CHAPTER V

ON AXIOMATICS

§ 1. On the Axiomatic Method.

1. On Axiomatics Strictly So Called.

In the preceding chapters, mention was often made of “axiomatics in the modern sense,” that is, of the method of constructing science—in our case, geometry, even Euclidean geometry—through purely logical deduction (i.e., deduction employing logical means alone) from a determinate, pre-established system of “first propositions,” which are called axioms.

In this construction of science there is not, nor can there be, any question concerning the truth of the axioms and conclusions, but only concerning the correctness of the reasoning from premises, which are either axioms or propositions already legitimately deduced from them, and concerning immunity from contradiction in conclusions correctly deduced.

The words employed are the same as those occurring in classical geometry; but they are employed merely as terms whose meaning is to be supposed unknown (above, Chapter I, § 5). Except, however, as was already said before, in Hilbert, the case of numbers, which are employed in their intuitive sense together with their proper attributes. Purely logical terms as well (“and,” “if,” etc.) are understood in an intuitive sense, together, as we shall say, with their logical properties; the same must be said of terms from the logic of relations. Absolute exclusion, however, is made of intuitive cognition of properly geometric terms. If in what follows we speak of “axiomatics” without qualification, this method is intended.

Because this method, perfectly elaborated by mathematicians, is proposed as the most perfect method of constructing science—every science—general epistemology will have to examine it: admitting what is sound, and rejecting or correcting whatever is erroneous. From this labor Scholastic epistemologists will not be able to withdraw themselves.

2. On the Method of Aristotle.

Aristotle, as was said, expounds a theory of science in general that at first sight is very similar to this one. He too constructed it according to the geometric data of his time; and Euclid in his Elements applied this theory [1].

Aristotle distinguishes between “understanding” strictly so called and “science” strictly so called (νοῦς and ἐπιστημή). Understanding provides the “principles” of science, while science provides the deduction alone, by means of the syllogism and other logical instruments, from those principles and from conclusions previously deduced. And “geometry” is a “science”; its function is not to establish its own principles. The Aristotelian theory of deduction must demand first principles. Because the value of conclusions depends upon the value of premises (the reasoning itself being supposed correct), first principles must be given; otherwise we would be driven either into a vicious circle or into a regress into infinity. Principles must be given that are known without geometric reasoning.

The sole function of geometry itself, therefore, is pure deduction from first principles, which nevertheless must be established from elsewhere; either they are “intelligible through themselves” (from the “terms alone”) or they must be established by another science. (Cf. above, Chapter I, § 1). The system of Aristotle therefore has a likeness to axiomatics; his system of “first” or “immediate” propositions can be called a system of axioms, which are the principles from which geometry, “science” itself, proceeds (cf. Gregorianum 1951, pp. 434 ff.). In this sense the system of Aristotle is also “axiomatic.”

It is diverse, however, from axiomatics in the modern sense. Aristotle, for science worthy of the name, demands not only correctness of reasoning, which is the proper function of science, but moreover the truth, and indeed the necessary truth, of all premises, ultimately therefore of the axioms. Then, and then only, is the reasoning a “demonstration.” And this truth must also be known as necessary in order that “science” may be had. Concerning this object the entire Posterior Analytics treats. It is clear that there the cognition of the meaning of terms is presupposed, both in the axioms and in the conclusions demonstratively deduced.

From this first diversity from modern axiomatics others flow, as we shall see, and those of the greatest moment.

3. Semi-Axiomatics.

A certain system, in some way intermediate, is also described, or at least indicated. Let geometry begin from a system of axioms, or “first propositions,” that are acknowledged and known as true and (it must be added: if they are not intuitively intelligible through themselves) are to be established by another science, which the distinguished Freudenthal calls “pre-geometry” [2]; other moderns, among whom Gonseth stands out, designate it by the name “la doctrine préalable” [the prior doctrine]; it is also called by the ancient name “dialectic,” which nevertheless does not seem to be identical with the dialectic of Plato. We today call that science rather “noetics.” For it inquires into the properties of the human intellect from its concrete activity, which constructs science, including geometric science; and this activity becomes known to the intellect from its concrete object (according to diverse modes of concretion). To this noetics, and indeed to its initial part (because the human intellect begins from the senses and the first things truly understood in sensitive data are properly arithmetic and geometric data), pertains that pre-geometric science. But it is clear that noetics extends itself much more broadly. In this science, therefore, attention is directed to the very activity of the human mind; but it is not psychology; for the activity of the human mind essentially has a twofold aspect: according to one aspect, it regards the subject itself, the soul, and if this is considered, we are engaged rather in psychology; but according to the other, intentional aspect, that activity regards the object; contact with the object is never broken and is primary; from the object the very noetic nature of the intellect is known; then we are engaged in the other science, in noetics.

This requirement of first principles for geometry is perfectly Aristotelian. But there are thinkers (thus the distinguished Freudenthal) who add another requirement, which agrees with the requirements of modern “axiomatics.” They demand, namely, that by that prior science an entire system of first propositions be established, in such a way that thereafter no “appeal to the phantasm” be permitted, and that no new first proposition be tacitly introduced (as Euclid rather often does). And this requirement is posited, as they say, in order that deductions may have perfect logical rigor. And the system will have to be complete in this manner, from the very beginning, so that the whole of geometry can be deduced from this system without the introduction of any new proposition read from the phantasm.

In the preceding chapters we attempted to establish diverse axioms through this noetics; but we did not construct a complete system of first propositions, or axioms. In what follows we shall attempt to establish that this is impossible. We attempted to develop noetics itself relative to diverse geometric problems and their modern and more ancient discussions.

The method that the distinguished Freudenthal proposes can rightly be called “semi-axiomatics”; it is intermediate between the method of Aristotle and Euclid on the one hand, and on the other the method that Hilbert champions. Axioms in semi-axiomatics have an intuitive geometric meaning, but then, in the deduction of science itself, we must abstract from this meaning; and this is demanded in order that “we may arrive at perfect rigor.”

§ 2. On the Foundation of the Possibility of Axiomatics.

In evaluating axiomatics, a certain general difficulty immediately arises, which nevertheless is easily solved. Someone will be able to say: if the meaning of geometric terms is ignored, or must be completely neglected, shall we be able to proceed logically in such a subject, through logical processes of the mind? This difficulty, as was said, is easily solved; already in the Middle Ages, indeed by Aristotle, it was already practically solved. The solution consists in that process which we previously called “symbolic abstraction” (see Théorie du jugement, Ch. VII, § 6, and above, Chapter IV, § 7).

1. On the Syllogism Expressed in Words.

This “symbolic abstraction” consists in this simple perception of the mind. In every classical science, the syllogism is enunciated in terms known to us with respect to their meaning, even their specific, concrete meaning. But to the syllogizing mind it is clear that the truth of the propositions of the syllogism must be distinguished from its correctness, and that the latter does not depend upon the former. This, since the time of Aristotle, has also been explicitly established. Hence we can neglect the meaning of terms, and attend to the correctness of the syllogism alone, inasmuch as it remains perspicuous to us in this “abstraction.” Therefore we can introduce other symbols in place of material terms, so that the correctness of consequence, even in the new symbolic formula, remains clear to the intuition of our mind. Let there be a concrete syllogism: every man is mortal, every philosopher is man, therefore every philosopher is mortal. If for the material, concrete terms we introduce symbols: “every B is A, now every C is B, therefore every C is A,” for the human mind (at least after practice) the correctness of the syllogism, the consequence, remains perfectly clear. Aristotle himself already applies this in enunciating the forms of syllogisms [3].

In the Middle Ages these symbols (letters) employed in expounding logical theory are called “transcendent terms” (termini transcendentes) in St. Albert the Great and in the author of the Summa totius logicae. (See above, Chapter IV, § 7, p. 148; Théorie du jugement, Ch. VII, § 6).

Conditions. In this “abstraction,” in which transcendent terms are put in place of words whose meaning is known, in order that the syllogism may be valid and be understood to be valid, two conditions must be fulfilled.

1. The terms must have a meaning, even if this remains unknown to us. In the example adduced, the three terms A, B, C each occur twice; in order that the consequence be valid, the same term in these two places must have the same meaning; it is supposed, therefore, to have a determinate meaning. And this meaning must be such that those terms signifying it can meaningfully be the subject and predicate of a proposition. Hence what is often said in this doctrine, “those terms have no meaning,” is erroneous.

2. Moreover, the structure of the symbolic proposition must be known to us, that is, the diversity of functions of subject and predicate; otherwise the consequence is not understood by us to be present; and in the very formula of the proposition this must be clear from the order, the structure, of the entire proposition; this will later be shown to have great weight. That order is decisive is evident already from the names of those terms; one is called the middle term, the others extremes.

It is not necessary, however, that these things have already been theoretically examined, that is, that “taught logic” (logica docens) precede. That such reasonings are valid, and perspicuous to us as valid, can already be established from the natural activity of the human mind. Therefore “taught logic” itself is also possible, which is constructed and examined theoretically; and that without a circle. How broadly logic constructed by means of Aristotelian symbols can be developed, we demonstrated in the work cited above. We naturally know, therefore, the valid logical structure of reasoning in every syllogism that is not too intricate. If it is such, we must invoke taught logic.

