Weyl Quotes

Besides language and music mathematics is one of the primary manifestations of the free creative power of the human mind.

The objective world is, it does not happen. Only to the gaze of my consciousness, crawling along the lifeline of my body, does a section of this world come to life as a fleeting image in space which continuously changes in time.

The goal of mathematics is the symbolic comprehension of the infinite with human, that is finite, means.

My work has always tried to unite the True with the Beautiful and when I had to choose one or the other, I usually chose the Beautiful.

Mathematics has the inhuman quality of starlight, brilliant and sharp, but cold.

Symmetry, as wide or as narrow as you may define its meaning, is one idea by which man through the ages has tried to comprehend and create or­der, beauty and perfection

The Greeks made Space the subject-matter of a science of supreme simplicity and certainty. Out of it grew, in the mind of classical antiquity, the idea of pure science. Geometry became one of the most powerful expressions of that sovereignty of the intellect that inspired the thought of those times. At a later epoch, when the intellectual despotism of the Church, which had been maintained through the Middle Ages, had crumbled, and a wave of scepticism threatened to sweep away all that had seemed most fixed, those who believed in Truth clung to Geometry as to a rock, and it was the highest ideal of every scientist to carry on his science 'more geometrico.

By refraining from reducing multiplication to addition we are enabled through these axioms to banish continuity, which is so difficult to fix precisely, from the logical structure of geometry.

Our generation is witness to a development of physical knowledge such as has not been seen since the days of Kepler, Galileo and Newton, and mathematics has scarcely ever experienced such a stormy epoch. Mathematical thought removes the spirit from its worldly haunts to solitude and renounces the unveiling of the secrets of Nature. But as recompense, mathematics is less bound to the course of worldly events than physics.

If Hilbert's view prevails over intuitionism, as appears to be the case, then I see in this a decisive defeat of the philosophical attitude of pure phenomenology, which thus proves to be insufficient for the understanding of creative science even in the area of cognition that is most primal and most readily open to evidence – mathematics.

Or, we may use Cartesian co-ordinate systems from the outset:

With mathematics we stand precisely at that intersection of bondage and freedom that is the essence of the human itself.

It is a tragic and strange fact , a superb malice of the creator , that man's mind is so immensely better suited for handling what is irrelevant than what is relevant to him.

Professor Weyl,* the mathematician, gave an excellent definition of symmetry, which is that a thing is symmetrical if there is something that you can do to it so that after you have finished doing it it looks the same as it did before. That is the sense in which we say that the laws of physics are symmetrical; that there are things we can do to the physical laws, or to our way of representing the physical laws, which make no difference, and leave everything unchanged in its effects. It is this aspect of physical laws that is going to concern us in this lecture.

Therefore, while the revolution was undeniably a transforming event, it was not about the “fundamental transformation” of American civil society itself, as President Barack Obama would proclaim about his own election. Moreover, its purpose and principles were the antithesis of and incompatible with the philosophies that undergird the modern Progressive Movement, such as those espoused by German philosophers Georg Wilhelm Friedrich Hegel and Karl Marx, and later American progressive intellectuals including Herbert Croly, Woodrow Wilson, John Dewey, and Walter Weyl, among others.

Brouwer's remark is simple but deep: we have here the creation of the “continuum,” which, although containing individual real numbers, does not dissolve into a set of real numbers as finished beings; we rather have a medium of free Becoming. We found ourselves in the domain of an ageold problem of thought, the problem of continuity, of change, and of Becoming.

Treason the only crime defined in the Constitution. Tyranny as under the Stuart and Tudor kings characterized by the elimination of political dissent under the laws of treason. Treason statutes which were many and unending, the instrument by which the monarch eliminated his opposition and also added to his wealth. The property of the executed traitor forfeited by his heirs because of the loathsomeness of his crime. The prosecution of treason, like witchcraft, an industry. Founding Fathers extremely sensitive to the establishment of a tyranny in this country by means of ambiguous treason law. Themselves traitors under British law. Under their formulation it became possible to be guilty of treason only against the nation, not the individual ruler or party. Treason was defined as an action rather than thought or speech. "Treason against the US shall consist only in levying war against them, or in adhering to their Enemies, giving them Aid & Comfort...No person shall be convicted of treason unless on the testimony of two witnesses to the same Overt act, or on Confession in Open Court." This definition, by members of the constitutional convention, intended that T could not be otherwise defined short of constitutional amendment. "The decision to impose constitutional safeguards on treason prosecutions formed part of a broad emerging American tradition of liberalism...No American has ever been executed for treason against his country," says Nathaniel Weyl, Treason the story of disloyalty and betrayal in American history, published in the year 1950. I say if this be treason make the most of it.

