ON THE ORIGIN OF GEOMETRIC KNOWLEDGE AND ITS PROBLEMATICS
We begin from a very concise exposition of the historical evolution of the problems which concern geometric knowledge, from Plato and Aristotle onwards. In what follows we will not presuppose the doctrine of Aristotle (and of the scholastics), we will only ask whether there are found in it data which contain the solution of noetic problems, even of modern problems. We do not even wish to exclude an interpretation of certain texts of Aristotle which differs from our own. But we are persuaded that we have found in Aristotle lights for the study of modern problems, and vice versa, in the latter, means for interpreting Aristotle.
The general epistemology of Aristotle is expounded especially in his Analytics, and for our special matter, especially in the Posterior Analytics. These principally treat of “science” in the strict sense (ἐπιστήμη) which constructs apodictic demonstrations, which is contradistinguished from “intellect” in the proper sense (νοῦς) which regards the first principles from which demonstrations take their beginning. The Prior Analytics study the means of demonstration, i.e., syllogisms according to their form only; such a syllogism, even a “valid” one, can be constructed from premises which are neither certain nor true. The Posterior Analytics have as their object demonstration strictly so called, whose premises are true, certain, necessary, i.e., a syllogism which produces science strictly so called. In this not only the form, but also the matter, the content, must be considered. Here is a paraphrase of the commentary of St. Thomas (Anal. Post. I lect. 6) expounding these differences.
Aristotle, in his investigations, proves that in demonstration, the analysis (resolution) which examines the premises of a conclusion cannot be prolonged to infinity, and that a strictly circular demonstration is also invalid. Analysis therefore must find primitive propositions, first principles, axioms, postulates, hypotheses. This is said of propositions insofar as they are elements of syllogisms.
Something similar must be said concerning the notions which are the elements of propositions: not all can be defined from more known ones; here too one must arrive at the first ones.
In such a way therefore Aristotle examines the general structure of sciences, which consists in the connections which are found between notions and first principles, and the notions and propositions (truths) thence derived by means of syllogism and definition. He does not examine this structure in the same way in which the special science itself (e.g., geometry) does so, namely to constitute its own proper (geometric) system; the object of the Analytics is not the object of the special science; its object is this very science and its origin and its structure. The Analytics therefore consider the very operations of the human mind; not indeed under a psychological respect, as they are acts of the human mind, but under a noetic respect of the ways by which these acts attain their objects. The Analytics therefore are the theory of knowledge; even in this their first part which examines the general structure of sciences.
But they also have another task, through which they no less constitute the theory of knowledge. For in this analysis of science we arrive at first principles which, if the conclusions must be apodictic, must themselves also be certain and necessary.
Whence the process of “judicative logic” by itself leads to the problem: whence arises in the human mind itself the certain knowledge of principles, whence is had the certain knowledge of their necessity. And this is the other part of the investigation of the Posterior Analytics.
Whence a twofold problem is considered in these books: 1º the problem of the structure of science, which arises from the connections between principles and those things which are derived from them, 2º the problem of first principles.
It seems clear that all these things are not considered in the abstract, i.e., in such a way that no special science is examined, and indeed that this is especially valid in the second problem, seeing that it primarily has to consider the matter of propositions, “because propositions per se and necessary are taken”.
Whence Aristotle in these investigations almost always has before his eyes, as a type of science: geometry; both when he treats of demonstrative science in general, and, as is self-evident, in very many cases in which he takes examples from geometry. St. Thomas already notices this at the beginning of his commentary; he says (Anal. Post. I lect, 1 n. 10):
“[Aristotle] manifests a premise proposition by induction. And first in demonstrative matters in which science is acquired. In these however the more principal are the mathematical sciences, because of their most certain mode of demonstration”.
