Abstract
Contrary to what is sometimes claimed, there is no real “turn” from philosophy of science to philosophy of culture in Cassirer’s intellectual development: it is the internal evolution of Geometry and its consequences in Physics that lead Cassirer to broaden an essentially unique perspective. Before Cassirer became intellectually active, the emergence of non-Euclidean geometries had triggered a crisis in Mathematics that had consequences in Natural sciences as well as in Philosophy. The concept of group of transformation in Geometry (Klein) and the subsequent use of Riemanian geometry in relativity theory (Einstein) were responses to this crisis in Mathematics and Physics, respectively. The philosophical response to the crisis prompted Cassirer’s original concept of a “Symbolic Form”. From a transcendental perspective, his approach would transform the concept of objectivity into that of the various modes of objectivation. Thus, Cassirer’s philosophical project can be stated as follows: since some modes of objectivation that pertain to Mathematics have also meaningful consequences in Physics, it is possible to philosophically consider the very notion of the variety of modes of objectification and their possible transfers as the key concepts of meaning formation in general, thus transforming the transcendental perspective into a semiotic one. According to Cassirer, this transformation becomes the goal of the philosophical enquiry and can be studied in all cultural forms: Language, Mythical thinking, Law, Art or Technology, to name a few.
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Notes
- 1.
Two concordant elements enable to justify this interpretation: (i) the publication in 1921 of his book about Einstein (E. Cassirer, Einstein’s Theory of Relativity, The Open Court Publishing Company, Chicago, 1923, ECW 10) in which he demonstrates the capital role played by non-Euclidean geometries and the concept of group in the constitution of the Einsteinian theory of relativity; and (ii) the autobiographical remarks provided in Cassirer’s last manuscript (ECN 8, pp. 187–188) bearing the date of his death (April 13, 1945) in which Cassirer mentions, on the one hand, the serious problem the very notion of non-Euclidean geometry represented for the Kantian student he was because it completely exceeded the framework of the Critique of Pure Reason and on the other hand, the solution which Felix Klein had produced with his use of the concept of group. Cassirer does not mention the date on which he read Klein’s works (whose most famous text, designated as the “Erlangen Program” is dated 1872), but he only cites the 1921 edition of Klein’s Collected Works. Even if Cassirer had already, during his student years, understood the solution Klein provided to the problem of the plurality of geometries, there are good reasons to believe that it is only around 1920–1921, the period during which he wrote his book on Einstein, that he understood the full epistemological, and more broadly, philosophical interest which could be derived from it. It is also not impossible that the interest he expressed already as a student regarding the concept of group was in fact an a posteriori reconstruction given that it was only much later, in 1921, that this interest manifested itself in his works.
- 2.
A last, posthumous, volume, regroups texts dating from 1928 to 1940 (The Philosophy of Symbolic Forms, vol. 4, trad. J. Krois & D. Verene, Yale University Press, 1996), is now published as the first volume in the series Ernst Cassirer Nachgelassene Manuskripte und Texte, Felix Meiner Verlag, 1995–2017 (ECN 1).
- 3.
E. Cassirer, Substance and Function, The Open Court Publishing Company, Chicago, 1923, pp. 268–269 (ECW 6, pp. 289–290): “The procedure of the “transcendental philosophy” can be directly compared at this point with that of geometry. Just as the geometrician selects for investigation those relations of a definite figure, which remain unchanged by certain transformations, so here the attempts is made to discover those universal elements of form, that persist through all change in the particular material content of experience”.
- 4.
Only the fourth and last volume of Das Erkenntnisproblem… has been translated into English (E. Cassirer, The Problem of Knowledge; Philosophy, Science, and History since Hegel, transl. by William H. Woglom & Charles W. Hendel, New Haven, Yale University Press, 1950). From now on, this translation will be quoted “E. Cassirer, The Problem of Knowledge…; vol. 4” followed by the reference to the German standard edition, i.e. ECW 5. Cf. E. Cassirer, The Problem of Knowledge…; vol. 4, p. 1 (ECW 5, p. 1): “Even in myth and religion all that is distinctive of man is associated with the miracle of knowledge. This miracle reveals the nature of man and his likeliness to God, yet in it man also realizes, in the deepest and most painful way, the very limitations of his nature. […]. To this religious pessimism the Greeks were the first to take a definitely opposite view. It was a decisive affirmation that knowledge is certainly possible to man. There was no longer that sense of a “fall” of man and of an estrangement from the ultimate ground of things, but on the contrary a conviction that knowledge is the one power which can sustain and unite man forever with the ultimate being”.