The three main mediaeval points of view regarding universals are designated by historians as realism, conceptualism, and nominalism. Essentially these same three doctrines reappear in twentieth-century surveys of the philosophy of mathematics under the new names logicism, intuitionism, and formalism. Realism, as the word is used in connection with the mediaeval controversy over universals, is the Platonic doctrine that universals or abstract entities have being independently of the mind; the mind may discover them but cannot create them. Logicism, represented by Frege, Russell, Whitehead, Church, and Carnap, condones the use of bound variables to refer to abstract entities known and unknown, specifiable and unspecifiable, indiscriminately. Conceptualism holds that there are universals but they are mind-made. Intuitionism, espoused in modern times in one form or another by Poincaré, Brouwer, Weyl, and others, countenances the use of bound variables to refer to abstract entities only when those entities are capable of being cooked up individually from ingredients specified in advance. As Fraenkel has put it, logicism holds that classes are discovered while intuitionism holds that they are invented—a fair statement indeed of the old opposition between realism and conceptualism. This opposition is no mere quibble; it makes an essential difference in the amount of classical mathematics to which one is willing to subscribe. Logicists, or realists, are able on their assumptions to get Cantor’s ascending orders of infinity; intuitionists are compelled to stop with the lowest order of infinity, and, as an indirect consequence, to abandon even some of the classical laws of real numbers. The modern controversy between logicism and intuitionism arose, in fact, from disagreements over infinity. Formalism, associated with the name of Hilbert, echoes intuitionism in deploring the logicist’s unbridled recourse to universals. But formalism also finds intuitionism unsatisfactory. This could happen for either of two opposite reasons. The formalist might, like the logicist, object to the crippling of classical mathematics; or he might, like the nominalists of old, object to admitting abstract entities at all, even in the restrained sense of mind-made entities. The upshot is the same: the formalist keeps classical mathematics as a play of insignificant notations. This play of notations can still be of utility—whatever utility it has already shown itself to have as a crutch for physicists and technologists. But utility need not imply significance, in any literal linguistic sense. Nor need the marked success of mathematicians in spinning out theorems, and in finding objective bases for agreement with one another’s results, imply significance. For an adequate basis for agreement among mathematicians can be found simply in the rules which govern the manipulation of the notations—these syntactical rules being, unlike the notations themselves, quite significant and intelligible.

Now, if on the one hand it is very satisfactory to be able to give a common ground in the theory of knowledge for the many varieties of statements concerning space, spatial configurations, and spatial relations which, taken together, constitute geometry, it must on the other hand be emphasised that this demonstrates very clearly with what little right mathematics may claim to expose the intuitional nature of space. Geometry contains no trace of that which makes the space of intuition what it is in virtue of its own entirely distinctive qualities which are not shared by “states of addition-machines” and “gas-mixtures” and “systems of solutions of linear equations”. It is left to metaphysics to make this “comprehensible” or indeed to show why and in what sense it is incomprehensible. We as mathematicians have reason to be proud of the wonderful insight into the knowledge of space which we gain, but, at the same time, we must recognise with humility that our conceptual theories enable us to grasp only one aspect of the nature of space, that which, moreover, is most formal and superficial.

We now come to the decisive step of mathematical abstraction: we forget about what the symbols stand for. ...[The mathematician] need not be idle; there are many operations which he may carry out with these symbols, without ever having to look at the things they stand for. Hermann Weyl, The Mathematical Way of Thinking

Two possibilities present themselves for the analytical treatment of metrical geometry.

the problem is then to develop a theory of invariance with respect to arbitrary linear transformations, in which, however, in contra-distinction to the case of affine geometry, we have a definite invariant quadratic form, viz. the metrical groundform once and for all as an absolute datum.

We must here follow the first course so as to be able to pass on later to generalisations which extend beyond the limits of Euclidean geometry.

What God has put asunder, man should not join together, (Pauli to Weyl)

Wheeler's brilliant student Richard Feynman had developed a unique approach to quantum mechanics, called 'sum over histories', that generalized Hamilton's least-action principle to the study of how photons transfer between electrons and other charged particles to generate the electromagnetic force. In creating a force, the photon acts as what is called an 'exchange particle'. (Its existence is required through Weyl's gauge theory of electromagnetism.) Unlike in classical mechanics, in which particles travel along unique paths, Feynman showed how in quantum interactions all possible paths are taken, weighted by their probabilities to create a net result.

No nation influenced American thinking more profoundly than Germany, W.E.B. DuBois, Charles Beard, Walter Weyl, Richard Ely, Richard Ely, Nicholas Murray Butler, and countless other founders of modern American liberalism were among the nine thousand Americans who studied in German universities during the nineteenth century. When the American Economic Association was formed, five of the six first officers had studied in Germany. At least twenty of its first twenty-six presidents had as well. In 1906 a professor at Yale polled the top 116 economists and social scientists in America; more than half had studied in Germany for at least a year. By their own testimony, these intellectuals felt "liberated" by the experience of studying in an intellectual environment predicated on the assumption that experts could mold society like clay. No European statesman loomed larger in the minds and hearts of American progressives than Otto von Bismarck. As inconvenient as it may be for those who have been taught "the continuity between Bismarck and Hitler", writes Eric Goldman, Bismarck's Germany was "a catalytic of American progressive thought". Bismarck's "top-down socialism", which delivered the eight-hour workday, healthcare, social insurance, and the like, was the gold standard for enlightened social policy. "Give the working-man the right to work as long as he is healthy; assure him care when he is sick; assure him maintenance when he is old", he famously told the Reichstag in 1862. Bismarck was the original "Third Way" figure who triangulated between both ends of the ideological spectrum. "A government must not waver once it has chosen its course. It must not look to the left or right but go forward", he proclaimed. Teddy Roosevelt's 1912 national Progressive Party platform conspicuously borrowed from the Prussian model. Twenty-five years earlier, the political scientist Woodrow Wilson wrote that Bismarck's welfare state was an "admirable system . . . the most studied and most nearly perfected" in the world.