What has been said regards so-called predicative propositions, which consist of a subject and a predicate, as is customary in classical logic. But the same holds for “relational” propositions considered in the logic of relations, which are scarcely examined in classical logic. In these reasonings too we have a natural intuition into the validity of reasoning, if the form is not too complicated.

And this is the foundation of the possibility of axiomatics. That preliminary difficulty is solved and is easily solved; we must, however, in order that the solution be valid, suppose that in such a mode of procedure the human mind primarily still operates and under this light attend to its structure; if this operation is moreover eliminated, and in its place purely external manipulation succeeds (as occurs in logistics), we shall have to speak in another way, as we shall see. Now let us attend to that operation of the mind.

2. On the Mental and Verbal Syllogism from a Noetic Perspective.

A) Syllogizing Activity is Intuitive.

Reasoning by syllogizing is primarily an activity of the human mind, and indeed a natural activity, which can certainly be perfected by art.

The syllogism itself says nothing about the mind itself and its operation, but only about the object of cognition. We intuit that from two determinate “dispositions of things” [4] there necessarily follows a third, other than those premises, and this we affirm. It is a natural activity of the human mind concerning its objects. The mind, therefore, in this its operation intuits a certain necessary connection between these dispositions of things, and therefore affirms.

This intuition of necessity is of the same genus as that which produces the affirmation: “a body is divisible through figure,” of which we spoke above. In such a geometric case, the intuition of the necessity of connection between subject and predicate through mere inspection arises from formal abstraction; there we immediately detect that the body’s “divisibility through figure” flows from this single attribute which we call extension (cf. above, Chapter II, § 4). It is an abstraction worthy of the name of this connection between two forms (extension and figurability) from the concrete entity, which contains many forms. Therefore we can say “a body, because extended, is figurable”; extension is the sole property of a body from which figurability flows.

In an analogous manner the mind is active in syllogizing. In the concrete syllogism mentioned above, which concludes to the mortality of philosophers, the human mind immediately sees that this conclusion necessarily flows from the disposition or order of notions or terms, precisely as they are subject or predicate respectively, in those propositions. Specific meaning itself, however, does not influence (just as color, etc., do not influence figurability) the correctness of the syllogism; hence the intuition of the consequence of the syllogism remains if this specific meaning is neglected and transition is made to transcendent terms, symbols; there, in opposition to the case of geometric truth, “symbolic abstraction” is justified.

B) Syllogizing Activity is Natural.

Above we were already saying that this activity is natural to the human mind, that is, the mind exercises it antecedently to every logical theory treating of this activity, and knows that it does so legitimately. And this logical theory can only be developed in reflection upon natural activity.

This is clear from the very history of logic. Aristotle, in constructing this theory, discovered the principles of this science in the syllogisms (not always explicitly formulated) that men used in the sciences, indeed in daily life. And Aristotle himself, in constructing his logical system, always had before his eyes the most perfect science existing at that time, geometry. Locke expresses this facetiously (though against logical theory) by saying: “God has not been so sparing to men to make them barely two-legged creatures, and left it to Aristotle to make them rational … He has given them a mind that can reason, without being instructed in methods of syllogizing” [5]. And even after Aristotle, men, learned and unlearned, who do not know logical theory, reason according to certain syllogistic moods (especially Barbara), and in general, not accidentally but per se, reason well. Below we shall also see very frequent specimens of reasoning in Barbara, where men do not suspect that such a syllogism is present.

Such correct reasoning is an activity proceeding from the “nature” (most strictly so called, Aristotelian “physis”) of the human mind; it is not the application of a general theory to particular cases. “Logical laws” are thus not “norms” imposed from without that the mind looks to in its operation in order to reason well; they are acts proceeding from its nature, they are its very life. Thus “physical laws” (which are very improperly so called) are not “norms of acting” for physical things; they are descriptions of their activity and passivity, that is, of their “nature.”

Those “logical laws” are not to be called psychological laws either, for they regard objects. They are often called “laws of thought” (German “Denkgesetze”), which name nevertheless must be interpreted as: “laws of what is thought,” or “of what is to be thought” (German “des Gedachten”) [6]. They are best called “noetic laws.” Therefore they are accurately expressed in terms regarding the objects themselves; even if they are expressed “purely” logically, that is, in such a way that transcendent, symbolic terms omitting the determination of meaning are explicitly employed, these terms always stand (supponunt) for the objects themselves, now unspecified. Thus above we already heard Aristotle affirming the correctness of the syllogism of the mood Barbara by a conditional proposition that affirms something universally concerning objects. It is a “logical law,” but it is expressed as a function of terms that stand for the objects themselves. Thus also in the case of such truths holds the adage of St. Thomas: from the object the act is known (and its noetic nature), and from the act the potency, that is, the noetic nature of the general activity of the syllogizing mind. The intuitive act begins from the concrete object; the intuition remains in the transition to the abstract object, that is, in symbolic abstraction.

In constructing science, this form, which can thus be abstracted, is present in determinate, concrete dispositions of things; hence logic, which the human mind naturally uses in constructing any science whatever, is form in the matter of such science; it is “logica utens” or logic “in use.” Hence St. Thomas identifies logic in use, in diverse sciences, not with the science of logic itself, but with the concrete, material science of which it is the form. Thus, if I mistake not, the text from Metaphysics IV, lect. 4, no. 577 is to be understood.

And thus the laws of logic in use are not “norms” properly so called; they are, as already said, descriptions of natural activity; they render the nature of the syllogizing mind—the noetic nature, I say, which regards the object—and therefore they are expressed in words that immediately regard objects.

Taught logic (logica docens), or theoretical logic, which is developed from this natural logic, discovers indeed logical, syllogistic laws that can also be and are employed as norms in judging the correctness of a complicated syllogism. In deducing this logic we certainly also use syllogistic reasonings; but these syllogisms pertain to natural logic, which is intuitively known (or are prior laws thus derived). It has been said [7] that theoretical logic necessarily contains a circle (which is not considered vicious). From what has been said, it is evident that this is not true. Logic could not be constructed without a circle, and indeed a vicious circle, if all logical laws, all syllogistic moods, even the mood Barbara itself, had to be demonstrated. That this cannot be done, but also that this is not required, has already rather often been said (cf. Théorie du jugement, Ch. VI, § 4).

We were saying “even the mood Barbara itself.” For this recurs again and again. If we wish to examine a certain concrete syllogism, in any science, with respect to its correctness, we inquire into its logical form; if this is one of the reasonings established in theoretical logic, it must be admitted. This is the truly normative function of theoretical logic. But this examination or its result (in the mind!) is a syllogism in Barbara, namely: Such a mode of reasoning is legitimate; this is the universal major. Now this reasoning is in that mode (this is the minor established from inspection of this syllogism). Therefore. Later we shall see again that this mental use of the mood Barbara occurs very frequently, in all sciences.

Thus in constructing science, in deducing, we can and do employ a twofold method indiscriminately. We can attend to the logical form of premises expressed in words (whether attending to, or attending less or not at all to specific meaning), and naturally intuit that from these premises such a conclusion follows. But we can also, and sometimes must, compare this form with a logical law deduced in theoretical logic.

In this “taught,” theoretical logic, we shall be able, even according to Aristotelian and medieval symbolism, to deduce a truly huge multitude of logical truths, as we demonstrated in the work cited above, Recherches de logique formelle. And this is to be marveled at: these truths, even very complex ones, which can be multiplied into infinity, retain this form indicating their origin from the contemplation of concrete objects, and manifest their relations to objects. This we already marveled at in op. cit., p. 315. They arise from the activity of the human mind, but they are not psychological, but noetic.

These logical laws treat universally of things themselves, of any disposition of things whatever; hence they have the same subject as metaphysics, which considers all being, according to the words of St. Thomas (In Boeth. de Trin., q. 6, a. 1): “both sciences (metaphysics and logic) are common, and in a certain way concerning the same subject” (text according to the Wyser recension). It is what Riehl (loc. cit.) expresses by saying “Denkgesetze sind Gesetze des Gedachten, des Gegenständlichen überhaupt” [Laws of thought are laws of what is thought, of the objective in general].

Thus also axiomatics strictly so called will be able to proceed; only this is required, that we do not attend to specific meaning. We use only natural or artificial logic. One thing must be added: if in the axiomatic method we arrive at a conclusion that is intuitively erroneous, it is not permissible to use this datum; only if it is in contradiction with an axiom or theorem previously deduced. Only this argument ex absurdo is permitted.

§ 3. On the Execution of the Axiomatic Program.

1. Preliminary Remarks.

From the foregoing it is permissible to conclude: the axiomatic method, which uses natural logic and, where needed, applies also intuitive taught logic, will be able from a presupposed system of axioms to deduce a body of theorems without attending to the meaning of the words expressing the matter of these propositions, the subject of the science (in the case of geometry, to geometric meaning); it will be able to construct a science, geometry, which in words is not distinguished from classical geometry, including non-Euclidean geometry.

