Ernst Cassirer

Alfred Korzybski to Charles Kay Ogden, June 1925

'In my experience and understanding, no one who is not familiar with this fundamental epistemological treatise* can appreciate either your Meaning of Meaning or my work.'

Quoted in Semantic Pioneers Revisited by W. Terrence Gordon
Et cetera Spring 1990, p. 57.

    * Substance and Function by E. Cassirer, Berlin 1910, English translation by the Swabeys, Chicago : Open Court 1923.  —   (WPT)

From THE PROBLEM OF KNOWLEDGE, 1940 by E. Cassirer

The first basic view of knowledge from which one can start out is that all knowing has to fulfill a representative function. It refers to a being and fulfills its role the more completely as it succeeds in expressing the definitive nature and essence of this being. To show the truth of an idea, therefore, is to specify the objective substrate to which it corresponds. Without that it would remain suspended in thin air, and might be no more than a subjective illusion. The correspondence between concept and thing, the adequation intellectus et rei, is thus the supreme postulate that must be complied with in every case—no matter whether we deal with mathematical or empirical conceptions. These differ according to their origin and content but not in respect to their mode of dependence upon their corresponding objects. Even mathematical thought can never be a mere cogitation, for no bounds could be set to such a process ; given free rein, it might pass over into pure fantasy. Hence even in pure mathematics we cannot speak of �free creations� of thought without endangering its content of truth. There must always be a fundamentum in re: it must be related to some given objective factual content with which its ideas correspond and which it will bring to expression.

The empirical sciences are concerned with the actuality of things, with their existence in spade and time&#!51;a reality that is only accessible to us through sensory perception. Mathematics does not move in this circle of sensory existence, yet its truth, too, is firmly anchored in original being ; in the reality or essence of things. This essential being can be apprehended only through concepts, yet abstract thinking in no way creates but only discovers it. In this sense even the pure form of mathematical reality is always related to a definite �matter,� though this is not a sensible but a wholly �Intelligible matter.�

This conception of the affair contrasts with another, which may be called the functional view of knowledge.¹⁵ Here, too, the relation of thought to object is the central point, but the theory establishes the possibility of this relation in another way and strikes out, as it were, in the opposite direction. For the object is not treated as a given fact but as a problem ; it serves as the goal of knowledge, not as its starting point. This lies rather in the realm that is immediately accessible to knowledge, the realm of its own operations. No matter whether we are concerned with the ideal or the real, the mathematical or the empirical, with nonsensuous or sensuous objects, the first question is always not what these are in their absolute nature or essence, but by what medium they are conveyed to us ; through what instrumentality of knowledge the knowing of them is made possible and achieved. This knowing is built up by stages; as the process advances the object becomes more and more sharply designated. We progress from general to particular, from abstract to concrete determinations, and try to show how the latter presuppose the former and are based upon them.   (pp. 61-62)

15. For the following I refer to the more detailed exposition in my Substansbegriff und Funktionsbegriff (Berlin, B. Cassirer, 1910) ; Substance and Function, Eng. Trans. By W. C. and M. C. Swabey (Chicago, Open Court Publishing Co., 1923).

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Frege . . . saw the only chance for a logically satisfying explanation of number in the closest possible association of its nature with the nature of the concept. He also had recourse not so much to the meaning of the concept as to its extension. Every concept denotes such extension in this wise : out of the totality of being a certain class of objects is selected that possess the characteristic given in the concept. Now it remains only to decide wherein consists the property of being that we want to express when we attribute number to certain things. For this it is only necessary to make clear what condition must be fulfilled before two sums can be declared to have the same "number." So the concept of numerical equivalence, and the criterion of the same, which is one-to-one correspondence, are prior to the number concept : "The number that belongs to the concept F," as Frege defines it, "is the content of the concept 'of number equal to the concept F.'"¹⁷ Independently of Frege, and by a very similar route and from identical intellectual motives, Russell arrived at his own definition of number as a "class of classes." Thus the number 2 is nothing other than the totality of all existing parts, the number 3 nothing other than all the triads. Russell also had a preference herewith for the extensional definition of the class over the intensional, seeing in the class only the aggregate of single and independent elements.¹⁸

Wholly different was the course of those mathematicians who took as their point of departure the ordinal number. They regarded "order" not merely as a property and, as it were, an appendix of number but as the truly fundamental and constitutive principle of number. For the numbers did not "exist," as is the case of those who started with sets and the power of sets as something given in themselves, which had the supplementary possibility of being arranged in a fixed order. On the contrary, the individual number represents nothing but a cut, a particular place in a universal order structure ; and this structure determines the character and essential nature of the number. This determination was thus transformed from an absolute into a purely relative one ; but this relativity could not possibly be regarded as a limitation upon the "objective" character of number, for it actually formed the only adequate expression of it. The relationship comprised all being and all the truth that we could sensibly attribute to number.