The reason is manifest. In the time of Aristotle geometry was the unique demonstrative science which constituted a systematic body of doctrine. Barely a quarter of a century after the death of the philosopher, Euclid wrote his Elements which then for almost twenty centuries were used by the human race as a manual book, almost perfect. But the things which are proposed there had already been discovered and in a certain manner systematically redacted in the time of Aristotle; the ultimate and greatest perfections had been discovered in the school of Plato, especially by Eudoxus and Theaetetus. It is not surprising that Aristotle, in examining the nature of apodictic science, always has geometry before his eyes.
The geometers themselves therefore, in the constitution of their science, made a resolution up to the first principles, which they already did not demonstrate. These principles, the endpoints of the resolution of geometers, were considered in the school of Plato as hypotheses (ὑποθέσεις) (Rep. VI 510 c) which were simply admitted by geometers; for they “these being posited, as manifest to anyone, think that no reason must be demanded concerning them”. But philosophers demanded and inquired into the reason for these; and in Plato it was precisely the task of the “dialectic method” to institute this inquiry; and thus “dialectic” destroys (ἀναιρεῖ) (Rep. VII 532 c) these hypotheses.
This doctrine, more evolved, as it seems, returns in Aristotle. He distinguishes (Anal. Post. I cap. 2) among the propositions which are the principles of sciences, those which are axioms (ἀξιώματα, common conceptions of the mind, dignities) from those which are either suppositions (ὑποθέσεις) or postulates (αἰτήματα, petitions). The difference can briefly be described thus: an axiom (ἀξίωμα) is evident to anyone; not so those other propositions.
A principle of a special science, if it agrees with the opinion of the learner, will be a supposition, if not, it will be a postulate 1. Even Plato had already spoken of the agreement (ὁμολογία) which is between the teacher and the learner (Rep. VII 533 c). But all these principles must be accepted by the geometer from another, higher science, whether they are proved in it by strictly so-called demonstration, or by the consideration of terms. This other science, which is assumed in place of Platonic dialectic and from which the special science receives such principles, according to St. Thomas is either metaphysics (Anal. Post. I lect. 5 n. 7 ; lect. 17 n. 4) or “natural philosophy” (ibid. lect. 5 n. 7) ; but it is always philosophy.
It is clear that in peripatetic philosophy the task of the philosopher is to determine concerning the first principles of geometry, and this will be even clearer from the following chapters; whether you attribute this task to metaphysics or to natural philosophy or to a part of philosophy distinct from these, which you could call the theory of knowledge or noetics.
The resolution of geometric propositions therefore leads to certain first propositions, and in the very making of this resolution almost everything will have to be left to the mathematicians themselves; only general things, which respect the general structure (as we were saying) of the science, will also have to be considered by the philosopher; but also the activity of the human mind in the individual operations which constitute the science; for these without any doubt pertain to the epistemologist. But in examining the value of principles the principal task falls upon philosophers. We can execute this for a twofold end. The first is this: that indeed all the foundations, as they say, of a complete geometric science be proven to be justified, so that it may indeed be apparent that an entire geometric science can be safely constructed upon these foundations; we shall not tend toward this plenitude in our readings, although we shall treat the principal questions.
But there is also another end of this philosophical investigation: we will find conclusions (and perhaps methods) which will have great value for the general theory of knowledge. One
we bring forward now, the rest to be seen in the course of the work. This theory certainly cannot be constructed without the judgment of the human mind being examined, in relation to its meaning, its origin, its value. But in order to judge the judgment itself it is not enough that we premise a certain general definition of judgment and resolve it; but indeed certain determinate judgments must be elicited — they must therefore arise in our mind — and these judgments must be considered as to origin and value 2. Just as we inspect these very necessary and universal judgments in concrete cases by mental intuition — examples will abound in the course of the exposition — so the universal theory of judgment itself shines forth from apt determinate judgments, which then are already considered not only as judgments of such a special science but as specimens of judgment itself as such, which reveal the nature of this operation of the human mind to the mind itself. Excellent such specimens will be provided by the examination of mathematical principles, if indeed it is true what we heard St. Thomas saying, that this science has a most certain mode of demonstration. For demonstration comprises not only rectitude of form, but also and especially the known necessity of matter, hence the necessity of principles. Whence it is wrong for a philosopher who applies himself to the theory of knowledge to neglect these principles.