- 5.
E. Cassirer, The Problem of Knowledge…; vol. 4, pp. 47–48 (ECW 5, pp. 53–54): “The Greeks were able to unearth this hidden wealth because the idea of measure lay at the heart of their view of the world and of all their thinking. Restricted to no particular sphere and not exhausted by any special application, this idea represented the very essence of thinking and of being. To discover the “limits and the proportions of things” was the task of all knowledge. But the concept extended far beyond this purely intellectual achievement, since it was the core not only of all cosmic but also of the human order, and lay at the center of ethics as well as of logic”.
- 6.
The role attributed to Greek skepticism as well as to French skepticism as seen in the tradition of Montaigne, for instance, plays an eminent role for Cassirer in this conclusion; cf. E. Cassirer, Das Erkenntnisproblem…, Band 1, ECW 2, pp. 143–168.
- 7.
E. Cassirer, Substance and Function, p. 21 (ECW 6, p. 20): “In opposition to the logic of the generic concept, which, as we saw, represents the point of view and influence of the concept of substance, there now appears the logic of the mathematical concept of function. However, the field of application of this form of logic is not confined to mathematics alone. On the contrary, it extends over into the field of the knowledge of nature; for the concept of function constitutes the general schema and model according to which the modern concept of nature has been molded in its progressive historical development”.
- 8.
E. Cassirer, Das Erkenntnisproblem…, Band 1, ECW 2, p. 27 for the particular role played by Plato in the philosophy of Nicholas of Cusa; more generally, on Plato’s role during the Renaissance, cf. Ernst Cassirer, Das Erkenntnisproblem…, Band 1, ECW 2, p. 65sq. Cf. also E. Cassirer, The Problem of Knowledge…; vol. 4, pp. 1–2 (ECW 5, p. 2): “To Galileo the “new science” of dynamics, which he founded, meant first of all the decisive confirmation of what Plato had sought and demanded in his theory of ideas. It showed that the whole Being is pervaded through and through with mathematical law and thanks to that is really accessible to human knowledge”.
- 9.
E. Cassirer, Das Erkenntnisproblem…, Band 1, ECW 2, pp. 439–441: “But from an historical point of view, Pascal’s doctrine in contrast constitutes a symptom of a general, inner flaw in the Cartesian system. The strict separation between Reason and Authority remains limited to the theoretical domain. (…). There is no any other moment in the history of modern philosophy where one cannot become more clearly aware of the conflict between the two paths, between the two directions of thinking and of the alternative that makes it necessary to choose between the two than in the system of Pascal”.
- 10.
E. Cassirer, Das Erkenntnisproblem…, Band 2, ECW 3, p. 624: “That the concept of an object would only present itself to the “Pure Understanding” certainly entails no contradiction and, from a purely logical point of view, cannot be disputed nor refuted; but this freedom from contradiction will here, just like in the case of all ontological concepts, be paid at the price of a complete lack of determined content”.
- 11.
Thus, what constituted for Cassirer the very originality of the modern project of knowledge, that is, the tendency to relate the mathematized usage of infinity to the category of subject without attributing it to the transcendental, a tendency which had been already discovered to be at work at the cusp of modernity with Nicholas of Cusa (1401–1464) became, for Kant, a project for the self-limitation of the powers of reason, likely to convert all which appears as an object of the world into a simple correlate of a function of knowledge. Cf. E. Cassirer, Das Erkenntnisproblem…, Band 2, ECW 3, p. 633: “Gradually, all the real properties of the “world” transmute into the methodical features of experience.” For Cassirer’s reflections regarding Nicholas of Cusa, cf. E. Cassirer, Das Erkenntnisproblem…, Band 1, ECW 2, pp. 17–50.
- 12.
