively occupied by the mobile be capable of a figure congruent with the figure to be moved, differing by position alone. And it must be observed that this must be possible in a continuous manner. But this property of extension, which is prerequired for the possibility of motion of an invariable figure and from which this possibility flows, is what was called above the “homogeneity, or uniformity, of extension as it is extended”; it is that on account of which extension as such is the principle of individuation. Because we naturally understand this proper attribute of extension, therefore and therefore alone do we naturally understand the possibility of motion of an invariable figure; it is present in our mind as a virtual judgment. This proper attribute, however, pertains to extension as such, and therefore the consideration of motion according to this prerequisite condition pertains to pure geometry [6].
The other respect according to which it pertains to geometry is this: motion essentially has “direction.” Let us consider the motion of a point along a determinate line AB. Really distinct is the motion from A to B, and the inverse motion. Moreover: that point can be moved in a plane, let us say through point P, along diverse lines that can be drawn through that point; and in each case the motions are diverse. This diversity results from a purely geometric datum, from the diverse direction of lines in a plane relative to one another. This notion of “direction,” therefore, cannot be neglected if we wish fully to understand extension as such; moreover, it pertains to the first notions that we have concerning extension. Therefore, this notion will have to be examined more diligently; we shall do this later; for now let us attend to two things.
First: direction is something qualitative; it is not indeed intensive, but neither is it purely quantitative. This is immediately clear from the fact that the difference of two predicamental quantities is a quantity of the same genus; the difference of two lengths is a length. But the difference of two directions is not a direction. It pertains indeed to quantity; for from that difference there immediately results something quantitative, which will later be able to be developed in such a way that we arrive at the notion of an angle; and angles can be added to and subtracted from one another; the difference of angles is an angle. Another thing I should like to observe now: velocity too has “direction”; it is not a pure intensity, but is what in physics is called a vector.
Aristotle in his treatise on time (Phys. IV 11, 219 a 10–18) already accurately explains that the direction of time, simultaneously with its continuity, arises from the direction of motion, and—what pertains to our subject—this arises from direction (θέσει) and continuity in magnitude. He says:
ἐπεὶ δὲ τὸ κινούμενον κινεῖται ἔκ τινος εἴς τι καὶ πᾶν μέγεθος συνεχές, ἀκολουθεῖ τῷ μεγέθει ἡ κίνησις· διὰ γὰρ τὸ τὸ μέγεθος εἶναι συνεχὲς καὶ ἡ κίνησίς ἐστιν συνεχής … (“Since that which is moved moves from something to something, and every magnitude is continuous, motion follows magnitude; for because magnitude is continuous, motion also is continuous…”)
First, therefore, continuity is found in extension, and thence this follows in motion. But the order of parts as well, the order of prior and posterior, which is already indicated above by the words ἔκ τινος εἴς τι (“from something to something”), has the same origin. He continues:
τὸ δὴ πρότερον καὶ ὕστερον ἐν τόπῳ πρῶτόν ἐστιν. ἐνταῦθα μὲν δὲ τῇ θέσει· ἐπεὶ δ’ ἐν τῷ μεγέθει ἔστι τὸ πρότερον καὶ ὕστερον, ἀνάγκη καὶ ἐν κινήσει εἶναι τὸ πρότερον καὶ ὕστερον, ἀνάλογον τοῖς ἐκεῖ (“The prior and posterior, then, is first in place. There indeed it is by position [θέσει]; but since in magnitude there is prior and posterior, it is necessary that in motion also there be prior and posterior, analogous to those there”) [7].
These are the things we were describing above; because time is treated here, attention is paid to the direction of one dimension, of one line alone.
Hence it seems we must say: to consider motion does not pertain to geometry insofar as it implies velocity, except insofar as this involves “direction”; for to understand this, the consideration of extension as it is extended is required, which is the foundation of the notion of direction. It does not pertain to geometry to deal with the efficient cause of motion itself. But purely geometric is the consideration of the possibility of continuous motion, of the possibility of motion of an invariable figure, of the possibility of directions of diverse motions.
Hence it is no wonder that among geometers motion is employed as a means of demonstration, indeed as a means of definition. Well known is the definition of a line as that which is caused (or described) by a point that is moved; thus from the motion of a line arises a surface, from the motion of a surface a solid. These can be admitted as descriptions, not as definitions strictly so called; for we would fall into the difficulties that we expounded earlier (Chapter III, § 1, no. 2). Already in the time of Aristotle these descriptions were in use. He says (De Anim. I 4, 409 a 3–5):
φασι κινηθεῖσαν γραμμὴν ἐπίπεδον ποιεῖν, στιγμὴν δὲ γραμμήν (“They say that a line moved makes a plane, and a point [moved makes] a line”) [8].
St. Thomas too knows this method of describing lines, surfaces, and solids, but he ascribes it to the imagination of geometers [9].
That this method can be useful for defining special lines we already saw earlier (Chapter III, § 2, no. 6 B). And it must be noted: “time,” which was introduced there as a “variable,” in the end is “eliminated” from the formulas; what remains is purely geometric; but this depends upon the velocity of the motions that are “compounded.”
The consideration of motion, therefore, without any doubt pertains to pure geometry; this explains the foundation and conditions of this notion. If extension is examined in its nature and simultaneously in this proper attribute through which it is the principle of individuation, it is evident that it grounds the possibility of motion of an invariable body, and grounds also the notion of constant direction and of those things that follow thence. Considering these things, we shall arrive at the axioms that regard the congruence of figures and that regard exact figures. Both points we shall investigate in what follows.
§ 3. ON THE CONGRUENCE OF FIGURES.¶
Euclid, therefore, in employing motion as a means of demonstrating congruence, does not exceed the limits of geometry, which considers extension as extended. Nor can the mode in which he does this be imputed to him as a fault, although this is sometimes done by those who do not well distinguish physical elements from purely geometric elements in motion. And on account of this, those difficulties pertain to our investigation, which is wholly engaged in examining the cognition of the human mind relative to the foundations of geometry.
The same B. Russell mentioned above mixes those diverse elements and thence takes occasion for critique (loc. cit.). I pass over for now the fact that he also intimates that a body transferred into a place where the temperature is different is by that very fact changed with respect to its dimensions. Another objection, however, can appear to be of greater moment.
He says: bodies that are so moved as to be applied to one another are supposed to be rigid bodies; and this is affirmed by not a few others. But, they add, in the nature of things no absolutely rigid bodies are given.
This objection would be entirely valid if the possibility of a congruent figure, perfectly and exactly equal to the first, namely to that which must be moved, had to be known by us just as physical laws must become known, namely from sensitive observation alone, aided, if the case requires, by instruments with which we measure. In such a case the intellect does nothing except render by concepts and experimental judgments those things that have been observed. But in our case the matter does not stand thus, and this diversity between our case and the case of a physical law must be carefully noted. Certainly experience, which teaches that bodies sensibly rigid exist—bodies, therefore, that are so moved that their extension and shape remain sensibly the same—can already suggest the notion of the motion of a body that, while it is moved, remains invariable relative to quantity and shape. But if we had nothing further, we would be engaged in a cognition that would not surpass experience either relative to necessity or relative to exactitude. But our geometric cognition de facto surpasses it. And the reason too is evident. That experience can be the occasion used by the intuition of our intellect, but its data are not the cause of our total intellective cognition; they are rather the “matter of the cause” (materia causae). And this in our case can easily be detected. The reason for affirming the possibility of motion of a body and of a geometric figure that are not changed in motion is understood by us—perhaps on the occasion of such experience—to be: the proper attribute of the extension in which they are moved, of receiving in each of its parts, which continuously succeed one another, such a figure as was present in another part; and that necessarily, and that exactly. Nothing else must be supposed, specifically nothing concerning the physical properties of bodies, concerning the question whether the efficient cause of motion deforms the shape of the mobile, whether this resists a small or a large or every force, whether it is rigid. The possibility of that motion is understood to be present on the part of the extension both of the mobile itself and of the “space” in which it is to be moved; we are dealing with a passive possibility. Hence whoever has clear insight into the origin of this intellective intuition, which sees the possibility of motion of an invariable figure flowing from the very nature of extension, must deny the assertion brought forward by the distinguished Poincaré: “If, therefore, there were no rigid bodies in nature, there would be no geometry” [10]. We understand, nevertheless, that even a non-rigid body can at least be moved without its extension and shape being changed, provided suitable forces are present. Nor do these words “provided suitable forces are present” imply a restriction; for only the passive part of motion has weight here; and this part is entirely intelligible to us, precisely on account of the understood nature of extension. The thesis that Poincaré posits completely equates our geometric intuition with the origin of purely experimental cognition; which, by the witness of our intuition as well, is false on this point too.
The manner in which Euclid and others use the motion of an invariable figure in their demonstration is justified in every respect from the foundation that is presupposed, namely from the proper attribute of extension through which it is the principle of individuation; from the fact that it is intelligible matter (materia intelligibilis) out of which figures can be made that differ by site alone, and therefore by number alone. Hence the question can arise whether, employing this foundation alone, without the intermediary of the translation of a figure, the same thesis can be demonstrated. And so the matter seems to stand.
For the Euclidean demonstration can be constructed without the notion of translation in this way. From point D (of triangle DEZ) there is given one and only one straight line in the direction DE; on it, according to the same direction toward E, there is given one point, and one only, namely E, such that segment DE is equal to side AB. On that side of this line that faces line DZ, there is given one, and only one, line that makes an angle (D) equal to angle A, namely the line DZ itself. On this line there is given again one, and only one, point (namely Z) such that segment DZ is equal to side AC. There is given again one straight line, and one only, that joins E and Z. Triangle DEZ is therefore the unique triangle that can be constructed in the position determined through the elements DE, D, and DZ, which are respectively equal to the elements AB, A, and AC. Now we apply the property of extension that is expressed by the principle of individuation, namely: in every position there is given a triangle that differs by number alone from triangle ABC, i.e., that is congruent to it. Because, however, from the given elements DE, D, and DZ only a single triangle is composed, namely DEZ, this will be congruent to triangle ABC, Q.E.D.
The individual elements we have used are found in the demonstration of Euclid; yet no translation has been employed; the theorem follows from the principle of individuation, which expresses a proper attribute (propria passio) of extension. And this principle seems always to be employed in similar argumentation as a virtual judgment. That by means of this principle, whose intuition is always and naturally blended with the operations of our mind, we are able to transcend the inexactitude of sensibility, appears in the course of the demonstration. Because, however, it is not explicitly affirmed, not even mentally, it must be said to be present there as a virtual judgment, which is detected by analysis and passes into a formal judgment. Other intuitions are also present; e.g., those on account of which we affirmed that only a single point, a single line, a single segment is present. And therefore the analysis of such a demonstration is useful and necessary for detecting the mode in which the human mind operates in geometric matters, for constituting, therefore, the doctrine concerning our geometric cognition.
After what we have seen relative to this demonstration of Euclid, perhaps we should advance further and say: Euclid himself in his demonstration did not intend to apply the translation of a figure (the triangle), but only to employ the principle of individuation, as we have done. We know, certainly, that historians do not so interpret Euclid; to us, however, it seems that his exposition admits this interpretation. For where he speaks of the “application” of triangle ABC to DEZ (ἐφαρμοζομένου τοῦ τριγώνου) and of the “placing” of point A at D (τιθεμένου τοῦ σημείου), we can understand his words in such a way that triangle ABC is not transferred, but that a triangle congruent to it, which according to the principle of individuation exists in every position, is constructed anew, actuated from its elements in the new site (DEZ); and the possibility of the actuation of the individual elements is again established from the same principle. This interpretation seems to be possible; if it is true, even in the demonstration of Euclid himself translation is not employed as a means of demonstration. And again it appears that the translation of an invariable figure as a means of demonstration is equivalent to its foundation, which is expressed by the principle of individuation.