All these things must be urged in our time more than before. To understand this, it will be helpful to very briefly narrate here the history of the problem of mathematical knowledge.
Up to almost the middle of the preceding century, the common opinion both of philosophers and of mathematicians — indeed of the entire human race — was this: that mathematical knowledge is ultimately drawn from the data of sensibility, whether from the external senses or from the imagination. In further explanation, to be sure, there was a diversity of opinions. For Plato, because of the imperfection of sensibility, the data of the senses were nothing but the occasion or preparation required so that the mind might recollect for itself those things which it had previously seen by mental intuition (See
Meno 82b-85b and elsewhere). And mathematical forms themselves, as it seems, could not be participated in by sensible things except imperfectly. Aristotle held that “mathematicals” (τὰ ἐξ ἀφαιρέσεως) are had by abstraction from sensitive data, both regarding notions and regarding first principles, which themselves are also said to be known by a certain induction 3. Not indeed so that the judgment of the mind only renders those things which the senses report, but, just as in the abstraction of ideas from sensitive data there is an operation of the agent intellect, so also in the origin of those judgments which are first principles, there is a similar operation which transcends sense and attains nature; in the data of the imagination the mind indeed inspects natures by mental intuition (Anal. Post. I 4, 77 b [^30]): these things (i.e. mathematics) are as it were to see by intellection (ταῦτα δ’ ἐστὶν οἷον ὁρᾶν τῇ νοήσει). Cf. De An. III 5, 432 a 5-9 (St.
Thomas lect. 13 ; Pirotta n. 791), ibid. III 6, 431a 14-16 ; 431 b 2 (St. Thomas lect. 12 ; Pirotta n. 770-772, n. 777).
By Kant a subjective form of external sensibility is fashioned for the explanation of the necessity of geometric knowledge.
Regarding this opinion we will say more later; now let it suffice to have noted that even in this opinion an intuition (not indeed an intellective one) through a phantasm is admitted, which must explain the origin of geometric knowledge.
These schools therefore agree in admitting the necessity and exactitude of mathematical knowledge. In the preceding century empiricism, especially as expounded by Stuart Mill, wholly admits the origin of mathematical knowledge from sensory data, but thereby wholly equates it to physical knowledge, so as to deny both the apodictic necessity and the omnimodal exactitude which the human race had always attributed to mathematical knowledge.
We will later be able to dispatch this empiricist theory with little trouble as inept; yet it well indicates two problems, the solution of which the epistemologist will have to find, if he wishes to vindicate necessity and exactitude for mathematical knowledge. This is very urgent today, not because
of that empiricist theory, but because of modern theories which have their origin from these difficulties. For these theories reject classical theories entirely and by that very fact deeply overturn the age-old conviction which the human race had concerning the value of geometric science.
The duty of examining these problems is therefore undoubtedly urgent.
Here are the two problems. The first is this: The data of sensibility, i.e., those things which are known by the senses and insofar as they are known only thus, are of themselves contingent; the senses only report facts; but the judgments of mathematics present themselves as absolutely necessary, both in arithmetic and in geometry. If those judgments must arise from sensitive data, whence comes that necessity?
This problem is found, as has been said, both in geometry and in arithmetic. But geometry alone carries with it another problem: sensory data regarding the continuum of imaginative intuition, and regarding the perceptions of the external senses — even with perfect instruments employed — are not exact. The senses do not perceive points without extension, but only small corpuscles which are not points; similarly they cannot perceive lines but only long bodies whose width is not zero. The senses cannot distinguish between a straight line and another which is only a little curved; they cannot decide whether three lines are almost or entirely exactly concurrent; nor can the imagination distinguish all these things. For all sensitive knowledge there is a “threshold of exactitude” which the senses cannot cross. And yet geometry contends that it proffers absolutely exact judgments concerning such things. It teaches that three median lines in a triangle concur entirely — “infinitely” — exactly. Whence this exactitude? In arithmetic this problem is not present; the senses can perfectly perceive and decide whether three or five objects are present and what results from the addition of these multitudes; there exactitude in sensitive data is not lacking.