It is not necessary here to mention the reflection by Kant (for example in the Prolegomena to Any Future Metaphysics That Will Be Able to Present Itself as a Science, § 13) regarding what remains non-geometrical in the perception of space, the experience of which being allowed for instance by non-congruent objects (symmetrical but not superimposable), because what is in question here is only the characterization of what is mathematizable in space (the case of non-congruent objects will finally be resolved geometrically by Legendre within an Euclidean framework; cf. Legendre, Éléments de géométrie, VI, proposition 25 (1794). It is only in the second edition of the Critique of Pure Reason, when it is question of exposing not the metaphysical concept of space but its transcendental counterpart, that the question of the Euclidean character of space and its necessity is posed. Cf. F. Pierobon, Kant et les mathématiques; la conception kantienne des mathématiques, Vrin, Paris, 2003, p. 110.
- 13.
Kant, Critique of pure Reason, 2nd edition, Transcendental Aesthetics, § 2: “Geometry is a science that determines the properties of space synthetically and yet a priori. What, then, must be the presentation of space be in order for such cognition of space to be possible? Space must originally be an intuition. For from a mere concept one cannot obtain propositions that go beyond the concept; but we do obtain such propositions in geometry. This intuition must, however, be encountered us a priori, i.e. prior to any perception of an object; hence this intuition must be pure rather than empirical. For geometric propositions are one and all apodeictic, i.e. linked with the consciousness of their necessity—e.g., the proposition that space has only three dimensions. but propositions of that sort cannot be empirical judgements or judgments of experience; nor can they be inferred from judgments” (Critique of Pure Reason, translated by Werner S. Pluhar, introduction by Patricia Kitcher, Hackett Publishing, 1996).
- 14.
For example, Pappus’s theorem (fourth century AD) can retrospectively be linked to projective geometry.
- 15.
In Substance and Function, p. 91 (ECW 6, p. 96): “Constancy and change thus appear as thoroughly correlative moments, definable only through each other. The geometrical “concept” gains its identical and determinate meaning only by indicating the definite group of changes with reference to which it is conceived. The permanence here in question denotes no absolute property of given objects, but is valid only relative to a certain intellectual operation, chosen as a system of reference”.
- 16.
As Cassirer remarks in Substance and Function, p. 88 (ECW 6, p. 93): “In this connection, projective geometry has with justice been said to be the universal “a priori” science of space, which is to be placed beside arithmetic in deductive rigor and purity. Space is here deduced merely in its most general form as th “possibility of coexistence” in general while no decision is made concerning its special axiomatic structure, in particular concerning the validity of the axioms of parallels”.
- 17.
In Euclid’s text, the fifth axiom is indeed formulated: “That, if a straight line falling on two straight lines makes the interior angles on the same side less than two right angles, the two straight lines, if produced indefinitely, meet on that side on which are the angles less than the two right angles”. (The Thirteen Books of Euclid's Elements, transl. T. L. Heath, University Press, Cambridge, 1908).
- 18.
Gauss, Riemann, Lobatchevski, Bolyai, Beltrami.
- 19.
Cf. Felix Klein, “A comparative Review of recent Researches in Geometry” (1872), English translation by Dr. M. W. Haskell and transcribed by N. C. Rughoonauth, Bull. New York Math. Soc. 2, (1892–1893), online at: http://arxiv.org/abs/0807.3161.
- 20.
Cf. Hans Wussing, The Genesis of the Abstract Group Concept: a Contribution to the History of the Origin of Abstract Group Theory, MIT Press, Cambridge, 1984, p. 198: “It is this very linking of two advances—the introduction of groups of geometric motions and of the question of their generators—that has produced the advance that we encounter in Klein’s studies of groups of isometries of regular polyedra. This advance enabled Klein not to apply to geometry the fundamental principle of permutation theory but also to work out the concept of a (discrete) group of transformations. The recognition and clarification of that concept’s potential to fuse geometry, algebra, and the theory of functions set in motion a far-reaching development”.
- 21.
E. Cassirer, The Problem of Knowledge…; vol. 4, pp. 42–43 (ECW 5, pp. 48–49): “Since its early development by Cauchy, Lagrange and Galois the range of these applications has broadened continuously and extended to the most various fields of mathematics. The group concept is not restricted either to mathematics or geometry or to number or dimension. In it we rise above any considerations of special elements of mathematical thought to a theory of operations. We are no longer concerned with the special content but rather with the very procedure of mathematics itself, for we have entered the realm of pure, “intellectual mathematics””.
- 22.