We can conclude: just as in abstracting the exact notions of surface, line, and point (as indivisibles either altogether or under a certain respect) from inexact sensitive data, our mind, from the understood nature of extension, not only conceives exact notions, but understands those objects to exist mathematically (Chapter III, § 2, no. 2), so also in the case of congruence, from understood extension as it is the principle of individuation, it not only conceives figures that are exactly equal, differing by site alone, but understands them to exist and from them can logically deduce other truths.
And what we saw earlier (Chapter III, § 2, no. 7) can be repeated for this case: it is manifest that we do not (freely) create these objects; we do not will congruent figures to exist, but we detect them as possible in extension, which is the intelligible matter (materia intelligibilis) out of which they can be made.
The principle of individuation, as it has been applied so far, whether with translation or without it, can demonstrate only “relative” exactitude. For it proves only, from the fact that a figure present in one site can also be actuated in any other whatever, that two figures that agree in certain elements also agree in others. Supposing that a triangle is given whose sides are AB and AC, between which is angle A, another triangle is possible such that DE = AB, A = D, DZ = AC; and it is proved that these two triangles also agree with each other in the other elements. But that a triangle ABC is possible, in which there is an exactly straight line AB and so forth, is not proved here, but supposed; yet once this is supposed, exact congruence is proved. Whence, if the principle of individuation is employed only thus, only relative or hypothetical exactitude is demonstrated. But the principle of individuation can again be employed to overcome the inexactitude of sensibility in demonstrating the possibility, the mathematical existence, of absolutely exact figures, such as the straight line. Concerning which matter we must now treat.
§ 4. ON EXACT FIGURES.¶
1. Preliminary Remarks.¶
Already in the second chapter, in “inexact geometry,” we were able to speak of points, lines, and surfaces—inexact, of course—and concerning these elements to enunciate as proper relations: points are situated on a line and on a surface, lines are situated on a surface; and conversely: a line passes through a point, a surface through a line. Concerning points constructed on the same line, proper relations manifest themselves to our mind, which are expressed by the word “between”; in a similar mode, a line can be situated between others constructed on the same surface. These notions and these proper attributes, as we saw in Chapter III, are preserved in exact geometry, which discovers true surfaces and lines and true indivisible points, and there lead to propositions that are verified only in it, e.g.: between two points infinitely many others can be constructed (Chapter III, § 2, no. 6). The axioms that pertain to this part constitute two series: the axioms of connection (Verknüpfung) and the axioms of order (Anordnung).
These notions and these axioms are preserved also in Hilbert, although what the meaning of these words is in his work cannot be asked. For the meaning must be drawn from intuition, which Hilbert indeed admits as a heuristic means, which nevertheless he then wishes to exclude, to the end that he may then proceed purely logically, through pure syllogisms, independently of all geometric intuition (Chapter I, § 5). His terms and axioms are therefore pure symbols, just as in minor logic, where the moods and figures of syllogisms are treated, symbols are introduced for subject and predicate. But it must be well noted: Hilbert indeed admits that the natural meaning of numbers is known and employed; moreover, notions such as this are supposed known: “an operation that can be repeated into infinity”; and perhaps others; e.g., in § 3 he begins to speak of a segment (Strecke) and to introduce it uses the expression “system of two points” (das System beider Punkte), where the word “system” seems to be supposed known.
Hilbert indeed in choosing (“creating”) notions and axioms follows intuition and remains closer to classical geometry than others, but he does not wish to use intuition as a foundation [11]. Yet in these matters three remarks seem to be of some, or even of great, moment. First: he enunciates as notions only, not as axioms, those things that express those “proper attributes” of which we have spoken; this he does in his “declarations.” Yet these are true judgments; e.g., the possible relations between points and lines are expressed by the words: “points are situated on a line,” and so forth. But here a judgment is present, at least a virtual one, which expresses that these relations are proper attributes (propriae passiones) of these objects, flowing from their nature. Those things that are then explicitly enunciated as axioms are nothing but specifications of these relations. In pure axiomatics, whose principal intention is to enunciate explicitly all axioms, these ought not to have been missing as axioms. And this is a confirmation of what we have often said, namely that virtual judgments are given in our mind, which are not affirmed there because we immediately proceed to inquire into further matters. An influence upon the affirmation of what we subsequently discover they certainly have.
The second remark is this: because we understand, in the strict sense, these proper attributes flowing from extension, therefore—for the intelligible is being—extension with its proper attributes is known as being, and these relations as real; and we can proceed in our investigation, considering extension in the classical manner as intelligible and real.
The third remark is: Relations such as these: “point A is situated on line a” and “line a passes through point A” are considered as altogether the same (or purely inverse); which does not seem to be true in every respect. For this second geometric relation implicitly seems to say more, at least in such a way as to demand a further determination; for diverse lines pass through the same point in a diverse manner. And this further determination seems to be of great moment, because it leads to the elaboration of an intuitive notion of great moment, which seems almost universally neglected, indeed at times despised, by geometers in the beginnings of geometry: to the notion of direction, of which mention was already made above. Concerning this and concerning those things that cohere with this we must now speak more broadly, first as it is found in sensitive data, next whether, like the other notions of which we have already treated, it can pass into exact geometry. With this notion cohere the notions of the straight and curved line and of the plane and curved surface.
2. On the Notion of Direction.¶
We were saying: the expressions a point is situated on a line and a line passes through a point do not in every respect say the same thing, because this second formula implicitly says more and demands a further determination that is not demanded in the first formula. Behold what we intend.
If only one line is considered, as was already said above, on which a point P is situated, the two expressions can seem equivalent. Nevertheless, two directions can already be distinguished in the line, which relative to each other are opposite or inverse. This direction coheres with the notion of possible motion through this point along this line. The mental actuation of motion teaches that two motions are possible that are really distinct from each other. Our mind intuitively recognizes this possibility as flowing from the nature of a line, an extension of one dimension. But simultaneously it intuitively recognizes this possibility as flowing from the nature of a line, from a proper attribute of this static extension, namely from the fact that this line has these two opposite directions according to the order, the position (θέσιν), of its parts. If we say only: a point is situated on this line, we do not attend to this; if we express the inverse relation: a line passes through a point, we can already attend to these opposite respects, inverse in direction.
These matters are of greater moment if we do not attend to a single dimension, the line, but consider this as it is situated in an extension of two or three dimensions, a surface or a solid (“space”). This will lead to the notion of “direction relative to that surrounding extension.”
Let two lines a and b pass through the same point P; this says more than the expression: P is situated on a and on b; for the passage is different if a and if b is considered. Even if a and b differ in no other respect, if therefore they are congruent according to both directions in the lines themselves, they nevertheless differ in the manner in which they pass through P, they differ in the direction of their passage; otherwise they would not be two. They differ in their passage through P, no longer as in the case of a single line, in which opposite directions must be distinguished, but relative to the surrounding extension in which P is situated; this extension requires at least two dimensions. The meaning of these words “manner of passage” or “direction of passage,” and the “existence” of those manners, is immediately clear to anyone inspecting a simple figure of two small segments of a line passing through the same point, which do not coincide. It is manifest: the expression “a and b pass through P” demands a further determination, which specifies the manner of passage more closely. It is necessary, therefore, to introduce the notion of the direction of lines, also relative to each other in an extension of at least two dimensions, and to examine it more accurately; for it is found in the first geometric data by anyone who attends, in the first proper attributes of extension, and is understood to flow from the nature of extension; the notion of direction cannot be suppressed; nor does it emerge only after a long evolution of geometry. That this notion is realized also in exact geometry, insofar as it considers lines without breadth, true lines, is immediately clear; under other respects too it can be developed to justified exactitude, as we shall immediately see; first, certain proper attributes of this notion must be examined.
The diversity of directions of lines relative to each other at the point P through which they pass implies a divergence or deviation of one relative to the other in the extension in which P is situated; this must be distinguished from direction itself, just as an effect from a cause. This divergence must now be examined more accurately. Let there be, in the same surface, a third line c that passes through the same point P; this, like b, has a divergence relative to a. It is immediately evident: this divergence can be greater or less than that which b has; in this sense: it begins to contain a greater or lesser part of the surface near the point P. Another proper attribute is: there can be given, relative to a, a deviation in one or in the other sense; the senses can be distinguished, e.g., by the words “on the right” and “on the left,” or by other words that suppose a similar opposition relative to each other. Suppose c to have a greater divergence relative to a than b; the divergence that c has can be considered as the sum of two divergences: b relative to a and c relative to b; divergences can be added and subtracted; (this no longer holds if the surface in which the three lines are situated is not the same).
Another proper attribute is very curious. Divergence in one sense can become greater (in the opposite sense it then becomes less), but it cannot grow beyond all limits; for in the end the divergence becomes zero, and the directions of the two lines at or through point P are the same. Thus this deviation of direction, which admits of measure, cannot, like extension itself and many figures in it, grow into infinity; therefore neither can the addition of divergences be repeated into infinity; at least in the proper sense. This notion of direction, therefore, and the notion of its divergence—which later will lead to the exact notion of an angle—pertain truly to the first notions, and their proper attributes to those things that are immediately manifest. All these things are read in the phantasm as necessary, flowing from the nature of lines situated in a surface; without the intuition of the phantasm they cannot be fully understood, neither with respect to the meaning of the notions nor with respect to the judgments that attribute mathematical existence to these notions and affirm their proper attributes as necessary; these judgments again are present as virtual judgments in the natural mode of procedure of the human mind and are elaborated only by philosophical analysis. They are again “constructions” whose intelligible matter (materia intelligibilis) is given, namely extension as such.
These things flow from what was said in § 3, namely: the possibility of motion of a rigid body, i.e., one that by motion alone is not intrinsically changed, supposes that the surrounding extension (this is “place”) is the principle of individuation, and therefore permits the congruence of figures in a continuous manner. These figures can then be so moved that they perfectly coincide. This can be applied to congruent lines that pass through the same point of the surrounding extension. In this case a special motion can be considered, that which consists in the revolution of a line, which is so rotated that this same point of the line always remains fixed. These things, which we know from experience as sensibly verified, are known, from the consideration of extension as it is the principle of individuation, also as exactly possible, as flowing from the “homogeneity” of extension as such. These lines, whose revolution is considered here, can be both sensibly curved and sensibly straight (and after the existence of lines that are exactly straight has been established, also such); these sensible lines are already naturally known by man.
And note moreover: congruent figures differ “by site alone” in the surrounding extension. Here, in the case of the revolution of a rigid line around one fixed point, we detect a new case of “difference in site alone,” which is found only in lines (and figures composed of lines), not in points. This difference is the difference of “direction in passing through the same point.” This we detect immediately, intuitively, after the possibility of congruence has been established as above. Two congruent lines can therefore differ from each other “by site alone” in a twofold mode. One mode is this which has now been described: two homologous points (i.e., points that divide the lines into parts, which, e.g., can be called left and right parts, in such a way that the two right parts and the two left parts of the two lines are congruent to each other) on both lines remain coincident with the same point of the surrounding extension, but the passage of the lines through this point is diverse. They can moreover differ in such a way that these homologous points coincide with diverse points of the surrounding extension. After introducing the comparison of directions through distant points (no. 3), we understand that in the second case it can happen that these two lines have specifically the same direction through the points of the surrounding extension occupied by the two homologous points of the lines. Then the lines can be said to differ “by position alone,” while having the same “orientation.”