Not everyone well describes this second problem; Kant seems to have not attended to it at all; in modern problematics it has the first role, although, as we shall see, not everyone has been able to express it clearly in every problem.
Modern theories therefore arise especially from the difficulties of the second problem; they speak little of the problem of necessity — except in one single case of which later; it seems to be considered as solved from the fact that they describe “pure” mathematics as a pure “creation”
of the human mind; in which opinion then also the problem of exactitude seems to disappear. But then arise, as is clear, other questions: the question of the real meaning of such a doctrine, created by the human mind, the question of its applicability to the real world, to which however they apply this doctrine.
Nor is it surprising that in this question the crude opinion of empiricism sometimes returns. Thus Einstein:
“How can it be that mathematics, which is nevertheless a product of the human mind independently of all experience, can be applied so perfectly to reality? Can therefore the human intellect without experience by its pure thought attain the properties of real things?
To this interrogation in my opinion it must be briefly answered: insofar as mathematical theses refer to reality, they are not certain; insofar as they are certain, they do not regard reality” 7.
That such opinions, even if you abstract from purely rhetorical exaggeration, demand a new examination of the foundations of mathematics, is clear. They presuppose a longer evolution of doctrines, which more or less deny faith to the imaginative intuition of the continuum, from which however geometric knowledge arises, nay rather they describe it as leading into errors.
This modern evolution has progressed especially along a twofold path; and each respects the problem of transition from the inexactitude of sensation to the exactitude of the mathematical intellect. The prior path leads to the so-called arithmetization of the continuum; the other to doctrines which have their origin from the critique of Euclid’s fifth postulate. Let us briefly view this evolution 8.
The Greeks deduced numbers from the consideration of the intuitive continuum. Aristotle, as is most well known, distinguished continuous quantity and discrete quantity, which has its origin from the division of the former. Multitude arises from division, multitude measured by one is number, an integer number, a natural number.
How the notion of a fractional number arose from the same division needs no explanation here. The Greeks, it seems, did not wish to consider fractions as numbers; whereas the Egyptians (and moderns) consider a fraction and indeed as a number, the Greeks admitted proportions between continua, lines, which relate as numbers; thus: if lines of different length can be brought to equality through multiplication by means of different numbers, then their proportion is the inverse of the proportion of these numbers.
But they also already discovered that there are given lines (e.g. the side and the diagonal of a square) which admitted no numeral proportion, which were incommensurable, irrational.
The difficulties which arose from these irrational (but real) proportions, Eudoxus splendidly solved through his theory of proportions (which is described in book V of Euclid’s Elements) 9.
Moderns, certainly from Descartes on, considered all these proportions as numbers and, just as the Greeks had a “continuum of proportions” which is the collection (insieme) of all
proportions in a continuum, so moderns have a continuum of numbers, a collection (insieme) of “real numbers,” which contains not only integers, but also fractional numbers, even irrational numbers (both algebraic and transcendental). By means of these numbers modern “analysis” could be constituted, which, through infinitesimal calculus, can subject the continuum, even continuous mutations, to calculation. But these numbers, this analysis is deduced from the intuitive continuum; these numbers are nothing but another name for the proportions of the Greeks.
In this point during the course of the preceding century difficulties were moved; the intuitive continuum, because of the defect of exactitude of sensitive perception, did not seem to be an apt medium from which the body of real numbers could be deduced; whence they attempted to deduce this in another way, from the pure arithmetic intuition of the series of integer numbers (1, 2, 3 … to infinity), which does not labor under a defect of inexactitude.