F. Klein, “A comparative Review of Recent Researches in Geometry” (1872), § 1: “The most essential idea required in the following discussion is that of a group of space-transformations. The combination of any number of transformations of space is always equivalent to a single transformation. If now a given system of transformations had the property that any transformation obtained by combining any transformations of the system belongs to that system, it shall be called a group of transformations. […]. Now there are space-transformations by which the geometric properties of configurations in space remain entirely unchanged. For geometric properties are, from their very idea, independent of the position occupied in space by the configuration in question, of its absolute magnitude, and finally of the sense in which parts are arranged. […]. The totality of all these transformations we designate as the principal group of space-transformations; geometric properties are not changed by the transformations of the principal group. And conversely, geometric properties are characterized by their remaining invariant under the transformations of the principal group. […]. This is the general problem, and it comprehends not alone ordinary geometry, boatels and in particular the more recent geometrical theories which we propose to discuss, and the different methods of treating manifoldness of n dimensions.
- 23.
E. Cassirer, The Philosophy of Symbolic Forms, vol. 3, p. 352 (ECW 13, pp. 405–406): “The theory of substitutions takes its place side by side with the theory of number developed in elementary arithmetic, and moreover it develops that the basic theorems of elementary arithmetic can be strictly deduced only on the basis of this theory. And from here the road leads to what has been said to be “perhaps the most characteristic concept in nineteenth-century mathematics” [note: Hermann Weyl, “Philosophie der Mathematik und Naturwissenschaft”, Handbuch der Philosophie, Munich & Berlin 1927, Pt II A, p. 23]. For investigations of groups of letter substitutions give rise to the general concept of group of operations and the new discipline of a theory of groups. Not only was the group theory an important addition to the system of mathematics, but it soon became increasingly plain that this was a new, far-reaching component of mathematical thought itself. Felix Klein’s famous ‘Erlanger Programm’ shows how it changes the inner form of geometry. Geometry now becomes subordinated to the theory of invariants as a special case”.
- 24.
“Reflections on the Concept of Group and the Theory of Perception”, translated in D. P. Verene, Symbol, Myth and Culture; Essays and Lectures of Ernst Cassirer 1935–1945, (1979), pp. 277–279 (ECN 8, pp. 187–188).
- 25.
E. Cassirer, The Problem of Knowledge…; vol. 4, p. 25 (ECW 5, p. 28): “In 1871, Felix Klein showed that the entire system thereof can be derived from Euclidean geometry, a fact that makes illusory every preference for one over the other on the score of absolute worth and proves that they share the same fate in respect to “truth”, since any contradiction in one is inevitably attended by similar contradictions in the others”.
- 26.
This was, for instance, the case with very famous philosophers such as Lotze who declared (ECN 8, p. 186): “[…] I plainy say that the whole of this speculation seems to me one huge coherent error [Hermann Lotze, Metaphysics, Oxford, Clarendon Press, 1887, I bk. 2, ch. 2, p. 276].” (quoted in Cassirer “Reflections on the Concept of Group and the Theory of Perception”, translated in D. P. Verene, Symbol, Myth and Culture; Essays and Lectures of Ernst Cassirer 1935–1945, (1979), p. 277) or Wundt who stated, in the strictest of Kantian orthodoxies (quoted in E. Cassirer, The Problem of Knowledge…; vol. 4, p. 28 (ECW 5, p. 32)): ““This question is on a par with that raised more than once in the older ontology: wether or not the real world is the best among all possible worlds. No one since Kant has hesitated to reply that the real world is the only one that exists, and that nothing at all can be said about the nature of worlds that do not exist” (W. Wundt, “der mathematische Raumbegriff”, Logik, I, p. 496, 2. Aufl. Stuttgart, Ferdinand Enke, 1893)”.
- 27.
E. Cassirer, The Problem of Knowledge…; vol. 4, p. 24 (ECW 5, pp. 26–27).
- 28.
E. Cassirer, The Philosophy of Symbolic Forms, vol. 3, pp. 352–353, (ECW 13, p. 406): “What links the various geometries is that each of them considers certain basic properties of spatial forms, which prove invariant in relation to certain transformations; what distinguishes them is the fact that each one of these geometries is characterized by a particular transformation group. [cf. F. Klein, “Erlanger Programm; Vergleichende Betrachtungen über neuere geometrische Forschungen”, Mathematische Annalen, 43, 1893 pp. 63–100]”.
