Logicians From Germany

"When I started to trace the later development of logic, the first thing I did was to look at Schröder's Vorlesungen über die Algebra der Logik, ...[whose] third volume is on the logic of relations (Algebra und Logik der Relative, 1895). The three volumes immediately became the best-known advanced logic text, and embody what any mathematician interested in the study of logic should have known, or at least have been acquainted with, in the 1890s. While, to my knowledge, no one except Frege ever published a single paper in Frege's notation, many famous logicians adopted Peirce-Schröder notation, and famous results and systems were published in it. Löwenheim stated and proved the Löwenheim theorem (later reproved and strengthened by Thoralf Skolem, whose name became attached to it together with Löwenheim's) in Peircian notation. In fact, there is no reference in Löwenheim's paper to any logic other than Peirce's. To cite another example, Zermelo presented his axioms for set theory in Peirce-Schröder notation, and not, as one might have expected, in Russell-Whitehead notation."

"[My aim is] to design logic as a calculating discipline, especially to give access to the exact handling of relative concepts, and, from then on, by emancipation from the routine claims of natural language, to withdraw any fertile soil from "cliché" in the field of philosophy as well. This should prepare the ground for a scientific universal language that, widely differing from linguistic efforts like Volapük [a universal language like Esperanto, very popular in Germany at the time], looks more like a sign language than like a sound language."

"This ideography is a "formula language", that is, a lingua characterica, a language written with special symbols, "for pure thought", that is, free from rhetorical embellishments, "modeled upon that of arithmetic", that is, constructed from specific symbols that are manipulated according to definite rules."

"Logic is an old subject, and since 1879 it has been a great one."

"Ein Philosoph, der keine Beziehung zur Geometrie hat, ist nur ein halber Philosoph, und ein Mathematiker, der keine philosophische Ader hat, ist nur ein halber Mathematiker."

"'Facts, facts, facts,' cries the scientist if he wants to emphasize the necessity of a firm foundation for science. What is a fact? A fact is a thought that is true. But the scientist will surely not recognize something which depends on men's varying states of mind to be the firm foundation of science."

"Equality gives rise to challenging questions which are not altogether easy to answer... a = a and a = b are obviously statements of differing cognitive value; a = a holds a priori and, according to Kant, is to be labeled analytic, while statements of the form a = b often contain very valuable extensions of our knowledge and cannot always be established a priori. The discovery that the rising sun is not new every morning, but always the same, was one of the most fertile astronomical discoveries. Even to-day the identification of a small planet or a comet is not always a matter of course. Now if we were to regard equality as a relation between that which the names 'a' and 'b' designate, it would seem that a = b could not differ from a = a (i.e. provided a = b is true). A relation would thereby be expressed of a thing to itself, and indeed one in which each thing stands to itself but to no other thing."

"Nur im Zusammenhange eines Satzes bedeuten die Wörter etwas. Es wird also darauf ankommen, den Sinn eines Satzes zu erklären, in dem ein Zahlwort vorkommt."

"Often it is only after immense intellectual effort, which may have continued over centuries, that humanity at last succeeds in achieving knowledge of a concept in its pure form, by stripping off the irrelevant accretions which veil it from the eye of the mind."

"From the medieval development of Aristotle's logic through Leibniz's Characteristica Universalis through Frege and Russell and up to the present development of symbolic logic, it could be argued that exactly the reverse [of Jacques Derrida's argument] is the case; that by emphasizing logic and rationality, philosophers have tended to emphasize written language as the more perspicuous vehicle of logical relations. Indeed, as far as the present era in philosophy is concerned, it wasn't until the 1950s that serious claims were made on behalf of the ordinary spoken vernacular languages, against the written ideal symbolic languages of mathematical logic. When Derrida makes sweeping claims about "the history of the world during an entire epoch," the effect is not so much apocalyptic as simply misinformed."

"The historical approach, with its aim of detecting how things began and arriving from these origins at a knowledge of their nature, is certainly perfectly legitimate; but it also has its limitations. If everything were in continual flux, and nothing maintained itself fixed for all time, there would no longer be any possibility of getting to know about the world, and everything would be plunged into confusion."