This system of axioms must necessarily satisfy the requirement of immunity from intrinsic contradiction; that is, it must be such as to exclude the possibility of arriving (in legitimate reasoning) at a conclusion contradicting an axiom, or another legitimately deduced conclusion. And this immunity from internal contradiction must be established, even for the future. A system of axioms arbitrarily chosen does not per se enjoy such immunity. In semi-axiomatics, this contradiction is excluded because the system of axioms is constructed intuitively from reality. And strict axiomaticians have discovered most ingenious methods to demonstrate this immunity, even for future conclusions.

There are thinkers who say that this requirement is not demanded so that a certain reality may correspond to the system (in real things contradiction cannot be present), but is posited only because otherwise any conclusion whatever could be deduced. For once a contradiction is deduced, two contradictories taken together constitute an absurdity; and according to the logical law “ex absurdo sequitur quodlibet” [from an absurdity anything follows], from that contradiction any proposition whatever would be legitimately deduced, according to this rule. And a system in which anything whatever can be demonstrated no longer has any weight or “interest.”

Concerning this extreme position we say only a single word. The word “follows” (sequitur) is used thus in the sense it has in the “material implication” of logistics, which establishes no necessary intrinsic relation between premises and conclusion. In the Middle Ages, by all thinkers so far as I know (except perhaps Pseudo-Scotus), the adage “ex absurdo sequitur quodlibet” was used in another sense, where “follows” has the sense of intrinsic consequence; and moreover “anything” (quodlibet) does not signify “any proposition whatever,” but only “both true and false” (cf. Théorie du jugement, Ch. IV, § 3 B).

If a part of science is constructed in an axiomatic mode, or rather if its parts are examined with respect to logical structure by means of axiomatics (cf. above, Chapter I, § 6), axiomatics can supply precious methods. But axiomaticians and semi-axiomaticians wish to deduce the entirety of geometry from a previously established system of axioms. Therefore they demand (not only immunity from intrinsic contradiction) but also a complete system of axioms. The accurate determination of this completeness, however, is a notorious crux of the axiomatic method. Concerning this we shall not treat. Let us call “complete” that geometry containing all notions (terms) and all theorems present and deduced in classical geometry (including non-Euclidean), and to be further deduced. Regarding this construction of science, axiomaticians greatly praise the rigor of their method, in opposition to the classical intuitive method, which is said to lack “rigor” on account of repeated (often tacit) “appeal to the geometric phantasm.” But in a noetic examination of this axiomatic construction, difficulties are detected that are very serious, and for the most part are not considered. We reduce them to three heads: there is a practical difficulty, a theoretical difficulty (these two are intrinsic), and a difficulty in the application of geometry, which is more extrinsic, but not entirely so.

2. Intrinsic Difficulties.

A) Practical Difficulty.

The practical difficulty consists in this, that man practically cannot apply the axiomatic method to constructing the entirety of geometry; it surpasses human powers. Let us take an example.

Let one read in Hilbert the demonstration of the very simple Theorem 5 (in 7th ed., pp. 5–8), which treats of the order of four points on a straight line, namely, of the relations expressed by the word “between” holding for these points. What geometric “between” signifies is ignored according to the axiomatic method. In the first edition this theorem was introduced as an axiom; later it was deduced from other axioms, also of order; but the demonstration occupies nearly two pages, and is illustrated by one figure. To anyone reading this demonstration it will be clear: if by such a method the entirety of geometry (as defined above) had to be deduced, this work would surpass human powers. Hence no one has done this, nor, we think, will anyone do it. The same holds of semi-axiomatics, which understands the geometric meaning of the word “between” but denies itself its intuitive use.

Let us ask the axiomatician: Do you admit such a theorem, established in classical geometry after a long deduction (let us say the existence of the five Platonic solids), on account of the argument constructed according to Hilbertian axiomatics, or on account of the classical intuitive deduction? The answer will probably be similar to that which the mathematician Hardy gave himself in a similar question regarding logistic demonstrations of mathematical truths according to Whitehead and Russell. He says: “I believe the prime Number Theorem because of the argument by which de la Vallée-Poussin proved it, but not because of the proof found in Principia Mathematica do I believe 2 + 2 = 4” [8].

We said that the demonstration of Theorem 5 in Hilbert is also illustrated by a geometric figure. This of course is not added so that something may be deduced from it, but serves merely as an aid. As a “heuristic aid,” they say; but this must also be understood in such a way as to be an aid to the reader to grasp the logical evolution of the argument. But this too, both of them, all the more reinforces the practical difficulty we are describing, which seems perhaps to return at every single step.

This method is introduced, they say, in order that the rigor of modern mathematical analysis may also be introduced into geometry. The rigor of deduction in nineteenth-century analysis, of that period sometimes called “l'ère de la rigueur de Weierstrass” [the era of Weierstrassian rigor], of the “epsilonic art” (ars epsilontica), is truly admirable (though in its very foundations it seems lacking). But if the practical difficulty (and the defect of elegance) becomes so great in geometry, we do not say in analysis, doubt can arise whether, even if theoretical rigor is present, it practically subsists. We say, if it is present theoretically; for a theoretical difficulty is also present, a doubt concerning right (dubium iuris); concerning this we must now treat.

B) Theoretical Difficulty.

That a difficulty concerning right is present is discovered immediately, though in a still vague way, if one compares the small number of axioms gathered in the prior system with the huge multitude of conclusions deduced thence, of constructions that ultimately find their justification in them, and thus demonstrate the mathematical “existence” of mathematical entities. Is the power of the simple syllogism so miraculous? The difficulty in the beginnings becomes less if one considers that geometric propositions, of axioms and of theorems, for the greater part are not purely “predicative” propositions, but rather “relational” ones; these indeed are often reduced or resolved into predicative ones, but this is done only very artificially, and thus a great complication is introduced that does not seem to correspond to the real process of our mind. But in these relational propositions, very many things cannot be understood and then affirmed without a new appeal to the phantasm. These propositions, however, have been little examined in classical logic, and logistics seems to avoid noetic logical problems; these without doubt ought to be more investigated by Scholastics. Let us recall the dictum of St. Thomas cited immediately above (Chapter V, § 2 B), which teaches that logic in use (as a habit) is identified with the special science that “uses” it.

And in such a case of “appeal” to the phantasm, the disposition of the thing looked upon is geometric rather than “logical”; and we shall have a new “first proposition,” immediate, besides the axioms. We repeat: here is present a great field of noetic investigations, and let these be commended to Scholastics. In such a deduction of “complete geometry,” are “appeals to the geometric phantasm” truly absent?

In order that an answer may be had, noetic inquiry must not be restricted to “pre-geometry,” or the “prior science,” but must extend also to the very deduction of geometry; the activities of the human mind in this very deduction must be examined; this noetics is not geometry, but a science whose object is geometry in its constitution, in its becoming (in suo fieri); it is a philosophical science, a habit of its own kind, which certainly not all mathematicians possess. This, therefore, is the question: in the very activity by which the human mind constructs geometry deductively, is every appeal to the geometric phantasm excluded, or can it be excluded?

Poincaré, who certainly does not despise the methods of modern axiomatics, already maintained that the “axioms of order” established by Hilbert, and topological elements in general, cannot be employed in geometry without a certain intuition of this kind (Dernières Pensées, pp. 93–95; cf. ibid., p. 271). And yet Poincaré, as we saw above (Chapter I, § 4), did not love geometric intuition. Does Hilbert in his deductions truly exclude every appeal to the geometric phantasm?

If anyone asserts this (and very many mathematicians seem to do so), he will have to demonstrate it from a noetic examination of the individual steps of such deduction; which, so far as I know, they do not do. Meanwhile it will be permissible, with Poincaré, to doubt the “rigor” of the deduction. But more must be said: such an appeal seems to be present; and that immediately. If in Hilbert (loc. cit.) we read the following theorem (6), which is a generalization of Theorem 5, and which (it is to be regretted) is not demonstrated but only enunciated, we observe these things: 1) a figure is proposed from which the meaning of the assertion is to be read; 2) also a series of symbols (letters) for the same end; these only regard the practical difficulty spoken of above; yet they are serious. But the theoretical difficulty is also not avoided.

Behold the enunciation of this Theorem 6: “In any given number of points of a single straight line, these can always be so designated with letters A, B, C, D, E, ….. K

that the point designated by B lies between A on the one hand and C, D, E, …. K on the other hand, further C between A, B on the one hand and D, E, ….. K on the other hand, then D between A, B, C on the one hand and E ….. K on the other, etc. Besides this arrangement

  A    B    C    D    E        K
        ──•────•────•────•────•────────•──
        

there is given only a single other that is of the same structure, namely the inverse” [9].

