Philosophers From Germany

"Now comes death, now one must hold one's head high!"

"Gustav Landauer wrote: “Uprising as basic law, change and overthrow as a rule for all times… that was the greatness and the holiness of the mosaic social order. We need that again: new rules and a spirit of change that does not fix things and laws definitively but declares itself permanent. The revolution should become part of our social order, the basic rule of our basic law”."

"Benares is holy. Europe, grown superficial, hardly understands such truths anymore. I feel nearer"

"Benares is holy. Europe, grown superficial, hardly understands such truths anymore.....I feel nearer here than I have ever done to the heart of the world; here I feel everyday as if soon, perhaps even today, I would receive the grace of supreme revelation...The atmosphere of devotion which hangs above the river is improbable in strength; stronger than in any church that I have ever visited. Every would be Christian priest would do well to sacrifice a year of his theological studies in order to spend his time on the Ganges; here he would discover what piety means."

"I have not found in Europe or America, poets, thinkers or popular leaders equal, or even comparable, to those of India today."

"Hinduism has produced the profoundest metaphysics that we know of."

"“This philosophical nation par excellence” says Count Keyserling, “has more Sanskrit words for philosophical and religious thought than are found in Greek, Latin and German combined.”"

"India has produced the profoundest metaphysics that we know of … the absolute superiority of India over the West in philosophy; poetry from the Mahabharata, containing the Bhagavad Gita, perhaps the most beautiful work of the literature of the world."

"Hinduism at its best has spoken the only relevant truth about the way to self-realization in the full sense of the word."

"Thus the unique formative power of Islam depends on the unique nature of their God. Allah deserves the name of the Master of Armies far more than Jehovah, far more than the Christian God. He is an autocrat in the sense of a general, not that of a tyrant And thus I appear to have it: the Mohammedan faith signifies, as the only one in the world, essentially military discipline. There is no question of right, no begging, no arguing, no crawling to and before God; here mere intention in prayer (Schirk) is a cardinal sin; man has to obey orders like a soldier. Now no one will deny that the form of consciousness of a well-drilled soldier ensures the greatest efficiency of all everywhere where execution and not thinking out of a problem is concerned. The Islamic world represents a single army with a unified, unbroken spirit. Such a spirit melts down all differences in the long runj it makes every one into a comrade.In Islam it has melted down all racial differences. The ritualism of this faith has a different significance from that of Hinduism and Catholicism. It is a question of making discipline objective. When the faithful perform their prayers at fixed hours in the mosque, kneeling there line upon line, when they all go through the same gesture simultaneously, this is not done, as in the case of Hinduism, as a means to self-realisation, but it is done in the spirit in which a Prussian soldier files past his Emperor. This fundamentally military attitude explains all the intrinsic advantages of a Mussulman. It explains simultaneously his fundamental failings: his lad: of progressiveness, his inadaptability, his lad: of inventive power. The soldier only has to obey his orders; the rest is Allah's business."

"The absolute superiority of India over the West in philosophy; poetry from the Mahabharata, containing the Bhagavad-Gita, “perhaps the most beautiful work of the literature of the world"."

"What we know is as nothing, if we do not love God properly in all things."

"Johann Herbart’s work on education and particularly mathematical psychology influenced me. I think mathematics is the pure instance of construct functioning—the model of human behaviour."

"In every page of David Hume, there is more to be learned than from Hegel's, Herbart's and Schleiermacher's complete philosophical works."

"The intention with which the educator is to approach his work, this practical reflection, provisionally detailed down to the measures which our present state of knowledge suggests we should choose, is to my mind the first half of pedagogics. But there must be a second in which the possibility of education is theoretically explained and presented with its limitations in the light of changing circumstances."

"It is of course a familiar precept that the teacher must try to arouse the interest of his pupils in all that he teaches. However, this precept is generally meant and understood to denote the idea that learning is the end and interest the means to attain it. I wish to reverse that relationship. Learning must serve the purpose of creating interest. Learning is transient, but interest must be lifelong."