Few describe this difficulty exactly; here are two most celebrated names of learned men, who speak with all clarity.
F. Klein in the book Anwendung der Differential- und Integralrechnung. Eine Revision der Prinzipien (which, as the second part of the title indicates, is entirely about this problem) expressly treats this difficulty. He explains well, in measurement by means of sensory perception (and also in the imagination), a threshold of exactitude is given beyond which we cannot perceive:
“In all these practical regions a threshold value of exactitude is given” 6.
But in the arithmetic definition of a real number (e.g. by means of a decimal fraction) no such threshold is given:
“In the ideal region of Arithmetic no finite threshold value is given, as in the empirical region, but the exactitude with which numbers are defined or at least considered as defined, is unlimited” 5.
We omit a certain difficulty which we have, as also the manner in which he then proceeds in the construction of analysis (and of “precise” abstract geometry, “Praezisionsmathemathik”).
Let us also hear the words of H. Poincaré:
“Intuition [of the continuum] cannot give us exactitude, nor even certitude; this was more and more animated” 10.
“We therefore have different kinds of intuition; first the appeal to the senses and to the imagination; then, generalization by means of induction … finally we have the intuition of pure number … The first two cannot give us certitude; I demonstrated this above by examples; but who would seriously doubt the third, who would doubt Arithmetic?
But in the analysis of today, if one wishes to bind himself to rigor, nothing is found but syllogism and the appeal to this intuition of pure number, which alone cannot deceive us. It can be said that absolute exactitude is attained today” 11.
In all these things Poincaré seems to neglect our intellective intuition of continuous extension; nor is he the only one among mathematicians who does this. It must be added however that elsewhere he does not in every respect reject our geometric intuition in rigorous science, where there will be intellective intuition. In Dernières Pensées he carps at a certain topological definition saying “this definition makes light of the intuitive origin of the notion of the continuum, and of all the riches which are contained in this notion. It belongs to the type of these definitions, which occur so frequently in mathematics, after they attempted to ‘arithmetize’ this science.
These definitions, irreproachable under a mathematical respect, as we were saying, cannot satisfy the philosopher” 12.
Because they neglect the intuitive origin of the notion of the continuum, and consequently its “riches”, they therefore fail under a philosophical respect; we will immediately add: and they render the application of analysis so constituted to real extensions difficult, not to say impossible. But let us also attend to what Poincaré then adds: “I do not wish to say that this arithmetization is a bad thing, I say it is not everything” 4. It will be very useful, but because it is not everything, it will not be surprising if it sometimes fails, and then, if it is considered as “everything”, it could lead to errors.
Analysis therefore, because it is constructed purely arithmetically from the series of integer numbers, already does not depend, as before, on the intuition of the continuum, but vice versa is used to construct a certain “abstract” geometry. And this is so, as we were hearing, because they deny faith to the intuition of the continuum on account of the inexactitude of sensory data. Indeed they think they have certain examples (we shall examine them later) which are deduced from analysis and are said to positively contradict imaginative intuition.
If these things are true, they constitute a most grave problem for the philosopher: whence could it be that the human mind for so many centuries was so firmly persuaded of the exact value and maximal certitude of geometry, deduced from such an intuition?
There are not lacking those who therefore affirm that this classical method of constructing geometry, notwithstanding the maximum genius of many who used it, is nothing but a dream, “a utopia”. Thus the distinguished “realist” mathematician E. Study, in the book titled Die realistische Weltansicht und die Lehre vom Raume (Braunschweig 1914 pag. 131) says:
“Decisive for judging the state of affairs seems to us to be the circumstance that a geometry truly independent from analysis, which the ancient ideal would properly demand, is apparent to be a dream” 13.
But the difficulty becomes even graver, if further examination teaches that this classical geometry cannot be constructed from analysis and dependently upon it. For in such a case mathematicians of maximum genius up to the middle of the preceding century would have erred, not only in method, but also in those things which they thought they had discovered.