The order of this series of symbols must be looked upon in order to understand what is said in the theorem concerning the order of points on a line, which arises from the unknown relation “between”; the theorem asserts that the order of points is designated by the intuitive order of the series of letters. In the relation of points, the meaning of the word “between” is neglected, but in the order of symbols, in whatever way it is described, it cannot be neglected. Therefore also those new terms can be introduced by Hilbert which he uses, which previously were neither named nor (implicitly) defined, namely the terms “on the one hand” and “on the other hand” (einerseits and andererseits), whose meaning must be understood intuitively and is taken from the order of letters (or even of words describing the abstract relation). But these terms then intuitively indicate the nature of the relations holding for letters, specific specimens of which in individual cases of applying the theorem must be looked upon in the phantasm, and understood in it as necessary; then there is always had an appeal to the phantasm of order between symbols or words. And what is of greater moment: the meaning of this intuitive relation expressed by the words “on the one hand” and “on the other hand” is entirely the same as the meaning of the relation “between” (it can be called “interity,” just as in English they say “betweenness”) attributed to points. For “betweenness” in a series of symbols is a geometric “betweenness,” because this series can only be looked upon “in space”; it is a spatial relation. If this intuition is called “logical,” it must be urged that that same order, even in symbols, is primarily and intuitively “geometric,” spatial, and remains such.

And wherever this theorem is applied, and that happens very frequently, whether directly or indirectly, an appeal is had to such a geometric phantasm; and in axiomatics this is done tacitly, as it seems. And let us observe: even if, as is done in axiomatics, this proposition is considered as a deduced theorem, its application in specific cases demands an appeal to a specifically determined phantasm, and thus a first, immediate proposition arises in the middle of deduction. Poincaré said well that in matters governed by “axioms of order” and in topological matters in general, appeal to intuition cannot be avoided. And in the axiomatic method, this appeal consequently is made tacitly. We of course do not say that deduction on that account is erroneous, nor even that this harms its rigor. But it certainly cannot be said that geometry, after the system of axioms is established, no longer appeals to intuition. And thus we better understand Euclid, who is often faulted because he explicitly establishes no axiom of order, but tacitly introduces those he needs. Nor is it a fault not explicitly to enunciate, but tacitly to employ, such judgments that legitimately result from the inspection of the phantasm. For they are virtual judgments, which the noeticist encounters so often, as we saw in what was expounded above in so many pages, and shall see even more often in what follows. If they are enunciated—which is not forbidden—they constitute first, immediate propositions. And thence seems to follow what was said above: it is impossible from the beginning to construct a complete system of axioms, complete in this sense, that it would render superfluous all subsequent appeal to geometric intuition. If the propositions that thus result are not to be called “axioms,” they are nevertheless first, immediate, intelligible through themselves, not deduced; and if the name “axiom” is denied to them, the same seems ought to be done regarding “axioms of order”; hence these are not enunciated by Euclid.

The conclusion of the enunciation of Theorem 6 also leads to the same conclusion. There it is said: “besides this arrangement (Bezeichnungsweise) there is given only a single other, which is of the same structure (Beschaffenheit), namely the inverse (die umgekehrte).” There the term “inverse” (umgekehrt) is supposed known as to its meaning. It is considered, of course, as a “logical” term signifying the intuitive order of letters arranged in a series. But in a manner similar to what was said above, it must be said: This arrangement of letters in a linear series is a spatial arrangement; its observation, and the intellective intuition associated with it, are geometric observation and intuition. Thus, however, we have not the introduction of logic into geometry, but the introduction of geometry into logic, precisely as the use of numbers in geometric axiomatics is not something logical, but something arithmetic in logic. And this arrangement in a series that can be inverted coheres with the notion of geometric direction—not the direction of a line relative to the surrounding extension, but along the line itself, which can be twofold so that one is inverse, opposite, to the other. In this too, this geometric intuition is not excluded, but included.

And we understand the position of Poincaré, spoken of above, who holds that in topological elements intuition cannot be excluded.

What we detect in the noetic examination of relational propositions holds in similar fashion of the logical instrument itself that we use in rigorous deduction, that is, of syllogistic reasoning. It is impossible for us to think without a phantasm. It is not necessary that this be a phantasm of the object itself (indeed objects are given that cannot be attained by any phantasm); a verbal phantasm, or one consisting of other symbols, will be able to suffice under certain conditions. In the axiomatic deduction of geometry we need such a phantasm, and not only for the exposition (for others) of our reasoning. In Hilbert (in his geometry), these are phantasms of words. In these the human mind, under certain conditions, can still have an intellective intuition of syllogistic consequence; without such intuition, deduction would already disappear. We observe two things: 1) there we continuously need a tacit appeal to those phantasms; tacit, for explicitly there is enunciated only what we intuit in the phantasm, not the correctness of the syllogistic form itself (and we do well, for to do this always is impossible; see Théorie du jugement, Ch. VI, § 4, 2nd ed., p. 201). But in order that this be done correctly, we must indeed intuit the order of words, which again is spatial; as was already said above, the very logical names “middle term” and “extreme terms” indicate this. In logic itself we need geometric intuition, absolutely, and more than elsewhere. We observe 2): With the best right, geometers are not accustomed to mention the correctness of their syllogisms (their form), just as they make no mention of the nature of the human mind’s activity in such deduction. Only logisticians sometimes place in a single body of “axioms” both purely logical (i.e., logistic) principles and principles regarding the matter of the science treated. But logisticians must employ other means of deduction, manipulations. And now it is marvelous: in such logistic deduction, as also in algebraic deductions and transformations, at individual steps, appeals, even tacit ones, to phantasms (and syllogisms!) can be indicated by the noeticist. Concerning these, more in the following section.

Pure axiomatics, therefore, even theoretically cannot suffice to construct the entirety of geometry. But the axiomatic method itself has its great utility in investigating the structural connections found between diverse theorems of this science (cf. above, Chapter I, § 6). In carrying out this investigation, however, the axiomatician will have to attend to first propositions that he perhaps assumes from intuition.

The difficulty flowing from application we shall consider in § 5; first, in § 4, we must treat of the appeal to the phantasm in sciences other than geometry.

§ 4. On the Appeal to the Phantasm in Other Sciences.

1. General Remarks.

Geometry, therefore, on account of difficulties both practical and theoretical, cannot be constructed in an axiomatic, or even semi-axiomatic, mode. For man, who in this life cannot think without a phantasm, if he does not have a phantasm of the object itself being considered, at least and always needs a phantasm of words or other signs; moreover, he needs the inspection of combinations of words or other symbols representing connections of objects. Indeed, both phantasms must have a spatial structure, in this sense therefore geometric. Now this “appeal to the phantasm” is a recourse to sensitive data, but such that with it there is associated an intellective intuition, which in individual cases in its own way must solve both the problem of necessity and, if present, the problem of exactitude. These things are established from the foregoing.

And we saw that Euclid is to be praised because even in the middle of deduction he admits recourse to the phantasm in cases determined in individual species. This we praised above (Chapter IV, § 7) where his first theorem was treated, in which he was vehemently carped at. In that case this recourse permits him to arrive at an exact conclusion, without an illegitimate transition from an inexact phantasm to exact intellection. Nor is he to be blamed because he does this tacitly.

In order, however, that this necessity of recourse to the phantasm, and its universality, may become more evident, let us see specimens from other sciences, among which are certain ones constructed today explicitly to the end of avoiding “appeal to the phantasm.”

2. In Establishing First Principles.

Above, in the first four chapters, we already saw abundantly that to establish first principles, axioms, appeal to phantasms is necessary, not only for geometry, but also for arithmetic, for combinatorial art. We can add, though we shall not develop it, that this is necessary to establish first principles in the theory of the localization of extended entities, of local motions, and consequently in the doctrine of time. (Concerning these cf. our Cosmologia, or Filosofia della natura inorganica; concerning “time as successive extension” cf. Gregorianum 1953, pp. 3–31). In these, besides the first proper attribute of extension (its figurability), another, equally immediately intelligible to us, becomes known and indeed independently of the first, namely the possibility of contact between diverse extended entities. These theories pertain to kinetics, mechanics, and in part to natural philosophy. This mutual independence of two proper attributes flowing from the same object seems worthy of attention; they are necessarily together, yet one does not flow from the other. Concerning these we briefly observe:

1. Localization is prior to motion according to place; this does not prevent admitting with Aristotle (Physics IV, ch. 1–2) that the notion of place, even as to its existence, becomes known to us after motion itself and from it. In our cognition, motion is prior and place posterior, both as to essence and as to existence.

2. These local relations are relations in the order of existence and between existential acts (cf. our article De connexionibus necessariis inter actus existentiales in the Appendix).

3. As was already said above, in order that successive contact, that is, local motion of a rigid body, be possible, it is required that that extension which is the medium of localization be the principle of individuation, as explained in the question concerning the possibility of congruent figures. This property of extension is therefore more fundamental, prerequired objectively prior to the local motion of a rigid body. Perhaps here too it can be said that the observation of motions of sensibly rigid bodies is prerequired, just as in Aristotle the observation of motion is prior to the notion of place; but in both cases objectively the order of priority and posteriority is inverse.

