CHAPTER VI¶
ON THE FUNDAMENTAL SUBJECT OF GEOMETRY
From the things we expounded above, various conclusions follow regarding the nature of this science which is geometry; and in these are present diverse elements that, as it seems to us, have importance for the general cognition of the activity of the human mind: a universal noetic importance, therefore. In sketching these we shall often have to return to what was treated above. We begin from a reiterated comparison of Euclidean and non-Euclidean geometry.
§ 1. Reiterated comparison of Euclidean and non-Euclidean geometry.¶
It has been said (cf. Gregorianum, 1951, pp. 432 and 451) that according to our exposition the “Euclidean structure” would be “more probable than any non-Euclidean one.” That does not aptly express our opinion. We contend that in extended entities, under certain conditions, a non-Euclidean structure can indeed be found, but that in those very extended entities the Euclidean structure is more fundamental, indeed uniquely fundamental, for this is the structure of extension as such.
Let us look again at certain matters concerning historical evolution. It is established: if (besides the other customary axioms) Euclid's Postulate V is admitted as a first proposition, Euclidean geometry follows, i.e., the extended entity in which this postulate holds has a Euclidean structure. If in place of Postulate V another, contradicting it, is assumed, the Euclidean structure is excluded: the postulate is required and suffices. Euclid (IV § 4 no. 9) applies Postulate V in only a single demonstration. Immediately he deduces a theorem that not only (in the manner described loc. cit.) contains the notion of the direction of a straight line, but moreover implies this truth: given a straight line, through any point of the extension (plane) a straight line can be drawn (which is now understood as a line of constant direction) that has the same (specifically identical) direction as the given straight line; in other words, in an extension that verifies Postulate V, around any point there is given (specifically) the same continuum of directions. And note moreover: Euclid (cf. IV § 4 no. 4) seems to employ for the definition of the straight line this constant direction, i.e., of its mode of passage (or site) through each of its points. For the rest, the very expression “straight line” seems to intimate this, and the only problem is the problem of its “existence.”
It is established, therefore: if in an extended entity Postulate V holds, it has the same continuum of directions at each of its points; its extension has this homogeneous structure relative to directions; it has “isotropy,” as is often said. Conversely, from our noetic investigation the converse proposition results: if an extended entity has the same continuum of directions at each of its points, in it Postulate V holds, and consequently Euclidean geometry. Hence we must say: such a structure is required and suffices for Euclidean geometry to hold, not only for congruence and consequently for the free mobility of a rigid body, but also for properties regarding direction. If, therefore, in a certain extension Postulate V does not hold, but in its place any non-Euclidean postulate whatever, in that extension there is lacking homogeneity relative to the continuum of directions through individual points. (Structures more differing from Euclidean structure can also be considered, namely extended entities that do not have constant Gaussian curvature, in which neither congruence in general is possible, nor consequently the motion of a rigid body).
Now, however, this structure of extension, whether Euclidean or non-Euclidean, is the fountain whence flow the diverse properties deduced either from Postulate V or from postulates assumed in its place. In the classical method and in the axiomatic method, these postulates are assumed as previously known to us, or at least as principles, so that deduction may be made from them. But they are not ontologically prior, since they themselves flow from that diverse structure, or nature, of the extension in which the deduced geometries hold. These structures, therefore, are fundamental. In order that the comparative value of diverse geometries may be recognized, we must therefore attend above all to this fundamental structure of the extended entity relative to directions.
And this can be done; indeed, it ought to be done. For this structure is accessible to us through itself, as was demonstrated in Chapter IV.
The notion of the direction of a line relative to the surrounding “space” belongs to the primitive notions, which are not capable of technical definition, but neither do they need it, because they are intelligible through themselves. And for direction this holds both for sensible lines and (if in them the first exactitude has already been detected) for exact, i.e., unidimensional lines; and then they can be lines either sensibly straight or even sensibly curved. This notion of direction relative to the surrounding medium, i.e., the notion of the mode of passage through a determinate point situated on the line, is a notion that holds for those lines of which we were speaking, whether sensibly straight or curved, and is truly intelligible, as is evident from the diverse first necessary judgments that we have intuitively concerning them. Let us recall certain ones: There are given modes of passage of lines through a common point that are different, even if the lines are congruent with one another; this difference is not like the difference between quantities (cf. IV § 2); for e.g. the difference between two lengths is a length, whereas the difference between two directions is not a direction, just as the difference between two colors is not a color. The notion of direction is qualitative, like the notion of color. It is, to be sure, a quality in quantity; hence the difference between the directions of these lines causes something quantitative: if they are situated in the same surface, they determine a greater or lesser part of this surface as boundaries. Thus the difference between two directions can be known to increase or decrease; and there we discover that marvelous fact, that this difference cannot increase into infinity (cf. IV § 4 no. 2). And from these intuitions, after establishing the existence of an exactly straight line, we can deduce the notion of an angle and of its measure.
This notion of direction truly belongs to the primitive notions and is truly understood by us, as is evident from these judgments, which are themselves primitive and necessary judgments. Just as in the “operative” definition of the notion of extension itself, which was justified above (II § 2 no. 3; V § 4 C note), so in this case of the notion of direction, i.e., of passage through the same point, we have an excellent such definition for this notion of direction, a clear intellection of it, which manifests itself in those first, necessary judgments.
This can then be applied (IV § 4 no. 4) to comparing the directions either of a single line or of diverse lines through diverse points, whence the notions of a sensibly straight line and a sensibly curved line can be defined as a function of the notion of direction. Concerning the notion itself of these lines, it is then established, not yet, however, concerning their existence, i.e., concerning the existence of specific straight and curved lines. In order that existence may be established, we need again the application of the principle of individuation, as was elaborated there in § 6.
Now we attend only to this: Of itself, extension has the same continuum of directions at each of its points; its points differ from one another “by site alone.” But through the introduction of a “constant curvature,” whether positive or negative, this homogeneity relative to continua of directions can be removed; and thus the Euclidean structure of this extension is removed. This “measure of curvature” can be slight, or can continuously increase, and thus arise structures that are the objects of diverse non-Euclidean geometries, which recede ever more and more from the initial Euclidean structure. Indeed, this curvature can be variable in a single extended entity; then there is removed not only homogeneity relative to directions, isotropy, but also relative to the general possibility of congruence of figures, and consequently to the motion of a rigid body (concerning these, IV § 5). Hence we say: in individual geometries that which is fundamental is the structure of the extension whose geometry it is; but among these structures themselves most, indeed uniquely fundamental is that whose “measure of curvature” is zero, the Euclidean, undeformed structure.