From this first insight springs that general reversal, that new total orientation of the problem of knowledge, which may be called the purely "functional" view, in contrast with the "conceptual realism" of Cantor, Frege, and Russell. To characterize it simply as nominalistic would be insufficient and misleading. Many of its advocates, to be sure, come close to doing so with their great emphasis on the fact that where numbers are concerned we are dealing not with a system of things but with a pure system of signs. But for a genuine mathematical nominalism, such as was first established in modern philosophy by Leibniz, all mathematical thinking and reasoning must stick to signs and always stay within that realm, whereas these signs in the functional sense are not empty but very meaningful signs. The meaning given them is not determined by looking out on the world of things but rather by considering the world of relations. The sings express the "being," the "duration," the objective value of certain relations, and hence they indicate "forms" rather than material things. The theory of number thus appears as one chapter in a general theory of forms, yet it loses nothing in its significance thereby but actually gains. Instead of becoming an idle game of signs it proves to be the terminus a quo, the very beginning and first matter of all objective knowledge ; the theory begins with pure number relation in which there is nothing given at the outset save the concepts of identifying and distinguishing. But number does not remain at this stage of pure conceptual identity and difference but adds thereto the concept of succession (sequence) of the serial order of what is given. On this one concept there is constructed, according to Leibniz and all mathematicians who have struck out in this direction, the whole of arithmetic.¹⁹ In this derivation number remains throughout in the domain of pure mathematics and in its foundations there must be no admixture of reference to the field of its application, the things that are counted. Yet it is by no means isolated thereby ; on the contrary, number stands at the threshold of a series of relationships which, further pursued and conceived more concretely, should lead finally to the determination of "reality" and be included in it.

Mathematical theory was not here concerned with a survey of the total system of these relational forms but had to be satisfied with making certain of the way to do so ad explaining the first steps in a logically unobjectionable way. In so doing, the ordinal theory started from an ordered series of discrete elements and then went on to show how it was possible thence to arrive at number as the expression of plurality ; the cardinal number of a set. That the same cardinal number is proper to every finite set, no matter what the arrangement of its individual elements, had to be shown by special proof.²⁰ The concept of 1 as the first member of the set and the general relation of ordering are of course presupposed herewith ; but even though it be assumed that this restricts the theory from the standpoint of formal logic, none of its epistemological value is lost. For unless one is willing to accept an infinite regress, the critical examination of knowledge must always come to a halt at a certain primal functions that are neither capable nor in need of an actual "deduction." Indeed it is difficult to understand why only logical identity and difference, necessary elements in the theory of set, are admitted as ultimate functions of this sort, whereas numerical unity and numerical difference are excluded at the outset from this realm. A truly satisfactory deduction of the latter from the former has not been successfully made in the theory of sets, and there remains always the suspicion of a concealed vicious circle in respect to all attempts in this direction.

The basic philosophical idea upon which the ordinal theory rests in essentials has been characterized in the simplest and most pregnant manner by Dedekind. For the construction of the whole realm of number nothing more is required than the �ability of the mind to relate things to one another.� On the basis of this single function, and without the admixture of any such alien notions as that of measurable quantities, it is possible to attain first to the series of so-called natural numbers and thence to all extensions of the number concept : to the originating of zero and of negative, fractional, irrational, and complex numbers. The number concept is and remains herewith �an immediate issue of the laws of pure thought� ; since the basic function of thinking is precisely �to relate things to things, to make one thing correspond with another or to represent one thing by another�—a function without which thinking would in general be impossible.²¹   (pp. 64-67)

17. Frege, Die Grundlagen der Arithmetik, S. 79 f.
18. Russell, The Principles of Mathematics (Cambridge, 1903), chap. Vi. [etc]
19. For Leibniz� attitude toward mathematical nominalism see �Meditations de cognitione, veritate et ideis,� Die philosophischen Schriften . . . , ed. Gerhardt, IV, 422 ff.
20. See, for example, Helmholz, �Zählen und Messen,� Philosophische Aufsätze zu Ed. Zellers, S. 32 ; and L. Kronecker, �Uuml;ber den Zahlbegriff,� ibid., S. 263 ff.
21. R. Dedekind, Was sind und was sollen die Zahlen? (2. unveränd. Aufl. Braunschweig, F. Vieweg & Sohn, 1893). S. vii ff.

