CHAPTER VII¶
ON EXTENSION — AS INTELLIGIBLE MATTER
§ 1. On the Construction of Geometric Figures¶
We were saying, therefore, that the fundamental subject of geometry is extension. This is not infrequently expressed by St. Thomas in this way: the “subject genus” (genus subiectum) or even simply the “subject” of geometry is magnitude (magnitudo; In I Anal. Post., lect. 15). Where there is discourse concerning such a subject of a science, there is often also added: the subject’s “proper passions or per se accidents” (propriae passiones vel per se accidentia). We added the word “fundamental” so that we might distinguish this from what may be called the secondary subjects of the science. For figures as well, which are traced in extension as boundaries in the division of an extended entity, are called (and rightly indeed) subjects of geometry; and their “proper passions” are likewise investigated in geometry.
The figures themselves, however, are primarily “passions” [proper attributes] of extension as such. And first indeed, from the division of the extended entity originally given to us (which is three-dimensional, as was expounded above in Chapter III, § 2), there arise through a successive process first surfaces, then lines, then points. And (with the exception of points) these figures, whether superficial or linear, can be distinguished into diverse species into infinity; and the “proper passions” of individual species are investigated. Figures, therefore, are on the one hand “passions” of their subject, originally of extension itself; but then on the other hand they are themselves subjects with their own proper passions: subjects of the science of geometry.
These things are excellently explained in the doctrine that St. Thomas delivers on this matter (In I Anal. Post., lect. 2, no. 5). In sciences “which are concerned with certain accidents, nothing prevents that which is assumed as a subject with respect to a certain passion from being assumed as a passion with respect to an anterior subject.” In this process, however, an infinite regress is not given: “for one must arrive at something first in that science which is so assumed as a subject that in no way is it a passion.” And our case is posed as an example: “as in the mathematical sciences, which are concerned with continuous or discrete quantity.” Continuous quantity is that which above we were calling extension, the fundamental object or subject of geometry [1]. And this distinction is subsequently clearly set forth.
Figures, therefore, which are deservedly assumed as subjects of their own proper passions and “per se accidents,” arise from extension, which contains them all in potency, but in potency alone. They must arise from it in act through construction. This must already be said where there is discourse concerning the first figures: concerning surfaces, lines, points, which are extended entities of two dimensions, or of one alone, or of none; they are made in or out of a given solid through any division of it whatever, which begets surfaces, whose divisions lead to lines, whose divisions lead to points as boundaries. These divisions are already constructions. But the actuation of a figure in extension is called by excellence “construction” if determinate exact, or “perfect,” figures are treated. In these cases, the construction of the figure in extension, as the matter out of which (materia ex qua), is the demonstration of their existence. Thus it is evident that such a figure, which is defined, in reality exists as a subject, that is, can be made through the indicated operation, first mathematically, and consequently (because to extension it belongs to be in the nature of things) also really; which does not imply that we are able to realize it in physical reality as well.
That among the Greeks construction was the demonstration of the mathematical existence of those subjects that are constructed according to demonstration is commonly admitted, unless we are mistaken, following the demonstration of this fact given by Zeuthen. The doctrine that holds that “to be able to be constructed” is the same as “to exist mathematically” is the doctrine of the Greeks for the geometric subject.
Yet this same doctrine belongs also to the patrimony of Scholastic noetics, although in modern theories of knowledge it is no longer remembered. Let us hear St. Thomas (in the same lecture): In the mathematical sciences, “those things which are first in the genus of quantity, such as unity and line and surface and other things of this kind,” are “supposed.” And he continues: “which being supposed, through demonstration certain other things are sought, such as the equilateral triangle, the square in geometric matters, and other things of this kind. Which demonstrations indeed are called quasi-operative, such as that: upon a given straight line to construct an equilateral triangle. Which being found, again concerning it certain passions are proved.”
We note: 1) The two exact figures, subjects of their own proper passions, which are named, the equilateral triangle and the square, are the most familiar specimens of figures whose “existence” must be proved, and is proved by Euclid (I, 1 and 46) through demonstrative construction. 2) These belong to “certain other things” that are not supposed, but are demonstrated in geometry itself through construction. 3) Such a demonstration is called “quasi-operative” (quasi operativa); where there already seems to occur the very term “operative,” which, after Bridgman, is so greatly urged in the “operational method”; and not undeservedly, provided it be done under due precautions, as was already said above (Chapter V, § 4, Note). Whether the term “construction,” which is employed (In Boeth. de Trin., q. 5, a. 1, ad 3) to indicate a specimen of “work” in the arts (and also in geometry), is to be understood strictly in this geometric sense is not established for us.
St. Thomas perfectly recognizes the “constructive character” of geometry with respect to subjects that are not merely defined, but whose “existence” must be demonstrated. And this is that which is demanded by Aristotle for subjects, which belongs to the too-neglected patrimony of Scholastic noetics in the Middle Ages.
Above (Chapter III, § 2, no. 7) we already said that geometry (in opposition to metaphysics, or to philosophy in general) was considered by Kant as a “constructive” science, which is commonly accepted by moderns. But these things are surely not an invention of Kant's genius; they are the Aristotelian heritage. But indeed this character of geometry is expressed in an exaggerated and erroneous manner by the opinion that affirms that the objects of geometry are created by the mind, indeed that this is done through free creation. In such an opinion there is neglected that which noetic analysis clearly demonstrates: the figure in geometry and its mathematical existence is not produced by the mind out of nothing, but its possibility is detected in the fundamental subject, in extension itself as potency. It is produced indeed by our activity, but it is produced, even in our mind, out of matter, which is extension, by whose properties our activity is bound. In extension itself, figures are not in act (this was supposed by B. Russell!) but in potency alone; and in order that this potency, that is, the possibility of these figures, may be detected, the human mind needs actuation, construction; according to the saying of Aristotle (Metaph. IX, 9, 1051a32) ποιοῦντες γιγνώσκουσιν (poiountes gignōskousin), in St. Thomas: “facientes cognoscunt” (“in making they know”; cf. Théorie du jugement, ch. X, §§ 4, 5). Therefore extension is the matter out of which figures are made, but it is intelligible matter (materia intelligibilis). Thus the fundamental subject of geometry is extension itself, and figures are its proper passions; and then they themselves become subjects of geometry with their own proper passions and “per se accidents.” What figures are possible, “exist,” does not therefore depend upon our activity, but upon the character of extension: a matter given to us, yet intelligible to us.