"Being true is different from being taken as true, whether by one or by many or everybody, and in no case is it to be reduced to it. There is no contradiction in something's being true which everybody takes to be false. I understand by 'laws of logic' not psychological laws of takings-to-be-true, but laws of truth. ...If being true is thus independent of being acknowledged by somebody or other, then the laws of truth are not psychological laws: they are boundary stones set in an eternal foundation, which our thought can overflow, but never displace. It is because of this that they have authority for our thought if it would attain truth. They do not bear the relation to thought that the laws of grammar bear to language; they do not make explicit the nature of our human thinking and change as it changes."

"Every good mathematician is at least half a philosopher, and every good philosopher is at least half a mathematician."

"Your discovery of the contradiction caused me the greatest surprise and, I would almost say, consternation, since it has shaken the basis on which I intended to build arithmetic."

"If the task of philosophy is to break the domination of words over the human mind [...], then my concept notation, being developed for these purposes, can be a useful instrument for philosophers [...] I believe the cause of logic has been advanced already by the invention of this concept notation."

"Without some affinity in human ideas art would certainly be impossible; but it can never be exactly determined how far the intentions of the poet are realized."

"If I compare arithmetic with a tree that unfolds upward into a multitude of techniques and theorems while its root drives into the depths, then it seems to me that the impetus of the root."

"A judgment, for me is not the mere grasping of a thought, but the admission of its truth."

"A scientist can hardly meet with anything more undesirable than to have the foundations give way just as the work is finished. I was put in this position by a letter from Mr. Bertrand Russell when the work was nearly through the press."

"We suppose, it would seem, that concepts grow in the individual mind like leaves on a tree, and we think to discover their nature by studying their growth; we seek to define them psychologically, in terms of the human mind. But this account makes everything subjective, and if we follow it through to the end, does away with truth. What is known as the history of concepts is really a history either of our knowledge of concepts or of the meanings of words."

"Gottlob Frege created modern logic including "for all," "there exists," and rules of proof. Leibniz and Boole had dealt only with what we now call "propositional logic" (that is, no "for all" or "there exists"). They also did not concern themselves with rules of proof, since their aim was to reach truth by pure calculation with symbols for the propositions. Frege took the opposite track: instead of trying to reduce logic to calculation, he tried to reduce mathematics to logic, including the concept of number."

"Our representative absolutist is Gottlob Frege, whose writings did as much as anything to revive the 'mathematizing' approaches of the Platonist tradition around 1900, and did so—quite explicitly—as a means of protecting philosophy from subordination to the facts of history and psychology. ...The Platonist strand in Descartes' philosophy was revived... by... Frege, who promulgated the original programme of 'conceptual analysis' in his Foundations of Arithmetic. ...Frege ...was rebelling ...against the tendency to telescope formal and prescriptive 'laws of thought', which were the proper concern of logic, with the empirical and descriptive 'laws of thinking', which were the business of cognitive psychologists... [W]e should ignore all merely empirical discoveries, whether about the development of understanding in the individual mind or about the historical evolution of our communal understanding. ...Philosophers must concern themselves with 'concepts' only as timeless, intellectual ideals, towards which the human mind struggles, at best, painfully and little by little. ...[A]ctual conceptions current in any existing community are philosophically significant only as an approximation to the eternal system of ideal 'concepts'. ...[A]ny actual, historical set of conceptions has a legitimate intellectual claim on us, only to the extent that it approximates that ideal."

"Bertrand Russell found Frege's famous error: Frege had overlooked what is now known as the Russell paradox. Namely, Frege's rules allowed one to define the class of x such that P(x) is true for any "concept" P. Frege's idea was that such a class was an object itself, the class of objects "falling under the concept P." Russell used this principle to define the class R of concepts that do not fall under themselves. This concept leads to a contradiction... argument: (1) if R falls under itself then it does not fall under itself; (2) this contradiction shows that it does not fall under itself; (3) therefore by definition it does fall under itself after all."