And it must be added, because this is sometimes denied: curvature that is not zero is curvature in the intuitive sense, because it arises from the diversity of the continuum of directions at individual points, and these directions have an intuitive and intuitively justified meaning. And as we saw at the beginning of this section: where in an extension there is present specific identity of the continuum of directions at individual points, there is present Euclidean structure; and conversely, where non-Euclidean structure is present, there this identity in continua of intuitive directions is lacking; there curvature in the intuitive sense is present.
And so we see: the fundamental subject of geometries is constituted not by possible figures whose properties are expressed by first principles, but by structures of extension, from which those very principles flow; and the Euclidean, undeformed structure of extension is fundamental relative to other structures. But the structure is fully known from the figures that are known through deduction to be possible in such an extension.
Note. Here inquiry would still have to be made into those cases in which “Gaussian curvature” does not coincide with curvature in the intuitive sense. Thus, e.g., an ordinary cylindrical or conical surface has Gaussian curvature whose value is zero; yet it has intuitive curvature, but such that such figures can be “applied” to a plane, on account of this value of Gaussian curvature. And similar corrections would have to be added to the consideration of the motion of a rigid body. Similar things must be said of figures on a “pseudosphere” whose motions are considered by Helmholtz.
§ 2. On extension as the fundamental subject of geometry.¶
A. On the extension of bodies.¶
We had arrived at this conclusion: “extension” and its structure is the fundamental subject of individual geometries, not figures traced in it. And the most fundamental subject, in a well-determined sense, is extension having a Euclidean structure, the extension that we called isomorphic, or homogeneous, or “amorphous,” which sometimes, if directions alone are treated, is called isotropic.
This position we must elucidate further on account of certain opinions that are often proposed, and this we shall do from these very opinions. For they say that that object is not the extension of bodies, but the extension of “space,” or rather is “space” itself. By this term is designated a certain receptacle of bodies, i.e., that in which bodies are placed, mutually touch or are distant from each other, and move. In that opinion, geometry would be a system of notions and propositions directly regarding this space; and thus this would be the fundamental subject of geometry. This opinion, however, cannot be sustained if we consider, as we already did in the preceding chapters, the mode by which the human mind arrives at geometric notions and principles, including the notion of extension itself.
That opinion is strongly championed by Poincaré where, in a very exaggerated sense, he maintains that geometry is not an experimental science; which negation of course also has a legitimate meaning. (Cf. above II § 3 nos. 1–3; § 4 nos. 2, 3; V § 5).
He says (La science et l'hypothèse ch. V no. 2 p. 92): “let us realize a material circle, measure its radius and circumference, and attempt to see whether the ratio of these two lengths is equal to the number \(\pi\); what have we done? We have made an experiment, not on the properties of space, but on the properties of the matter from which we made this round thing, and of that of which the meter with which we measure consists” [1]. And so he continues through many pages.
From all that was said above, it is clear that it is not such an experiment from which geometry has its origin. Indeed geometry, where it is applied, serves to avoid such experiments. True indeed is what Poincaré asserts, that this experiment is made not concerning space, but concerning the properties of bodies; but so much the more must we urge that we are dealing here not with any property whatever, but with a single one, which we subsequently call “extension”; and it must be strongly urged that we are dealing here with nexuses between extended entities that are known by us as necessary; in this the experiment does not fail, though it does in the exactitude of relations.
These things are then developed further by Poincaré, and in the end he arrives at the question that is of greater, indeed of decisive moment. He says (no. 7 p. 101): “perhaps it is said: if experiments bear upon bodies, they bear at least upon the geometric properties of bodies” (“direz-vous que, si les expériences portent sur les corps, elles portent du moins sur les propriétés géométriques des corps ?”). In reality he thus touches at long last the problem itself. But its solution he does not seem to touch. For he continues: “and first, what is intended by the words ‘geometric property’ of bodies? I suppose we are dealing here with the relations of bodies to space, etc.” (“je suppose qu’il s’agit des rapports des corps avec l’espace etc.”). And this, although declared as a sufficient answer, is then developed further.
What we have cited suffices, after all that we saw above (see especially II §§ 3, 4; V § 5 A), for us to detect the true solution. If someone asks: what is the definition of the term “geometric property of bodies,” the answer is already found from the very beginning of our exposition, where the problem of necessity was treated. The initial experiment is this: in a body we can trace a figure (which indicates a division of the body). What is then affirmed as a property of bodies, according to Descartes's formula, is: “a body is figurable.” But with this experiment there is associated an intellective intuition, which sees this relation to be necessary and therefore universal. Indeed, this intuition proceeds from the fact that we simultaneously understand that the property of “figurability” flows uniquely from a determinate property of the body, uniquely, i.e., from no other (such as color, etc.), and this property we subsequently call by the name “extension.” We saw rather often that this is a genuine “operative definition” of extension. And this is more evident from a similar indeed but contrasting case, where in similar fashion we affirm a certain property: “this body is electrifiable through friction.” But this property of this body becomes known to us only as a purely experimental fact; we do not see it to be necessary; and similarly (or on account of this reason) from what property of the body “electrifiability” flows completely escapes us from this experiment alone (this property seems first to have been discovered in the body called in Greek ἤλεκτρον; hence the name). Both affirmations are had from experiment, but in one case, the “geometric case,” with observation there is associated an abstractive intellective intuition, in the other, in the “physical case,” this is lacking. Hence the immense noetic diversity between the two experiments. If someone neglects this datum of our consciousness, and this occurs in Poincaré (and so many others!), errors will infallibly follow.