"Leibniz foreshadowed the entire doctrine of the unconscious, but Herbart actually began it. Wundt was to appeal first to unconscious inference in order to explain perception, and then to apperception. Fechner was to take from Herbart the notion of the measurement of the magnitude..."

"Among the reforms necessary for the triumph of true refinement and true morality, which ought to be our earnest aim, is the Dietetic one, which, if not the weightiest of all (allerwichtigste), yet, undoubtedly, is one of the weightiest. Still is the ‘civilised’ world stained and defiled by the remains of a horrible barbarity; while the old-world revolting practice of slaughter of animals and feeding on their corpses still is in so universal vogue, that men have not the faculty even of recognising it as such, as otherwise they would recognise it; and aversion from this horror provokes censure of such eccentricity, and amazement at any manifestation of tendency to reform, as at something absurd and ridiculous — nay, arouses even bitterness and hate. To extirpate this barbarism is a task, the accomplishment of which lies in the closest relationship with the most important principles of humaneness, morality, æsthetics, and physiology. A foundation for real culture — a thorough civilising and refining of humanity — is clearly impossible so long as an organised system of murder and of corpse-eating (organiserten Mord-und-Leichenfratz System) prevails by recognised custom."

"Dedekind's language in introducing irrational numbers leaves a little to be desired. He introduces the irrational α as corresponding to the cut and defined by the cut. But he is not too clear of where α comes from. He should say that... α is no more than the cut. ...Heinrich Weber told Dedekind this, and in a letter of 1888 Dedekind replied that... α is not the cut itself but something distinct, which corresponds to the cut and brings about the cut. Likewise, while the rational numbers generate cuts, they are not the same as the cuts. He says we have the mental power to create such concepts."

"Julius Wilhelm Richard Dedekind stands out as one of the most prominent contributors of the 19th century to the theory of algebraic numbers. He wrote various important memoirs on the binomial equation and on the theory of modular and Abelian functions, but is best known for his treatises Was sind und was sollen die Zahlen? (1888) and Stetigkeit und irrationale Zahlen (1872). In the latter work he set forth his idea of the Schnitt (cut) in relation to irrational numbers,—an idea he had in mind as early as 1858."

"The tacit assumption on which analytic geometry operated was that it was possible to represent the points on a line... by means of numbers. This assumption is... equivalent to the assertion that a perfect correspondence can be established... The great success of analytic geometry... gave this assumption an irresistible pragmatic force. It was essential to include this principle... But how? Under such circumstances mathematics proceeds by fiat. It bridges the chasm between intuition and reason by a convenient postulate. ...The very vagueness of all intuition renders such a substitution... highly acceptable. ... On the one hand there was the logically consistent concept of a real number and its aggregate, the arithmetic continuum; on the other hand, the vague notions of the point and its aggregate, the linear continuum. All that was necessary was to declare the identity of the two... to assert that: It is possible to assign to any point on a line a unique real number, and, conversely, any real number can be represented in a unique manner by a point on a line. This is the famous Dedekind-Cantor axiom."

"The beauty of [ Eudoxus' ] theory of proportions [ expounded in Book V of Euclid's Elements ] was its adaptability to this new climate. ...The length \sqrt2 is determined by the two sets of positive rationalsL_\sqrt2 = \{r: r^2 < 2\}, \qquad U_\sqrt2 = \{r: r^2 > 2\}Dedekind... decided to let \sqrt2 be this pair of sets! In general, let any partition of the positive rationals into sets L, U such that any member of L is less than any member of U be a positive real number. This idea, now known as the Dedekind cut, is more than just a twist of Eudoxus; it gives a complete and uniform construction of all real numbers, or points on a line, using just the discrete, finally resolving the fundamental conflict in Greek mathematics."

"The above comparison of the domain R of rational numbers with a straight line has led to the recognition of the existence of gaps, of a certain incompleteness or discontinuity of the former, while we ascribe to the straight line completeness, absence of gaps, or continuity. In what then does this continuity consist? Everything must depend on the answer to this question, and only through it shall we obtain a scientific basis for the investigation of all continuous domains."