"I hope I may claim in the present work to have made it probable that the laws of arithmetic are analytic judgments and consequently a priori. Arithmetic thus becomes simply a development of logic, and every proposition of arithmetic a law of logic, albeit a derivative one. To apply arithmetic in the physical sciences is to bring logic to bear on observed facts; calculation becomes deduction."

"Is it always permissible to speak of the extension of a concept, of a class? And if not, how do we recognize the exceptional cases? Can we always infer from the extension of one concept's coinciding with that of a second, that every object which falls under the first concept also falls under the second?"

"The ideal of strictly scientific method in mathematics which I have tried to realise here, and which perhaps might be named after Euclid I should like to describe in the following way... The novelty of this book does not lie in the content of the theorems but in the development of the proofs and the foundations on which they are based... With this book I accomplish an object which I had in view in my Begriffsschrift of 1879 and which I announced in my Grundlagen der Arithmetik. I am here trying to prove the opinion on the concept of number that I expressed in the book last mentioned."

"It really is worth the trouble to invent a new symbol if we can thus remove not a few logical difficulties and ensure the rigour of the proofs. But many mathematicians seem to have so little feeling for logical purity and accuracy that they will use a word to mean three or four different things, sooner than make the frightful decision to invent a new word."

"Now it is time for us to realize that, in his Grundzüge, Kronecker did not merely intend to give his own treatment of the basic problems of ideal-theory which form the main subject of Dedekind's life-work. His aim was a higher own. He was, in fact, attempting to describe and to initiate a new branch of mathematics, which would contain both number-theory and algebraic geometry as special cases. This grandiose conception has been allowed to fade out of our sight, partly because of the intrinsic difficulties of carrying it out, partly owing to historical accidents and to the temporary successes of the partisans of purity and of Dedekind. It will be the main purpose of this lecture to try to rescue it from oblivion, to revive it, and to describe the few modern results which may be considered as belonging to the Kroneckerian program."

"Es handelt sich um meinen liebsten Jugendtraum, nämlich um den Nachweis, dass die Abel’schen Gleichungen mit Quadratwurzeln rationaler Zahlen durch die Transformations-Gleichungen elliptischer Functionen mit singularen Moduln grade so erschöpft werden, wie die ganzzahligen Abel’schen Gleichungen durch die Kreisteilungsgleichungen."

"I saw the Emperor—this world-soul—riding out of the city on reconnaissance. It is indeed a wonderful sensation to see such an individual, who, concentrated here at a single point, astride a horse, reaches out over the world and masters it."

"Die Architektur ist dann die Kristallisation, die Skulptur die organische Figuration der Materie in ihrer sinnlich-räumlichen Totalität; die Malerei die gefärbte Fläche und Linie; während in der Musik der Raum überhaupt zu dem in sich erfüllten Punkt der Zeit übergeht; bis das äußere Material endlich in der Poesie ganz zur Wertlosigkeit herabgesetzt ist."

"You know my present occupation and why I am pursuing it. You know also that I always had a penchant for politics. But this interest has been weakened by journalism far more than it has found sustenance in it. I have to look at political news from a different point of view from that of the reader. The important thing for the reader is content. For me a news item has interest as an article filling a page. But the diminished enjoyment afforded by the satisfaction of my political curiosity has its compensations. In the first place, income. I have convinced myself by experience of the truth of the biblical text which I have made my guiding light: “Strive ye first after food and clothing, and the Kingdom of God will fall to you as well’’ [reversal of Matthew 6:33]. The second advantage is that a journalist is himself an object of curiosity and almost of envy, in that everybody wants to know what he is holding secretly [in petto]—which, according to the universal persuasion, is surely the best part. But just between us, I never know more than what appears in my newspaper, and often not even that much."