More even we discovered subsequently, namely those things regarding the problem of exactitude; how this is solved in geometric cases alone, not in physical cases, was also expounded at length above. There it also became clear that in geometric cases themselves the direct transition from inexact sensitive data to establishing the exactitude of relations is not always, indeed is rather rarely, legitimate. This can be done directly in few cases; above we saw legitimacy in establishing the indivisibility of boundaries of extended entities, insofar as they lie between the extended parts of which they are boundaries; next, in cases depending upon the intellection of extension as it is the principle of individuation; which are the cases of the congruence of figures and the cases of the existence of certain exact figures. (We did not treat of certain others, specifically of the comparison of lengths of lines of diverse curvature).
Once these first principles of the exactitude of figures are acquired, they can be applied in evaluating other sensitive data. Just as in axiomatics the supposed exactitude of the axioms passes through reasoning itself into the conclusions, so this justified exactitude of principles passes into the cases to which they can be applied. In the example that Poincaré poses, concerning the determination of the number \(\pi\) from the comparison of lengths of radius and circumference, we certainly cannot read this value for this determinate case, still less for the universal case of the proportion between the lengths of the circumference and its radius, whether in Euclidean or non-Euclidean geometry. In this sense Poincaré says excellently that geometry is not a science derived from experience; but he wrongly omits the consideration of the twofold problem, of necessity and exactitude, by which “geometric cases” are distinguished from “physical cases”; and he omits the consideration of cases in which these problems can be solved directly.
If these cases are duly considered, we find a clear answer to his question: what do you intend by “geometric properties of bodies.” They are not their “relations to space,” but are relations detected in individual bodies themselves, and indeed as necessary and absolutely exact relations, notwithstanding the imperfection of the data of the senses, which can only report facts and inexact connections. And these relations, necessary and exact, are so detected that from them, through intellective intuition, we have the definition, i.e., the strict determination, of extension. For these properties of bodies themselves, which are rightly affirmed in first propositions, are properties of that which in bodies we call their extension. And this extension so defined is the fundamental subject of the science that develops these first propositions by deduction, geometry — hence these properties of bodies are called their geometric properties — and, if the strictly fundamental subject is sought, this extension reveals itself as of itself homogeneous, isotropic, Euclidean. And these geometric relations become known antecedently to the notion of space and independently of it.
B) On space.¶
Not every use of the term “space,” nor every use of the notions corresponding to this term (for it is not a single notion), is to be despised or carped at. Poincaré supposed (and very many seem to do so with him) that geometric properties are “relations of bodies with space,” “des relations des corps avec l’espace”; and this was proposed by Poincaré as the definition of a “geometric property.” Geometric properties seem to have to be drawn and justified from properties of space. But in truth the matter stands in the inverse way.
How is this notion of “space in which bodies are placed” introduced, and how consequently is it to be interpreted? It is a philosophical question, indeed a question that in philosophy, both systematically and historically, is and was of the highest moment [2].
From what is expounded in these books, it is clear: 1) in reality a body is prior by nature to the place in which it is, prior also to its motion according to place; but also 2) before every notion of place, and a fortiori before every notion of “space in which bodies are placed,” there is present and understood in the human mind the notion of the extension of bodies. (According to Aristotle, man from the cognition of motion arrives at the notion of place, Phys. IV 1, 4). Indeed, this notion of extension belongs to the first things understood by the human mind; indeed according to St. Thomas it is (with the first arithmetic principles) the very first thing that is fully understood by man. For the knowledge of them is according to him the origin of all human science, not only of mathematical science (cf. Théorie du Jugement, Introduction, ch. V § 2, § 5, in 2nd ed. pp. 1, 164, 188). And this notion not only — this was abundantly expounded above — leads, after the understood “figurability” of bodies flowing from their extension, to the principles of geometry, but moreover, after another necessary relation is understood, as will immediately be said, to the principles regarding local relations. Extension as it is the origin, the fundamental subject, of geometry is often called “space”; this use is rather an abuse (cf. above II § 2 no. 2, IV § 2, against Russell, cf. also Gregorianum, 1951, p. 441; Filosof. della nat. inorg. pp. 131 ff.). Concerning this use of the word “space” there is no need for us to say more; its meaning is the same as the meaning of the word “extension.”
There are, however, other properties of the extension of bodies that lead to notions and principles regarding the “collocation of bodies”; and these principles and notions will be able to lead to the notion of “space in which bodies are placed and move”; which notion nevertheless must be cautiously handled.
The first among these properties was already briefly indicated above (V § 4 no. 2 p. 176). From daily experience it is very well known to us as a fact: two bodies can touch each other, coincide according to surfaces, and then they are “present” to each other; they can touch a third intermediate body, or a series of such bodies, and then a “distance” results between these extreme bodies. And thus from mutual contacts a system of bodies, indeed the entire universe, can be, and is, composed according to its “topography.”
In this consideration one thing above all is clear: this possibility of contacts and of what results is not only known to us as a fact, but is truly understood; the possibility of contact is understood to flow from the extension of bodies, and from it alone (not e.g. from color), and from it necessarily; in the same way as figurability is intuitively understood to flow from extension. Now let us attend to a few things; one is that we understand it to be the same property, extension itself, from which figurability and from which, let us say, “locability” flows; here we see that from a single subject two distinct proper attributes can flow, which are then affirmed by principles independent of each other; this seems to have noetic weight. The other is: in the division of extension, we first understand that a figure results, and then we can inquire into the properties of this figure; and we find them. Thus we understand what results from contacts of bodies. And this manifests itself as a greater extended system, whose extension is understood to have a likeness to the extension of a single body, and is subject to the same “geometric laws.” (So great is the likeness that from the confusion of both objects the celebrated philosophical question of the “continuous” and the “contiguous” arose. Concerning this question cf. Cosmologia Bk. I ch. I art. 1, in 4th ed. pp. 7–40; Filos. d. nat. inorg. ch. II § 1 pp. 57–72). On account of this likeness, therefore, it is understood without any question that the same geometry holds of this composite extended entity as of the extension of a single body of the same magnitude. If the extended entities that thus touch each other have a Euclidean structure, the system that arises is understood to be able to be increased into infinity.