If congruent figures are composed of many lines, these considerations can be applied to the homologous lines in them. A twofold diversity “in site alone” is also found in these figures. One, where two homologous points coincide with the same point of the surrounding extension, will be able to be called diversity in “orientation” alone; the other will be diversity in “position”; in the general case of two congruent figures, both diversities are simultaneously present.
3. On the Continuum of Directions.¶
Divergence, which according to what was said above admits of measure, can be so small as to escape sense. The same must be said of the diversity of directions; the directions of passages through one point can be varied in a continuous manner; hence around any point whatever there is given an entire continuum of directions, a continuous sheaf (fascis continuus), whose apex is the point itself. But on that account too, in considering divergence and its measure, certain things can be found that pose anew the problem of exactitude. That at individual points there is given a continuum of directions, however, seems established from what has been said.
It is also immediately understood without further examination that around any point whatever of a surface there can be given only a continuum of directions that is already found in the extension of three dimensions in which the surface is the boundary of parts; not this entire continuum is found in the surface, but only a section of it; those directions that pass through this boundary, namely the surface, are not found in the surface, whereas the others are, at least at the very point P that is considered. Hence it is that at the individual points of an extension of three dimensions there are given three-dimensional continua of directions, while in surfaces there are given only two-dimensional ones. (It must be added: it can happen that at one point of a surface a different continuum of directions out of the three-dimensional continuum is preserved than at another; as will immediately be expounded more fully).
For in another way too continuity must be considered in this problem. We began our analysis by considering two small segments of lines (even sensibly curved ones) that, intersecting at point P, pass through that point according to diverse directions. These segments can be taken smaller, for they are divisible. If previously segments PA and PB were considered, now PC and PD can be taken, in which C is between P and A, and D between P and B; and this process can be repeated, and indeed into infinity, and indeed on both sides of the line passing through P. Proceeding thus, however, we already approach a more exact intellective consideration of the direction of lines at point P, i.e., of the manner in which they pass through that point. In order to have an altogether exact consideration, whose possibility is intuitively manifest to our mind, we must consider the limit, i.e., the manner in which the mobile points C and D, on lines PA and PB, coincide with P. And this “diversity in the manner of coinciding” for the mobile points is the true and exact diversity of directions of both lines at point P, of the manner in which they pass through that point.
This direction is considered in the infinitesimal calculus, and there it is evident that even a straight line and a curved line can pass through in the same manner, i.e., according to the same direction. Nor is it surprising that the same mode of procedure is employed there where mention is made of the velocity of a mobile by which it passes through point P (which is considered in mechanics), and where mention is made of contact between diverse curves or of determining the direction of the tangent to a curve; not surprising, we say, that there the same mode of procedure, differentiation or an equivalent mode, is employed; for the direction of motion or of velocity, as it is a vector, has its origin from the geometric direction of a line.
The accurate notion of direction in infinitesimals directs our attention to another notion, which we can thus subject to exact analysis: two lines can pass through the same point P in the same manner, according to the same direction (thus diverse curves and a straight line that have a point of contact at P); at P, or through P, they have the same direction, but then at other points they will have diverse directions. This notion then invites us to take a twofold further step: to compare the directions of lines at diverse points, and also the parts of one line with one another.
4. Curves and Straight Lines.¶
In inexact geometry mention can already be made of curved and straight “lines,” which are known sensibly; as is manifest, we now intend to speak only of sensibly curved and sensibly straight lines. But this distinction must also be sustained if we pass to lines worthy of the name, which lack breadth; they will be exact lines, but their straightness or curvature is taken in a sense still inexact, as sensibly known. Yet to these lines we can apply the elements of exactitude that we recently discovered.
To accomplish this, let us recall that the directions of lines with respect to their similarity or diversity are also understood of lines as they pass through diverse, distant points; even in sensible geometry. Segments of lines distant from each other that are traced on a surface, such as on this page, can be compared under the respect of direction. In one part let there be present a segment that tends from the left side toward the right, in another a segment drawn from the bottom upward. These directions are without doubt diverse. But two segments can also be given, on the right and left sides, that both tend upward in (almost) the same manner; and intermediate cases are given. Let us look again at the first case, in which the directions are certainly diverse; on one line let there be point P, on the other point Q. We already know that through P there passes an entire continuum of directions, in such a way that lines can be given that, beginning from the first, are varied in a continuous manner according to direction until finally the direction opposite to the first is attained. Among these directions only one will be able to be present (if one is present, which will have to be examined) that is exactly similar to the direction of the other line through Q. This direction will be perfectly and exactly similar; it will be specifically the same. Concerning the notion this is already established, not yet concerning mathematical existence.
What we were saying concerning diverse lines can be applied to distinct parts of the same line. These too, relative to direction, can be compared with one another, and will have diverse directions or specifically the same, perfectly similar direction.
We can use these considerations to define the notions of a straight and a curved line. A line will be straight if the direction according to which it passes through one point is preserved further in it as specifically the same, if through any subsequent or preceding point whatever it continues to pass with the same direction, without deviation. If at one point there is an abrupt change of direction, it will be a “broken line”; if the direction is varied in a continuous manner, it will be a curved line. That such lines, only sensibly exact, exist, experience teaches; whether exact lines exist mathematically must be investigated; for thus in this case the problem of exactitude returns. Concerning the notions it is now established, concerning mathematical existence not yet; or as Aristotle speaks: it is established concerning the “what it is” (quid est), but not yet concerning the “whether it is” (an est); and in a question concerning a “proper attribute” (proprium), such as the question concerning “straightness” and concerning “curvature,” it must first be established concerning the “what it is.” We observe at the end that the mode of procedure we have followed to establish these exact notions and to pose the problem concerning their existence seems in every respect to correspond to the natural process of the human mind, “le cheminement de la pensée”; we were instituting nothing other than an analysis of this mode of procedure, beginning from sensitive data.
This too seems to be the notion that Euclid intends in his definition of a straight line, concerning which there has been so much dispute; this at least seems to us to be very probable, for on account of the obscurity of the words it can scarcely be established with certainty concerning his intention. The definition, however, is this (Def. IV, Elementa, ed. Heiberg, I, pp. 2 and 3): “εὐθεῖα γραμμή ἐστιν, ἥτις ἐξ ἴσου τοῖς ἐφ' ἑαυτῆς σημείοις κεῖται. A straight line is that which lies evenly with the points situated on itself.” The expression “to lie evenly” (ex aequo iacere) seems truly to admit this interpretation: “to pass through the points in the same manner, i.e., according to the same direction, without deviation.” Geminus also, in Proclus (In Euclid. Comment., ed. Friedlein, p. 185), from the straight line so understood either as a property or as a definition, infers that from any point to any point one straight line can be drawn; he says:
“ἑπόμενόν ἐστι τῷ ῥύσιν εἶναι τοῦ σημείου τὴν γραμμὴν καὶ τὴν εὐθεῖαν ὁμαλὴν καὶ ἀπαρέγκλιτον ῥύσιν,” i.e., [this property] “follows upon the fact that a line is the flux of a point, and a straight line an even flux without deviation.”
5. On the Surface.¶
Lines sensibly straight and curved are therefore given, and concerning their definition, by means of the primitive notion of direction, even if they are supposed exact, it is already established; we shall have to investigate further whether lines exist mathematically that are exactly straight. Similarly must it be said concerning surfaces. For from sensitive data it immediately becomes known that surfaces exist that are curved; that have a concave face and an opposite, convex one. Surfaces are also given that, sensibly at least, are neither concave nor convex, but plane. In curved surfaces sometimes only sensibly straight lines can be drawn (this will remain true in exact geometry as well), never, however, through every single point and according to any direction whatever. Nor is a straight line that joins two arbitrary points of the surface situated in its entirety in the surface, but on the concave side it passes outside it.
A sensibly plane surface, which is given in our experience, is such that through any point whatever there passes an entire continuum (sheaf) of sensibly straight lines, which are situated wholly in the surface; every straight line that joins two arbitrary points of the surface is situated in its entirety in it; this surface is concave on neither side. These connections, certainly, are verified only sensibly [12].
Whether in exact geometry the straight line, the plane, and their connections, according to the matters we briefly touched upon, truly exist mathematically must now be inquired. In other words: we must now investigate whether in these cases, which also imply qualities of figures, the transition from the imperfection of sensibility to the exactitude of the intellect can be justified from a means that permits the transition. Yet meanwhile it must be observed again: in the natural process of the human mind, according to which geometry has been constructed, virtual judgments will be able to be found such that they already contain this justification. If in this problem too, as in the preceding ones, the matter stands thus, the mind already in that natural process immediately from sensitive data, verified only sensibly, legitimately deduced exact geometry, without having explicitly affirmed those judgments. Then the office of the philosopher is to inquire into and consider and explicitly enunciate these virtual judgments; this office will perhaps be difficult not in itself, but only on account of historical circumstances.
The principal matters that must be inquired into, as follows from what was said above, are roughly these: Whether in distant lines that can have a sensibly similar direction, this similarity can exist exactly; whether in extension a constant direction, and consequently [13] the straight line that sensibly exists, is also given exactly; whether to sensibly plane surfaces there also correspond exact planes, in which every exact straight line of which two points are situated in them is contained in its entirety in them; whether the plane is such that around its individual points continua of directions are found that differ among themselves by number alone.
Above it was said that the definition of a straight line posited by Euclid not without probability signifies its constant direction. Similarly it is not improbable, as it seems to us, that the proper attributes of a plane surface found above are intended (as exact, of course) in his definition of a plane, which otherwise would be most obscure. Behold his definition (Def. VII, in ed. Heiberg, pp. 2 and 3): “ἐπίπεδος ἐπιφάνειά ἐστιν, ἥτις ἐξ ἴσου ταῖς ἐφ' ἑαυτῆς εὐθείαις κεῖται, A plane surface is that which lies evenly with the straight lines situated on itself.”
6. On the Existence of the Straight Line and the Plane.¶
We must now inquire whether exact figures, in the first place the straight line and the plane surface, whose notions we drew from inexact sensitive data, also truly exist mathematically; whether in extension as such, uniform, in “space,” they are possible. The problem, therefore, is whether and by what means in this point the human intellect by affirming can transcend the defect of exactitude of sensibility.
Above we already saw that this inexactitude has already been overcome in these points: 1° in recognizing the existence of indivisibles under a certain respect or even simply, i.e., in recognizing the existence of surface, line, and point in the exact sense; 2° in the analysis of the continuity of extension as such, which teaches divisibility into infinity and the impossibility of composing an extended entity out of unextended entities; 3° in recognizing extension, as it is extended, as homogeneous, uniform (or unformed) [14], i.e., as the principle of individuation; from which immediately flows the possibility of congruent figures and the possibility of deducing total congruence if partial congruence is given; from which flows the possibility of motion of an invariable figure; 4° in the analysis of the notion of direction, where the analysis of the continuity of extension leads to recognizing infinitesimal direction and the existence of a continuum of directions around a point in a surface or in “space.”