New Haven : Yale University Press,
London : Oxford University Press,
1950.

From PLATONIC RENAISSANCE IN ENGLAND, 1934 by Ernst Cassirer

This is a translation of Die platonische Renaissance in England und die Schule von Cambridge, which was originally published in 1932 by B. G. Treubner in Leipzig and Berlin as a contribution to the 'Studies of the Warburg Library.'

The task of rendering the German work into English was begun nearly two decades ago in gratitude for the author's friendship and intellectual guidance during the year 1933-4 at Oxford University. In that year Professor Cassirer went into exile and commenced the Odyssey that ended only with his death in 1945. At Oxford with characteristic patience and cheerfulness he resumed, in an alien tongue, his interrupted studies. It was typical of the man that he chose as the subject of his first lecture at All souls College the Greek origins of the idea of justice. Similarly, throughout the year, neither his personal remarks nor his lectures reflected in any way the threats and indignities that reached him daily from his homeland. On the contrary, his words and actions were full of that devotion to reasonableness and tolerance, to justice and truth, which is part of the precious heritage of the modern world from the English Platonists discussed in this book.

* * *

This translator has benefited by the criticisms of several specialists who have been kind enough to read the manuscript. Above all, it was read and approved by the author, who graciously responded to all my queries regarding the meanings of words and passages and the sources of unannounced quota ions. I owe a special debt of gratitude to Dr. Walter H. Freeman for generous assistance in translating the Latin and Greek cited by the author.

* * *

JAMES P. PETTEGROVE

New Jersey State Teachers College
Montclair,

12 September 1953

(Nelson 1953)
University of Texas Press : Austin 1953, pp. v-vii

Selected bibliographic

Author Cassirer, Ernst, 1874-1945. Title Substanzbegriff und Funktionsbegriff : Untersuchungen �ber die Grundfragen der Erkenntniskritik / Ernst Cassirer ; Text und Anmerkungen bearbeitet von Reinold Schm�cker. Publisher Hamburg : F. Meiner, c2000. Description ix, 395 p. ; 25 cm. Series Gesammelte Werke, Hamburger Ausgabe / Ernst Cassirer ;Bd. 6 Series Cassirer, Ernst,1874-1945.Works.1998 ;Bd. 6. Note Includes bibliographical references (p. [381]-391) and index. ISBN 3787314067

Author Cassirer, Ernst, 1874-1945. Title Substanzbegriff und Funktionsbegriff. Untersuchungen über d. Grundfragen d. Erkenntniskritik. Edition (Reprograf. Nachdr. d. [Ausg.] Berlin, 1910. 3., unveränd. Aufl.) Publisher Darmstadt : Wissenschaftliche Buchges, 1969. Description xv, 459 p. 21 cm. Note Includes bibliographical references.

Author Cassirer, Ernst, 1874-1945. Uniform Title [ Substanzbegriff und Funktionsbegriff. English] Title Substance and function, and Einstein's theory of relativity / by Ernst Cassirer; authorized translation by William Curtis Swabey, PH.D., and Marie Collins Swabey, PH.D. Publisher Chicago : Open court publishing company, 1923. Description xii, 465 p. ; 24 cm. Note "The first part of the present book, Substanzbegriff und funktionsbegriff, was published in 1910, while the second part, which we have called the supplement, Zur Einstein'schen relativit�tstheorie, appeared in 1921." Note Includes bibliographical references (p. 457-460).

Author Cassirer, Ernst, 1874-1945. Title Substanzbegriff und Funktionsbegriff : Untersuchungen �ber die Grundfragen der Erkenntniskritik / von Ernst Cassirer. Publisher Berlin : B. Cassirer, 1910.