The objective possibility of motion of a rigid body flows from that property of extension from which the possibility of congruent figures flows, and in our cognition of the very nature of extension, as it is the principle of individuation, this priority is also known. Hence above (Chapter IV, § 3) we had to reject the dictum of Poincaré: “if, therefore, there were no rigid bodies, there would be no geometry.”

3. In Scientific Deduction.

Now, however, our discourse is not concerning first principles, axioms, but concerning the mental processes by which scientific deduction itself, science strictly so called, is constructed. For in developing deduction itself, first judgments are also found, in this sense that they are “immediate,” not themselves deduced, intelligible through themselves from appeal to sensitive data, though not in the pregnant sense that they are universals from which many other things are derived. They can be judgments regarding dispositions of things in the lowest species or “individual species,” as philosophers say with Aristotle. Such judgments occur in geometry, as we saw, but also in other sciences: in arithmetic, in logistics, in logic itself, in other sciences. Concerning these now.

A) On Arithmetic.

We say: in arithmetic deduction such recourse to the phantasm recurs again and again, and cannot be avoided. Let there be a very elementary algebraic equation: ax*² = *bx + c (1). To calculate the “value” of the symbol x (to solve the equation), we have universal rules permitting us to transform such an equation, supposed valid, into another, also valid. Such a rule is: the translation of a term (or complex of terms) from one member of an equation to the other, with sign changed. Thus from that given equation, supposed valid, follows this other, also valid: ax*² — *bx — c = 0 (2).

We ask: what occurs noetically in our mind while we perform such an operation, and then affirm what results? And noetic analysis says: the application of the rule of “translation” supposes this universal truth (the “major”): every equation resulting by translation from any valid equation is itself valid. We add in the mind the “minor”: now from the (valid) equation ax*² = *bx + c by translation there arises ax*² — *bx — c = 0. Therefore this is valid (affirmed).

We observe first: the process of the mind here consists in a syllogism in Barbara, more accurately called “of type Barbara.”

We say rather that this reasoning pertains to the “type” Barbara; for it is not in the classical mood indicated by this name, but very similar to it. For its major is triply universal. If fully enunciated, it must be said: any equation resulting by any “translation” from any valid equation is valid. And the minor will be: this equation (2) by this (determinate) translation results from that valid equation (1). The conclusion immediately resulting is: (2) is valid. It must be noted that the pronouns “this,” etc., do not indicate singulars, but specifics in the lowest, individual species; they are therefore universals. Note moreover: the first “any” from the major is determined not independently, but dependently upon the determinations of the other two “any’s” to “this” or “that,” and that both specifically and individually. Hence the major is more accurately called: doubly universal. Such syllogisms and their correct consequences are immediately understood by the human mind, no less than the classical syllogism in Barbara. These pertain to the treasury of logic “in use,” but have not yet been examined, and do not yet pertain to taught logic.

Naturally, therefore, we intuit in such a mental syllogism the correctness of the consequence and the truth of the consequent.

Hence we have what we first observe: in every such process there occurs in the mind a syllogism (of type Barbara); it is not expressed in words, but mentally directs the act of affirming the conclusion; it does this, as was said above (Chapter V, § 2 B), not as an extrinsic norm, but as an intellective activity of the mind itself, as “nature.” It is therefore a syllogism in “exercised act” (actu exercito); our noetic investigation does not introduce the syllogism, but detects its presence; it causes it to pass from exercised act into “signified act” (actum signatum).

And such an operation seems to be present in virtually every step of this arithmetic science. Other operations in which the same is found are so-called “substitutions” in algebraic formulas. The use of the syllogism of type Barbara seems to be immense.

The second thing to be observed in this operation regards the minor of this syllogism. Now that minor was: by the indicated translation from (1) there arises (2). We ask: how is this necessary and universal proposition established in the individual species? Noetic analysis manifests this: we must carry out the operation of “translation”; which requires an external manipulation, whether actual (on paper) or represented in the phantasm, and the empirical inspection of what results. But with this empirical inspection, with this appeal to the phantasm, as with the activity of manipulation itself, there is associated an intuition of the mind, understanding what results to be necessary. And we observe: this necessity, even in our activity, is known intuitively by us. Hence we do not say: in this experiment, at this hour, in this place equation (2) results, but it necessarily results; of itself always and everywhere the same equation in the individual species results [10]. It is a universal proposition because we are dealing with “necessary matter” and with our necessary activity concerning this matter. And these necessities are known naturally by us, without reasoning, through naturally formal intuitive abstraction.

Under the respect of abstraction, the case is entirely the same as that of axioms, which are first principles, “dignities.” That minor is also a first proposition, in this sense that it is immediate, known without a syllogistic middle, from appeal to intuition. But it is not an axiom, because not many other things are deduced from it: it is employed only once.

In this simplest case, rapid inspection of the phantasm suffices; in complicated cases, this inspection can demand greater care and circumspection. But the noetic elements entering there remain the same.

The second remark, therefore, will be this: in virtually every step of such arithmetic deduction, the human mind must necessarily appeal to the phantasm, to know truths expressed by first propositions. And thus there is present the explanation of this marvelous fact of the fecundity of logical deduction from few axioms; other first, that is, immediate, propositions are found in virtually every step, into infinity.

Here it must be carefully noted regarding a proposition called “universal”: Often this term is explained as signifying a proposition that must be admitted by every human intellect. This meaning is not primary. A proposition is “universal” because it affirms something concerning every object of which the proposition is affirmed. And therefore it must be admitted by every intellect, not vice versa. Because the disposition affirmed is in itself necessary, therefore it is objective, and therefore to be admitted by every intellect. In noetics, cognitive operations are essentially considered as regarding the object, not the knowing subject. In our case we are dealing with universality in the individual species; therefore its occurring cases differ from one another by number alone.

Let us resume: In virtually every step of arithmetic deduction there is present a syllogism in exercised act, and for its minor there is required an appeal to intellective intuition in the phantasm. And we add: in this recourse to sensitive data there is present again an element of spatial order, a geometric element. Why is this excluded in geometry itself?

Both elements present in our mind in each of those steps of arithmetic deduction are applied tacitly; yet this is considered by no one, even among those who carp at this in geometry, as a defect of logical rigor. This “taciturnity” can be removed by noetic inquiry, which, as already said, makes a “signified act” out of an “exercised act.” Hence it is also evident that these exercised acts cannot be called “unconscious.” For they intelligibly direct the operations of our mind; and therefore in reality no defect of rigor is present there. The same two elements are found in another very frequent operation regarding algebraic formulas, namely in “substitution.” Similar specimens of this we shall find in logistics itself.

B) Logistics.

This will perhaps be surprising, but in reality in logistic methods there are elements that must constantly make appeal to sensitive data, although they are destined to avoid this, and above all, as it seems, to avoid this “taciturnity.”

Logistic deduction, even that which constructs this art itself, begins also from a system of axioms. In order that deduction may occur, “rules of operation” must also necessarily be introduced, according to which the logistician transforms formulas already given into others; in this logistic deduction consists; these rules are truly extrinsic norms. In this “logical calculus” those rules have the same function as in classical science logical laws have, which are, as was said, the natural understanding activity of the human mind, perhaps perfected by logical art. According to logistic requirements (at least more recently), these rules are formulated in express words, and when they are applied, this is symbolically indicated. The operations directed by these rules have as objects the given symbols alone.

Primitive and derived rules are distinguished. We name two that seem to be principal: (legitimate) “substitution” and the rule of consequence (or of the deductive schema). Applying those legitimate rules in effecting transformations—in deducing in a logistic mode—we must appeal to intuition, sensitive and intellective simultaneously, in the same way as in the algebraic operations described above. And in different steps of deduction we have different intuitions according to individual species. And then, as in the case of the algebraic formula, we have a syllogism of type Barbara, and an implicit or explicit judgment determinate in the individual species, not deduced. And this is found even among those logisticians who, inveighing against Aristotle, describe the syllogism as the “summit of stupidity,” “the apotheosis of stupidity.”

Let us see a specimen of substitution, and indeed that which constitutes the first step in the deduction of the first theorem (*2.01) from the work Principia Mathematica (p. 100). This theorem, expressed in words, enunciates: “if p implies not-p, then p is false” (p is the symbol of a proposition). The demonstration begins through the substitution of the symbol not-p for the symbol p in the axiom called “tautology,” which says: “p or p implies p” (*1.2). From this substitution there results (1): “not-p or not-p implies not-p.” The affirmation of this result is nothing other than the application of the general primitive rule of legitimate substitution (A proposition resulting by such legitimate substitution from a valid, or true, proposition is valid). But the very content of the result, that is, the specific resulting proposition, is not known except through real manipulation and intuition into sensitive data. For we ask: whence do we know that formula (1) and not another results by the indicated substitution into the formula of the axiom? It is a case similar to the algebraic case expounded above. It is not seen in the general rule; this affirms only in a universal mode the legitimacy of substitution; that thence a specific result may be had, it must be further determined—this is done in individual applications of the general rule—and inspection of sensitive data, appeal to intuition, is necessary. In our case, in the very first step of deduction, this operation is very simple, but cases can also be given that are very complicated. That sensitive intuition is simultaneous with an intellective intuition of necessity, as in the algebraic case; the result, even as to its specific “content” or “determination,” is known as necessary, not contingent. Then from the application of the general rule of legitimate substitution enunciated above, we affirm the resulting formula, with its specific content; it is a necessary, not a contingent judgment.