Contact between bodies, which we were considering so far, was supposed to be fixed contact; but there is understood, again intuitively, the possibility of flowing contact, from which flows the motion of bodies (necessarily relative to each other), and this intuition is the origin of the science of kinematics, which is not purely geometric (because it inquires into the efficient cause of motion and into its duration), yet which is very closely related to geometry. One problem we name here that pertains to the meaning of the word “space.” For the question arose, already from antiquity, concerning the possibility of motion in a plenum; by some that possibility is denied, but systems have also been constructed that explain the possibility of motion in a plenum (See systems historically proposed in Filos. d. nat. inorg. pp. 137 ff.).
Such an integral extended system, which can be called the “topography of the universe,” is sometimes called “space,” and without doubt geometry holds of it, but because it is extended, it holds of its extension. Geometric properties are not defined through relations of bodies to this space, but vice versa, to this space are rightly attributed geometric properties of extended bodies, on account of the extension resulting from contacts. And concerning the structure of this space, questions can meaningfully be posed.
The word “space,” however, is used par excellence in another sense, namely as a certain receptacle of bodies, which is prerequired that they may exist, be placed, and move; and that in such a way that this space is an extended entity that of itself necessarily exists, and consequently, even if all bodies are supposed destroyed, necessarily remains; just as in our imagination bodies placed there can disappear, but not the imagined space in which they were placed. This “space” is recognized as something unreal, as a “being of reason” (“ens rationis”), as the Scholastics say, “ein Unding,” as Kant used to say. It is clear that the extension of a real body and its “geometric properties” cannot be derived, neither in reality nor in our cognition, from this “space.” The relations, insofar as they are present, are inverse, i.e., to this “space” are attributed properties that belong to a receptacle of bodies, so that by means of these the real properties, of localization and motion, may be aptly described. Hence this “space” is aptly called by Scholastics an “ens rationis” with the addition “cum fundamento in re,” because it can serve for such descriptions of real relations. But it is clear that this unreal space cannot serve either to constitute or to define “geometric properties,” as Poincaré established. (Concerning these, see more in the works named above in the chapters indicated there).
The intuitions that we describe briefly in this section (and elsewhere at length and breadth) as the origin of our cognition of relations of collocation and local motion belong to the ancient doctrine of St. Thomas; and this seems to be the only one justified in the analysis of those things that we, even today after so great an evolution of the sciences, detect in our mind. But after St. Thomas it seems gradually given over to oblivion, although these intuitions direct the ordinary activity of the mind; given over to oblivion not only among very many Scholastics, but also, and in general, among modern philosophers and physicists.
Yet Einstein discovered them anew in his meditations on this subject. According to Einstein too, the first thing known in this subject, antecedently to the notion of space, is the extended body itself; after the notion of the body is established, in experience we necessarily find local relations (Lagen-Beziehungen) between them. And these relations are relations of mutual contact. And what we describe (deuten) as spatial relations between bodies is nothing other than this; first is the concept of the body, then comes the concept of spatial relations, and in the third place only the concept of space. And this is precisely what was established above regarding this subject [3].
And thus we return perfectly to the doctrine of St. Thomas; even in this subject akin to geometry, the fundamental subject is the extension of bodies, intuitively intelligible to us from experimental data, from the phantasm.
Hence it seems manifest that the use of the word “space” both for “extension,” the fundamental subject of geometry, and for the “prerequisite receptacle in which bodies must be placed and move” is to be carped at (cf. above II § 2 no. 2); it is rather an abuse, capable of leading into errors. Let us insist, therefore, on this: the fundamental subject of geometry is: the extension of bodies.
C) On the relative independence of the notion of extension from experience.¶
The origin of geometry is therefore empirical; it depends upon experience, i.e., upon sensitive data. But for geometry (and for arithmetic, etc.) it is characteristic that it is not, as holds for a “physical case,” purely experimental in its principles; for there is present also a certain independence from sensible experience; and this is of the highest noetic moment, both in constructing geometry itself, and in applying it, and above all relative to general noetics. For — and this is precisely the difference from “physical cases,” as has often been said — with this experience there is associated the intuition of the necessity of that nexus given by experience. These things, if considered, demonstrate also a certain independence, of the highest moment, from sensitive data; they demonstrate the autonomy of purely intellective intuition itself, both as to the necessity and as to the exactitude of the relations affirmed. Relative to exactitude, this independence or autonomy is had directly in certain cases only, but through deduction is then derived to others. These things are worthy of special consideration. (Concerning these cf. Théorie du Jugement ch. VI especially § 5).
This manifests itself even in pre-scientific reflection. Let us compare the first geometric affirmation with a similar one in a physical case. Observing the figure that we cause in a body, we affirm “the body is figurable”; observing the attraction of small particles by a glass rod after its friction, we affirm “the body is electrifiable.” To someone asking why we affirm this, in the second case we answer: “because I see this” (and solely because I see this, and this motive must be examined more accurately). In the first case we answer: “because I understand this.” We can advance further. In the first case we can add: “the body is figurable, because extended”; in the second case we can add nothing of the kind; to someone asking “why is it electrifiable” an answer cannot immediately be given. And therefore in the first case we know that we affirm a necessary connection; in the second case the connection is not known as necessary, it can be contingent. And because the senses report facts alone, not necessity, therefore it is already clear that in the geometric case, besides sensitive data, something must be present that is owed to the intellect alone. This diversity between the “physical case” and the “geometric case,” or between “geometric sensitive experience” and “physical sensitive experience,” we detect already in pre-scientific reflection.
In another way also this difference manifests itself in pre-scientific life. The required geometric experiment can already be validly performed in the imagination alone. If in the imagination we divide a body by a figure (a small line suffices), we can affirm: a body, because extended, is figurable, is necessarily figurable, therefore universally figurable. The experiment is purely mental. This no longer holds in the “physical case”: a (glass) rod in an external experiment through friction is electrifiable, i.e., it attracts small particles. Such an experiment can also easily be performed in the imagination alone; too easily. For we imagine very well a copper rod after friction attracting small particles. And if we then affirm “it is electrifiable through friction,” we err. In the physical case, in reality the only motive having value (which must be examined) is: because I see it.