The first, most general genera of figures that become known to us are surfaces and lines (and points); in the production of these we first detect the very notion of extension intelligibly, as the fountain whence alone and necessarily figurability of bodies flows. From sensitive data these become known to us in an inexact manner; but quickly, in reflection upon these data, we detect their exactitude in the thing itself (not in the phantasm), that is, exactitude relative to indivisibility according to one “dimension,” or according to two, or all-embracing (in the point).
Then in that very extension out of which more specified figures (though in sensitive data inexact) are produced and become known to us, we observe — and it is again a proper attribute of extension — that figures (inexactly equal to one another) can have a diverse “position” and moreover a diverse “orientation” in extension, which “orientation” has its origin in the “direction” of lines, which is again intelligible to us as a proper attribute of lines that can “pass” through a point in diverse modes (concerning “position” and “orientation” cf. above Chapter IV, § 4, no. 2). These notions can subsequently be polished to perfect exactitude (as to “what they are”), and finally, from the reflexive cognition of extension as it is the principle of individuation (also as to orientation), can be detected as “existing” in that exactitude. Thus, as was expounded above, we arrive at the cognition of the existence (not of quiddity alone), and indeed “everywhere” in extension, of a perfectly straight line, and of a perfect circle. The mathematical existence of these figures is understood from the known nature of extension, and indeed as “existence everywhere and in every orientation” (for straight lines). In realized extension they are constructed by means of instruments called “straightedge and compass” (regula et circinus); in physical reality these constructions are inexact, but their possibility, as well as the possibility of perfect instruments and their use, is perfectly understood and employed in mental construction. For mentally constructing more complicated subjects (figures), that is, for proving their mathematical existence (in potency in the fundamental subject, extension itself), these elementary constructions are employed in a “quasi-operative demonstration,” as St. Thomas says. And thus, for example, the Greeks already demonstrated the existence of the five Platonic solids, and the non-existence or impossibility of other “regular solids.”
The legitimacy of elementary constructions in Euclid is expressed through the first three postulates: “1) that it be postulated to draw a straight line from any point to any point; 2) and to produce a terminated straight line in directum continuously; 3) that with any center and radius a circle be described” (according to Heiberg's version). For the execution of the first two constructions the “straightedge” is employed; for the third, the “compass.” These postulates are “supposed” by the geometer as legitimated, but they can be established in pre-geometric science, as was done above.
Other means of demonstrating the existence of determinate figures can then certainly be introduced. One was indicated above, where a line is constructed by means of the motion of a point composed of two motions, subsequently “eliminating” “time,” provided the existence of the motions is established (the example in Chapter III, § 2, no. 6 C was employed for another purpose).
It is clear how geometry is a “constructive science of its own (secondary) subjects”; it presupposes the understood character of the fundamental subject, extension itself, from which it is clear what secondary subjects are possible, exist mathematically. Out of extension figures are constructed, not vice versa extension as a system out of figures, as seems to be done in Hilbert, who nevertheless does not explain what he intends by the word “system.” Surely we do not arrive at the notion of “extension” in that way.
§ 2. On the Twofold Intelligibility of Intelligible Matter¶
What we were saying in the preceding section explains in general the intelligibility of constructions that demonstrate the mathematical “existence” of figures, even exact ones. This intelligibility of constructions (operations, therefore) has a twofold aspect; it directly regards that which this intelligible matter can undergo or suffer (subire sive pati potest), but simultaneously it also reveals certain intelligible, necessary aspects in our very activity and that of our instruments, whether real or ideal. This second intelligibility arises from the intelligibility of the very passivity of matter. Concerning both, certain points are touched upon here.
1. On the Intelligibility of the Passivity of Intelligible Matter¶
Both intelligible matters (called by St. Thomas “continuous quantity” and “discrete quantity”) we briefly review here.
A) On Arithmetic Matters¶
We begin from arithmetic matters, recalling what we expounded above in Chapter II.
a) In every multitude we understand that it can be divided into lesser multitudes or into units (“elements” of the multitude), in which divisions the elements remain what they are; only their union is dissolved; and these lesser multitudes or units are what are understood necessarily to result. Behold what we first understand, expressed by saying they “can be divided”: it is the passivity of this matter, which is “discrete quantity.”
b) Conversely, multitudes as such can be added to one another; and a greater multitude results. And as was already said above (Chapter II, § 1, no. 4): these things belong to first intelligibles, which are so clear that they are not accustomed to be enunciated, although known they are presupposed. It is again something passive simultaneously with that which results from “undergoing” (ex « pati »).
c) We can compare diverse multitudes to one another with respect to their “quantity.” And this we can do by constructing a one-to-one “correspondence” (correspondentia bi-univoca) between the elements of the multitudes we compare. Multitudes are thus found to be either equal or unequal, and in this case we find which is greater and which lesser. Our operation does not produce that equality or inequality, but permits us to recognize them. And we understand that the multitude can be subjected to this operation, and from this “undergoing” arises a necessary, intelligible result.
d) These things are applied in the “enumeration” of multitudes; this we do by establishing a correspondence of the elements of the multitude to be computed with the elements of the series of natural numbers. Thus we determine the species of the multitude, that is, we do not produce that species, but detect it. This we applied above (Chapter II, § 1, no. 4) in inquiring what the species of multitude is that results from the addition of specifically determinate multitudes; subsequently in multiplication this is applied. All these are therefore passive properties of multitudes; and they are so clearly known by pre-scientific man that, when people wish to express the clarity of some truth, they do so with the words “you can count it on your fingers.”
e) In the combinatorial art (above, Chapter II, § 1, no. 4), which fundamentally inquires into the possible orderings of the elements of a multitude, the same holds. We ask “how can the elements of this multitude be ordered?”; and this passive aspect is again intelligible through itself. Let us note moreover that in inquiring into the first principles of this art we need also the consideration of the spatial order of elements, which enters there and in its own way also regards passivity first.
f) These things can be extended further. Above (Chapter V, § 4, no. 3) we considered operations performed in developing the sciences, in arithmetic (algebra), but also in logistics. There specifically two transformations of formulas are considered, which we called “transposition of a term” and “substitution.” And there the same returns: we understand first and spontaneously that formulas can undergo (pati) such transformations, and simultaneously we understand that other, specifically determinate formulas necessarily result. And thus here, as in the prior cases, we have in these formulas an intelligible matter whose passive aspect is of itself intelligible.