The things we found relative to the exact notion of direction and relative to extension as it is the principle of individuation, if taken together, will also permit a legitimate transition from the inexactitude of sensibility to knowing the mathematical existence of exact figures.
A) On the Straight Line.¶
Let us begin from the straight line. We saw how the idea of the direction of a line arises from the intuition of lines that pass through one point in a diverse manner. We saw also how, in order to conceive direction exactly, it is necessary to pass to the limit through an infinitesimal consideration. Lines that, in this passage to the limit, preserve a diverse direction, truly pass through that point according to a diverse direction; others touch each other at this point and there have the same direction. In these notions we have already transcended the defect of exactitude of sensibility, employing the ideas of a truly unidimensional line, and of divisibility, which follows continuity, both in dividing the lines themselves and in considering the continuous diversity of directions.
Yet we saw moreover that the diversity of directions can be apprehended in lines as they pass through distant points; there we apprehend at least the notion (not the existence) of directions that are exactly similar. And applying these considerations to diverse parts of a single line, we also apprehend the notion of a straight line, not yet its existence.
That both, however, also exist mathematically, extension as it is the principle of individuation again demands. In extension as such, which is not yet determined and bounded through a figure, in pure “space,” any part whatever and any point whatever is distinguished from others of its species by number alone; according to the structure of the surrounding extension, it is specifically the same with every other point. Hence every direction that exists at one point is repeated at each of the others as specifically the same; at the individual points specifically the same continuum of directions exists. Hence it also immediately results that lines exist that in each of their parts have the same direction, according to all possible directions. Straight lines therefore exist, and indeed in such a way that at the individual points a continuous sheaf of straight lines can be constructed.
We say that in extended “space” the matter stands thus; for again, as in the case of congruence, an extension must be supposed that is not yet determined through a figure. Thus in a spherical surface at the individual points (except those that are opposed according to a diameter) the same continuum of directions is not present; on the contrary, at each point a different and ever different one is present. In such a surface, on account of its curvature, straight lines cannot be given; indeed, in this does their curvature consist. Nor can lines be given there whose curvature is less than that which great circles have. The exactly infinitesimal directions that at one point of such a surface are situated in it, at other points (except one) are indeed present in the space in which that surface is a figure, but pass through the surface.
One intuitive truth that regards the connection of point and straight line must be touched upon here. A straight line, as it is a figure, can be moved in every way in uniform space. Let it be so moved that one point remains fixed. It is manifest that the line can be so rotated that it attains any other point whatever in space; whence we understand: through any point whatever to any other whatever at least one straight line can be drawn; whether only one, is not yet established.
B) On the Plane Surface.¶
The first figure whose existence was found by us was the surface, as the boundary of the parts that arise by the first division of the extension originally given to us (which then revealed itself as an extension of three dimensions). This figure was already immediately recognized as exact in this sense, that according to one dimension present in the original extension it is indivisible, permitting a passage in an indivisible from one part into the other (Chapter III, § 2, no. 2, A). This figure must be examined with respect to other elements of exactitude, and specifically whether a plane surface, whose notion has already been established, also exists mathematically; whether after the “what it is” (quid est) of this proper attribute of surfaces (namely, “planeness”) it can also be established concerning the “whether it is” (an est).
In a surface sensibly and clearly curved, these things are observed: it does not equally face the two parts of extension between which it is the boundary; toward one part it is concave, toward the other it is by that very fact convex. On the contrary, a sensibly plane surface faces both parts in the same manner; from that datum arises the concept of an exactly plane surface. That it exists we immediately understand from the uniformity of extension as such, of “space”; again, because its parts are distinguished from one another by no distinction except a numerical one, therefore a boundary can be given between two parts that faces them in perfectly the same manner. And indeed according to a twofold respect: 1° in such a way that this is verified in one part of the surface that we consider, and 2° in such a way that the same can be verified in the same manner in each of the other parts, which thus are distinguished from one another only numerically. The first is verified also in the infinitesimal parts of a sphere, the second is not.
Yet this diversity and identity of respects toward the one and the other part of space can be determined more closely, even in exact geometry. The individual points of a surface are, as was said above, the apices of a continuum of directions that are found around those points in the surface. As a point is situated in space, which is divided through the surface, it is the apex of a three-dimensional continuum of directions, which is split through the surface into two parts (besides those directions that are present in the surface itself). All these directions pass through the surface, but one class penetrates the surface, let us say from left toward right, while the other contains the inverse directions that pass from right toward left. (Lines that penetrate through the same point belong individually, according to one direction in the line itself, to one class, and according to the opposite direction to the other). These things too are preserved in exact geometry, which can also consider directions whose passage cannot be sensibly observed. By means of these concepts we can determine more closely that diversity and identity of respects.
Let there be a sensibly curved surface, whose concave face faces the right part, and whose convex face therefore faces the left part of space; let there be a sensibly straight line (which can be an exactly straight line, for we already know that it exists) and vertical. This line, in the direction that tends upward, first penetrates the surface from left toward right, then, at a greater height, from the right side toward the left. The same direction in one line, therefore, is related in an inverse manner to the surface at one and at another point of it. This is what we were saying: the surface does not in the same manner face the one and the other part of space; and this remains true in exact geometry: relative to the same directions (of this straight line), the surface at diverse points is related in an opposite manner.
A plane surface, therefore, which in the same manner faces both parts of the space that it divides, will be that which relative to directions that are exactly similar is related in the same manner. And on account of the uniformity of extension as such, of space, such a surface exists in exact geometry as well. In such a surface the same directions pass through it from one part into the other, the same in the inverse manner, the same are situated in it; and indeed at the individual points.
7. On the Connections of Straight Line and Plane [15].¶
What has been said in the last place regards the connections between the exact straight line and the exact plane and immediately leads to diverse axioms. Only certain ones are to be expounded here.
The first thing that immediately follows is: a straight line that has two points in common with a plane cannot between those points be situated outside it. For the same direction of the straight line at one point would pertain to the directions that tend from one part of space into the other, while at the other point it would pertain to the directions that tend in the inverse manner; the surface would no longer be plane. The direction of the straight line is therefore situated in the plane itself, and consequently the entire straight line.
In the plane itself, moreover, through two points only a single straight line can be drawn; (that one is always present has already been established earlier). For if two were supposed, at one of those points they would have diverse directions, which in the individual straight lines are constant; constantly, therefore, would the directions differ. Yet at those two points the directions would be opposed in a diverse manner; at one the first would differ from the other, e.g., by a deviation toward the right, at the other toward the left. At least one direction would not be constant. Unique, therefore, is the straight line that joins two points, and in the plane in which those points are situated, the entire straight line lies.
There follow certain considerations that employ the motion of figures, whose legitimacy was established earlier; we already employed it in considering a straight line that is moved in such a way that one of its points remains fixed.
A plane that contains a straight line can be rotated around it. In that motion it attains a single position in which it contains a determinate point situated outside it; for the rest, this point is arbitrary. Hence: a straight line and a point situated outside it determine a plane. Hence it is understood that a plane can be described or arise thus: through the motion of a straight line, drawn through any point whatever, which is so moved that it always falls upon another straight line. The same plane is described through a similar motion of a straight line from any point whatever along any straight line whatever, provided they are situated in the first plane. Thus it is evident that all points of a plane are equivalent among themselves, likewise all straight lines; not only relative to the parts of space that the plane bounds—that we already knew—but also relative to the plane itself. And perhaps we do not err in saying that this is the intention of Euclid in his definition that we cited above: “a plane surface is that which lies evenly with the straight lines situated on itself.” A consequence of the greatest moment is: The plane is homogeneous, uniform, i.e., it is the principle of individuation for all figures situated in it.
Hence it immediately follows that circles exist that can be described in a plane from any point whatever as center and with any radius whatever. The continuity of a circle is best understood from the continuity of the motion of the point that describes it.
Hence the notion of an angle can now be determined. Already in inexact geometry we saw that from the diversity of direction of lines that are drawn in one surface through the same point, there result “divergences” that, relative to one another, can be called greater and lesser. Now this notion of divergence can be polished in exact geometry into the notion of an angle, a quantitative notion that admits of greater and lesser and can be measured exactly. Straight lines that pass through the same point in a plane, according as they differ more or less in direction, contain between themselves a greater or lesser part of the indefinite plane. These parts of the plane can be compared with one another; they can, e.g., be measured exactly by means of a circle whose center is that point, and whose radius has a unit of length; the sectors of this circle comprehended within the straight lines, or also the relative length of the arcs, are the exact measure of the divergence, which will now be called an “angle.” Consequently, they are the measure of the diversity of directions, just as a quantitative effect is the measure of a qualitative cause.
Let us note especially what is essential in this entire analysis: because the plane is the principle of individuation, at any point whatever an exactly similar diversity of directions causes angles that are exactly equal, congruent.
Hence there results a consequence of the greatest moment. In a plane there exist straight lines, not coinciding, that have an exactly similar direction; let us call these straight lines: parallels. A parallel, drawn in space through a point outside a given straight line, is situated in the plane determined through that point and that straight line, for its direction cannot intersect the plane that contains that direction. Through any point whatever outside a given straight line, there always exists one parallel; also one only; for a straight line diverse from this, which passes through the same point, by that very fact has another direction. This is the principle of the unique parallel. Yet parallel straight lines, according to what we saw above concerning the plane as the principle of individuation, with any straight line whatever intersecting them, because they have the same diversity of directions, make the same corresponding angles.
And this consequence constitutes Euclidean geometry in the pregnant sense, as opposed to modern, non-Euclidean geometries.
8. Again on the Notion of Direction.¶
This conclusion of the greatest moment follows from the very notion of direction, as has already been observed by many. Yet they did not subject the notion to analysis, as we have done. Hence, and hence alone, it has also come about that mathematicians have raised a difficulty against the introduction of this notion. Before we proceed in our investigation, this difficulty must be examined.
One mode of proposing the difficulty we already encountered above in the distinguished Helmholtz. It comes down to this: we define direction by means of the straight line; if, therefore, we define the straight line, as has been done in our exposition, by means of direction, we are engaged in a vicious circle.
These things would be true if de facto we defined direction thus. But we do not do this; we define it as the “manner of passing” of a line, even a curved one, through a point; and this is a truly primitive idea. For the notion of direction (still inexact), as is manifest from what has been said, arises immediately in the mind contemplating the simplest figure; likewise the notion of a straight line, in opposition to curves, as the notion of a line with a constant direction, is conceived without any difficulty. There no difficulty lies hidden. It arises only if, after this question concerning the “what it is” (quid est), there arises the problem “whether it is” (an est), both for the notion of direction and for the derived notion of the straight line. And to solve this difficulty there is required the intuition of extension as it is the principle of individuation and the development and application of this notion. And, as seems to follow from the foregoing, the philosophical analysis of these notions and of this principle demonstrates that the human mind applies the same mode of procedure, yet without analysis, using only virtual judgments. In such a mode of procedure no vicious circle is present, nor is one latently present there. And analysis demonstrates that the difficulty is not present in the notions, as the difficulty raised intimates, but in intuiting the mathematical existence of exact objects; we are dealing with the problem of exactitude, not in the notions, but in existence.
More copious and better are the considerations found in Killing, to whom we are also referred in B. Russell [16]. He says: “The angle is the measure of the diversity of direction of two straight lines; consequently, the angles that two parallels make with the same straight line are equal” [17].
He attributes this method to Leibniz; but he calls it worse than the mode of procedure of Euclid, because the latter indicates the difficulty that the former clouds over.