"In connection with Kant we must here begin by speaking of Jacobi, whose philosophy is contemporaneous with that of Kant; in both of these the advance beyond the preceding period is very evident. The result in the two cases is much the same, although both the starting point and the method of progression are somewhat different. In Jacobi's case the stimulus was given mainly by French philosophy, with which he was very conversant, and also by German metaphysics, while Kant began rather from the English side, that is, from the skepticism of Hume. Jacobi, in that negative attitude which he preserved as well as Kant, kept before him the objective aspect of the method of knowledge, and specially considered it, for he declared knowledge to be in its content incapable of recognizing the Absolute: the truth must be concrete, present, but not finite. Kant does not consider the content, but took the view of knowledge being subjective; and for this reason he declared it to be incapable of recognizing absolute existence. To Kant knowledge is thus a knowledge of phenomena only, not because the categories are merely limited and finite, but because they are subjective. To Jacobi, on the other hand, the chief point is that the categories are not merely subjective, but that they themselves are conditioned. This is an essential difference between the two points of view, even if they both arrive at the same result."

"This is Bruno's fundamental idea. He says: ”To recognize this unity of form and matter in all things, is what reason is striving to attain to. But in order to penetrate to this unity, in order to investigate all the secrets of Nature, we must search into the opposed and contradictory extremes of things, the maximum and the minimum” It is in these very extremes that they are intelligible, and become united in the Notion; and this union of them is infinite Nature. “To find the point of union is not the greatest matter; but to develop from, the same its very opposite, this is the real and the deepest secret of the art” It is saying much if we speak of knowing the development of the Idea as a necessity of determinations; we shall see later how Bruno proceeded to do this. He represents the original principle, which is elsewhere known as the form, under the Notion of the minimum, which is at the same time the maximum One, which at the same time is All; the universe is this One in All."

"Poetry is the universal art of the spirit which has become free in itself and which is not tied down for its realization to external sensuous material; instead, it launches out exclusively in the inner space and the inner time of ideas and feelings."

"Hegel ... destroyed the illusion of the subject's being-in-itself and showed that the subject is itself an aspect of social objectivity. ... However, ... we must ask this question: is this objectivity which we have shown to be a necessary condition and which subsumes abstract subjectivity in fact the higher factor? Does it not rather remain precisely what Hegel reproached it with being in his youth, namely pure externality, the coercive collective? Does not the retreat to this supposedly higher authority signify the regression of the subject, which had earlier won its freedom only with the greatest efforts, with infinite pains?"

"Aristotle (De Anima, I. 1) makes in the first place the general remark that it appears as if the soul must, on the one hand, be regarded in its freedom as independent and as separable from the body, since in thinking it is independent; and, on the other hand, since in the emotions it appears to be united with the body and not separate, it must also be looked on as being inseparable from it; for the emotions show themselves as materialized Notions (λόγοι έννοια), as material modes of what is spiritual. With this a twofold method of considering the soul, also known to Aristotle, comes into play, namely the purely rational or logical view, on the one hand, and, on the other hand, the physical or physiological; these we still see practiced side by side. According to the one view, anger, for instance, is looked on as an eager desire for retaliation or the like; according to the other view it is the surging upward of the heartblood and the warm element in man. The former is the rational, the latter the material view of anger; just as one man may define a house as a shelter against wind, rain, and other destructive agencies, while another defines it as consisting of wood and stone; that is to say, the former gives the determination and the form, or the purpose of the thing, while the latter specifies the material it is made of, and its necessary conditions."

"Viewed in the light of such facts, we may conclude slavery to have been the occasion of the increase of human feeling among the Negroes. The doctrine which we deduce from this condition of slavery among the Negroes, and which constitutes the only side of the question that has an interest for our inquiry, is that which we deduce from the Idea: viz., that the “Natural condition” itself is one of absolute and thorough injustice – contravention of the Right and Just. Every intermediate grade between this and the realization of a rational State retains – as might be expected – elements and aspects of injustice; therefore we find slavery even in the Greek and Roman States, as we do serfdom down to the latest times. But thus existing in a State, slavery is itself a phase of advance from the merely isolated sensual existence – a phase of education – a mode of becoming participant in a higher morality and the culture connected with it. Slavery is in and for itself injustice, for the essence of humanity is Freedom; but for this man must be matured. The gradual abolition of slavery is therefore wiser and more equitable than its sudden removal."