"In the preceding section attention was called to the fact that every point p of the straight line produces a separation of the same into two portions such that every point of one portion lies to the left of every point of the other. I find the essence of continuity in the converse, i.e., in the following principle: "If all points of the straight line fall into two classes such that every point of the first class lies to the left of every point of the second class, then there exists one and only one point which produces this division of all points into two classes, this severing of the straight line into two portions." ...every one will at once grant the truth of this statement; the majority of my readers will be very much disappointed in learning that by this commonplace remark the secret of continuity is to be revealed."

"Just as negative and fractional rational numbers are formed by a new creation, and as the laws of operating with these numbers must and can be reduced to the laws of operating with positive integers, so we must endeavor completely to define irrational numbers by means of the rational numbers alone. The question only remains how to do this."

"Although the real theory might have been less useful than the complex in obtaining properties of special functions, its significance for the development of mathematics as a whole has been incomparably greater. It was in the real variable that the necessity for a rigorous theory of the number system of analysis was first recognized. ...the reconstruction of the real number system by Weierstrass in the 1860's and by Dedekind and Cantor in the 1870's led in the last three decades of the nineteenth century, to a profound reconsideration of the nature of all mathematical reasoning. This in turn initiated some of the most searching examinations of all deductive reasoning since the days or Aristotle. Thus the theory of the functions of a real variable since the 1870's has increasingly acquired more than merely a local interest: its problems, solved and unsolved, are significant in fields far distant from technical mathematics."

"If a is any definite number, then all numbers of the system R fall into two classes, A1 and A2, each of which contains infinitely many individuals; the first class A1 comprises all numbers a1 that are < a, the second class A2 comprises all numbers a2 that are > a; the number a itself may be assigned at pleasure to the first or second class, being respectively the greatest number of the first class or the least of the second. In every case the separation of the system R into the two classes A1, A2 is such that every number of the first class A1 is less than every number of the second class A2."

"The way in which the irrational numbers are usually introduced is based directly upon the conception of extensive magnitudes—which itself is nowhere carefully defined—and explains number as the result of measuring such a magnitude by another of the same kind. Instead of this I demand that arithmetic shall be developed out of itself."

"That such comparisons with non-arithmetic notions have furnished the immediate occasion for the extension of the number-concept may, in a general way, be granted (though this was certainly not the case in the introduction of complex numbers); but this surely is no sufficient ground for introducing these foreign notions into arithmetic, the science of numbers."

"In discussing the notion of the approach of a variable magnitude to a fixed limiting value, [Dedekind] had recourse, as had Cauchy before him, to the evidence of the geometry of continuous magnitude. ...Dedekind's approach was somewhat different from that of Weierstrass, Méray, Heine, and Cantor in that, instead of considering in what manner the irrationals are to be defined so as to avoid the vicious circle of Cauchy, he asked himself... what is the nature of continuity? ...The philosophy and mathematics of Leibniz had led him to agree with Galileo that continuity was a property concerning conjunctive aggregation, rather than a unity or coincidence of parts. Leibniz had regarded a set as forming a continuum if between any two elements there was always another element of the set. ...Ernst Mach likewise regarded this property of denseness of an assemblage as constituting its continuity, but... rational numbers... possess the property of denseness and yet do not constitute a continuum. Dedekind...found the essence of... continuity, not by a vague hang-togetherness, but in the nature of the division of the line by a point. ...in any division of the points into two classes such that every point of the one is to the left of every point of the other, there is one and only one point which produces this division. This is not true of the ordered system of rational numbers."

"I regard the whole of arithmetic as a necessary, or at least natural, consequence of the simplest arithmetic act, that of counting, and counting itself as nothing else than the successive creation of the infinite series of positive integers in which each individual is defined by the one immediately preceding; the simplest act is the passing from an already-formed individual to the consecutive new one to be formed. The chain of these numbers forms in itself an exceedingly useful instrument for the human mind; it presents an inexhaustible wealth of remarkable laws obtained by the introduction of the four fundamental operations of arithmetic."