To effect this, he begins by saying that the idea of direction is held by the authors who employ it as primitive, because they do not give a definition. And it is excellently considered thus, provided this notion is genetically developed and polished to exactitude. Next, the distinguished Killing, as is right, urges the connection that the idea of direction has with the idea of an angle. Yet he thinks that in this way a means of recognizing equal diversities of directions cannot be indicated without the entire theory of parallels being presupposed. That this entire theory is involved in a suitable means of recognizing this equality is true; but he thinks that the equality of angles is not a suitable means, because it would presuppose this theory. And he intends to say that this theory is gratuitously presupposed; in which he is mistaken.
What the distinguished author intends is more clearly understood (simultaneously with his error) from the things that follow next. For he wishes, in order to expound the idea of Leibniz more fully, to introduce this definition of direction: “Two straight lines have a similar or dissimilar direction if with a straight line intersecting both they make equal or unequal angles” [18]. If this is truly employed as a definition, what he adds is true, namely that this does not lead to the goal, i.e., to deducing the Euclidean theory of parallels; and therefore this theory would be gratuitously presupposed. For as he well continues:
“It can only be said: those straight lines have a similar or dissimilar direction relative to a determinate third straight line; but then it is uncertain whether two straight lines that make equal angles with a determinate straight line are also intersected by any straight line whatever under equal angles” [19]. These things are excellently and most truly said on the supposition of that proposition as a definition; if, however, it is a proposition that follows from the exact notion of direction established from elsewhere, they are no longer true. For from this follows the universal proposition: any straight line whatever makes equal corresponding angles with two straight lines of the same direction.
And we expound this author and his difficulty at such length especially because from it one understands better that in our mode of procedure—which genetically discovers also the existence (with respect to the “whether it is,” an est) of the exact notions: of direction, of the straight line as a line of constant direction, of the plane as the principle of individuation, of the angle as the measure of the diversity of direction—we immediately arrive at the universal proposition: any straight line whatever makes equal corresponding angles with two straight lines of the same direction (parallels); just as there universally follows the inverse proposition: if one straight line with two others, situated in the same plane, makes equal corresponding angles, these straight lines are parallels.
And concerning this entire process it can be said: these things are indeed evident from experience in an approximative manner, but from the means employed of arriving at exact intellective cognition, it is evident that it is exact and necessary. The ultimate means of attaining exactitude is the intuition of extension as it is the principle of individuation. This principle implies not only specifically the same continuum of directions at the individual points, but moreover the equality of the angles that result from specifically the same difference of directions. A special case of this principle is Postulate IV of Euclid concerning the equality of right angles. And all the means employed demand indeed, in order to be explicitly detected, an accurate philosophical analysis, yet these very means in constructing classical geometry were employed implicitly and naturally, as virtual judgments. Therefore, this analysis manifests the mode of procedure of the human mind, le cheminement de la pensée [20].
9. On Direction in Euclid.¶
Rightly, therefore, is it said that the consequence we drew above from the elaborated notion and mathematical existence of direction contains the Euclidean theory of parallels. Euclid himself certainly does not use that notion in his theory; hence also his very definition of parallels is different (Def. XXIII): “parallel lines are straight lines that, being situated in the same plane and produced in both directions into infinity, in neither direction meet.” He uses, therefore, a notion that pertains to connections (the absence of concurrence) for the definition; hence also the famous Postulate V enunciates something concerning such a connection. Yet the same theory of parallels is deduced thence.
Hence it is not surprising that the notion of direction, and indeed of constant direction, as a proper attribute of the straight line, is found immediately after Postulate V has been employed (in Proposition 30, immediately after 29, in which the first and sole application of the postulate is found). For this proposition establishes: if straight line a is parallel to straight line b, and this to straight line c, a and c also are parallels. That the notion of (constant) direction as a proper attribute of the straight line is contained there is established thus according to the logic of relations. This relation (of parallelism), according to that proposition, is transitive. Yet concerning the same relation there hold: if a is parallel to b, b also will be parallel to a; it is a symmetric relation. Moreover: straight line a can be called parallel to itself; it is a reflexive relation. Parallelism pertains, therefore, to the same genus of relations to which the relation of equality pertains. For if in quantities a = b and b = c, it follows: a = c. Likewise: if a = b, it follows: b = a. Likewise it can be said: a = a. Both relations, therefore, of parallelism and of equality, are transitive, symmetric, and reflexive. Just as the relation of equality supposes in its terms some proper attribute as a foundation, so the similar relation of parallelism supposes in its terms, namely straight lines, some proper attribute, and this is direction itself. Thus, therefore, in the mode of procedure of Euclid there appears again the notion of a constant and unique direction in a straight line and the existence of a perfectly similar direction at distant points of a plane. If “direction” is defined thus, then we cannot use that notion to define the straight line without a circle; and on this supposition, but on this alone, Helmholtz and others would speak rightly. Yet thence it is also manifest that this notion does not arise thus in the human mind; it apprehends it directly, as sensibly verified in lines; and this notion is elaborated to exactitude by other means than through the circumlocutions of the definition and postulate of Euclid; it is already had from the fact that extension is known as the principle of individuation. And this is a principle that certainly is not accustomed to be enunciated, but that is always present in the human mind while it considers the proper attributes of extension. Whence whoever examines the mode of knowing of the human mind cannot neglect these things.
Relative to what has been said above, let two remarks follow. The first is this: from the things we were saying according to the logic of relations it does not follow that the relations of equality of quantities and of parallelism of direction of lines are altogether identified with each other. Let another relation serve as an example, which in its own way too is transitive, symmetric, and reflexive, namely the relation of perfect similarity between objects of qualities, e.g., of colors. There too it holds: if a is similar to b, and b similar to c, a is similar to c; simultaneously: if a is similar to b, b is similar to a; and moreover it can be said that a is similar to itself. Although, therefore, the relation of equality of quantities and of perfect similarity between qualities have the same three properties, nevertheless their foundations, quantity and color, are diverse and known as diverse. In the same way the foundation of the relation of parallelism, namely the direction of a straight line, which is characteristic for Euclidean geometry, is diverse, and known to us as diverse, from the foundations of those other relations. Which certain thinkers, applying the doctrine of the logic of relations, sometimes seem to forget.
The second remark is this. In the theory of the unique parallel there returns the intuitive notion of direction, of constant direction, of the straight line that has this direction as a proper attribute. If, therefore, an extension exists that does not admit the Euclidean theory of the unique parallel, it does not admit a constant direction, it does not admit lines that are truly straight; it will therefore be an extension that in the proper and intuitive sense is: curved. Mathematical philosophers who determine curvature (which is defined according to Gauss) through intrinsic measures in an extension (e.g., in a surface) and do not look to the things that can be outside it, sometimes say that this curvature is not curvature worthy of the name, in the intuitive sense. If measures are considered in the customary and unchanged sense, they seem to err. For in that extension there is lacking “straightness,” which implies a constant direction; it is truly curved. And thus it necessarily supposes an extension of more dimensions, in which the directions lacking in the extension that is measured are present; and this extension is curved in that higher one in the proper sense. For the rest: surfaces are given that have a “Gaussian” curvature of “value zero,” which therefore can be applied to a plane, yet which have an “intuitive” and true curvature (cylindrical and conical surfaces).
§ 5. On Euclidean and Non-Euclidean Geometry.¶
From our analysis there follows Euclidean geometry, and indeed as necessarily unique; which in our times can be held for a defect, indeed for an error. For besides Euclidean geometry, non-Euclidean geometry as well, and indeed more than one, must be admitted as true. These are indeed sometimes called by mathematicians—so at least it is done in the Netherlands—“pathological geometries”; yet this “pathology” does not regard the minds of the geometers who cultivate them, but rather the objects concerning which they treat. And these things are perhaps said rather in jest; for mathematicians are accustomed to attribute the same value to non-Euclidean geometry as to Euclidean. Indeed, they say that anyone who calls Euclidean geometry uniquely necessary ought no longer to be read. The distinguished F. Hausdorff says:
“Mathematics is now permitted to lay aside unexamined into the archives every a priori construction that claims to deduce Euclidean space with its special properties as necessary, without arbitrary presuppositions” [21].
These things were written in the year 1904; yet even later similar statements are brought forward. Nevertheless, today the genuine character of non-Euclidean geometry is better known, and we too, after our deduction, attribute to non-Euclidean geometry a true and real meaning, indeed a meaning that extends more widely than that which the cited author alone vindicates for it. Only we shall not be able to attribute the same realistic value to both geometries: Euclidean and non-Euclidean; and this not only in the practical order (as Poincaré did), but especially in the theoretical order and relative to the theory of cognition of the human mind. (See below, Chapter VI, § 1).
Non-Euclidean geometry has indeed a true and real meaning, indeed it has several meanings. First among them is a certain logical meaning, and from it diverse real meanings can be derived; yet there is none among them that could equate it to Euclidean geometry. Now we must show what errors we do not commit in asserting the necessity of the unique Euclidean geometry as the science of extension as it is extended, which was the conclusion from what has been expounded above.
We observe first. The error that is often attributed to those who assert the necessity of Euclidean geometry is this: they endeavor to demonstrate Postulate V from the (sole) other axioms that Euclid, whether explicitly or implicitly, admits. I do not know whether anyone truly exists who has endeavored to do this; if he exists, he certainly errs. For it has been demonstrated by mathematicians that those other axioms, united with a certain proposition denying Postulate V, cannot lead to a contradiction. This is properly that primary, purely logical meaning of non-Euclidean geometry. Hence it is certain that Postulate V is altogether independent of those other axioms, and therefore cannot be demonstrated from them alone. We in what has been expounded above did not endeavor to do this; we indeed established the postulate, but by employing other means, primitive intuitions, besides the other axioms of Euclid.
From which it is also immediately evident that we do not commit another error that is often attributed to those who vindicate a unique position for Euclidean geometry. We do not say that Postulate V itself is immediately evident. Concerning this point certain considerations must at least be sketched here.
What was, in the course of history, the difficulty that could be raised against Postulate V, proposed as a first principle? It is this: the postulate affirms that two straight lines in one plane falling upon a third concur on that side of the plane, divided through this third line, where the sum of the interior angles that they make with the third is below two right angles. This proposition, to anyone inspecting a figure in which this sum is far below this value, is certainly immediately clear. Yet if it is only a very little below this value—and, on account of continuity, the difference can decrease into infinity down to zero—on account of the inexactitude of the senses it can no longer be immediately read intellectively in the figure; add that the point through which one of the two straight lines, almost parallel to the other, passes can be very distant from the other; and this distance can grow into infinity. It is manifest: it cannot in every case be established from mere inspection that those two straight lines only fail to concur in the case in which that sum of the angles attains two right angles. From mere inspection it is not established that this is not verified earlier. The exact proposition that affirms this—and this is Postulate V—cannot immediately through abstractive intuition be understood as true.
It is strange that so few mathematicians who treat of Postulate V accurately expound this very difficulty. They are accustomed to employ expressions such as this: Postulate V is less evident than the other axioms of Euclid [22]. They ought to say: in its exactitude it is immediately altogether non-evident. — For the rest, as follows from our exposition, the same must be said concerning the axiom of the unique straight line through two points; this too in its exactitude cannot be read immediately from a figure; in order to become evident, it requires another means of overcoming the inexactitude of sensibility.