B) On Geometric Matters¶
In these we shall find similar things.
The first thing that we understand in extension was always, both in the first and in subsequent analysis, that which extension can undergo (subire) and that which results from that “undergoing” (ex illo « pati »).
a) Thus the first intelligible, which led to the very “operative definition” of the notion of extension, was the “figurability” of a body through division, which in extended entities given to us leads first to surfaces as boundaries, then to lines, then to points. This last result, the point, now no longer undergoes (patitur) division. The original extension was three-dimensional, and through successive division there arise figures of two dimensions, then of one, then of zero dimensions, and the figures that result are known exactly through the intellect as lacking one dimension, or two, or three, which were present in the original extended entity. And this exactitude of the things that are made is the first that is detected.
b) Next we understand extension as the principle of individuation, and this too is something passive. A figure that can be constructed in one part of extension can be repeated in any other part, that is, in another “position,” so that these figures differ from one another by site alone, by number alone. This is the foundation of the passive possibility of exactly congruent figures.
And the same passive proper attribute of extension, expressed by the formula “it is the principle of individuation,” leads, after reflection upon the primitive notion of the “mode by which a line passes through a point,” to the recognition of the existence of specifically the same continuum of directions, and consequently to the existence of exact lines, of the plane, and of their connections.
c) There is also another proper attribute of an extended entity as it is extended, which is intelligibly known independently of the first (figurability through exact boundaries introduced through divisions); this proper attribute can be expressed: two extended entities can have common boundaries, that is, one can be “touched” by another according to boundaries. This too is a primitive proper attribute, intelligible through itself and intelligibly flowing from extension itself. And it is a passive proper attribute: extended entities can undergo this.
That which results is also intelligible: it is a greater extended entity that is like one continuous extended entity; thus from such contact of two cubes it is understood that a body arises which is like a parallelepiped. From this other things follow that regard “magnitude”; extensions can be added to one another, extension can be “multiplied”; it is understood as the subject of these arithmetic operations. The two primitive proper attributes are correlative, that is, relatively opposed to each other, like to be divided and to be multiplied. But all are understood as passive proper attributes of extension.
d) As was said above (Chapter V, § 4, no. 2), from this second primitive proper attribute, from the possibility of contact, both permanent and fluent, in the order of actual existence there intelligibly flow those things that regard the localization of bodies and their local motion. And analytical reflection upon problems of localization and local motion is without doubt of the highest noetic moment in cosmology, and, as is historically evident, consequently in general noetics.
And this intelligibility of the passivity of that matter which is extension proceeds in all these cases from our abstraction (which we call intuitive formal abstraction) through which we discern in bodies extension as the property from which alone, in the midst of other properties of bodies, these proper passions flow.
2. On the Intelligibility of Our Activity¶
In all these constructive operations of ours, the first intelligibility that we find is situated in the very passivity of intelligible matter, whether of continuous or discrete quantity. But thence there results also a certain intelligibility per se in our constructing activity.
In general, the influx of efficient causality itself we do not perceive. An exception will have to be made for our conscious life, sensitive and intellective. But the influx itself of transeunt causality into another external thing we certainly do not perceive, and therefore here it is not understood through itself [2].
Certainly we understand the principle of causality itself. If something is changed (a passion is perceived), there must necessarily be presupposed (as prior by nature) an efficient cause that is other than the being moved itself; everything that is moved is moved by another (omne quod movetur ab alio movetur). And we understand that this efficient cause, this other entity, must actually exist, and be of such a nature as to be apt for producing that determinate effect, that “becoming” (illud « fieri »). Yet the principle of causality itself does not indicate what that other agent is which is the efficient cause or contains it, nor in every case does it directly reveal its nature (cf. Gregorianum, 1953, pp. 617–619, and below in the Appendix). Thus in general we do not understand through itself the determinate nature of activity.
But in our cases, in the activity that corresponds to the passivity of intelligible matter, certain aspects of activity are intelligible to us through themselves. These we shall briefly indicate.
A) In Arithmetic Matters¶
In the cases expounded above, these are immediately found: In the division of multitudes, in their addition and multiplication, in establishing a one-to-one correspondence between elements of multitudes, in their enumeration, in the diverse orderings of elements of a multitude and their computation (cf. above, Chapter II, § 1), next in the transformation of formulas (above, Chapter V, § 4, no. 3), in all these operations we immediately and through ourselves understand how we ought to direct our operating activity, whether mentally or even actually, so that the passivity, the potentiality, of intelligible matter may be actuated; activity directed in such a manner necessarily leads to the actuation of the effect. If we operate mentally alone, a multitude represented in the phantasm suffices; if we operate actually, the actual existence of this matter is required, according to the adage “it is moved, therefore it exists” (« movetur ergo existit »; cf. Gregorianum, art. cit., and below, Appendix, Part I). And observe: even in a purely mental operation concerning a represented multitude, this must be represented or thought of as existing.