- 29.
In his last manuscript (April 1945), Cassirer described the two directions in question by mentioning on the one hand Hermann Weyl’s book Gruppentheorie und Quantenmechanik [The Theory of Groups and Quantum Mechanics] and, on the other hand, Gestalt Theory as a theory which sought to establish new foundations for a theory of perception, particularly with respect to “perceptual constancy”, using the concept of invariant by transformation inherited from group theory. For these two points, see below.
- 30.
E. Cassirer, The Problem of Knowledge…; vol. 4, p. 33, (ECW 5, p. 38): “In passing from one geometry to another the peculiar change in meaning is constantly seen”.
- 31.
E. Cassirer, The Problem of Knowledge…; vol. 4, p. 22, (ECW 5, p. 24): “The whole character of mathematics appeared radically changed by this view, and axioms that had been regarded for centuries at the supreme example of eternal truth now seemed to belong to an entirely different kind of knowledge. In the words of Leibniz, the ‘eternal verities’ had apparently become merely ‘truths of fact’. It was obvious that no simple mathematical question had been raised here; the whole problem of the truth of mathematics, even of the meaning of truth itself, was placed in an entirely new light.”.
- 32.
It was the same reason for which the Vienna circle defended the logicist thesis according to which the properly conceptual content of geometry could be fully reduced to logical propositions, its experimental content being limited to empirical facts.
- 33.
E. Cassirer, “Kant und die moderne Mathematik”, Kant-Studien, 12, 1907, p. 46 (ECW 9, pp. 79–80): “Even if [experience] is unable to ground and legitimate mathematical concepts by itself alone, it can contribute to determine them more closely and to make a selection between various possible principles that we could consider the apex of our deductions with an equal logical right. If we were to deny experience its function of selection, we would take the risk of depriving the fundamental concepts of geometry of all explicit sense; we would have then no means to make any distinction at all between various complex structures, in so far as are fulfilled all the conditions that we have registered in the axioms”.
- 34.
E. Cassirer, “Kant und die moderne Mathematik”, Kant-Studien, 12, [1907], p. 27, (ECW 9, p. 61): “Thus from now on Geometry doesn’t possess, compared to the general logical axioms, independent and unprovable theorems: what is usually called the ‘geometric Axioms’ are rather only implicit definitions”.
- 35.
These five groups comprise the axioms of incidence, order, congruence, parallels and continuity.
- 36.
E. Cassirer, The Problem of Knowledge…; vol. 4, p. 26 (ECW 5, p. 29): “The single elements receive their roles, and hence their significance, only as they fit together into a connected system; thus they are defined through one another, not independently of one another. Hilbert has expressed and clarified this feature of mathematical thinking with the greatest precision in his theory of “implicit definition”. It is clear that for this theory any given geometry can be from the first nothing but a certain system of order and relations, whose character is determined by principles governing the relationships, and not by the intrinsic nature of the figures entering into it. The same system can acquire as many and as highly varied elements as one wishes without in any respect losing its identity. Hence the points, straight lines, and planes of Euclidean geometry can be replaced in an endless number of ways by other and entirely different objects without the least change in the content and truth of the corresponding theorems”.
- 37.
For example, when Hilbert responds to Frege: “Every axiom contributes something to the definition, and hence every new axiom changes the concept. A ‘point’ in Euclidean, non-Euclidean, Archimedean and non-Archimedean geometry is something different in each case. […]. In thinking of my points I think of some system of things, e.g. the system: love, law, chimney-sweep… and then assume all my axioms as relations between these things, then my propositions, e.g. Pythagoras’ theorem, are also valid for these things.” G. Frege, Philosophical and Mathematical Correspondence, ed. By G. Gabriel, H. Hermes, F. Kambartel, C. Thiel, A. Veraart, trans. By H. Kaal, The University of Chicago Press, 1980, p. 40.
- 38.