On the contrary, the most celebrated F. Klein, as he is accustomed to do, expounds the difficulty with complete clarity [23]; from him we have borrowed the formulation of the problem as it has been described.
From his exposition it is manifest: we are dealing with our problem, with the problem of geometric exactitude, which arises from sensible data. Hence it is also clear: directly through intuitive abstraction it cannot be solved; hence a means of indirect solution must be sought; which we endeavored to do above.
If this means were not present or were not found, it would have to be said: it is not certain that an extension (a plane) by its nature is Euclidean; it can happen that extensions are given that are not Euclidean, in which those two straight lines are no longer concurrent before the value of that sum of the two angles attains two right angles [24]. In the same extension, certainly, the position of Euclid and this other position are not simultaneously verified; for they are contradictory to each other. Yet a Euclidean plane could be given, and others, non-Euclidean in this sense. On this supposition it is also evident that diverse extensions of this second genus could be given, which are so related to one another that the limiting position, in which those two straight lines no longer concur, is attained earlier or later. Wherefore, if these extensions exist, they constitute an entire class that contains infinitely many species, which differ from one another through their structure. This difference can be expressed by a certain factor, constant in the individual species, continuously variable in the transition to another species; if this factor is sufficiently small, the extension determined by it does not sensibly differ, in a suitably chosen region, from a Euclidean extension; this is the geometry that Lobachevsky discovered. The problem posed is again the problem of exactitude, and direct intuition of Postulate V cannot solve it.
We were saying that diverse extensions are possible; from what has been said so far, only in this sense, that the absence of internal contradiction, in the sense described, is established in those extensions of the other genus. Yet on this supposition the position of those mathematicians is understood who, like Gauss, wished to decide by experiment whether the real world is Euclidean or perhaps not.
These things must be said if another means of deciding the problem is not present, if Postulate V cannot be deduced as a necessary truth from others that are indeed immediately evident. In the course of history diverse means have been proposed; and hence it is clear that at least those who proposed these did not commit the error mentioned earlier; they did not wish to demonstrate the postulate from the other axioms alone. Yet another error was present: the means proposed labored under the same defect as the postulate itself; nor were these means, on account of the inexactitude of the senses, sufficient. We indicate two. The first means is the proposition that affirms that unequal triangles exist whose sides are proportional. This is truly equivalent to Postulate V. Yet neither is this proposition immediately evident; for the senses, certainly, such triangles can be given; but thence, on account of the inexactitude of sensibility, it does not follow that that proportionality is exactly verified. The second is that which is attributed to Clavius; as a definition it is assumed: let the locus of points equidistant from a straight line be a parallel. This locus is indeed sensibly a straight line; but that it is exactly straight cannot be established from intuition alone; again on account of the defect of exactitude in sensible data, which cannot be overcome directly.
In place of these notions we have used the notion of direction; yet this too has already been introduced more than once to decide our problem; among philosophers, e.g., by Renouvier. And there indeed this is immediately evident: through a point situated outside a straight line, only a single line of the exact same direction can be drawn. Yet if this notion is not genetically explained and its existence, simultaneously with the existence of the straight line and the plane, and of their connections, is not understood and examined by means of the principle of individuation, it is subject to the difficulties that we heard brought forward earlier by mathematicians. Hence we examined this notion and these existences with all care. And in our mode of developing the notion of exact direction, those difficulties, as we saw, do not subsist; and the entire process seems to solve the problem. Whence we have not committed this third error either; we have not used, in place of Postulate V, another that is indeed equivalent to it, but that labors under the same defect, namely the absence of immediate evidence.
What, therefore, must be said concerning those extensions, described by Klein, that can be sensibly Euclidean, yet in reality, in their internal and exact structure, are not Euclidean? We certainly do not deny that they can exist. Indeed, their existence was already discovered by Beltrami in his “pseudosphere,” and this figure was elaborated by Helmholtz as an example of non-Euclidean geometry, in which Postulate V does not hold; if its curvature is sufficiently small, in a suitably chosen finite part, it cannot be sensibly distinguished from a Euclidean plane. Nevertheless, this geometry is not to be equated to Euclidean geometry. It is a special case of this very geometry; such a “plane” is a curved Euclidean surface; and, as has already been demonstrated above, this curvature must be understood in the intuitive sense; it supposes other dimensions, according to which the figure is curved. This has already been said by many; yet in order that it be fully demonstrated, we require the notion of exact direction, fully elaborated. Hence this notion, which we have shown to be primitive, is of such great moment [25].
One thing we briefly add: in the manner described, non-Euclidean geometry of two dimensions is real; whether in a similar manner three-dimensional non-Euclidean geometry is real depends evidently upon the solution of the problem whether a “space” that has more than three dimensions is possible. That such a geometry does not contain an internal contradiction, in such a way that from the posited axioms contradictory propositions cannot follow if we proceed purely logically, is already established.
In this sense, therefore, Euclidean geometry is necessary and unique. Alongside it there can indeed be constructed a geometry that uses the same terms and the same axioms as classical geometry, with the exception of Postulate V alone, in whose place another, contradicting it, is posited. This geometry logically does not lead to a contradiction; indeed, extensions can be given, sensibly not differing from Euclidean ones, in which they are verified. Yet by that very fact the exact meaning of the terms and axioms that are retained has been changed. They can indeed have a real meaning, indeed a geometric meaning; thus in the case considered above their meaning is closest to the classical meaning of the notions, and can sensibly coincide with it. Yet this case deals only with a Euclidean extension curved according to a further dimension. Which cannot be surprising, because it deals with an extension that precisely relative to constant directions is no longer the principle of individuation.
Thus Euclidean geometry can be unique and nevertheless admit non-Euclidean geometry, indeed non-Euclidean extensions.
§ 6. On Non-Euclidean Geometry Relative to the Cognition of the Human Mind.¶
From what has been said before, it is easy to solve the difficulty that is raised against the theory of cognition, indeed against the very value of the cognition of the human mind, which through so many centuries contended that extension is necessarily Euclidean. This difficulty is accustomed to be opposed to the Kantian theory of cognition; certain of the neo-Kantians (such as Cassirer) recognize it, while others endeavor to refute it. Yet if the difficulty is rightly sustained, it holds not so much against the Kantian theory as against all theories that recognize classical geometry as a necessary science; and those who raise it intend rather to affirm this. Indeed, as has been said, it would hold against the very character of the human mind, insofar as it admits necessary cognition; for the mind itself through so many centuries would have led man into error.
Against the theory of Kant we can rather argue in another way from geometry. For Kant in his theory undeservedly thinks that Euclidean geometry is justified; hence, if he admitted it as unique, he did this rather on account of reasons sought not from his own theory, but from the natural conviction of the human race. This was already noted above (Chapter III, § 2, no. 5). Kantian extensions, as was said there, lack exactitude; and Kant himself altogether ignored the problem of exactitude (Chapter I, § 5); yet from this problem arises the question concerning non-Euclidean geometry. From this ignorance it also seems to come about that Kant recognized only sensitive intuition, not intellective intuition; the latter, however, is evidently proved from the overcoming of the problem of exactitude. To have briefly intimated these things may suffice.
If, however, the difficulty of the moderns can be urged against the very character of the human mind, it must certainly be solved by every other theory of cognition that admits real, not merely phenomenal, objects of geometry, whether these objects are found in bodies themselves, as Aristotle holds, or exist separated, as Plato wills. For these theories admitted Euclidean geometry as necessary and unique.
From what has been said above, the objection is easily solved. First we ask: what did the geometers constructing classical geometry intend? Using the terms “straight line” and “plane,” they intended to speak of those exact figures whose notion is easily found, and whose mathematical existence, according to all that it implies, becomes manifest from the analysis given above. And if anyone by the terms he uses intends to treat of these exact objects, he rightly asserts that these objects necessarily and uniquely obey classical geometry; there no error is present; of these objects no other geometry holds; modern critique has demonstrated this. That analysis is indeed long if all the elements are to be expounded separately and clearly; yet the analysis itself also shows that those elements of exactitude are always at hand, in virtual judgments, to the human mind operating naturally; the sole difficulty is found in elaborating them; (concerning this matter, more in the following section). The human race, therefore, did not err in asserting Euclidean geometry as unique; nor in the (practical) exclusion of other geometries that are implicitly contained there. For it intended to treat of the straight line and the plane in the sense described; and the things that in the new geometries are described by these terms are other objects.
Yet 2°, according to what has been expounded above, even more must be said: these other objects are not excluded by classical geometry. For those objects, even those that are sensibly similar to the objects of classical geometry (and these are what is dealt with in the problem of exactitude), are nothing but a part of the objects of classical geometry; they are in reality curved objects in a higher space. Therefore it must more truly be said: in classical geometry they were not excluded, but were not yet examined; therefore they pertain rather to those things that in the course of the evolution of geometry are found as new; prior ignorance of them certainly does not constitute an error.
Yet among the newly discovered objects, non-Euclidean extensions have something very particular; and hence the origin of our difficulty is understood. It is this: the properties of non-Euclidean extensions can be deduced from axioms that are expressed in altogether the same terms as the Euclidean axioms of the straight line and the plane, with the exception of Postulate V alone, in whose place Lobachevsky was able to posit another, contradicting it. This is a very wonderful property, and its discovery is a work of the greatest merit. Hence it is also understood why geometers have endeavored to equate the new geometry to Euclidean geometry; hence arises the opinion that raises a difficulty against the classical theories of cognition, which hold Euclidean geometry as necessary and unique. Thus it is understood why this difficulty can arise, while simultaneously it is understood that it does not hold.
We say that the other axioms, namely those that precede Postulate V, are in both geometries expressed in altogether the same terms. Hence, as has already been said, it does not follow that they have the same meaning. We have already seen that classical geometry attributes to the terms “straight line” and “plane” the exact meaning that we explained earlier; these words, however, cannot retain the same meaning in the new geometries. For from these exact concepts Postulate V necessarily flows, as has been shown. Not on that account can it also be said that this postulate logically, by means of a syllogism, follows from the other axioms. Only this must be said: these other axioms do not yet express all the proper attributes of the “straight line” and the “plane” that we intuitively know concerning these objects; in order that these proper attributes may be expressed, we require Postulate V, or something equivalent to it. In this sense, and in this sense alone, can we approve what Poincaré says, namely that Postulate V contains (rather, completes) the definition of the “straight line”; we add: and of the “plane.” In this sense, namely, do we admit this: the things that we intuitively understand (by means of the notion of direction, for this is necessary) relative to the straight line, we do not yet fully express by the other axioms; in order that a full expression of them may be had, the addition of Postulate V is required.
Let this, therefore, be the conclusion from this section: the entire enigma of the possibility of new geometries that seem capable of being constituted alongside classical geometry, in such a way that they can be equated to Euclidean geometry, and from which at first sight it seems to follow that the human race erred in admitting classical geometry as unique and necessary—this entire enigma consists in this: those geometries can be deduced from a system of propositions that are expressed in the same words as the Euclidean axioms, with one or another excepted that contradict certain axioms. This truth is certainly most worthy of attention, and its recognition constitutes an admirable progress, yet it contains no mental calamity [26].
§ 7. On Classical Geometry Relative to the Philosophy of Cognition.¶
Classical geometry having been vindicated from the errors that are undeservedly attributed to it, we can safely use it in order to understand from its structure the mode of procedure of the human mind. This consideration pertains to the philosophy of cognition, which must above all be examined in the most suitable object, namely in that which is the very “proper object” of the human mind. The things that have been found in our analysis constitute a field of broad investigation; certain general points at least we shall now endeavor to indicate.