In individual cases of this activity there is found something that is directly and through itself intelligible, according to which activity must correspond to the effect it causes; but that effect is an effect (form) in intelligible matter; hence it is that the correspondence itself in the activity is intelligible. In pure physics, activity that acts, but not on intelligible matter, must also have such a correspondence from the very principle of causality, but this escapes us, and only after perhaps long inquiry becomes known to us. In our cases, therefore, activity is immediately understood because it deals with intelligible matter. Concerning this matter too, activities can be given that as to part are contingent, or whose necessity escapes us. Suppose that from an urn containing marbles (as is accustomed to be considered in probability theory) I draw a handful of marbles; counting them, I find that I drew twelve; repeating the experiment, I have only eleven. If, calculating the sum of the numbers \(7\) and \(5\), on one occasion I find \(12\) and then \(11\), at least in one of these calculations I erred; I know that I am engaged in necessary matter and in activity that of itself is necessary. In the first case, the outcome without error in the repeated experiment can be diverse; for there is present a contingent element in the activity by which I grasp marbles with my hand; and it is the “capacity” of the hand that grasps; this can easily be varied, and this we know. Not every operation concerning necessary matter is necessary. (Certainly if in the repeated experiment all things in the activity are the same in both experiments, and this is possible, the resulting effect will be the same).
That operations of ours concerning arithmetic matters are given which are necessary and whose nature is intelligible to us through itself will be clear. Only this must be observed: our activity, through itself infallible, for example, the simple enumeration of a multitude, can err per accidens; if we are fatigued, or for another reason insufficiently attentive, an operation of itself infallible is badly performed by us on account of an accidental indisposition. And such an error, which theoretically is surely not tragic, can therefore through repetition, with all care given to execution, be recognized and corrected as an error. A human faculty of itself infallible can per accidens err.
We have therefore activities of themselves infallible, which if actuated, not mentally alone but also actually, presuppose a corresponding intelligible matter that actually exists; hence, in making they know (facientes cognoscunt; cf. Théorie du jugement, ch. X, § 5).
B) In Geometric Matters¶
In our activity concerning arithmetic matters there are found, therefore, elements that of themselves necessarily lead to a determinate effect, which is therefore intelligible to our mind. Nor are those activities merely understood as necessary, of themselves non-contingent, but moreover they are understood to lead to exactly determinate results; under this aspect as well they are intelligible.
In geometric matters we also find necessity, known to us, in the intelligible matter which is extension (above, Chapter II, § 2). In the same matter we detect also the existence of exact figures that can be constructed in it, although, in opposition to arithmetic matters, the senses cannot attain exactitude (above, Chapters III and IV).
It is to be expected that in our activity concerning geometric matters, relative to this our activity itself, an analogy is present with activity concerning arithmetic matters. Not perfect likeness is to be expected, but analogy, on account of the difference found there relative to the senses; which can be a source of difficulties.
Relative to the first element of exactitude there is no difficulty. The activity dividing a solid body, that is, of three dimensions, produces as boundaries surfaces, which are figures of two dimensions, lacking all depth; their division produces as boundary a line, a figure of one dimension, lacking all breadth; and its division produces as boundary a point, all-embracingly indivisible, lacking dimension. The dividing activity produces not mentally alone, but also really, such an exact effect (cf. above, Chapter III, § 2, no. 2 C).
But difficulties arise relative to the further determination of these figures, relative to the production of a figure that is exactly congruent to a given figure, to the production of a figure antecedently perfectly determined. Let us take the simplest examples: an exactly plane or spherical surface, or a straight line or perfect circle. If mental construction is treated, these difficulties do not yet press. For mental construction consists in an “operative demonstration” that demonstrates the mathematical existence of these figures, employing the knowledge of what intelligible matter can undergo (pati), and from these manifests the geometric structure of the activity required to perform the operations, even if motion of figures enters there.
But if real construction in real extended entities is treated, a twofold difficulty arises. For external human activity itself has a nature that is partly sensitive; and therefore it too participates in the defect of exactitude found in sensitive cognition. Then the instruments it uses to perform constructions are affected more or less with the same defect of exactitude. On account of physical laws, the very passivity of intelligible matter (for the extended entity out of which and in which the figure is constructed has a physical “nature”) cannot always be perfectly actuated. And in similar fashion the perfect exactitude of real instruments cannot be attained, although it is understood to be able to be present in ideal ones.
Yet simultaneously we understand that these difficulties can be overcome in the sense that the defect of exactitude can be more or less corrected; real construction can be an “approximation” to ideally exact and known construction.
Above (§ 1) we named the elementary constructions of the Greeks, according to which a straight line is drawn between two points, or a terminated straight line is produced continuously in a straight line, and with any center or radius a circle is described. Concerning passive possibility it is already established (for pure extension). For practical execution there are employed as instruments the “straightedge” and “compass.” The straightedge is applied to two points or to the line to be produced (this possibility of application is established from the second proper attribute of extension). Our execution of the application, and the real instrument itself, lack exactitude, but simultaneously we understand that real execution can approach, more or less, to ideal operation. In similar fashion in constructing a circle by means of a compass. Our activity, even as to exactitude, contains elements intelligible to us.
These things are elementary, but similar things we find in more complicated operations, in innumerable cases in daily life, in developing the physical and technical sciences. To name one example from these: in constructing approximately exact surfaces in our instruments or in physical objects, such as the surface of a plane mirror or prism, or a spherical surface, we use operations, that is, activity whose geometric elements are accessible to us.
That the proper noetic moment of this fact of our cognition may shine forth more, we can again compare this case with a purely physical case of which we were already treating above, with the electrification of a glass rod by means of friction. After friction that rod attracts small particles, a power it lacked before friction. Inquiring into the efficient cause of this mutation, very quickly we admit that it is situated in the activity of friction; but this physical activity we do not intuitively understand, as in geometric cases above we understood the geometric aspect of our activity, just as in this very case we understand the motion, whence friction arises, from our activity. The geometric elements in the activity we understand.
And this intellection proceeds from the fact that relative to geometric elements, we enjoy the faculty of abstracting intuitively formally. But not every physical causal influx escapes us in non-geometric matters; for similar things, though in lesser degree, are found in certain non-geometric matters, in comparing qualities, in considering their intensity; concerning these we shall add certain matters below. (See Cosmologia on the cognition of qualities and intensities, and in the 4th ed., Note XI on proper sensibles).
These things are entirely neglected in noetics; undeservedly, as it seems to us.