And indeed, in the case which we suppose, the matter seems to hold thus. Certainly, if it is a question of approximative application (Approximationsmathematik according to F. Klein) there is not a great difficulty; for this only treats of experimental extended things, as they are known by the senses by means of inexact sensory perception. But if it is a question of the properties of abstract extension, of the extended as extended, about which classical geometry argued, this will not be able, as it seems to us, to be deduced from analysis, without all the same difficulties appearing again in this very application. And thus we have profited nothing.
If this is true, must we not affirm that the human intellect neither knows this extension perfectly, that which the human race has hitherto always believed? What specifically must be said of the Aristotelian theory of knowledge, which firmly thought that this science arises from sensitive data?
Something else is added. A little after the beginning of this century, in this very proud arithmetic analysis difficulties arose, which according to some (thus Weyl, Brouwer and others) lead to a true “crisis” in mathematics. And there is one or another (e.g., the distinguished O. Hölder) who thinks the difficulties cannot be solved except by a return to the consideration of the intuitive continuum; which seems to us to be true. All the more urgent is the task of inquiring into the mode of overcoming the inexactitude of sensitive data; which we shall attempt to do in what follows.
Another path by which the modern evolution of the philosophy of mathematics has progressed was the critique of Euclid’s V postulate, or the postulate of the unique parallel. The principle which is expressed by this postulate is certainly not immediately evident, it cannot be immediately abstracted from sensitive data; again because of the defect of exactitude in these data, as we will see later in its place.
(It is surprising: very many mathematicians, who call this principle “less evident” than other axioms, seem to be unable to indicate the root of this defect of evidence). It pertains therefore to those principles which according to Aristotle are not axioms (ἀξιώματα) but postulates (αἰτήματα); which accordingly are admitted in geometry as hypotheses (ὑποθέσεις) but must be explained and justified from another science, a “preliminary science”, by philosophy; this, if you wish, you could call “dialectic”.
Already in antiquity, as Proclus testifies, difficulties were moved regarding this postulate and, in place of it, others were proposed, which nevertheless labored under the same defect. The critique of the preceding century led to this conclusion, at least now generally admitted by mathematicians: besides classical, Euclidean geometry, another is possible, which rejects the V postulate and introduces another in its place, opposite to it. Many — not of course all — theorems deduced from it contradict classical theses; but internal contradiction in the individual geometric systems is absent nor will it be able to be found in the future. Thence they conclude: for many centuries Euclidean geometry seemed to the human mind to be the uniquely possible, necessary one; this now appears to be false; next to it other systems are present, which are no less possible. We fall into a problem which must be solved by the theory of knowledge.
At first glance the necessity of geometry seems to be denied; and indeed there are some who in this sense already number geometry among pure physical sciences, which in a similar way lack necessity perceived by us. This is without any doubt false, as can easily be shown; the individual systems are intrinsically necessary and are known by us as such. But this comparison between geometric and physical sciences will be worthy of our attention.
Yet another most grave problem remains: the entire human race and all mathematicians, even those who were not content about the V postulate, acknowledged Euclidean geometry as the unique necessary one; if it is an error, how to explain it; if it is not an error, what regarding the severe modern critique? It is clear: again it is a question of a grave problem, which has momentum in the general theory of knowledge. Do not Aristotle and St. Thomas, when they wish to propose an absolutely certain thesis as an example, choose theorems from Euclidean geometry?