And these things recur again and again, in virtually every step of logistic deduction, which is a certain combinatorial art according to determinate rules, primitive and deduced.

And let us observe further: these phantasms or sensitive data in general contain not only a multitude of symbols, but also their spatial, geometric order. In non-geometric sciences themselves, in deduction itself, recourse to geometric phantasms is necessary; and these methods are greatly praised on account of their deductive “rigor.” Why, then, in the name of rigor is this recourse excluded in geometry itself?

We note moreover: in logistics, according to Whitehead and Russell, and in other systems too, arithmetic numbers themselves are not supposed known; their notions and the propositions holding of them must be discovered in a logistic mode, after immense prior labor. Yet in hundreds of formulas preceding, they are present, e.g., in the system of punctuation, and these systems of dots must be understood and employed in their natural sense, and there enters either the recognition of geometric figure, or the order of natural numbers, or rather both. Those things are employed intuitively which after much labor are said to be known at long last.

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Recapitulation.

In the foregoing we saw: purely axiomatic geometry cannot be constructed. For we need, at least in topological elements, in individual specific cases, an appeal to intuition, which is an intellective intuition in sensitive data having a geometric structure.

The syllogism itself, which is the means of deduction in axiomatics, in order to be a means of rigorous deduction, requires that its correctness, its rigor therefore, be understood, by the inspection of the human mind into the structure of propositions expressed in words; if this structure is correct, the intellect intuitively sees the correctness of reasoning in these very words (this was the foundation of the possibility of axiomatics); but this intuition is a recourse to spatial order, in its own way geometric.

In order that the universality of the necessity of such intuition might shine forth more brightly, we considered cases of external operations that play a principal role in algebraic deduction. We found there, in virtually every step: 1) that mentally a syllogism of type Barbara intervenes; 2) that the minor of this syllogism is a proposition (specific in the individual species) that is immediate, arisen from a similar intuition in sensitive data (which in their own way are geometric).

In these arithmetic calculations, the intuitive sense of terms (of numbers, of their addition, etc.) is always present (perhaps in a vague way), and this does not impede rigor. But the same is found in logistic deduction, where abstraction must also be made from such meaning. And yet there too, in individual steps, the same manifests itself: progress occurs by means of a mental syllogism (of type Barbara), whose minor is a specifically individual, immediate proposition.

We can inquire concerning other sciences; there in general it seems admitted that we attend to the meaning of words; there axiomatics is not applied, nor, except in special cases, operation upon symbols. But there nevertheless certain things seem to approach axiomatics. Concerning which a few words at least must be said here.

C) Deduction in Other Sciences.

In sciences that are not mathematical or similar in method to mathematics, that similar appeals to intuition are found in the middle of deduction is not surprising. We can perhaps marvel more if we find there a more or less strict imitation of the method that in modern axiomatics is the sole method; that is, in the sciences, especially if we use syllogisms explicitly expressed in words, we proceed in a certain way axiomatically, not attending, or attending less, to the material meaning of the words, which are signs of concepts, but looking only to the logical structure of propositions occurring in the science being developed (whether for others or for ourselves). What the axiomatic method intends to do systematically, we in the classical method do practically, more or less.

We do it, however, in such a way that, unlike in axiomatics, we are always prepared to form anew, to “realize,” the very notions of which those words are signs. And in our soul there is present, partly by nature, partly from critical habit, a certain schematism that by its nature tends to cause this connection between words and notions, of which they are signs, and simultaneously the reasons for affirming, to revive.

A sign of this we find in the classical Scholastic method of disputation. If someone in such a disputation solves an objection, not by simply denying, he says “distinguo”; and thus he appeals to the meaning of the word, which can be ambiguous. In one sense the proposition (expressed in words) is recognized as justified, but in another sense not so. And this method is nothing other than the mode of operating of natural man, perfected by art.

An extreme specimen in this context is that which Aristotle describes in passing (Posterior Analytics I, 12, 77b30–33) to illustrate the noetic difference between a term (expressed by voice) and the notion of the mind itself, especially in mathematics, where we as it were see notions with our mind. Such a realization of the notion, on the occasion of an ambiguous word, expels ambiguity. Such an equivocal word is the word “cycle” (cyclus), which is used of the geometric circle, but also of the cycles of poems after Homer. He says: “is every cycle a figure? If a diagram is constructed, it is clear. What then? Are epic poems a cycle? It is clear that they are not.” It is an extreme example, but less extreme ones occur frequently. And in the classical method, where that “distinguo” occurs, it is an appeal to intellective intuition, to rectify incorrect reasoning.

At this point we encounter something that at first sight is very strange. The axiomatician, to save the “rigor” of deduction, does so by introducing rigid terms, which are always used in the same sense without possible equivocation. And to attain this, he sacrifices, and does so very explicitly, meaning itself, that is, our cognition of the notions of which terms are signs. The classical Philosopher, to restore rigor, proceeds in the inverse manner: he appeals to the true, determinate meaning of words that was obscured or too much neglected; through recourse to intuition he causes intellection to revive. And thus, if this is well done, that rigor is restored is clearer than light. Who does better? The axiomatician who sacrifices our cognition to purely formal deductive rigor, and thus, if science is cognition, sacrifices science worthy of the name itself, or the classical thinker, who wishes to restore meaning, that is, cognition, that is, science? The axiomatician of course can intend (not all do this) to find again the meaning of his complete system, that is, to apply his construction to a certain reality; indeed, such reality can direct him in construction. Concerning practical and theoretical difficulty we treated sufficiently above; concerning the difficulty of this application we shall treat in the following section.

Not on that account must everything comprehended in axiomatics be rejected. And the very method of disputation described above is a sign that in the classical method of constructing sciences, which uses and must use syllogisms expressed in words, elements of axiomatics are present, and must be considered in noetic examination. They are present, for often in constructing deductions, in diverse reasonings, man can attend to words alone, which preserve the logical structure of reasoning so that the conclusion, expressed in words of course, is recognized as legitimate and therefore, because premises are supposed true, as also true. This can be done and is done, perhaps too much done.

Man in the sciences intends to attain reality, but on account of the imperfection of the intellect, which must proceed from imperfect cognition to more perfect cognition, he needs also the movement consisting in reasoning, which must use the syllogistic process. These syllogisms are primarily mental and lead to conclusions that are mental judgments, not mere oral propositions. Thus alone does he arrive at new scientific cognitions.

But in this mental labor he is greatly aided by terms, by propositions, by syllogisms expressed in words. He needs them not only to communicate science to others, but also, indeed in a large way, that his own mental activity may be facilitated. But because we wish to understand, we must always return to the understanding of the meaning of our discourses; this we intend to do not only at the end, through the application of the complete theory, but rather often, indeed regularly, in the very progress of reasoning.

Here, however, the question arises: whence comes this great utility of verbal expression, even if the human mind reasons for itself alone, not yet thinking of means of communicating its reasonings to others?

Behold certain points perhaps worthy of attention. If a certain new notion, a new relation dawns upon the human mind, this initially occurs in a rather vague mode; our intuition becomes clearer, becomes firmer, if we attempt to express it in words; the formula seems to have greater firmness, “consistency.” And if this process is repeated, the intuition of the mind itself becomes easier, habitual. This perhaps Aristotle intended to say in his famous exposition of the last chapter of the Posterior Analytics.

This we can already recognize where we are dealing with intuitions to which a phantasm corresponds, with mathematical intuitions. Let us take, e.g., the mathematical judgment that arises if in a physical body we trace a figure; we can express this thus: a body is divisible through figure. We intuit the necessity of this disposition of the thing; and indeed because in this case we have the faculty of abstracting formally and intuitively: a body because extended is divisible through figure. In a similar case in the mind itself, this precise nexus, between extension and divisibility through figure, is in the intellective representation itself of the body (with the figure traced) simultaneously with other nexuses of the body with its attributes (and of attributes among themselves, at least materially, i.e., through the intermediate body itself), and therefore an activity of abstraction is required that that formal nexus may be perceived; naturally, of course, the human mind exercises this activity, but active operation is required. In the verbal expression, however, “a body because extended is divisible,” even more in this: “the extended is divisible,” this abstraction has already been made; its effect, the separation of this nexus from other attributes that are of no importance, is already present in the formula itself; and therefore it now has greater stability, firmness, consistency: and the proposition itself teaches other men more easily what they ought to attend to in the phantasm. By verbal expression (e.g., because the extended is divisible) we fix our attention and that of others on what must be attended to, and which without such admonition is not sufficiently attended to (or not without more intense effort) and is neglected. Almost as if by words we fix attention on special parts of a certain painting, we understand it clearly; if not, this occurs only in a vague way. But words do not give the intuition of what is important, but arouse it, and efforts are spared.