§ 3. On the Origin of General Noetics¶
1. General Survey¶
The things we have seen permit us to determine in what sense the noetics of geometry (and of arithmetic) can and ought to be called the origin of general noetics (origo noeticae generalis).
The term “origin” we derive from the capital text of St. Thomas, who says: “All our cognition originally consists in the knowledge of first indemonstrable principles. The cognition of these, however, arises in us from sense, as is clear at the end of the Posterior Analytics. Therefore our science arises from sense” (Q. D. de Veritate, q. 10, a. 6, "sed contra" 2; cf. Théorie du jugement, Introduction, ch. I, § 3 B, ch. V, §§ 3, 5, ch. VII, § 5, in the 2nd ed., pp. 1, 25, 164, 188, 227).
There our intellective cognition is treated, not sensitive cognition; for this cognition is said to proceed from sensitive cognition; and that is established from the entire article. Nor is it treated merely of principles that are notions, but also and especially of judgments; for these are indemonstrable principles. And we are expressly taught that these first indemonstrable principles are the origin of all our intellective cognition, and therefore also of that which is not directly had from sensitive data, such as noetics itself and metaphysics. These are born from those mathematical principles as from an origin; not certainly in such a way that they are deduced from them through syllogistic demonstration; how they are born from them will have to be investigated further, and indeed in constructing noetics itself.
For noetics is the science that intends to examine the nature of intellection itself, of noesis. And the consideration of mathematical principles not only de facto is the first part of noetics, but is necessarily so if our intellective cognition ought to be derived from the senses.
The first part of noetics cannot consist in the consideration of abstract “understanding” (ipsum « intelligere ») itself, nor in the consideration of judgments concerning “physical cases.” Noetics examines understanding itself, and indeed first the “act,” and only afterwards the intellect itself eliciting the act; from the act the potency is known. He who examines the intellect must know “what understanding is,” and that he may know this, the mind must elicit an act of understanding and notice “this is understanding.” But, as St. Thomas often says, “to understand is to understand something” (« intelligere est intelligere aliquid »); which seems to be most clear; understanding without an added object has no meaning. That term “something” (aliquid) does not signify the notion “something” that is accustomed to be enumerated among the transcendentals, but stands for a determinate disposition of a thing, for a specified intelligible. Yet the first object of this kind (in the intellect proceeding from potency to act) cannot be the act of understanding itself, because thus an infinite regress would be supposed. It must be something in the external world, presented by sense (sensation itself, which can lead to a certain principle “I feel, therefore I am” [« sentio ergo sum »], is without doubt not something primitive). Our first understanding understands something that is an external object. That we may know the intellect itself, we must know its act of understanding; that we may know this, there is prerequired a genuine intellection of an external object; and thus the act is known from the object, from this external object understood.
These objects, however, as known by us, are easily divided into two classes; one contains what above we often called “mathematical cases,” the other contains “physical cases.” Let us resume the specimens posed above. In a given body we observe that a figure can be traced, and we affirm: “this body is figurable”; in another specimen we observe that a body after friction acquires the power of attracting small particles, and we affirm: “this body is electrifiable through friction.” If we inquire nothing further, the cases seem to be entirely similar. In further reflection, however, upon those things found in our mind, we find between the two cases a capital discrimination. For to someone asking “why do you affirm this?” in the first case we answer “because I understand this” (« quia hoc intelligo »), in the other “because I see this with my eyes” (« quia hoc video oculis »). In the first case we know the necessity of the nexus between the subject body and its figurability; in the second case cognition of the necessity of the nexus itself is lacking; that is, in the first case we understand the nexus and therefore affirm it; in the second, intellection of the nexus itself is lacking, and we affirm it only because we observe it by sense. Thus an immense noetic discrimination manifests itself between two entire classes of cognitions present in our mind relative to external objects, between “mathematical cases” (not geometric alone, but arithmetic and similar ones as well) and “physical cases.”
2. On Mathematical Cases¶
We do not yet contend that these are critically justified; we say only: a man who begins to institute reflection upon his intellective cognitions finds these things in his mind as facts perceived by the mind. In the first class he finds that he truly understands the nexus he affirms (there he really experiences what understanding is); in the second class, however, he finds that he lacks an intellective intuition of this nexus. And therefore, beginning the construction of noetics, that is, of the science of the nature of his own intellection, he will first have to direct his attention to the first class, in which he experiences what understanding is, and to the second class only (at the beginning at least) insofar as, from opposition to the first, this is more illustrated.
Even more he detects; for immediately he finds also the reason for this discrimination between the two classes of his cognitions. For in the first case he simultaneously understands that figurability flows not from color or any other attribute of the body except a single one (which subsequently we call extension), and from this flows necessarily. In the second case it is not at all evident from what attribute of the body electrifiability flows. And this cognition we express by saying: in the first case man has intuitive formal abstraction, in the other he lacks it. And this is known, as we already expounded above, even to pre-philosophical man, but in actu exercito, not in actu signato. And noetic reflection does nothing else than bring this cognition to actus signatus.
And this is the fundamental diversity on account of which noetics must distinguish two classes of our cognitions, even ordinary ones, which we find in our mind. We were describing one specimen of the mathematical class, and indeed the first of all, unless we are mistaken, namely, the figurability of bodies. It is worthwhile briefly to describe or enumerate certain others. Above we were finding another example almost equally primitive: bodies can touch each other extrinsically, and thus arises a system of bodies that is like one body out of which that system, one by contact alone, could arise. The moment of this most simple intuition, of the possibility of contact and of what results from contact, was described above (Chapter V, § 4, no. 2). We ask: whence in bodies does this possibility flow? Immediately we understand, truly understand, that it arises from extension, and from it alone, and from it necessarily. And now attend: this extension is entirely identical with that which we first detected from figurability; to say this explicitly is not superfluous, it has noetic moment (Chapter VII, § 2, no. 1 B c).
And from the same extension again are understood to flow the properties of figures that arise from the division of extension (or from addition through contact): such as the modes of passage of lines through points, in which primitive properties are intuitively recognized, the properties of localization and local motion. All these are intuitively detected from the same formal abstraction.