We have found a certain axiom in classical geometry that is not immediately evident, which nevertheless is posited without demonstration in the beginnings of geometry: Postulate V or its equivalent. And from our analysis it has been established that this axiom must be sustained; other geometries, insofar as they are truly geometries, sciences concerning extended being as such, are not to be equated to Euclidean geometry, they are truly parts of it, and follow therefore from Postulate V if the terms are taken in their natural sense. We posit, therefore, in the beginnings of geometry, a certain truth that is not immediately evident, which can and must be justified through mental operations that are instituted before the very beginnings of geometry.
Properly speaking, we have found more than one truth of this genus. To the same genus pertain: the axiom that establishes that two points always determine one straight line (and this straight line, if it is “terminated,” can be “produced continuously in a straight line”); likewise: the axiom that establishes that in a plane from “any center and radius a circle can be described”; and these three are the first “postulates” of Euclid. His fourth postulate as well, which affirms that “all right angles are equal to one another,” is of the same character; something similar to this postulate we found in showing that equal corresponding angles arise in every part of any plane whatever if a straight line falls upon other straight lines parallel to each other. The mathematical existence too—not the notions themselves—of the straight line and the plane pertain to these truths. All these regard the problem of exactitude, and indeed especially its second part. Out of the axioms that are employed in geometry, only certain ones, pertaining to “order” and to “connection,” such as those that regard the notion “between” and the site of a point on a line, are immediately evident; and these up to the preceding century were not accustomed to be explicitly enunciated.
It is manifest, therefore, that in classical geometry certain truths are employed as “first principles” that are not immediately evident, that require deduction and can be deduced. Observing this, we remember the things that we were taught earlier (§ 1) by Aristotle. Let us resume them according to the explanation of St. Thomas. In Anal. Post. I 2 and 10, Aristotle distinguishes “axioms” (dignitates) from “hypotheses” (suppositiones) and “postulates” (petitiones or postulata). To the first class pertain those principles that have this as proper, “that it is not only necessary that they be true through themselves (per se), but it is also necessary that it be seen that they are true through themselves. For no one can opine the contraries of them” (St. Thomas, lect. 19, no. 2; Aristotle has at 76 b 23: ὃ ἀνάγκη εἶναι δι’ αὑτὸ καὶ δοκεῖν ἀνάγκη). Suppositions and postulates have this in common, “that although they are demonstrable, nevertheless the demonstrator takes them without demonstrating them, and especially because they are not demonstrable through his own science, but through another” (ibid., lect. 19, no. 4; Aristotle, 76 b 27: δεικτὰ ὄντα λαμβάνει αὐτὸς μὴ δείξας). Among themselves, however, they differ “because if indeed such a proposition is probable to the learner to whom the demonstration is made, it is called a supposition … If, however, he is neither of the same opinion nor of the contrary, it is necessary that the demonstrator request this from him, and then it is called a petition [postulate]. If, however, he is of the contrary opinion, then it will be a question” (St. Thomas, ibid.; Aristotle, ibid., 28: ταῦτ’, ἐὰν μὲν δοκοῦντα λαμβάνῃ τῷ μανθάνοντι, ὑποτίθεται… ἂν δὲ ἢ μηδεμιᾶς ἐνούσης δόξης ἢ καὶ ἐναντίας ἐνούσης λαμβάνῃ τὸ αὐτὸ, αἰτεῖται). This doctrine especially regards a master teaching a disciple; the difficulties [27] and the difference indicated here we neglect; for the rest, it is manifest that this division, in diverse cases of “disciples,” leads to diverse outcomes. Yet in general the things we found above are described thus.
To the class of “dignities” (dignitates) will pertain the truths that regard the divisibility of extended entities, next the existence of indivisibles, and certain axioms of order and of connections, which therefore in antiquity were not even enunciated. Euclid too does not enunciate them and on that account is sharply carped at by certain authors; and indeed for the greatest part undeservedly, indeed erroneously. Among them there again stands out B. Russell, from whose critique we already saw certain points above. Relative to the very first proposition of the Elements he writes thus [28]: “There is no evidence whatever that the circles which we are told to construct intersect, and if they do not, the whole proposition fails. Euclid's problems are often regarded as existence-theorems, and from this point of view, it is plain, the assumption that the circles in question intersect is precisely the same as the assumption that there is an equilateral triangle on a given base.”
Euclid is carped at because without argument he assumes that those two circles intersect each other; which he manifestly does, for he speaks simply of “the point C, in which the circles intersect each other.” Next, Russell (with many others) affirms that Euclid in this proposition wishes only to demonstrate the mathematical existence of an equilateral triangle (upon a straight line of any length whatever in any site whatever, as a side). [29] That this is most true seems so to us as well; for the rest, St. Thomas already understands the intention of Euclid in the same way (see Anal. Post. I, lect. 2, no. 5). And therefore Euclid is said, in Russell, to have been able with equal right (i.e., without a justifying foundation) to suppose the existence of this triangle directly, without this argumentation or construction. This assertion, however, is erroneous; it ignores the problem of exactitude or the suitable mode of solving it. For, from the sole intuition of a sensibly equilateral triangle, we cannot rightly affirm that an exactly equilateral triangle exists mathematically; this direct transition to the affirmation of exactitude is manifestly illegitimate. Yet that those circles, constructed in that manner (whose existence is established from Postulate III), intersect each other is a truth that we can immediately, from sensible intuition, affirm by an intellective intuition. Hence Euclid, in place of the prior exact abstraction that could not be justified, with the best right posits the other, which to one who understands is immediately manifest. This second introduces an axiom that pertains to “connections.” Hence it is manifest that Russell wrongly identifies the two cases. And this example illustrates the case of Postulate V and similar ones; in these postulates too, which are truly “petitions” and not manifest through themselves, we can through other means overcome the inexactitude of sensibility, not, however, through direct abstraction. And this is the first fruit of the epistemological examination of the argumentation of Euclid.
There is present a second as well. Does Euclid sin in not explicitly and previously enunciating this axiom (of the existence of the point of intersection of those circles) which he uses? This is often said by extreme logicians; undeservedly, however, do they demand this everywhere. First it must be said: if this is demanded, Euclid ought to have premised a general proposition (namely: two circles intersect each other if the distance of their centers is less than the sum of their radii and greater than their difference); yet this proposition is not an axiom. And the truth that Euclid uses to establish the existence of the point of intersection becomes immediately evident, for this special case, at the moment when the circles are described; nor does the evidence become greater if this is explicitly said. Whence his mode of procedure seems to be more conformable to the nature of the mind, in which that intuition is present, determining the further progress in reasoning as a virtual judgment. We intuitively understand that those circles intersect each other; without explicitly affirming this, we use, in developing the argument, the existence of the point of intersection.
Next, there is present another virtual judgment as well. Because the point of intersection C is given, it must have the proper attributes of a point; namely: it can be joined through a straight line with any other point whatever (this was Postulate I); again: without explicitly affirming this (or referring us to Postulate I), Euclid immediately proceeds by saying: “to the points A, B let straight lines CA, CB be drawn.” And thus there is present again a virtual judgment, which is not expressed, yet which determines the progress in the argument. The mode of arguing of Euclid is an excellent example of the mode of procedure of the human mind.
Such as is the intuitive affirmation of the intersection of those two circles, such are the principles that can neither be demonstrated nor require demonstration. Yet the question is especially concerning the other geometric principles, which can indeed be proved, but in geometry itself are not proved, but supposed. How do these differ noetically from geometric conclusions, i.e., from those propositions of geometry itself that are indeed demonstrated in it? In order that we may detect this difference, certain considerations must be sampled beforehand that regard the diversity that Aristotle and the Scholastics are accustomed to establish between science strictly so called, which is the habit of conclusions, and understanding (intellectus) strictly so called, which is the habit of immediately evident principles. Science strictly so called, by means of the syllogism, deduces conclusions from premises; it uses a term that is middle between the extreme terms of the conclusion; understanding strictly so called understands without a middle term between the extremes of the principle. In order that a syllogism be truly a demonstration, it is required that the premises be necessary and that the form of the syllogism be correct, i.e., there is required the necessity of the connection, by means of the middle term, between the premises; there is required both the necessity of the matter, which is the necessity of the principles, and the necessity of the very form of the syllogism, the necessity of the consequence.
Yet to one who attends well to each necessity, there manifests itself a certain wonderful diversity, which is not accustomed to be examined and nevertheless is of the greatest moment. For we can express the individual propositions, in order to urge their form alone, by means of symbols, e.g., of letters; then we use “transcendent terms” (termini transcendentes), as St. Albert the Great says [30]; so Aristotle already did, and since him this use has become universal. Thus we can express a syllogism: every S is M, every M is P, therefore every S is P. This formula expresses the necessity of the connection that is between the conclusion and the premises taken together, and therefore between S and P, by means of the middle M. The necessity of the form, or of the consequence of the syllogism, is thus expressed.
Yet now to one who attends, this is manifest: the necessity of this formal connection, or consequence, remains after the transition to the use of symbols, which has already been effected in the formula. We can express this intuition by a symbolic formula as well: If S is M and M is P, S is P. This expresses the necessary connection between the consequent and the antecedent—a necessity, I say, whose intellective intuition is preserved even if we use these symbols and ignore their specific meaning [31]. (Cf. below, Chapter V, § 2).
Hence it follows: in constructing a syllogism whose form is correct, it is no longer necessary that we attend to the meaning of the terms in order to affirm the truth and necessity of the conclusion. It suffices that the terms have a meaning (even one unknown to us), next that the connection between the extremes of the premises be necessary (without this necessity now being seen), next that the terms that occur several times have the same meaning in the individual cases and that the functions of subject and predicate be distinguished. In a simple syllogism, therefore, attention must be paid that M have the same meaning in both premises, and that to the terms S and P in the conclusion the same meaning be attributed that they had in the premises.
The matter does not stand thus in the understanding of principles. Evident is this truth: every extended entity is divisible; yet if we pass to the use of transcendent terms: every A is B, the intuition of necessity immediately disappears, even if we intend to express the same principle. This disparity between the syllogism and a simple principle is of the greatest moment. We cannot establish a principle if we do not intuit the necessity flowing from the meaning; hence the Scholastics were accustomed to say: principles are known “when the terms are understood” (intellectis terminis). Yet we can know a new conclusion as necessary, provided the form of the syllogism—which we always intuit even in transcendent terms—is correct, provided we hold in memory that the premises—in the meaning that the terms have—are necessary; hence it is not necessary that we intuit the ultimate necessity in concluding.
Hence it comes about that the work of science strictly so called—the science of conclusions—already differs greatly from the work of understanding strictly so called; for the former the intuition of the rectitude of the syllogistic form with the memory of the necessity of the premises suffices, whereas for understanding there is always required the intuition of this very necessity. Hence: in constructing science strictly so called, we are accustomed especially to use the intuition of the rectitude of the syllogistic form, of the consequence. Indeed, hence is understood the endeavor of what is called axiomatics to reduce the operations of science to these intuitions alone, for having which it suffices to suppose the axioms, without taking care concerning their meaning. The former is already found in the doctrine of Aristotle, who distinguishes science from the understanding of principles, in whom therefore the beginnings of a certain axiomatics are present; the latter is fully had in modern axiomatics, which, as we were taught by Hilbert (Chapter I, § 6; cf. Chapter IV, § 4, no. 1), abstracts from the meaning of the terms [32]. Nevertheless, Aristotle differs from the moderns; for although in science we especially use the intuition of the syllogistic form, we nevertheless do not neglect the meaning of the terms, and thus a certain intellective intuition, greater or lesser or even total, is always present; and the understanding of the meaning of the terms and propositions is never lacking; these things we already intimated earlier (Chapter I, § 5), and we shall see them again in Chapter V.