This case is of course still very simple and easy. In others, even where mathematical matters are treated, it will be more difficult; then it will be even more necessary to propose a formula in which the effect of the abstraction made is expressed in a stable manner and its use rendered easier.

In cases where we are dealing with the disposition of things “of which there is no phantasm” (St. Thomas), concerning those, namely, that “are in our soul,” the utility, indeed the necessity, of employing verbal formulas having “stability” will be even greater. In order that we may truly understand such dispositions of things, we must arouse them in our mind, just as to understand in act in geometric matters we must inspect the phantasm; and such a vital action is already more dynamic than static. Moreover, such an operation, even a purely mental one, in this life, in order to be aroused, needs us to begin from sensitive data that permit a transition to purely mental operations. In such a process, therefore, if we wish to run through it fully (and this in the beginning is absolutely necessary), we must expend much greater, and more complicated, labor. But then we shall be able to express what results in stable words; and employ such a proposition in a syllogism, in which we attend to the words and their order alone, and we can arrive at a conclusion. This use of the verbal syllogism therefore becomes much easier; that syllogism is a phantasm without which man cannot understand. And this simplification explains, if I mistake not, the fertility of the syllogism.

In the end, however, we tend toward the understanding of the specific meaning of conclusions, and we direct the gaze of the mind to the dispositions of things, not only to their signs, that is, words, from the very nature of our mind. And in the ultimate perfection of cognition, toward which the human mind necessarily tends by its nature, we arrive through “resolution all the way to first principles,” according to the words of St. Thomas, intuitively known, and then we have an intuition also of the connection of truths, in which the human intellect, after laborious movement, arrives at the “rest” (quies) of the intuition of the whole (cf. Théorie du jugement, Ch. I, § 4, pp. 37 ff.).

The human mind, therefore, in its natural mode of deducing, undoubtedly needs elements of procedure elaborated in axiomatics, but which there are pushed to an extreme no longer justified.

Note on Neo-Positivism.

In modern neo-positivism, at least according to the direction followed since Bridgman, appeal to intuitive data in notions, and then also in principles, is again urged; this is customarily called the “operative” or “operational” method. The problem of necessity, which is present in this method too, is tacitly considered as solved, nor undeservedly; but the problem of exactitude, even in the physical part, which in this method is of decisive moment, is not considered where it ought to be considered; in expositions concerning operations “mentally” or “conceptually” or “theoretically” or “in principle” alone possible, this problem cannot be neglected and yet is neglected; moreover, the fundamental datum is neglected: our cognition of “intelligible matter” first attains the “passivity” of this matter, and consequently the activity of our operations; cf. below, Chapter VII, § 2. Hence arise great deviations of this otherwise excellent method. But these things no longer directly pertain to our subject (cf. Théorie du jugement, 2nd ed., pp. 242–244).

§ 5. ON THE APPLICATION OF GEOMETRY [11].

A) On the Application of Classical Geometry.

Classical geometry, as a science deductively constructed, was applied to physical extensions since antiquity: not only to astronomy, but also to physical science concerning sublunary objects, to kinematics, to mechanics, to acoustics, to optics, and more recently to field theory. This application, which especially since Newton’s work Philosophiae naturalis principia mathematica had splendid success, contains no difficulty. That extension (“space”) whose laws were established in geometry was discovered through abstraction in physical objects, and was realized in them. It was applied to that from which it was abstracted.

In this application, the necessity of relations in physical extensions is easily known, through the same formal abstraction. Regarding exactitude a difficulty is present, which nevertheless is easily solved. In order that physical extensions may be quantitatively known, we need measurements, which on account of diverse causes (imperfection of the senses, imperfection of instruments, influx of physical laws) can never be perfectly exact. The difficulty is solved from the fact that we thus have truly approximate values, and according to geometric laws themselves the meaning of the notion itself and the degree of approximation will be able to be determined. Elsewhere (Théorie du jugement, Ch. IX, § 3) we already said that in the application of classical geometry to physical extensions the cognition of necessity remains, but the cognition of exactitude must be sacrificed; in its place, however, we have approximation worthy of the name. (Cf. also above, Chapter II, § 2, no. 6, “on mathematical experimental judgments” of the mathematician Stahl, and Chapter I, § 4, Klein's “Approximationsmathematik and Praezisionsmathematik”).

Thus the mathematical science of physical extensions, “Newtonian physics,” was constructed since his celebrated work.

B) On the Applicability of Axiomatically Constructed Geometry.

If, however, we wish to apply axiomatically constructed geometry, greater difficulties arise. If we wish to apply it, we must first attribute a determinate meaning to the words that axiomatic geometry uses—to subjects, such as points, lines, etc., and to relations, such as “incidence,” “betweenness,” etc.—so that determinate physical objects from our experience correspond to them.

The notion of extension itself does not seem to be expressed in these words; it will be that in which the systems are possible of which Hilbert speaks (above, Chapter I, § 6); through relations these systems are formed. Generally it will be possible to speak thus: to the terms must be attributed the meaning they have in classical geometry.

Next, however, it must be examined whether the axioms, which now have an intuitive meaning and affirm real relations between subjects, are verified in physical extensions. Internal non-contradiction in the axiomatic system alone no longer suffices; without such examination, unknown proper attributes could be given in extended entities that would imply a contradiction (this seems to be postulated in the last axiom of Hilbert, but this too would have to be verified; but how?).

In the verification of axioms (and the same holds if a certain theorem is directly examined), the problems of necessity and exactitude return in their entirety. In physical extensions, all dispositions of things that verify the individual axioms must be empirically observed. And because geometric propositions are to be held as necessary, these empirically known dispositions must be necessary; a disposition that was not necessary would not verify the axioms; moreover, these empirically known dispositions must also be known as necessary. There is required, as in the system of Aristotle expounded above, empirical observation with which the understanding of necessity is associated; that experience is not sufficient which is had in physical cases, in which de facto the constancy of facts can perhaps be established through many experiments, but not their necessity, for the intuition of a necessary connection is not present; intuitive abstraction is lacking there, of which above (Chapter II, § 2, no. 3). The same abstraction is therefore required as in classical geometry, in order that the application may be justified on the part of necessity.

From what has been said it results: in order that axiomatically constructed geometry can be applied to physical extensions, the necessity of the relations affirmed in the axioms must be established from observation, for the meaning now attributed to the terms. Concerning this matter it is indeed established, as we saw in our analysis (above, Chapter II). But the same observation, associated with the understanding of necessity, and the same intuitive formal abstraction are the beginning of classically constructed geometry. If these were not valid, neither would the application of axiomatics be legitimate, relative to necessity.

Even greater is the difficulty if the problem of exactitude is considered, from which at long last the critique leading to the construction of axiomatics arose. At first sight it seems to be the same as that which we indicated above, treating of the application of classical geometry to physical extensions in the Newtonian manner; for exactitude in measurements must be sacrificed in this too. But in classical geometry the solution is easy and classical: what we know imperfectly, inexactly from experience, we can compare with exact figures and their exact relations, known from abstract geometry; for we are dealing with the same extension. And thus we can judge concerning the approximation of empirical data to exact objects. And we can attribute the causes of the defect of exactitude to the means of our observation, objective and subjective; there we can rightly speak of experiments “conceptually” or “in principle” possible; this expression has meaning.

Not so in the application of axiomatically constructed geometry. If we wish to apply this directly to physical extensions, we find discrepancies that cannot immediately be explained in such a way that cognition drawn from the senses appears to be an approximation to absolutely exact data.

Let us take an axiomatic proposition (in Hilbert it is a theorem, in another system it will be able to be considered an axiom): “between two points on a line there always exists a third.” In the line of experience, even in the phantasm, this is not verified; there we arrive at minima sensibilia, which for the senses are indivisible; and that proposition implies that the third intermediate point causes a division or is a division. Is that proposition verified “approximately” in an experimental line? That must be denied; the geometric line is divisible into infinity, the sensible line only down to a minimum. And thus an infinite discrepancy seems to be present between the physical line and the geometric line; this cannot be called an approximation. And such cases can be multiplied.

Directly, therefore, axiomatic geometry (after the introduction of the “meaning” of symbols) will not be able to be applied to physical extensions with the classical correction. Indirectly, of course, the application will again be able to be justified, for the difficulty even here has a solution. For from physical objects we can abstract the notion of pure extension (as realized in those objects, of course; cf. Cosmologia, 4th ed., p. 40; Théorie du jugement, Ch. X). To this extension, as represented intellectively (for the necessity of relations is known) in the mind, axiomatic geometric propositions can meaningfully be “applied.” In that extension, objects concerning which axiomatic geometric propositions hold exactly can be mentally constructed, actuated, that is, they exist mathematically. And because this extension is found in physical objects, therefore axiomatic geometry will also hold concerning this extension, realized in the nature of things. And because there they are observed imperfectly, without mathematical exactitude, they can therefore be justified as an approximation worthy of the name; concerning these we truly have “Approximationsmathematik” [12].