E. Cassirer, The Problem of Knowledge…; vol. 4, p. 25 (ECW 5, p. 28): “In his essay On So-called Non-Euclidean Geometry, published in 1871, Felix Klein showed that the entire system thereof can be derived from Euclidean geometry, a fact that makes illusory every preference for one over the other on the score of absolute worth and proves that they share the same fate in respect to “truth”, since any contradiction in one is inevitably attended by similar contradictions in the others. His proof was completed and made still more rigorous by Hilbert, who showed in his Foundations of Geometry that the theorems of the various systems are reflected not only in one another but in the purely analytical theory of real numbers, so that every contradiction in them must also appear in this theory”.
- 39.
E. Cassirer, The Philosophy of Symbolic Forms, vol. 3, pp. 475–476 (ECW 13, p. 555): “For here it is not a matter of disclosing the ultimate, absolute elements of reality, in the contemplation of which thought may rest as it were, but of a never-ending process through which the relatively necessary takes the place of the relatively accidental and the relatively invariable that of a the relatively variable. We can never claim that this process has attained to the ultimate invariants of experience, which would then replace the immutable facticity of “things”; we can never claim to grasp these invariants with our hands so to speak Rather, the possibility must always be held open that a new synthesis will instate itself and that the universal constants, in terms of which we have signalized the “nature” of certain large realms of physical objects, will come close together and prove themselves to be special cases of an overarching lawfulness”.
- 40.
What is traditionally called in retrospect “Hilbert’s programme” (see for example Georg Kreisel, “Hilbert’s Programme”, Dialectica, Volume 12, Issue 3–4, Dec. 1958, pp. 346–372) was developed during the 1920s, using an approach said to be “meta-mathematical” which exclusively concerned the form (as in written form) of mathematical statements and consisted in an attempt to demonstrate the internal coherence of the whole axiomatic approach by reducing the question of this coherence to the sole coherence of integers. Cassirer allows for (without adhering to) this specifically Hilbertian conception of the sign even if he does not follow the ulterior development of Hilbert’s program as such concerning the internal coherence of formal axiomatics nor of the controversies that such a program sparked. Cassirer cites for example Hilbert’s article of 1922 “Neubegründung der Mathematik” [“New Foundations of Mathematics”], Gesammelte Abhandlungen, Band III, Springer Verlag, 1970, pp. 156–177 in E. Cassirer, The Philosophy of Symbolic Forms, vol. 3, pp. 379–380 (ECW 13, p. 437): “In diametrical opposition to Frege and Dedekind, writes Hilbert in summing up his fundamental point of view, “I find the objects of the theory of numbers in the signs themselves, whose form we can recognize universally and surely, independently of place and time and of the special conditions attending the production of the signs as well as of insignificant differences in their elaboration. Here lies the firm philosophical orientation which I regard as requisite to the grounding of pure mathematics, as to all scientific thinking, and communication. ‘In the beginning,’ we may say here, ‘was the sign’”.”
- 41.
Cassirer’s allusion to Hilbert in volume 3 of The Philosophy of Symbolic Forms pertains only to the status conferred by Hilbert to the sign in mathematics and which he compares to Leibniz. Cf. E. Cassirer, The Philosophy of Symbolic Forms, vol. 3, p. 379 (ECW 13, p. 437): “It is a critical authority of this sort that Hilbert strives to create in his theory of proof. Here the fundamental idea of Leibniz’ “universal characteristic” is resumed and given pregnant and acute expression. The process of verification is shifted from the sphere of content to that of symbolic thinking. As precondition for the use of logical inferences and for the practice of logical operations, certain sensuous and intuitive characters must always be given to us. It is in them that our thinking first gains a sure guiding thread, which it must follow if it wishes to remain free from error”.
- 42.
This expression designates the often heated debates between the years 1910 and 1930 opposing the advocates of three theses concerning the nature of the number with respect to a new arithmetics of infinity elaborated by Cantor: Frege’s and Russell’s logicism, Hilbert’s formalism and Brouwer’s and Weyl’s intuitionism. I will return to this.
- 43.
Henri Poincaré, Science and Hypothesis, New York, Dover Publications, 1952.
- 44.
E. Cassirer, The Problem of Knowledge…; vol. 4, pp. 42–43 (ECW 5, pp. 48–49): “The group concept is not restricted either to mathematics or geometry or to number or dimension. In it we rise above any consideration of special elements of mathematical thought to a theory of operations. We are no longer concerned with the special content but rather with the very procedure of mathematics itself, for we have entered the realm of pure, “intellectual mathematics”. Poincaré appears to have been the first thinker to draw the consequences of this for the problem of geometry, and the result was a categorical denial of geometrical empiricism. For if the theory of the group is what must be introduced into the definition of geometry, and if each geometry can be designated as a theory of invariants in respect to a certain group, then a pure a priori element has entered into the conceptual definitions”.