"Addition is the combination of any arbitrary repetitions of the above-mentioned simplest act into a single act; from it in a similar way arises multiplication. While the performance of these two operations is always possible, that of the inverse operations, subtraction and division, proves to be limited. Whatever the immediate occasion may have been, whatever comparisons or analogies with experience, or intuition, may have led thereto; it is certainly true that just this limitation in performing the indirect operations has in each case been the real motive for a new creative act; thus negative and fractional numbers have been created by the human mind; and in the system of all rational numbers there has been gained an instrument of infinitely greater perfection. This system, which I shall denote by R, possesses first of all a completeness and self-containedness which I have designated... as characteristic of a body of numbers [Zahlkőrper] and which consists in this, that the four fundamental operations are always performable with any two individuals in R, i.e., the result is always an individual of R, the single case of division by the number zero being excepted."

"As professor in the Polytechnic School [autumn of 1858] in Zurich I found myself for the first time obliged to lecture upon the elements of the differential calculus and felt, more keenly than ever before, the lack of a really scientific foundation for arithmetic. In discussing the notion of the approach of a variable magnitude to a fixed limiting value, and especially in proving the theorem that every magnitude which grows continually, but not beyond all limits, must certainly approach a limiting value, I had recourse to geometric evidences. Even now such resort to geometric intuition in a first presentation of the differential calculus, I regard as exceedingly useful, from the didactic standpoint, and indeed indispensable, if one does not wish to lose too much time. But that this form of introduction into the differential calculus can make no claim to being scientific, no one will deny. For myself this feeling of dissatisfaction was so overpowering that I made the fixed resolve to keep meditating on the question till I should find a purely arithmetic and perfectly rigorous foundation for the principles of infinitesimal analysis."

"The statement is so frequently made that the differential calculus deals with continuous magnitude, and yet an explanation of this continuity is nowhere given; even the most rigorous expositions of the differential calculus do not base their proofs upon continuity but, with more or less consciousness of the fact, they either appeal to geometric notions or those suggested by geometry, or depend upon theorems which are never established in a purely arithmetic manner. Among these, for example, belongs the above mentioned theorem, and a more careful investigation convinced me that this theorem, or any one equivalent to it, can be regarded in some way as a sufficient basis for infinitesimal analysis. It then only remained to discover its true origin in the elements of arithmetic and thus at the same time to secure a real definition of the essence of continuity. I succeeded Nov. 24, 1858."

"What advantage will be gained by even a purely abstract definition of real numbers of a higher type, I am as yet unable to see, conceiving as I do of the domain of real numbers as complete in itself."

"The system R forms a well-arranged domain of one dimension extending to infinity on two opposite sides. What is meant by this is sufficiently indicated by my use of expressions borrowed from geometric ideas; but just for this reason it will be necessary to bring out clearly the corresponding purely arithmetic properties in order to avoid even the appearance as if arithmetic were in need of ideas foreign to it."

"If a, c are two different numbers, there are infinitely many different numbers lying between a, c."

"The modern theory of functions of one real variable was first worked out by H. Hankel, Dedekind, G. Cantor, Dini, and Heine, and then carried further, principally, by Weierstrass, Schwarz, Du Bois-Reymond, Thomae, and Darboux. Hankel established the principle of the condensation of singularities; Dedekind and Cantor gave definitions for irrational numbers..."