From this critique evolved an entire science, a part of “judicative logic,” which they call Axiomatics. Above there was talk of intrinsic necessity and immunity from intrinsic contradiction, which belongs to individual geometric systems, even non-Euclidean ones. So that it may be certain of these properties, indeed the first principles of such a science must be enunciated, each and every one. This done, one system e.g. Euclidean geometry can be enunciated as a massive conditional proposition. Let us indicate the principles with the letters A, B … and F, then the conclusions with letters P, Q, R … Entire geometry can now be enunciated thus: If A and B … and F, then P and Q and R and … This proposition, as truly conditional, enunciates intrinsic necessity. And this science will be: hypothetico-deductive. Many axiomaticians wish pure geometry to be nothing but such a hypothetico-deductive system. The fundamental hypothesis i.e. the series of axioms A, B … F, is no longer examined relative to the real value of the propositions; indeed according to extreme axiomaticians, not even relative to their meaning. Most well-known are the words with which the distinguished Hilbert begins the exposition in the first chapter of the book Die Grundlagen der Geometrie:
“We think of three different systems of things: the things of the f i r s t system we call points and we indicate them with the letters A, B, C, … ; the things of the s e c o n d system we call straight (lines) and we indicate them with the letters a, b, c, … ; the things of the t h i r d system we call flat (surfaces) and we indicate them with the letters α, β, γ, …” 14.
Therefore neither value nor meaning imports but they only require that from the premised hypothesis, according to pure formal logic, deduction be made by correct syllogisms.
This position at first glance can seem sufficiently Aristotelian; for also according to the philosopher, science strictly so called — insofar as it is opposed to the intellect of principles — purely regards logically deduced conclusions. Yet we shall see in its place that a massive discrimination is present.
Now, if one does not care about the meaning of axioms, it is clear, there can be no talk of the evidence of axioms.
Whence neither can we be certain a priori that from such a system of axioms no contradiction will follow. Whence axiomatics devotes itself to finding methods to prove that no contradiction will follow.
This position leads both to problems and to precious results. The first problem is the same which always returns, if we abandon the natural way according to which the human mind proceeds (“le cheminement de la pensée”). The human mind thinks it knows objective extension optimally and that a most certain science can be constructed about such an object. Such an axiomatic, non-contradictory system does not without further ado attain this object; for application to be made, the entire classical problem of transition from inexact (and perhaps contingent) sensitive data to exact and necessary science seems to return. For classical geometry was not reinvented solely through a hypothetico-deductive system.
But axiomatics has also discovered precious results. See: as has already been said, it begins from a system of “axioms” concerning whose evidence it supposes nothing, and therefore without any doubt a system could be established from which quickly or after longer deduction a contradiction would follow. That the deduction be safe it is therefore required that this be guarded against. And they investigated and found methods of demonstrating that a determinate system is immune from contradiction even in future conclusions. Such methods, sometimes most ingenious, are without doubt a precious evolution of logic.
Moreover these methods permit inquiries into particularities of the structure of a science, just as Aristotle did in the Posterior Analytics regarding the general structure. Not all conclusions exact the entire series of axioms for their deduction; and it could be determined in diverse cases how a system of axioms can be distinguished into diverse parts, which flow into diverse parts of the science. Thus the structure of a science can be more distinctly known. We have a further evolution of the doctrine of Aristotle. Coherent with this question and with the problem of non-contradiction are methods of inquiring into the true mutual independence of the axioms which are proposed; and these also bring gain for “judicative logic”.
And this entire method of proving non-contradiction seems to lead to a problem which coheres with ontology, namely with the theory of possibles. Described as possibles are: objects whose notes bear no contradiction to each other. Can this theory be applied to systems which according to axiomatics are immune from contradiction, e.g., to non-Euclidean geometry?
And in general to the problem of “mathematical existence” about which also Aristotle and St. Thomas have a theory, still to be evolved?
From this brief conspectus it is allowed to conclude: in this part of philosophy we see with great sorrow the absence of investigations of scholastics. Without doubt this part of “judicative logic”, which science had been so felicitously begun by Aristotle, especially after the modern critique of these questions, must be cultivated again; it is a question of the gravest problems of the theory of knowledge, not only special but also general, of problems of logic and of problems, regarding the structure and activity of science, epistemological.