And it must also be attended to: This noetic difference between the two cases does not arise from the fact that in the “geometric case” our imagination is necessitated, while in the other case not. For as was already indicated above (III § 2 no. 5 A), in certain geometric cases we find a necessity in the phantasm that is denied in the judgment of the mind. If the imagination proposes to us a line dividing a surface into two parts, this line in the phantasm has breadth, and that necessarily, for without breadth the imagination cannot present it. We affirm nevertheless in the intellect that the line, the boundary between two parts of a surface, lacks breadth, necessarily. The intellect surpasses sensitive data, not only relative to necessity (they report facts alone), but also relative to exactitude, according to which it corrects sensitive data. And the intellect attributes necessity to the exact relations it detects, not because the phantasm is necessitated, but because it detects this necessity in the object presented to it deficiently by the phantasm as to exactitude, notwithstanding the imagination.
In the intellect, therefore, there is present a certain autonomy relative to sensitive data, a certain independence from them. In “geometric cases,” in opposition to “physical cases,” the intellect enjoys this autonomy. What in those cases is the function of sensibility? Its sole function is: to present the object (the concrete relation of the extended body to the figure traced in it) to the intellect, which cannot think without a phantasm; this complex object is then assumed into a concept of the mind, also complex; and in the understood (conceived) object the mind detects what it then affirms, justifies, by saying “so it is.” There remains a certain dependence upon sensitive data, upon experiment, but solely in this sense, that experience supplies the matter concerning which the mind judges. If the same complex concept, the same disposition of the thing, were present in the mind without being drawn from sense, or without man being conscious of this origin, the judgment “so it is” would remain the same, and justified from the sole content of the concept, autonomous and in this case in every respect independent of sense. This we simply “ascertain” (“constatamus”) through simple reflection upon the activity of our mind.
From what we have discovered, we can decisively judge concerning diverse considerations proposed by various thinkers in this matter of the dependence of geometry upon experience.
Often, in investigating the problem of how and in what degree geometric truths depend upon sensitive data, upon the “phantasm,” empirical psychological inquiries are described concerning the mode in which human sensations arise and how sensible images of the same object in diverse circumstances can be diverse. (Poincaré also does this in the work cited above and often elsewhere). As is evident, in these investigations and discussions, notions regarding extension and following from extension (of dimensions, distances, and so forth) are repeatedly employed as already known. In these investigations great value seems also to be attributed to the mode in which, from the combination of actions of diverse senses, sight especially and touch, images arise. And thus much contingency and much inexactitude is detected in them, which, if geometry must have its origin from experience, seems in similar fashion to depress its value.
These things, although very useful and of great value for psychological science, seem to contribute absolutely nothing to our problem. Noetic investigation, which begins from that moment in which in a body its attribute which will subsequently be called “extension” reveals itself as intelligible in itself (not on account of its origin), leads to this conclusion: geometry does not depend upon sensitive, experimental data except as upon the matter concerning which the intellect judges by a fully autonomous right; if this matter, as it is in the intellect, in the concept, did not proceed from the senses, the principles of geometry would remain just as they are; for geometric judgments are justified from intellective intuition alone: 1) of the necessity of the connection in the conceived disposition of the thing, 2) of the exactitude of those relations recognized in concepts, notwithstanding its absence in sensitive images. And consequently this passes into conclusions, although in these deductions a continuous appeal to the phantasm, which as to exactitude must be corrected, is required. In no way is there required a cognition of the mode in which the senses operate, still less theories explaining these sensitive operations as to mode and value. Conversely, such an inquiry into the activity of the senses and its noetic value can only be instituted after the cognition of the nature and value of the operations of the intellect, to which they supply matter. (Concerning these cf. Théorie du Jugement Ch. IV § 5).
Investigation into the geometric activity of the intellect begins, as was said, not from the consideration of what is prerequired in the senses (although we know this precedes), but from that intellective representation in which in the represented body we detect the first necessary relation, the first intelligibility, namely the relation of dependence of “figurability” upon another property of the represented body, which we call “extension.” There we simultaneously detect “what extension is” and a certain necessary proper attribute of it: there the origin of geometry is had, in the intellect, and this relation to figurability is the first “geometric property.” Here the intellect indeed depends upon sensitive data, but only as upon the matter concerning which it acts, as a disposition of the thing that it judges by autonomous right from its intelligibility alone.
These things also become clearer from the comparison with “physical cases,” which are opposed to “geometric cases” precisely because in them that relative independence from sensitive data is lacking. This enormous noetic diversity between the two types of science is an immediate datum of our consciousness; if we judge concerning the origin of both cognitions, physical and mathematical, from sensitive experience in the same way, we sacrifice that most precious datum and simultaneously sacrifice the relative independence of geometric relations from sensitive data; which constitutes a grave error. In the application of geometry to physical extensions, exactitude of course cannot be saved; but even there a distinction of the greatest moment must be made.
It cannot be saved insofar as measurements of figures and distances, even exact in themselves, cannot be taken exactly by us, who depend upon the senses and upon physical laws (which can impede exactitude of measurement); we must fail in measuring what can be measured, and in itself can be perfectly measured. But in all cases where geometric laws themselves are treated, we know that these laws hold exactly, not only in themselves, for “abstract” extension, but also for real extension in the corporeal world. Hence in the construction of physical science, both experimental and theoretical, the influence of this noetic diversity between “geometric matters” and “physical matters” remains. In that construction we apply geometric (and arithmetic, etc.) laws and physical laws already discovered previously. We know that the former laws hold exactly, and need no correction, whether for large physical objects or for small or even minimal physical objects. For they are extended entities. Concerning a purely physical law, however, which regards other properties of bodies besides extension (and multitudes), which therefore becomes known to us from experience alone, without an associated intellective intuition [4], even if this law is expressed by an exact mathematical formula, we do not know whether it holds exactly of determinate objects, for there the intellect does not overcome the inexactitude of sensitive data. And thus it can happen that a law holding rather accurately for medium values needs major or minor corrections for extreme values, indeed in minimal things e.g. totally fails. The geometric law of the proportionality of a spherical surface to the square of its radius holds with perfect exactitude both for the greatest and for the smallest things; inexactitude there regards only our cognition of the objects to which it is applied. But a similar law of Newtonian attraction concerning a similar (inverse) proportionality of attractive force perhaps (indeed probably) is only approximately exact, indeed perhaps at minimal distances totally fails (cf. Cosmologia 4th ed. pp. 593 ff.); in this “physical case,” in such a supposition nothing absurd is admitted. To suppose, however, that a geometric law fails in minimal things is absurd; for concerning “extension,” the fundamental subject of geometry, it is established that it remains divisible into infinity into “extended entities” that have the properties of “extension”; in geometric matters we “understand,” we understand also exactitude; in “physical cases,” from sensitive data alone, we do not understand; that is precisely the profound diversity between the two “cases.” And thus there truly remains a noetic diversity between these two classes of cognitions arising from sensitive experience.