Moreover in these, besides the necessity of relations, we detect also their exactitudes, which are lacking in the sensitive data whence they are drawn, and are understood to be necessary even relative to their exactitude. For justifying this exactitude (in the described cases) there serves another property, of the very same extension always, which we express by saying that it is the principle of individuation. And thus we find that the intellect relative to the same objects surpasses sense in a twofold respect, in necessity, in exactitude; to recognize these without doubt belongs to the science we call noetics. And this necessity, at least as to exactitude, shines forth to us, not from a necessity to which imagination can be subject, but is present in the object; we often affirm it against the data of our imagination, as in the case of a surface without depth, a line without breadth.
In a word we recall here that similar noetic relations are found in arithmetic and similar matters, and there primitive intuitions, immediate propositions, intelligible through themselves and immediately, are multiplied into infinity (above, Chapter V, § 4).
Moreover let us recall that in our very activities relative to these objects necessity is found (§ 2, no. 2), and, at least in mental operations, exactitude. These too are noetic acquisitions originating from the consideration of “mathematical cases,” through “resolution to first principles” (resolutio ad prima principia), as St. Thomas is accustomed to say. Thus “quantity,” whether continuous or discrete, that is, extension and multitude, reveals itself as intelligible matter, in passivity and in certain of our activities at least relative to it.
Let us briefly resume. To understand is to understand “something”; this “something” designates a determinate, specified object. In origin this object cannot yet be “understanding itself” in general; it must be an external object received from sense. And among these the first intelligibilities found in our mind are primitive mathematical principles, not axioms alone, but also other first, that is, immediate propositions that are applied in mathematical deduction. But they are not “physical” principles. Noetics, which examines our intellection, must know “what understanding is,” and this it originally experiences in “mathematical cases.”
This is the “knowledge of first indemonstrable principles” in which “all our cognition originally consists” and whose “cognition in us arises from sense.”
Mathematical science itself is derived from these first principles through syllogistic reasoning, but in it there does not consist “all our cognition.” Yet from the same principles other sciences also are born, metaphysics and general noetics itself, not through syllogistic reasoning, however, but through further reflection upon those things that occur in our mind when in sensitive data it truly understands a necessary nexus; then it knows what “to understand” is, and moreover that intelligibility carries with it “being” itself (ipsum « esse »; objective in the sense of realism, which can thus be defined through an operative definition); this “being” (esse) belongs to the intelligible, and to the intelligible alone (in itself). In this critical reflection our mind acquires the very notion of its intellection, and in it knows its own noetic nature. It detects the justification of its affirmation “so it is” (« ita est »), it detects the relations between being and the intellect: the intelligible is being (intelligibile est ens), which is the fundamental principle of general noetics. Moreover it detects the converse principle: being (of itself) is intelligible (ens [de se] est intelligibile); and thus it enters into the region of metaphysics. In such a manner these sciences are born from first mathematical principles.
Intellection itself does not have a representation in sense (as mathematics and physics do), but its nature is known from the intellection of mathematical entities whose dispositions of the thing are assumed from the senses. In this sense St. Thomas (like Leibniz after him, to whom this adage is accustomed to be attributed), to the well-known adage “nothing is in the intellect which was not first in sense,” adds “except the intellect itself” (« nisi ipse intellectus »; Q. D. de Malo, q. 6, a. 1, ad 18; cf. Théorie du jugement, ch. VII, § 1, 2nd ed., pp. 214 ff.). But these things are expounded at length and breadth in Théorie du jugement, ch. VII, and elsewhere.
One thing must be urged again and again: in our exposition there is evident the decisive moment of that which we called “intuitive formal abstraction,” which is the cause of our primitive intellective intuitions. Some years ago, to the question “whether Aristotelian abstraction can justify affirmation,” it was answered: this cannot be done. But in the exposition of this Aristotelian abstraction, not even mention was made of this intuitive abstraction in mathematics. And yet Aristotle calls mathematical objects by excellence τὰ ἐξ ἀφαιρέσεως (ta ex aphaireseōs), “objects from abstraction.” It is no wonder that in such an omission justification is not detected in Aristotelian abstraction.
This judgment concerning the value of Aristotelian abstraction is surely erroneous.
Neglect of the consideration of this intuitive formal abstraction (which reveals the nexus between forms that is therefore known as necessary, intelligible) does not always lead into error; which is not wonderful, for even if it is not explicitly considered, this cognition is nevertheless always present in actu exercito; it is the very life, the nature, of our mind (today often called its dynamism) which understands all these immediate principles “naturally” and intuitively, “naturally” understands that it understands, not “dogmatically,” but by a natural critique in actu exercito.
We add: it is often said that a single example of a “mathematical case” suffices for all these things to be understood. And it is true, we do not need a multitude of such cases, from whose comparison our cognition would be induced; our abstraction is not of such a nature, for it is intuitive. But on the other hand it must be said: much exercise of the sensitive faculties is prerequired, so that sensitive data may become apt from which we may intuitively abstract; which is described by Aristotle in the famous last chapter of the Posterior Analytics and at the beginning of the Metaphysics. In similar fashion, much exercise of “resolution to first principles” is required so that the processes that the human mind exercises by its nature may be clearly detected and formulated, so that the judgments of the human mind may be brought from actus exercitus (from a virtual state, as was rather often said above) into actus signatus. Even in this case there is required a true “habit” of this resolution. Therefore such frequent and diligent exercise of “noetic resolution” is greatly to be commended, indeed seems to be necessary, in order that general noetics may be worthily constructed. And the first objects of this exercise are those which we called “mathematical cases.”
Thus in the noetic analysis of intuitions that we have relative to “mathematical cases,” not only does the nature of geometric and arithmetic cognition itself manifest itself to us in actu signato, but moreover from it, through reflection that is instituted further (it is the term of St. Thomas), we detect its objective relations (in the realistic sense) to “being” itself (ipsum « esse »), to entity. Only armed with these intuitions can we construct further noetics, even of the cognition of “physical cases.”