That conclusions are necessary, therefore, we always understand, even if we have used premises whose necessity we no longer intuit, but understood earlier and hold in memory. A certain intuition of necessity is always required, yet in the work of constructing a science the intuition of the consequence suffices, which we understand in the syllogistic form expressed in words, even if the meaning of the words is hidden from us. Hence the great utility of the syllogism is evident.
What we were saying regards the necessity of propositions. Yet in our case—the case of geometry—from what has been expounded above concerning the problem of exactitude, there follows an even much greater difference between science and the understanding of principles. For in this case, in order to understand a principle intuitively, it is required that we transcend the inexactitude of the senses; sometimes this is easier, sometimes difficult, as is abundantly evident from what has been expounded above; this difficulty is proper to geometry. Yet behold: this difficulty in constructing science strictly so called already disappears. Attending to the syllogistic form, in a syllogism expressed in words, we certainly require the intuition of this form as it lies concrete in the terms and propositions, yet through this intuition the problem of exactitude no longer has to be overcome. For the form of the syllogism, expressed in words, like the first arithmetical data (cf. Chapter I, § 3; Chapter II, § 1; Chapter III, § 1, no. 1), already is exact; hence there by the intellect only the necessity of the connection, i.e., of the consequence, has to be recognized; the problem of exactitude has entirely disappeared. For if the exactitude of the notions and connections is already present and affirmed in the premises, originally therefore in the axioms, by that very fact it is transferred into the conclusions. Hence, in the operations of science strictly so called, the difficulty of overcoming inexactitude is no longer found. Hence the greatest problem that arises in the noetic explanation of geometric cognition, the problem, I say, of exactitude, no longer presents itself in the syllogistic deduction of theorems from premises; this problem in its entirety is situated in the noetic explanation of the origin of the axioms themselves, from which syllogistic deduction takes its beginning. In geometry, therefore, there is specifically the greatest distinction between the understanding of principles and science strictly so called, the science of conclusions.
Hence it is also better understood why modern axiomatics arose among mathematicians; the axioms—more or less the same as are present in classical science—are purely supposed, and they restrict mathematical science strictly so called to syllogistic deduction alone. Yet if the realistic value and the truth of the axioms have not thus been placed in safety beforehand, by that very fact there arise, as is evident, at least two problems: 1° whether the supposed system of axioms is immune from internal contradiction, i.e., whether contradiction is excluded in correct syllogistic deduction; 2° if this matter is established, there arises the problem whether the “science” so constructed can truly be applied to real extension; not only to physical extensions, but also to pure extension, τὰ ἐξ ἀφαιρέσεως (“the things from abstraction”) according to Aristotle; in other words: whether this science is truly geometry.
The classical theory, as it is understood by Aristotle in his theory of cognition, at first sight, as has already been said, is quite similar to axiomatics. For it separates the understanding of principles from the science of conclusions. Yet there is a great difference. For it refers the examination of the principles to another science, philosophy, and thus wishes to justify them beforehand. Yet moreover in the science itself, in the deduction itself, it differs from modern axiomatics. As we have seen: it wishes so to deduce science that in scientific deduction it always holds in mind and understands the meaning of the terms and relations. This had already been said concerning the element of necessity; this must now be repeated concerning the element of exactitude. Nor on that account in this mode of procedure will the problem of exactitude have to be solved anew in the deduction itself, although it understands that meaning in the phantasm. For, as is abundantly evident from what has been said above: the exact notions are very easily conceived, and likewise the concept of exact relations is very easy; the difficulties—now lesser, now greater—were only in detecting the mathematical existence of these notions and relations in extension. Yet behold, in deducing conclusions, by means of the syllogism, from exact axioms already established, and indeed in a deduction such that we always keep the meaning of the words present, we require only the exact notions, concerning whose existence it is already established. For concerning the existence of the exact objects and first connections it is already established from the study of the principles: concerning the existence of indivisibles, of surface, line, and point; concerning the existence of congruent figures; concerning the existence of the straight line, the plane, and parallelism. And this existence had already been affirmed in the axioms, and this legitimate affirmation legitimately passes also into the conclusions, without the problem of exactitude arising again; for to establish legitimate conclusions we require only the intuition in the syllogistic form, which, as has been expounded, is free from the problem of exactitude. To these matters we shall return in Chapter V.
Whence even in the Aristotelian conception of geometry there remains a vast distinction between the noetic difficulties of the cognition of principles and of conclusions, and their examination is legitimately attributed to diverse sciences.
The science that justifies the principles, in Plato was dialectic, in Aristotle was metaphysics, in St. Thomas was either metaphysics or natural philosophy; let us say in general: it is philosophy (see above, Chapter I, § 1). And such a science is today again admitted under the name of “pre-geometric” or “preliminary” (préalable) science or “dialectic.” Let us call it “noetics.”
A certain preliminary difficulty will be able to arise here from the things we have seen. In this preliminary, philosophical science too there occur formal syllogisms; and concerning these it will be possible to say what has been said concerning geometric syllogisms: they suppose only the intuition of the syllogistic form, even expressed in words, and share in the facility of this intuition; therefore they do not differ from geometric syllogisms.
Let us say first: there sometimes without doubt occur such syllogisms, which scarcely or not even scarcely differ from purely scientific syllogisms. Hence too it is that the system of axioms posited as the beginning of geometry is not in every respect prescribed by nature; in one and another way a suitable system can be chosen. Thus we, from the analysis of the notion of direction, arrived at another definition and other axioms of parallelism than are those that occur in Euclid. Our system is better adapted to solving modern difficulties, and moreover seems to be more natural; the axiom of Euclid, however—from which immediately, as was said above, the entire theory of parallels and the very notion of the direction of a straight line results—is very suitably posited on account of its simple content—it treats, as has been said, of a certain connection—yet it offered a handle to those difficulties.
Next, however, it must be said: even if here and there some formal syllogism occurs, indeed perhaps is necessary, nevertheless the entire process is not formally syllogistic; often it is non-syllogistic. Often its “movement” is the inverse of the syllogistic movement (cf. Chapter II, § 3, no. 3). It consists rather in an analysis, not indeed of terms—this leads only to analytic judgments in the Kantian sense, which do not have great value—but in an analysis of the very processes of the human mind. This method is truly analysis; for it inquires into those operations by which the human mind in its natural mode of procedure—antecedently to philosophy—constructs geometry, and indeed in such a way that this is justified as a critical science. Difficulties arise only if, in the philosophical examination of the origin of this science, those processes that are truly present there and justify the science are not found; if the analysis has been badly instituted. Briefly we indicate here certain points concerning the nature of these processes.
Among them are certain ones that are not formal syllogisms, but virtual only; just as virtual judgments, so also virtual syllogisms without doubt occur in the mode of procedure of the human mind.
A virtual judgment is present where the mind, having intuited a necessary connection between S and M, does not explicitly affirm it: “S is M,” but proceeding to further things, intuits the necessity of the connection between M and P; from both intuitions, gathered into one, there results the cognition of the necessity of the (indirect) connection between S and P; and this can immediately be affirmed. In this movement of the mind two virtual judgments are present, while only the third connection is formally affirmed. This process constitutes a virtual syllogism, whose existence, simultaneously with the existence of virtual judgments, is thus detected. The affirmation that follows after such a syllogism is an actual judgment. We can certainly then by reflecting construct the syllogism explicitly, for all its materials are present, yet what is contained in the natural movement of the mind is an operation that is closest to the operation that brings forth a first principle; with the syllogism it agrees only in this, that it contains a progressive movement of the mind. That this movement can be rather the inverse of the syllogistic movement, we already saw earlier (Chapter II, § 3, no. 3). For first we detect a specific truth (that two numbers can be added, and that from their addition a number results); the reason why these things are affirmed is detected to flow from a certain formal connection, which can be enunciated by a causal proposition, and from this a more universal, generic truth is understood. The syllogistic movement leads from a generic truth to a specific one. That the same things are verified in many processes that we found in examining geometric principles, the reader will easily be able to inspect.
Relative to these processes of the human mind, this can still be observed: quite often in order to find principles we must consult sensitive data, “inspect in the phantasm,” as St. Thomas says; no longer in the phantasm of words, as can be done in a syllogism, but in the phantasm of things. Yet with this sensitive intuition there is always, as we saw, associated an intellective intuition, which justifies both necessity and exactitude. Thus we detect: the divisibility of extension as it is extended, from the inspected nature of extension; the existence of indivisibles, of surface, line, and point, from understood continuity; its divisibility into infinity, from the same nature of continuity; the existence of congruent figures, and then of exact ones, from the understood nature of extension as it is the principle of individuation. These intellective intuitions, which are always conjoined simultaneously with sensitive intuition, in analysis are accurately detected and explicitly affirmed; yet in the natural process of the human mind, as is evident from the same analysis, they are present without being explicitly affirmed, and indeed as motives determining assent to the principles. The principles, therefore, that are thus affirmed as necessary and exact suppose both the inspection of the phantasm—and thus are altogether like the absolutely first principles—and abstraction, from which is had the intellective intuition of the very nature of extension. These processes, however, as is manifest, are not syllogistic; they are direct, intuitive-abstractive processes; they are of another genus than the syllogistic processes that constitute science strictly so called, the science of conclusions. It would take long to examine this in particulars; there a broad field of noetic investigation lies open. More in Chapter V.
If, therefore, this very natural mode of procedure of the mind is subjected to analysis, this no longer pertains to geometric science, but rather to the theory of cognition, to philosophy. This noetic science, however, can be reduced to metaphysics; for in all these cases it is a question first of knowing the essential relations of the nature of extension in general, and of the nature of diverse extensions in species, and next of recognizing these relations and their foundations as beings, on account of their intelligibility. And thus this noetic analysis can be reduced to metaphysics; or, because it deals with extension, to natural philosophy. Certainly it is a philosophical discipline. And the analysis is an “operative analysis.”
In the preceding sections we have touched upon several points that seem to demand a fuller investigation; concerning these we shall treat in the following chapter, which will deal with certain axiomatic questions. One remark must already be added here. We have spoken of symbolic expressions of propositions and of the syllogism, which preserve the form of the consequence in such a way that we can read its necessity in the formulation itself. This possibility would not matter much for making a deduction if, as we did above, we employ only a purely “predicative” proposition, “S is P.” If, however, we use also symbolic expressions of relations (between two and especially between several subjects), an ample deduction becomes possible. Above (Chapter IV, § 4, no. 9) we found that diverse real relations have similar logical properties; the relations of quantitative equality, of qualitative similarity, and of similarity of direction have this in common, that they are transitive, symmetric, and reflexive. These relations seem to agree in logical properties, yet in reality they are diverse and known as diverse. In the theory of Euclid, the notion of direction is detected only through this logical process; in our method, in the inverse manner, we arrive from the notion of direction at the justification of the postulate. The study of these relations will be able to detect elements that are suitable for illustrating the doctrine of analogy. Other relations differ in the logical properties themselves. All these things, therefore, are worthy of deeper investigation.