§ 3. On successive extension [5].¶
1. On the notion itself and its proper attributes.¶
The fundamental subject of geometry is the extension of bodies. Besides this, however, there is given also another extension, which can be called successive, fluent, dynamic extension; and it is the duration of the motion of bodies. The extension of bodies, as opposed to duration, is called permanent, fixed, static. Between these two extensions there is found a strict analogy; they have common properties, and hence geometric propositions are given that are immediately understood to hold for fluent extension. But there are also given great discrepancies between the two extensions. It will be helpful to transcribe here certain matters that are found partly more elaborated in the cited article.
In local motion a twofold extension is found (besides the extension of the mobile itself). One is that which is the extension of the trajectory, of the path that the mobile traverses; this is the extension of place, permanent, static, which is investigated in the theory of place. The other is the fluent extension of motion. The former is extrinsic to motion itself; the other is intrinsic to it, and is called its duration. A certain dependence of the second extension (fluent) upon the first (permanent) is given. According to the doctrine of Aristotle, that act which is motion draws its continuity and order of parts from “magnitude,” which is the path, the trajectory. Nevertheless, this dependence being presupposed, a certain independence must also be recognized; duration is not determined by the length of the trajectory alone; it is varied also, in an inverse manner, according to the velocity of motion. Indeed, if we are dealing with motion of alteration, qualitative and continuous, with the continuous intension and remission of a certain quality, there too an extension strictly so called, fluent extension, i.e., duration, must be recognized, although intensity is not quantity properly so called, extensive quantity.
In both cases duration is quantity strictly so called, extensive quantity, and has its proper attributes. Durations can be added to one another, and this addition is a quantitative operation strictly so called: from it a greater duration results, just as from the addition of a line to a line a greater line arises. Thus in oscillatory motion, the total motion, composed of many contiguous oscillations, has a duration that is the sum of the durations of the individual oscillations. And if each of these oscillations is congruent with one another (this is realized in the most modern oscillatory clocks with great, not of course with “infinite,” exactitude), the duration of the motion composed of n oscillations can be calculated by arithmetically multiplying by the number n the value of the duration of a single one. Thus in an inverse manner: the difference of two durations, even in alteration, is a duration; although the difference between two intensities is not an intensity. These are principles intelligible through themselves regarding duration, fluent extension.
The possibility of mutually congruent motions, which was presupposed above, is established from the fact that this fluent extension is understood to be the “principle of individuation” for “figures” possible in it, just as permanent extension is for geometric figures (cf. above IV, §§ 2, 3 and Gregor. 1951, 447–450. Cf. also our work Filosofia della natura inorganica, Brescia, 1949, p. 207).
This agreement between the extension of geometric lines and fluent extension proceeds from the fact that both are extension worthy of the name; hence they have common proper attributes.
There are given, however, discrepancies also between the two extensions.
The fundamental difference seems to be this: Duration is a purely existential extension; it is the extension of that fluent being (esse fluens) which is motion; which is fluent existence, does not have it. If we are dealing with local motion, even the static “where” (“ubi”) itself is already an existential act (cf. Cosmol. nos. 45, 47, 59 ad 3, 162, and Note VI), but a static one, to which motion corresponds as a fluent act. But the extension of place itself, from which the “where” is taken, is the permanent extension of a body, of a real coordinate system; this extension is a determination in the order of essence, of essence realized of course. This extension, as was already said, is extrinsic to motion; fluent extension is intrinsic to motion; it is duration, which in no way is found directly in any quiddity, but is purely existential. This difference between the two extensions seems to be the most fundamental. Let us see other discrepancies.
The extension of a body, known to us from data of the senses, is three-dimensional (see above III § 2 nos. 2–4). On the contrary, a duration that would have two or more dimensions lacks intelligibility in itself; it is not in reality, because it is unintelligible. Hence the uniqueness of dimension in duration stands in another way than in the geometric straight line. The straight line is a boundary in a plane, which is a boundary of a given three-dimensional extension. Hence an immense difference flows between the two unidimensional extensions. The geometric straight line in a plane can have diverse directions relative to other straight lines; a straight line passing through a determinate point, simultaneously with others, differs from them in its very mode of passage; it differs in direction in the plane. Concerning this direction we have several principles intelligible to us through themselves, immediately, intuitively, which belong to the foundations of geometry, and not without danger of errors are neglected (Cf. above IV § 4 nos. 2, 3, VI § 1, and Gregor. 1949, 448). Thus, e.g., from inspection of the phantasm we immediately understand: the diversity between directions through a single point (the modes of passage through this point) can be varied; the difference between the direction of a stable line and another variable line can increase, but not into infinity; after one “revolution,” the variable direction falls back into the first direction. Direction itself is not a quantum strictly so called, for the difference between two directions is not itself a direction; yet it causes something quantitative, namely an angle; around individual points there is present a continuous sheaf of lines, there is present what we call a “continuum of directions,” and indeed through itself specifically the same continuum at individual points. These things manifest the intelligibility of the notion of direction, although this notion, as primitive, cannot be defined.
All these things are lacking in that unidimensional dimension which is duration; two durations that at one of their points would make an “angle” with each other are an absurdity. Durations are not boundaries in a certain “duration” of several dimensions; duration is not an extension situated in a higher extension.