3. On Physical Cases¶
The reason for the affirmation of geometric relations was: “because I understand”; in physical relations it was: “because I see.” In both cases the matter of affirmation is drawn from the senses, but the reason for the prior affirmation is taken from the sole intellection of this matter, the reason for the second also from the testimony of sense. We say “also,” for in the very terms of the affirming proposition there must be present something that we understand (so St. Thomas; cf. Théorie du jugement, ch. IV, § 4, and ch. X, § 1); but the nexus itself that is affirmed does not manifest itself as intelligible; therefore precisely as the reason for affirmation there is given: “because I see.” There is required therefore, and indeed prior, a critical justification of the value of the testimony of sense. This part of noetics presupposes the “origin” of noetics and its further evolution that we were describing. This part of noetics, which examines the value of the testimony of sense, we expounded at length in the work Théorie du jugement, chs. VIII–XII, and here we do not repeat the analysis. Only certain special points may be expounded here.
In “mathematical cases,” from understood quiddity (ex quidditate) we intuitively infer the very “being” (ipsum « esse ») that belongs (competit) to this quiddity; and that esse is realistic being, which can thus be “operatively defined”; according to this esse quiddity can be actuated, or is actuated. Conversely, in “physical cases” in general, from the actual being (ex esse actuali) of a relation we infer (by force of the principle expressed “being is intelligible”) that that nexus between quiddities must be intelligible, whether it be intelligible in itself or from another, that is, from a cause. That this “being” (esse), proposed to the mind by sense alone, may rightly be affirmed, there is required the justification of the testimony of sense, as is expounded in the cited chapters from Théorie du jugement.
In the noetic analysis that manifests this justification, as an ultimately decisive element (which regards actuality itself) we find certain necessary, intelligible activities of ours. Such activities, which presuppose the actuality of intelligible matter, the subject of activity, if the activity itself is exercised in act and not mentally alone, we were finding above (Chapter VII, § 2, no. 2).
St. Thomas introduces the consideration of such activity in many texts, in which he attributes to the human artisan cognition of his work, as to its actual esse, on account of his activity of this kind, which in such matter necessarily produces such an effect. And this doctrine he applies to explain that knowledge of God which regards the actual existence of created things and is called the knowledge of vision (scientia visionis). And he himself applies this theory to solve cases in which a difficulty arises concerning the actuality itself of the matter that receives activity; Aristotle also seems to do this (Théorie du jugement, ch. X, § 5).
It was said that in “physical cases,” in opposition to “mathematical cases,” an intuitive abstraction of the nexus that is affirmed is lacking. That is in general true. There are nevertheless certain physical relations that do not regard extension or multitude, but qualities, which nevertheless permit such an abstraction. These cases regard the division of a genus into species and the distinction of genera from each other (red and green necessarily belong to the same genus, color, distinct from the genus of sounds), but especially the properties of the “intensity” of such qualities, belonging to the subalternate genus of sensible qualities (qualitates patibiles). In these we have an intuitive intellection relative to the “degrees” of intensity, from which we know the objectivity of this genus with its proper attributes; concerning these at length in Cosmologia.
In this matter of intensities we detect an analogy between a geometric line and intensity. There is present a likeness in the ordering of “points” in both objects; but there is present also a fundamental discrimination: a line can be composed of smaller pre-existing lines and divided into lines existing after division; not so intensity relative to lesser intensities. Another likeness consists in this, that according to both objects motion can be given, and this motion, even if it is a motion of alteration, has an extension strictly so called, fluent extension (cf. above, Chapter VI, § 3, and Gregorianum, 1953, pp. 3–19). But there is also another discrepancy of great moment: a line is a unidimensional extended entity situated in an extended entity (e.g., in a plane) of multiple dimensions, but the duration of motion is a unidimensional extended entity that is not situated in a higher extension (ibid.).
Hence we have also the answer to a difficulty someone might have. It can be asked: if the duration of motion is an extension worthy of the name and we understand it intuitively, why concerning duration itself do we not have a most ample science, as we have in geometry relative to the permanent extension of bodies? The answer is at hand: because it is a unidimensional extension that is not situated as a boundary in a superior extension. If permanent extension were in such a manner of one dimension, geometry itself would likewise lack such amplitude.
We have therefore, even relative to qualities, especially relative to their intensities, certain intuitions originating from intuitive abstraction that lead to intellective cognition of relations intelligible through themselves; these flow from understood intensity, just as geometric relations flow from the understood extension of bodies. Hence already previously we reduced such processes of the mind, which in constructing physics are employed naturally, and in cosmology in actu signato, to the second degree of abstraction, which is accustomed to be attributed to mathematical objects alone (cf. Cosmologia, 4th ed., Note XI on proper sensibles and Gregorianum, 1948, pp. 295–303).
These abstractions, which contribute so much to constructing the natural sciences, without doubt must be considered in special noetics, which ought to judge concerning the nature of these sciences, but they are most frequently neglected, if not despised.
Furthermore, in constructing those sciences physicists, already for centuries, have employed methods that under another aspect are similar to intuitive formal abstraction, and which therefore can more or less supply its defect in our mind relative to these objects; they are found first in experimental “mathematical physics,” which in the midst of complex physical phenomena is able to detect those “forms” between which constant experimental relations, “physical laws,” hold. In mathematics, in intelligible matter, we detect these forms intuitively; “le mathématique se détache du fond du réel,” said Meyerson. Thus in physics the experimental art imitates natural abstraction in mathematics (concerning this comparison cf. Théorie du jugement, ch. XI, § 3, no. 2 and no. 3 C and D).
But on account of their origin these laws remain “phenomenal” in this sense, that the necessity of the constant nexus affirmed by the law is concluded from constancy, but is not yet understood by us, although in itself it must be intelligible. To attain this intuition as well, “explanatory physical theory” is introduced, which, beginning from hypothesis, attempts to “explain,” that is, to make intelligible, experimental laws. Where it must be observed: it happens that in such a theory, besides elements containing intuitively intelligible relations, others are also employed that are still experimental laws, but better known (most frequently and in various circumstances established); then theories of a mixed noetic type arise; these without doubt are objects most worthy of deeper noetic examination. But such an inquiry is not, or almost not, instituted by noeticists, though demanded by their philosophical science.