"The Vienna Circle was a discussion group of philosophically interested specialists who came together in 1923 and from 1925 to 1936 met regularly once a week in an institute of Vienna University. These gatherings were conducted by Moritz Schlick, the physicist and philosopher who was appointed professor of the philosophy of inductive sciences in 1922. Over the years, members included Hans Hahn, Otto Neurath, Philipp Frank, Viktor Kraft, Herbert Feigl, Friedrich Waismann, Rudolf Carnap, Kurt Godel, Karl Menger, Bela Juhos and others. There was no conscious aim of radically revising traditional views on the task and place of philosophy, but the members were on the whole well aware that current findings of research into the foundations of logic, mathematics and the natural sciences had important philosophic consequences. Among subjects for discussion were Wittgenstein's Tractatus, the possibility of reducing all concepts of science to what is directly given in experience, the setting up of a criterion of meaningfulness for non-logical utterances, the character of the basic propositions of empirical science, and the devising of a meta-language for the syntactic analysis of scientific language systems."

"The members of the Vienna Circle (Moritz Schlick, Rudolf Carnap, , Hans Hahn, , Fritz Waismann, Kurt Godel, Otto Neurath and others) are working out a ‘Logical Empiricism’. Following Mach and Poincare, but above all Russell and Wittgenstein, all the sciences are treated uniformly. Carnap’s Logischer Aufbau der Welt (1928) shows in which direction future systematic work will move. Wittgenstein’s Tractatus Logico- Philosophicus (1921) clarified, among other things, the position of logic and mathematics; besides the statements that make additions to what is meaningful, there are the ‘tautologies’ that show us which transformations are possible within language. By its syntax the language of science excludes anything that is meaningless from the very beginning."

"Schlick ( [Wende] p.8 ) interprets Wittgenstein's position as follows: philosophy "is that activity by which the meaning of propositions is established or discovered" ; it is a question of "what the propositions actually mean. The content, soul, and spirit of science naturally consist in what is ultimately meant by its sentences; the philosophical activity of rendering significant is thus the alpha and omega of all scientific knowledge"."

"The 'physical' does not mean any particular kind of reality, but a particular kind of denoting reality, namely a system of concepts in the natural sciences which is necessary for the cognition of reality. 'The physical' should not be interpreted wrongly as an attribute of one part of reality, but not of the other ; it is rather a word denoting a kind of conceptual construction, as, e.g., the markers 'geographical' or 'mathematical', which denote not any distinct properties of real things, but always merely a manner of presenting them by means of ideas."

"Philosophy is not a system of propositions, and not a science."

"Philosophy... is that activity by which the meaning of propositions is established or discovered. Philosophy elucidates propositions, science verifies them. In the latter we are concerned with the truth of statements, but in the former with what they actually mean."

"If we take an unprejudiced view of the processes of consciousness, free from all the so-called association rules and theories, we see at once that an idea is no more an even relatively constant thing than is a feeling or emotion or volitional process. There exist only changing and transient ideational processes ; there are no permanent ideas that return again and disappear again."

"He aims at being a sort of Napoleon of the intellectual world. Unfortunately he will never have a Waterloo, for he is a Napoleon without genius and with no central idea which, if defeated, brings down the whole fabric in ruin."

"Throughout the nineteenth century, apart from the division in theoretical sciences and arts, classifiers attempted to divide the sciences into two groups. Already they had before them the examples of Francis Bacon (speculative and descriptive) and Hobbes (quantitative and qualitative). For Coleridge, the sciences were either pure (Grammar, Logic, Rhetoric, Mathematics, Metaphysics) or mixed. Arthur Schopenhauer’s similar groups were called pure and empirical, Wilhelm Wundt in 1887 called them formal and empirical, Globot mathematical and theoretical, and the St. Louis Congress of Arts and Sciences (1904) normative and physical. made similar division of the sciences into abstract and concrete"

"The whole task of psychology can therefore be summed up in these two problems : (1) What are the elements of consciousness ? (2) What combinations do these elements undergo and what laws govern these combinations ?"

"In Aristotle the mind, regarded as the principle of life, divides into nutrition, sensation, and faculty of thought, corresponding to the inner most important stages in the succession of vital phenomena."

"From the standpoint of observation, then, we must regard it as a highly probable hypothesis that the beginnings of the mental life date from as far back as the beginnings of life at large."