There is also another difference between the two unidimensional extensions; and this would be present even if our world were so unidimensional that its extension were not a boundary in a higher extension. And this difference is of the greatest moment for understanding the topological structure of duration. In this linear world two directions would still be present. There would no longer be present diverse directions of which we spoke above, namely diverse directions relative to a plane; but in the single line itself there are still present two opposite directions; if we call one “to the right,” the other will be “to the left”; thus the road situated between Athens and Thebes has a direction from the first city to the second, and vice versa; and motions really diverse according to opposite directions are possible. This opposition between two opposite directions is not present in duration. We can of course mentally return into the past, but in real duration such a real return is impossible; duration flows necessarily in a single direction alone, from past to future; and this belongs to first, intuitively intelligible truths. Motions can be reversed; their durations not so; the duration of circular motion itself is not circular, but “straight.” Duration is necessarily irreversible. And from these things, in the midst of analogy between the two extensions, other discrepancies flow.
Every quantity — even quantity of virtue — is measured by a “unit” that is “homogeneous” with the quantity to be measured. Aristotle (Met. X c. 1 1053 a 25) calls the unit of measure a “meter”; and rightly demands such a meter to be συγγενές [connatural] to the quantity to be measured; the measure must be in the same genus as the measured; it must be “homogeneous.” Next: the meter is by its nature that by which the quantity of a thing is known, ᾧ τὸ ποσὸν γιγνώσκεται (1052 b 20, St. Thomas lect. 2 no. 1938). We ask: what quantity is thus made known? Not the genus itself; cognition of this is presupposed; measure and measured must be of the same genus, homogeneous, and this must be foreknown [6]. The operation of measuring then reveals the species (the most special, ἄτομον, individual species) of the concrete quantity being measured. On account of the confusion sometimes present among moderns, this must be explicitly recalled. Indeed it will also have to be observed: this quantitative species is not effected by the operation of measuring, but is detected by it in the object where it was already present; our cognition of it, of course, is thus effected.
And it is expressed by means of numbers, i.e., by the number (integral or fractional) of units whose sum is equal to the quantity to be measured. We observe that this determination, insofar as it must be effected through observations of the senses and physical instruments, cannot be perfectly exact.
In considering the mode by which this operation of measuring is performed, there appears again a certain agreement with discrepancy between duration and the extension of a geometric line. Such a length (a line in a body or the distance between two points) is measured directly through the application of a measure (a meter or its multiple or part) to the line to be measured. This operation supposes the knowledge of a certain proper attribute of extension as such: two extended entities can be applied to each other through contact. It is again a certain proper attribute of extension immediately understood by us, and indeed independently of its first proper attribute, which is the divisibility or figurability of the extended; moreover we understand: that such contact may be had, the existence of two extended entities is required; moreover we know that the mere existence of two bodies does not suffice for this, does not per se imply contact; for this contact can be varied, can be had according to different parts (cf. above VI § 2 B, and moreover what we expounded relative to the contact of bodies in Cosmol. when treating of “ubi”). All these things belong to truths “intelligible through themselves.”
In similar fashion, two determinate durations can be “together” (totally or partially) or not, and in this second case they are “outside each other,” one after the other: and this is understood to flow as a proper attribute from the very nature of this fluent extension as such. In chronometry this “co-presence” (“simultas”) or “simultaneity,” as in the case of the line to be measured, is effected through the application of a determinate duration (which is realized in the motion of a clock, a chronometer) and known to us according to its quantity, to the duration of the motion to be measured. This is the likeness between these two cases.
But discriminations are present. This simultaneity of durations in measuring the duration of a certain motion is obtained from the fact that we initiate the motion of the chronometer (or note the beginning of its part) for the moment in which the motion to be measured begins; then both durations are understood to flow simultaneously and to be equal while they flow. And in similar fashion we act at the end of the motion to be measured; the two past durations were simultaneous and equal. But in measuring a line we must attend (and this we almost automatically understand and do) to ensuring that the lengths (of the measure and of the measured) truly touch each other, coincide, i.e., do not make an angle with each other, nor even differ by opposite direction. To this there is no need to attend in measuring duration; durations, by the very fact that they begin together, in the same moment, coincide, nor can they differ in direction. Add: the same is true if only in the middle of duration momentary coincidence is observed; then on the part of reality there is present coincidence of fluent extensions both in the past, already since the origin of at least one motion, and for the future. These things are true because a “deviation of durations” is impossible. Thus we understand two durations to “penetrate” each other, to be “together” (and not merely parallel); these intellective intuitions direct our operations without our instituting an explicit reflection concerning them; they are present in actu exercito; they belong to first truths intelligible through themselves.
On the part of reality, antecedently to our observation of which it is the object, the beginning alone of those existences called durations of motions (we add: and durations of rest of things that can move) determines their relations insofar as they are durations. All that begin together, whether of near bodies or of bodies distant from each other, penetrate each other, and this penetration ceases, and ceases only, when one or the other fluent existence ceases. If one such existence begins after another, from this moment (in the prior existence) they are together; these relations uniquely depend upon concrete existences.
These things are connected with another proper attribute in which duration differs from a geometric line, fluent existence from static existing extension. It was already said above what flows from the fundamental difference: local contact between existing bodies can be varied; it is not determined by the mere existence of those static extended entities; given their existences, a further determination is required, namely that contact itself, and this can be varied; hence the possibility of local motion, motion in the category “ubi.” On the contrary, one duration cannot touch one part of another duration, and then, this contact being broken, touch another part; “what is done cannot be undone” (“quod factum est infectum fieri nequit”). In the case of durations, contacts are irrevocably determined by the beginnings alone of each duration; no further determination is required, indeed none is possible; the unique determination is the concrete existence of both. The extension of a body — hence the body itself — can be changed with respect to order to others; first there is present order A, B, C, then A, C, B; such a thing is impossible in the order of durations. “There is no motion in the category ‘quando,’” say the Aristotelians.
The notion of simultaneity is a primitive notion. Hence it cannot be defined by a technical definition; but certainly through the operative method. For it is perfectly understood, like the notion of extension itself, from concrete empirical data (in concretion with many other attributes of bodies and their motions) through intuitive formal abstraction. In the same way arithmetic is understood; in the same way the notion of the extension of bodies itself is understood; this extension too is a primitive notion, it cannot be defined by means of other notions, yet in the midst of those empirical data it is perfectly understood by us, and as understood has perfect “meaning”; what other meaning could there be of the term “meaning” itself? Such an object retains its meaning, even if its specific quantity cannot be perfectly measured, on account of the defect of our observation. And then other proper attributes of this notion of “extension” (besides divisibility, addition, and so forth) we immediately understand by a similar intuition, among which are “exact” proper attributes, and thus other primitive notions arise (cf. for the notion of “direction” what was said above). So stands also the notion of simultaneity.