And also in this theoretical-explanatory method we can, indeed ought to, employ a method that in its own way is similar to intuitive formal abstraction; namely, the method that we called the “elimination of superfluous elements of physical theory,” whose utility we attempted to demonstrate in Cosmologia. This method is, as we often said, purely logical, but similar to intuitive abstraction (see also Théorie du jugement, ch. XI, § 3, no. 3 D).
Above (Chapter V, § 5 A) it was already said that in the application of geometry to physical things, our cognition of the necessity of relations certainly remains, but its exactitude perishes. For the cognition of this depends upon measurements that, on account of the imperfection of the senses and instruments and the influx of physical laws, lack perfect exactitude. But we can know that from these sources we have an approximate cognition, whose value, between certain limits, can also be determined.
In the application of a physical law that is expressed by a mathematical formula in itself exact, that the same holds is not wonderful. But in the case of a physical law known from experiments, it can happen that the law itself, expressed by an exact formula, is not exact. Such a formula can be corrected through more exact experiments, in opposition to a formula that expresses geometric relations in physical things. Thus Newton’s venerable law of attraction seems to need correction (relative to dependence according to the inverse square of the distance); but the law expressing that a spherical surface is proportional to the square of the radius cannot be corrected. Indeed it can happen that an experimental physical law, rather accurate for ordinary magnitude, in extremes, whether toward magnitude or toward smallness, totally fails; this can be supposed without any absurdity (cf. Théorie du jugement, ch. XI, § 3 B, and ch. IX, § 3, in 2nd ed., pp. 319 and 262–263; 1st ed., pp. 323 and 265–266; R. a. J., pp. 269 and 223, and Gregorianum, 1953, p. 30).
This is not wonderful; it follows from those things we established relative to the problems of necessity and exactitude. But wonderful indeed, and very wonderful, is that the opposite can obtain: from physical observations not perfectly exact, a perfectly exact law can be derived, like arithmetic applied to (not overly large) multitudes. The most admirable specimen will be this, which we previously described in this way (Gregorianum, 1953, p. 23): in its latest evolution, “atomic theory has arrived at this: that it can order the chemical elements, with their proper attributes, according to the order of natural numbers 1, 2, 3, ….. 92 …..”. To each species its proper attribute is assigned through a number. Here we spontaneously remember the medieval adage drawn from Aristotle: “species are like numbers” (« species sunt sicut numeri »). Such an outcome of theory without any doubt invites noeticists to examine diligently under a noetic aspect the long process that leads to such a splendid term.
Certain points we can indicate. The first step toward this end was already made by Dalton through his stoichiometric law of small integral numbers. And he presupposed (not tacitly alone) that in stoichiometric laws we are dealing with “pure” bodies, concerning which in the nature of things these laws are established with the exactitude of small integral numbers. In the magnificent evolution of atomic theory, whose summit we were naming, from the first definition of “pure bodies” (which considers them as “hyletropic bodies,” and is the beginning of an operative definition) of which stoichiometric laws hold, there are present and return again and again all those cases (and similar ones) in which we were finding Aristotelian intuitive formal abstraction, both relative to arithmetic and geometric elements, and relative to quantitative experimental methods, and also theoretical-explanatory ones, for in that long evolution theoretical, explanatory considerations also enter. In Cosmologia we systematically and at length examined the evolution of atomic theory in order to address the cosmological problem: how mixtures are beings; and many noetic elements are there explicitly and implicitly (but not unconsciously) considered; but to consider these explicitly and systematically in order to build noetics is a work of great moment. And there the noeticist will have to attend to the problems of necessity and exactitude of which we were treating, and always again to intuitive formal abstraction.
One point here we think must briefly be added. Intuitive abstraction itself of our cognition of necessary relations (not only between forms, but) also between existential acts themselves is detected in our mind [3]. Such an intuition is principally enunciated in the principle, already ancient, but which acquired celebrity in the Cartesian formula “I think, therefore I am” (« cogito ergo sum »). In that principle there is intuitively affirmed a connection between the actuality of thought and the actuality of the thinking subject, between accidental esse and substantial esse, which is prior by nature. But similar intuitions we detect in our mind relative to connections in physical things. Certain ones we indicate here. If we use formulas similar to the Cartesian formula, they can be enunciated:
“This is moved, therefore it exists” (« Hoc movetur ergo existit »). In this case both existential acts, motion and prior substantial esse, are in the same subject.
But the actuality of a distinct subject is also thus necessarily presupposed and can be known: “This body is transferred, therefore place exists” (« Hoc corpus transfertur, ergo locus existit »). Or where the principle of causality enters: “This is moved, therefore a moving cause exists” (« Hoc movetur, ergo causa movens existit »). This cause too is in general in a distinct subject.
Another intuition, which seems to have moment in modern cosmology, can be expressed: “Two bodies are in local relation to each other, therefore they exist simultaneously (in time)” (« Duo corpora sunt in relatione locali ad invicem, ergo simul existunt »). And again this coexistence in duration is necessarily prerequired before localization.
To have indicated these things for now may suffice. In the Appendix the patient reader will be able to read an ample analysis of these intuitions; their likeness (in abstraction) with “geometric cases,” simultaneously with their discrimination.
From our entire exposition the capital moment of our mathematical intuitive formal abstraction will be manifest. Even for noetics itself. Its consideration is necessarily required as the origin of general noetics itself. It is found not only in the beginnings of the sciences (and of noetics itself), but is required always again in entire deductions, and this manifests itself explicitly from noetic analysis. In other (mixed) deductions it is present simultaneously with cognitions of nexuses of which we do not have intuition.
It seems therefore to be of the highest moment in constructing noetics itself and in developing it. And that the noetics of geometry itself may be well understood, and all that coheres with it, there will be necessary a full intellection of both fundamental problems in it, the problem, I say, of necessity and of exactitude.