"That condition which man terms the life of man in unity with nature, and in which man meets with God in nature because he finds his satisfaction there, has ceased to exist. The unity of man with the world is for this end broken, that it may be restored in a higher unity, that the world, as an intelligible world, may be received into God. The relation of man to God thereby reveals itself in the way provided for our salvation in worship, but more particularly it likewise shows itself in Philosophy; and that with the express consciousness of the aim that the individual should render himself capable of belonging to this intelligible world. The manner in which man represents to himself his relation to God is more particularly determined by the manner in which man represents to himself God. What is now often said, that man need not know God, and may yet have the knowledge of this relation, is false. Since God is the First, He determines the relation, and therefore in order to know what is the truth of the relation, man must know God. Since therefore thought goes so far as to deny the natural, what we are now concerned with is not to seek truth in any existing mode, but from our inner Being to go forth again to a true objective, which derives its determination from the intrinsic nature of thought."

"We assert then that nothing has been accomplished without interest on the part of the actors; and — if interest be called passion, inasmuch as the whole individuality, to the neglect of all other actual or possible interests and claims, is devoted to an object with every fibre of volition, concentrating all its desires and powers upon it — we may affirm absolutely that nothing great in the World has been accomplished without passion."

"The first remark we have to make, and which — though already presented more than once — cannot be too often repeated when the occasion seems to call for it, — is that what we call principle, aim, destiny, or the nature and idea of Spirit, is something merely general and abstract. Principle — Plan of Existence — Law — is a hidden, undeveloped essence, which as such — however true in itself — is not completely real."

"This final aim is God's purpose with the world; but God is the absolutely perfect Being, and can, therefore, will nothing but himself."

"Very similar were the views expressed by Raymundus of Sabunde or Sabeyde, a Spaniard of the fifteenth century, and professor at Toulouse about the year 1437. In his theologia natural is, which he handled in a speculative spirit, he dealt with the Nature of things, and with the revelation of God in Nature and in the history of the God-man. He sought to prove to unbelievers the Being, the trinity, the incarnation, the life, and the revelation of God in Nature, and in the history of the God-man, basing his arguments on Reason. From the contemplation of Nature he rises to God; and in the same way he reaches morality from; observation of man's inner nature. This purer and simpler style must be set off against the other, if we are to do justice to the Scholastic theologians in their turn."

"Hegel's complex thought was brought into focus by power "concentrated in a point," which made new beginnings possible. When political reaction followed Napoleonic innovations and Prussian reform , Hegel sought to convert philosophy into a political weapon. He succeeded in politicizing philosophy; his lectures satisfied the striving toward power and "relevance" that was inherent in the University of Berlin-and in much of modern intellectual life. Hegel expressed, first of all, the supreme self-confidence of the thinking man in the value of his thought. Everything became relative to historical context because his own capacity for seeing the whole picture was assumed to be absolute. Accepting the romantic belief that truth was revealed in the peculiarities of history rather than in a static natural order, Hegel nevertheless simultaneously pressed the Enlightenment idea that all was rational. His method applied reason to precisely those phenomena that most interested the romantic mind : art, philosophy, and religion. He had begun as a student of theology, in search of a theodicy, a justification of the ways of God to man ; he ended up instead creating a new God : the "World Spirit." Just as Hegel saw his chair of philosophy giving overall coherence to the intellectual variety of the new university, so the World Spirit provided a unifying rationale to the historical process. Just as Berlin University was the dynamo for regenerating German society, so Hegel's philosophy was its source of dynamism."