It is also a question of completely constituting the foundations of geometry; but we shall not specially attend to this; the first problems will exact our attention, because there especially it is a question of first principles, of a purely philosophical matter. We will not need considerations of high mathematics; mathematical elements will suffice.
If anyone should think that Aristotle, where he speaks of postulates, thought of the proposition which is the famous V postulate of Euclid or similar to it, we would not contradict him; although we can in no way prove it. ↩
See our work La théorie du jugement d’après St. Thomas d’Aquin, cited above. ↩
More on these things later. Cf. temporarily GEYSER, Die Erkenntnistheorie des Aristoteles, ch. VI and XII; O. HAMELIN, Le Système d’Aristote, pp. 258 sq., 234 sq.; W. D. Ross, Aristotle, pp. 38-41, 54, 217; furthermore those things which we wrote in the article “De origine primorum principiorum scientiae” in Gregorianum XIV (1933) pp. 153-184. You will also find this article in the appendix to the 2nd ed. of Th. d. J. ↩
“I do not wish to say that this ‘arithmetization’ of mathematics is a bad thing, I say that it is not everything”, ibid. ↩ ↩2
“In the ideal field of arithmetic there is no finite threshold value, as in the empirical field, but rather the exactness with which the numbers are defined or at least viewed as defined, is unlimited” (op. cit. p. 11). ↩ ↩2
Op. cit. in ed. 2 (Leipzig 1907) p. 7. “In all these practical fields there is a threshold value for accuracy.” ↩ ↩2
A. EINSTEIN, Geometrie und Erfahrung, Berlin 1921: “How is it possible that mathematics, which after all is a product of human thought independent of all experience, fits so excellently the objects of reality? Can human reason then, without experience, through mere thought, fathom the properties of real things? To this, in my view, the brief answer is: in so far as the propositions of mathematics refer to reality, they are not certain, and in so far as they are certain, they do not refer to reality.” Let the reader apply this to the mathematical proposition “2 X 2 = 4”; insofar as this proposition refers to real things it is not certain, insofar as it is certain it does not respect real things! ↩
See our Cosmologia ed. 4 note III pp. 446-455, VII pp. 471-482. ↩
See H. HASSE and H. SCHOLZ, “Die Grundlagenkrisis der Griechischen Mathematik” in Kantstudien (1928) pp. 4-34; H. SCHOLZ, “Warum haben die Griechen die Irrazionalzahlen nich aufgebaut?” ibid. pp. 35-72. ↩
In the work La valeur de la science p. 17. “Intuition can give us neither rigor nor even certainty; this has been noticed more and more.” ↩
“We therefore have several kinds of intuitions; first the appeal to the senses and to the imagination; then, generalization by induction … finally we have the intuition of pure number … The first two cannot give us certainty, I have proved this above by examples; but who would seriously doubt the third, who would doubt Arithmetic? Now, in the Analysis of today, when one wants to take the trouble to be rigorous, there are no longer anything but syllogisms or appeals to this intuition of pure number, the only one which cannot deceive us. It can be said that today absolute rigor is attained” (pp. 22 sq.). ↩
Op. cit. p. 65. “This definition makes light of the intuitive origin of the notion of the continuum, and of all the riches which this notion conceals. It falls into the type of these definitions which have become so frequent in Mathematics, since one tends to ‘arithmetize’ this science. These definitions, irreproachable, we have said it, from the mathematical point of view, could not satisfy the philosopher.” ↩
“Decisive for the assessment of the situation seems to us to be the circumstance that an essentially analysis-independent geometry, as the ancient ideal would properly demand, has turned out to be a Utopia.” The underlining is of the distinguished Study himself. ↩
“We think of three different systems of things: the things of the f i r s t system we call points and designate them A, B, C, … ; the things of the s e c o n d system we call straight lines and designate them a, b, c, … ; the things of the t h i r d system we call planes and designate them α, β, γ…” Op. cit. ed. 7 (1930) p. 2. ↩