In order, however, that this solution of the difficulty may be truly valid, it is required that the application to “abstract extension” itself be justified. But in the analysis of conditions it appears that in this application there must be found all those things that are also required relative to the problem of exactitude, in order that the first propositions of classical geometry may be justified. And thus axiomatically constructed geometry is nothing other than a superfluous means to discovering that geometry which can be applied to physical extensions in such a way that the cognition resulting from measurements is truly approximative.

There are mathematicians who contend that that burden of applying geometry to physical extensions does not fall upon them, the mathematicians; indeed, who say that application to intuitive abstract extension no longer pertains to pure mathematics. (They also affirm this if analytic geometry is constructed arithmetically). But such mathematicians teach their doctrine to their disciples the physicists, who are supposed to apply it and do apply it, and use geometric propositions introduced into physics since Newton. This at least they will have to justify; and this cannot be done without reconstructing geometry in the classical manner. It does not suffice to say jestingly with Einstein (cf. above, Chapter I, § 3): “insofar as mathematical propositions refer to reality, they are not certain; insofar as they are certain, they do not refer to reality”; and note that in this dictum no distinction is made between the problem of necessity and the problem of exactitude, which two precisely in the question of application differ so greatly.

In such a proud, quasi-aristocratic separation of axiomatically constructed geometry from physical extension, a purely experimental object, other inconveniences also seem to arise. If they do not wish to care concerning the meaning of geometry after its construction, they now have a science that knows nothing real except a series of formal consequences in a symbolic world. What is worse: as we saw above (Chapter V, § 2 A), geometric words must have a meaning, otherwise the syllogistic discourse employed will have no value; and axiomaticians, like Hilbert, employ that discourse; but the office of determining meaning is left to others. Someone will be able to advance further and say that he employs syllogistic rules not as they are modes of operating of the human mind, which understands that correct consequence holds for reality, but only as freely chosen rules of manipulating symbols, which in this case are words. In such a position, the entirety of “abstract” geometry (though it is not truly abstract; it is an artificial construction, or creation as they say, and in this case a construction out of words that have no meaning) is nothing but a pure game with symbols (I do not say a children’s game, for it is very difficult). The reality considered there is the very order of symbols, which are no longer signs of any reality, but the very reality upon which they operate. This game, however, seems no longer worthy of the name “science.” And moreover observe that in such manipulation, as in algebra and in logistics, in individual steps the operations are directed in exercised act by mental syllogisms, which they also wish to avoid.

Physicists, therefore, will be able to expect no aid from such a mathematician. It cannot happen, nevertheless, that they do not attempt to construct geometry. For since the first attempts, even pre-scientific ones, to know nature, man sees a huge noetic discrimination between the relations he observes regarding the extension of bodies and their multitude (numbers) on the one hand, and physical properties on the other hand, in this sense: observing a body in which he traces a figure, he affirms “this body is figurable”; observing a body that after friction attracts small particles, he affirms “this body is electrifiable.” These two cases noetically differ from each other by the whole heavens (toto caelo), and this does not remain unknown. In the first case, man intellectively intuits the necessity of the connection, and that because through abstractive intuition he understands that “figurability” flows from the extension of the body, and from it alone, and from it necessarily. In the other case, this is totally lacking. In such a case, the physicist from repeated and varied experiments will be able to know a “physical law”; he will be able firmly to believe that necessity is present there, but he does not understand the connection. In order to arrive at this, the physicist must employ the method of inquiring into this intellection by means of a hypothesis, an explanatory physical theory.

It is clear that such a physicist tends toward developing his physical science concerning the body as it is extended; but the method will be different by the whole heavens: he begins from intellection and will attempt to elaborate it deductively. He will attempt to become a geometer, to construct this apodictic part of his science; he will construct classical geometry in the classical manner. The human mind itself, which wishes to understand, attempts from those data of experience to construct an apodictic science.

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Geometry axiomatically constructed seems to labor under diverse defects: to construct the entirety of geometry thus is practically impossible; to construct it without incessant recourses to phantasms, which will always have to procure new “immediate propositions,” is impossible even theoretically; and this recourse demands inspection of an order that is itself geometric. Finally, in application to physical extensions—and these demand deductive geometric considerations—it seems to fail.

The axiomatic method can nevertheless have its utility (cf. above, Chapter I, § 6): not indeed total axiomatics, but its parts (and its special methods), especially for investigating the logical structure of sciences. But to expound that is not for this place.

All these things, however, seem to invite the Scholastic philosopher, above all the noeticist, to develop more broadly the matters briefly touched upon here.

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[1]

Thus Thomas Heath in his famous work A History of Greek Mathematics, Oxford, 1921, I, pp. 335 ff. The same historian affirms there (concerning which doubts are sometimes raised, or which is even denied): “Aristotle was no doubt a competent mathematician, though he does not seem to have specialized in mathematics.”

[2]

Cf. the discussion between the distinguished Freudenthal and the author in Gregorianum, 1951: H. Freudenthal, De fontibus geometriae in intuitione et abstractione quaerendis, pp. 425–433, and the author’s De noetica geometriae, responsum ad animadversiones Cl. Fr., pp. 434–452.

[3]

Thus he expresses the form of the syllogism “Barbara” and affirms its value by saying: “if A is predicated of all B, and B of all C, necessarily A is predicated of all C,” which is thus reduced to the “schema” of the syllogism: “all B is A, now all C is B, therefore all C is A.” Prior Analytics I, 4, 25b37–39. Concerning the further development of the symbolism of Aristotle cf. our work Recherches de logique formelle. La structure du système des syllogismes et des sorites. La logique des notions « au moins » et « tout au plus », Rome, 1947.

[4]

Concerning the term “dispositio rei” [disposition of the thing] in St. Thomas (which is equivalent to the modern German term “Sachverhalt”), see Théorie du jugement, Ch. II, § 5.

[5]

J. Locke, An Essay Concerning Human Understanding, Bk. IV, ch. 17, § 4 (in Fraser ed., II, pp. 390 ff.): “God has not been so sparing to men to make them barely two-legged creatures, and left it to Aristotle to make them rational … He has given them a mind that can reason, without being instructed in methods of syllogizing.” Leibniz in his reply to Locke (Nouveaux Essais, in Gerhardt ed., V, pp. 458 ff.) extols the merit of Aristotle, but as is clear admits this thesis. Similarly Hegel, Wissenschaft der Logik, II, 1, k. 3, Anmerkung, Werke (ed. 1834), pp. 142 ff.

[6]

Thus well A. Riehl, Logik und Erkenntnistheorie, in Kultur der Gegenwart, Abt. VI, 3rd ed. (1921), p. 71.

[7]

G. Stammler in Begriff, Urteil, Schluss, Halle-Saale, 1928, pp. 229, 245.

[8]

“I believe the prime Number Theorem because of de la Vallée-Poussin's proof of it, but I do not believe that 2 + 2 = 4 because of the proof in Principia Mathematica.” G. H. Hardy, Mathematical Proof, in the journal Mind, 38 (1929), p. 17.

[9]

“Sind irgendeine endliche Anzahl von Punkten einer Geraden gegeben, so lassen sich dieselben stets in der Weise mit A, B, C, D, E, ….. K bezeichnen, dasz der mit B bezeichnete Punkt zwischen A einerseits und C, D, E, ….. K andererseits, ferner C zwischen A, B einerseits und D, E, …. K andererseits, sodann D zwischen A, B, C einerseits, und E, ….. K andererseits liegt. Ausser dieser Bezeichnungsweise gibt es nur noch die umgekehrte Bezeichnungsweise K ….. E, D, C, B, A, die von der nämlichen Beschaffenheit ist.” Hilbert, Grundlagen der Geometrie, 7th ed. (1930), p. 8.

[10]

These things were already observed by the mathematician Hardy in the article we cited above (Mind, 38, p. 12): “If Hilbert has made the Hilbert mathematics with a particular sheet of paper, and I copy them on another sheet, have I made a new mathematics? Surely it is the same mathematics, and that even if he writes in pencil and I in ink, and his marks are black while mine are red.” (“Si Hilbert construxit mathematicam hilbertianam in particulari folio papyraceo, et ego transcribo illam in alio folio, num construxi novam mathematicam? Absque dubio est eadem mathematica, etiam si ipse utitur stilo plumbeo et ego atramento, et eius signa sunt nigra et mea rubra.”) Hardy by these words wishes to indicate the necessity of these operations and observations, and the cause of this necessity, which is intuitive formal abstraction.

[11]

Cf. our communication already cited above, Pour une philosophie de la connaissance de l'étendue physique, in Gregorianum, 1949, pp. 193–203.

[12]

How physical proper attributes of objects can sometimes impede the realization of exactitude in bodies, we explained for one case in Cosmologia, in the note "de ente extenso physico," in 4th ed., pp. 438–445.