- 45.
E. Cassirer, The Problem of Knowledge…; vol. 4, p. 45 (ECW 5, p. 52): “The autonomous nature of mathematical thought may now be recognized in full measure, and we may see convincing proof of this autonomy precisely in the possibility of setting up a plurality of entirely independent systems of axioms”.
- 46.
E. Cassirer, The Problem of Knowledge…; vol. 4, p. 45 (ECW 5, p. 52): “[…] here the modern concept of axioms differs characteristically from the ancient. They are no longer assertions about contents that have absolute certainty, whether it be conceived as purely intuitive or rational. They are rather proposals of thought that make it ready for action—thought devices which must be so broadly and inclusively conceived as to be open to every concrete application that one wishes to make of them in knowledge”.
- 47.
E. Cassirer, Einstein’s Theory of Relativity, pp. 430–431 (ECW 10, p. 94): “No measurement, as Poincaré objects with justice, is concerned with space itself, but always only with the empirically given and physical objects in space. No experiment therefore can teach us anything about the ideal structures, about the straight line and the circle, that pure geometry takes as a basis; what it gives us is always only knowledge of the relations of material things and processes”.
- 48.
Pierre Duhem, The Aim and Structure of Physical Theory, New York, Atheneum, 1962; reprint Princeton, Princeton University Press, 1991.
- 49.
E. Cassirer, The Problem of Knowledge…; vol. 4, pp. 111–113 (ECW 5, p. 128–133).
- 50.
E. Cassirer, The Problem of Knowledge…; vol. 4, pp. 111–112 (ECW 5, p. 129): “Every judgement concerning an individual case, in so far as it purports to be a proposition in physics, already includes a whole system of physics. It is not true, therefore, that the science consists of two strata, as it were: simple observation and the results of measurement being in the one, theories built upon these in the other. Observation and measurement prior to all theory and independent of its assumptions, are impossible”.
- 51.
E. Cassirer, Einstein’s Theory of Relativity, pp. 432–433 (ECW 10, p. 96): “The reality which alone it [space] can express is not that of things, but that of laws and relations. And now we can ask, epistemologically, only one question: whether there can be established an exact relation and coordination between the symbols of non-Euclidean geometry and the empirical manifold of spatio-temporal “events.” If physics answers this question affirmatively, then epistemology has no ground for answering it negatively. […]. If it is seen thus, that the determination of this element as is done in Euclidean geometry, does not suffice for the mastery of certain problems of knowledge of nature then nothing can prevent us, from a methodological standpoint, from replacing it by another measure, in so far as the latter proves to be necessary and fruitful physically”.
- 52.
Cf. E. Cassirer, Substance and Function, p. 43 (ECW 6, p. 44): “Leibniz whose entire thought was concentrated upon the idea of a “universal characteristic”, clearly pointed out in opposition to the formalistic theories of his time, the fact which is essential here. The “basis” of the truth lies, as he says, never in the symbols but in the objective relations between ideas”.
- 53.
E. Cassirer, The Problem of Knowledge…; vol. 4, p. 114 (ECW 5, p. 133): “Here a particular symbol can never be set over against a particular object and compared in respect to its similarity. All that is required is that the order of the symbols be arranged so as to express the order of phenomena”.
- 54.
E. Cassirer, The Problem of Knowledge…; vol. 4, p. 112 (ECW 5, p. 129): “The scientific statement would not describe the ocular, auditory, or tactile sensations of an individual observer in a particular physical laboratory but would state an objective fact of quite another sort—a fact that could be made known, of course, only if an appropriate language, a system of definite symbols, had been created for its communication”.
- 55.
Cassirer borrows Kant’s metaphor of the alphabetical readability of phenomena; E. Cassirer, Einstein’s Theory of Relativity, p. 434 (ECW 10, p. 97): “What Kant says of the concepts of the understanding in general, that they only serve "to make letters out of phenomena so that we can read them as experiences” holds in particular of the concepts of space. They are only the letters, which we must make into words and propositions, if we would use them as expressions of the laws of experience”.