2. Conclusions.¶
In concluding, we propose certain matters having noetic weight, even in universal noetics.
A) Exactitude in chronometric relations.¶
In measuring duration, the same problems of exactitude return as in measuring static extension. Under due conditions we can understand that there are given durations perfectly equal, or having exactly determined proportions, and that there is given also the simultaneity of events with perfect exactitude, even if we cannot attain this in the execution of measurement. (Cf. Cosmologia 4th ed. pp. 257–259; 532–541; Filos. d. nat. inorg. pp. 214–222; Acta 2nd Thomistic Congr. Rome 1937 pp. 367–374).
These things have value above all in evaluating the neo-positivistic position, even in its latest evolution. Those exactitudes are fully understood that can be defined by an operative definition, which simultaneously with exactitude contains necessity, and therefore permit experiments that are “mentally,” “conceptually,” “in principle” possible, which regard those things that, as we can say, can be measured, although we cannot measure them.
We indicate one experiment of this kind, which as to exactitude is only mentally possible, and which is fundamental in this subject. Measurements of duration are made by means of oscillatory clocks, whose individual oscillations are supposed to have the same duration. As was already said, this cannot be realized with perfect exactitude. Yet this measurement, even in neo-positivism, in the operative method, even in theories, they employ without noetic scruple. And rightly indeed. For such instruments, absolutely exact, are “conceptually,” “in principle” possible; for as was already said above, they are truly understood, even in their exactitude. But what in this consideration is supposed, and in neo-positivism is often tacitly supposed? The individual oscillations themselves, of themselves in perfectly equal circumstances (and these of themselves are possible), are supposed to have perfectly the same duration. There is supposed what in the cited article (p. 21) we called: “the principle of specific indifference of chronology” (and “of topography”).
B) A particular specimen of analogy.¶
The notions of the extension of bodies and of successive duration are therefore analogous to each other, as is abundantly clear from what was said above. But it seems to be an analogy sui generis, concerning which, so far as I know, Scholastics are not accustomed to treat. Ordinarily, if two notions are analogous, only one is the proper concept of the thing whose notion it is; the other is a concept of the thing it represents, not proper but analogous only.
Here another case is present. Of both extensions, both static and dynamic, we have a proper concept, and these concepts do not perfectly agree with each other, but partially only, although each concept is a primitive concept; they agree, for they have common proper attributes, and simultaneously these two primitive concepts differ from each other, for they have also discrepant proper attributes; and each is a proper concept of its object. And the fundamental diversity seems to be this: one is a concept of a quidditative object, in the essential order, whereas the other, the concept of fluent extension, is a concept (a concept in the true sense of the name!) in the order of existence.
Moreover, a similar case of analogy is already found in permanent extension itself. Let us recall what we were saying above (III § 2 no. 3) concerning the origin of our notion of “dimensions” of extended entities, from the process of division of the extension given to us, namely of a body. This extension is divided by an extended surface, which in its turn is divided by an extended line, whose division leads to an unextended point, and thus to the end of this process. This is the origin of the notions of three-dimensional, two-dimensional, unidimensional extension; and simultaneously of the notion of “dimension” itself. These notions, each and every one, are truly primitive, perfectly determined by this process as by an operative definition, through which they are truly understood (have meaning), as is evident from the first principles we affirm concerning them. And they are understood to agree in certain proper attributes affirmed by those first judgments, such as “the extended is divisible through figure”; but in this very thing they are also understood to differ from one another in those three cases (in the case of a line, the “figure” is a simple point). Concerning each of these objects we have proper concepts, which belong to first intelligibles, which simultaneously agree and differ; they are therefore analogous; and each object is extended, but each extended in its own way.
C) On the chronotope.¶
Greater is the discrepancy between the notions of permanent extension on the one hand, and the notion of fluent extension, duration, of which we also have a proper concept, on the other hand; among themselves the three prior concepts agree more than each with the concept of duration. It is not to be wondered at: permanent extension, of bodies and of boundaries of a body, is a determination in the quidditative order, of essence; fluent extension, on the contrary, is a determination, an act, in the pure order of existence; the former is understood as something quidditative (which of course can exist), the latter is understood as something (we must use this word) merely existential, like permanent “ipsum esse.” And in this understood reality we can detect something particular, not yet examined by philosophers: an existential act, a fluent becoming (like the “being at rest” of something that can move) can have a “mode” of being, or rather we should say: “is a certain mode of being,” which is analogous to a certain determination in the quidditative order, namely to continuous quantity. Here is to be applied the theory of St. Thomas, which holds that the diversity of “modes of being” proceeds from quidditative diversity, through which “esse” itself is determined by something else, not as potency by act, but as act by potency [7].
What is said of extension itself holds of its dimensions. The single dimension of duration is a determination in the existential order; the dimensions of the extension of a body, even the single dimension of a line, are in the quidditative order; hence the great diversity between duration and line expounded above. The concept of dimension has an intuitive origin in the process of division of extensions according to the operative method. Dimension as verified in the three concepts of the extension of bodies seems to be a univocal notion; but dimension said of these (even of linear extension) and of duration is known through concepts analogous to each other. Concerning “dimension” holds what we were saying concerning “extension”; in a body it is a notion in the quidditative order, in duration in the existential order.
Hence we can evaluate the attempts (from the early period of the theory of relativity, not yet abandoned by all) to constitute a real four-dimensional extended entity (universe), of which one would be “time,” as a dimension entirely to be equated with the three others; according to the formula first enunciated by Minkowski. If this construction is described as something real, a metaphysical monster is posited. If it is posited merely as a means of calculation, as a “graphical representation,” on account of the established analogy it will be able to be admitted; we can use “beings of reason” with a foundation in reality (“entia rationis cum fundamento in re”) to describe, and also to calculate, real events. Concerning these cf. Filos. d. nat. inorg. pp. 221 ff. and Cosmolog. 4th ed. p. 259.