"The Democratical State is not Patriarchal, does not rest on a still unreflecting, undeveloped confidence, but implies laws, with the consciousness of their being founded on an equitable and moral basis, and the recognition of these laws as positive. At the time of the Kings, no political life had as yet made its appearance in Hellas; there are, therefore, only slight traces of Legislation. But in the interval from the Trojan War till near the time of Cyrus, its necessity was felt. The first Lawgivers are known under the name of The Seven Sages, a title which at that time did not imply any such character as that of the Sophists teachers of wisdom, designedly [and systematically] proclaiming the Bight and True but merely thinking men, whose thinking stopped short of Science, properly so called. They were practical politicians; the good counsels which two of them Thales of Miletus and Bias of Priene gave to the Ionian cities, have been already mentioned. Thus Solon was commissioned by the Athenians to give them laws, as those then in operation no longer sufficed. Solon gave the Athenians a constitution by which all obtained equal rights, yet not so as to render the Democracy a quite abstract one. The main point in Democracy is moral disposition. Virtue is the basis of Democracy, remarks Montesquieu; and this sentiment is as important as it is true in reference to the idea of Democracy commonly entertained. The Substance, [the Principle] of Justice, the common weal, the general interest, is the main consideration; but it is so only as Custom, in the form of Objective Will, so that morality properly so called subjective conviction and intention has not yet manifested itself. Law exists, and is in point of substance, the Law of Freedom, rational [in its form and purport,] and valid because it is Law, i.e. without ulterior sanction. As in Beauty the Natural element its sensuous coefficient remains, so also in this customary morality, laws assume the form of a necessity of Nature."

"Universal History exhibits the gradation in the development of that principle whose substantial purport is the consciousness of Freedom. The analysis of the successive grades, in their abstract form, belongs to Logic; in their concrete aspect to the Philosophy of Spirit. Here it is sufficient to state that the first step in the process presents that immersion of Spirit in Nature which has been already referred to ; the second shows it as advancing to the consciousness of its freedom. But this initial separation from Nature is imperfect and partial, since it is derived immediately from the merely natural state, is consequently related to it, and is still encumbered with it as an essentially connected element. The third step is the elevation of the soul from this still limited and special form of freedom to its pure universal form ; that state in which the spiritual essence attains the consciousness and feeling of itself. These grades are the ground-principles of the general process; but how each of them on the other hand involves within itself a process of formation, constituting the links in a dialectic of transition, to particularise this must be preserved for the sequel. Here we have only to indicate that Spirit begins with a germ of infinite possibility, but only possibility, containing its substantial existence in an undeveloped form, as the object and goal which it reaches only in its resultant full reality. In actual existence Progress appears as an advancing from the imperfect to the more perfect; but the former must not be understood abstractly as only the imperfect, but as something which involves the very opposite of itself the so-called perfect as a germ or impulse. So reflectively, at least possibility points to something destined to become actual; the Aristotelian δύναμιςis also potentia, power and might. Thus the Imperfect, as involving its opposite, is a contradiction, which certainly exists, but which is continually annulled and solved; the instinctive movement the inherent impulse in the life of the soul to break through the rind of mere nature, sensuousness, and that which is alien to it, and to attain to the light of consciousness, i. e. to itself."

"On the stage on which we are observing it, — Universal History — Spirit displays itself in its most concrete reality."

"The destiny of the spiritual World, and, — since this is the substantial World, while the physical remains subordinate to it, or, in the language of speculation, has no truth as against the spiritual, — the final cause of the World at large, we allege to be the consciousness of its own freedom on the part of Spirit, and ipso facto, the reality of that freedom."

"Aims, principles, &c., have a place in our thoughts, in our subjective design only; but not yet in the sphere of reality. That which exists for itself only, is a possibility, a potentiality; but has not yet emerged into Existence. A second element must be introduced in order to produce actuality — viz. actuation, realization; and whose motive power is the Will — the activity of man in the widest sense."

"It is a matter of perfect indifference where a thing originated; the only question is: "Is it true in and for itself?" Many think that by pronouncing a doctrine to be Neo-Platonic, they have ipso facto banished it from Christianity. Whether a Christian doctrine stands exactly thus or thus in the Bible, the point to which the exegetical scholars of modern times devote all their attention is not the only question. The Letter kills, the Spirit makes alive: this they say themselves, yet pervert the sentiment by taking the Understanding for the Spirit."