- 56.
E. Cassirer, Einstein’s Theory of Relativity, p. 434 (ECW 10, p. 97): “The particular geometrical truths or particular axioms, such as the principle of parallels, can never be compared with particular experiences, but we can always only compare with the whole of physical experience the whole of a definite system of axioms”.
- 57.
E. Cassirer, The Problem of Knowledge; vol. 4, p. 44 (ECW 5, p. 50): “For example, we derive from experience the presupposition that it is possible for an object to move freely in all directions without any change in shape, and accordingly we eliminate from the possible geometries a definite group for which this assumption does not hold good. […]. Experience is not the means of proving geometrical truths, though it can well serve as “occasional” cause, in furnishing a motive for and invitation to the development of certain aspects of these truths and the choice of one above the others”.
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E. Cassirer, Einstein’s Theory of Relativity, pp. 439–440 (ECW 10, p. 104): “The step beyond him [Kant], that we have now to make on the basis of the results of the general theory of relativity, consists in the insight that geometrical axioms and laws of other than Euclidean form can enter into this determination of the understanding, in which the empirical and physical world arises for us, and that the admission of such axioms not only does not destroy the unity of the world, i.e., the unity of our experiential concept of a total order of phenomena, but first truly grounds it from a new angle, since in this way the particular laws of nature, with which we have to calculate in space-time determination, are ultimately brought to the unity of a supreme principle, that of the universal postulate of relativity. The renunciation of intuitive simplicity in the picture of the world thus contains the guarantee of its greater intellectual and systematic completeness”.
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E. Cassirer, Einstein’s Theory of Relativity, p. 440 (ECW 10, p. 105): “A doctrine, which originally grew up merely in the immanent progress of pure mathematical speculation, in the ideal transformation of the hypotheses that lie at the basis of geometry, now serves directly as the form into which the laws of nature are poured. The same functions, that were previously established as expressing the metrical properties of non-Euclidean space, give the equations of the field of gravitation”.
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Bernhard Riemann, “Über die Hypothesen, welche der Geometrie zu Grunde liegen”, Abhandlungen der Königlichen Gesellschaft der Wissenschaften zu Göttingen, 1867, Band 13, pp. 133–150; English transl. “On the Hypotheses which lie at the Bases of Geometry”, trans. William K. Clifford, Nature, vol. VIII, Nos. 183, 184, pp. 14–17, 36, 37.
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E. Cassirer, Einstein’s Theory of Relativity, p. 440 (ECW 10, p. 105): “Riemann, in setting up his theory, referred to its future physical meaning in prophetic words of which one is often reminded in the discussion of the general theory of relativity”.
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E. Cassirer, Einstein’s Theory of Relativity, p. 440 (ECW 10, p. 106): “Instead of regarding “space” as a self-existent real, which must be explained and deduced from “binding forces” like other realities, we ask now rather whether the a priori function, the universal ideal relation, that we call “space” involves possible formulations and among them such as are proper to offer an exact and exhaustive account of certain physical relations, of certain “fields of force”. The development of the general theory of relativity has answered this question in the affirmative; it has shown what appeared to Riemann as a geometrical hypothesis, as a mere possibility of thought, to be an organ for the knowledge of reality”.
- 63.
Commenting on Herder’s work, Cassirer points out the following in E. Cassirer, Freiheit und Form; Studien zur Deutschen Geistesgeschichte, ECW 7, pp. 124–125: “Thus restriction and “deprivation” do not mean a lack per se any longer, but rather the necessary condition for any individual perfection. A certain deprivation from knowledges, dispositions and virtues determines just as well as a “positive” quality the place from which its efficiency can and must come from in a particular nation and at a particular time. This very deprivation is immediately fruitful and inspiring as well for it is immediately distinctive and typical. The force of limitation does not simply resist the force of perfection but is rather another expression for it; it is only their combination that grants everything particular the determination of its being and its achievement”.
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Lassègue, J. (2020). The Epistemological Situation of Cassirer. In: Cassirer’s Transformation: From a Transcendental to a Semiotic Philosophy of Forms. Studies in Applied Philosophy, Epistemology and Rational Ethics, vol 55. Springer, Cham. https://doi.org/10.1007/978-3-030-42905-8_1
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