when placed in two positions successively, realises this idea of the equality of two portions of space ; by a rigid body we mean one which, however it be moved or treated, can always be made to appear the same to us as before, if we take up the appropriate position with respect to it. I shall

Among the leading intellectual proponents of Roosevelt’s form of liberalism were the three brilliant young founders of The New Republic, Herbert Croly, Walter Lippmann, and Walter Weyl—all slightly older friends of Adolf Berle’s. In 1909 Croly published a Progressive Era manifesto called The Promise of American Life. “The net result of the industrial expansion of the United States since the Civil War,” Croly wrote, “has been the establishment in the heart of the American economic and social system of certain glaring inequalities of condition and power … The rich men and big corporations have become too wealthy and powerful for their official standing in American life.” He asserted that the way to solve the problem was to reorient the country from the tradition of Thomas Jefferson (rural, decentralized) to the tradition of Alexander Hamilton (urban, financially adept). Weyl, in The New Democracy (1913), wrote that the country had been taken over by a “plutocracy” that had rendered the traditional forms of American democracy impotent; government had to restore the balance and “enormously increase the extent of regulation.” To

Among the leading intellectual proponents of Roosevelt’s form of liberalism were the three brilliant young founders of The New Republic, Herbert Croly, Walter Lippmann, and Walter Weyl—all slightly older friends of Adolf Berle’s. In 1909 Croly published a Progressive Era manifesto called The Promise of American Life. “The net result of the industrial expansion of the United States since the Civil War,” Croly wrote, “has been the establishment in the heart of the American economic and social system of certain glaring inequalities of condition and power … The rich men and big corporations have become too wealthy and powerful for their official standing in American life.” He asserted that the way to solve the problem was to reorient the country from the tradition of Thomas Jefferson (rural, decentralized) to the tradition of Alexander Hamilton (urban, financially adept). Weyl, in The New Democracy (1913), wrote that the country had been taken over by a “plutocracy” that had rendered the traditional forms of American democracy impotent; government had to restore the balance and “enormously increase the extent of regulation.” To liberals of this kind, these were problems of nation-threatening severity, requiring radical modernization that would eliminate the trace elements of rural nineteenth-century America. Lippmann, in Drift and Mastery (1914), argued that William Jennings Bryan (“the true Don Quixote of our politics”) and his followers were fruitlessly at war with “the economic conditions which had upset the old life of the prairies, made new demands on democracy, introduced specialization and science, had destroyed village loyalties, frustrated private ambitions, and created the impersonal relationships of the modern world.” A larger, more powerful, more technical central government, staffed by a new class of trained experts, was the only plausible way to fight the dominance of big business. The leading Clash of the Titans liberals were from New York City, but even William Allen White, the celebrated (in part for being anti-Bryan) small-town Kansas editor who was a leading Progressive and one of their allies, wrote, in 1909, that “the day of the rule of the captain of industry is rapidly passing in America.” Now the country needed “captains of two opposing groups—capitalism and democracy” to reset the

necessity of character as the chief factor in any man's success—a teaching in which I now believe as sincerely as ever, for all the laws that the wit of man can devise will never make a man a worthy citizen unless he has within himself the right stuff, unless he has self-reliance, energy, courage, the power of insisting on his own rights and the sympathy that makes him regardful of the rights of others. All this individual morality I was taught by the books I read at home and the books I studied at Harvard. But there was almost no teaching of the need for collective action, and of the fact that in addition to, not as a substitute for, individual responsibility, there is a collective responsibility. Books such as Herbert Croly's "Promise of American Life" and Walter E. Weyl's "New Democracy" would generally at that time have been treated either as unintelligible or else as pure heresy. The teaching which I received was genuinely democratic in one way. It was not so democratic in another. I grew into manhood thoroughly imbued with the feeling that a man must be respected for what he made of himself. But I had also, consciously or unconsciously, been taught that socially and industrially pretty much the whole duty of the man lay in thus making the best of himself; that he should be honest in his dealings with others and charitable in the old-fashioned way to the unfortunate; but that it was no part of his business to join with others in trying to make things better for the many by curbing the abnormal and excessive development of individualism in a few. Now I do not mean that this training was by any means all bad. On the contrary, the insistence upon individual responsibility was, and is, and always will be, a prime necessity. Teaching of the kind I absorbed from both my text-books and my surroundings is a healthy anti-scorbutic to the sentimentality which by complacently excusing the individual for all his shortcomings would finally hopelessly weaken the spring of moral purpose. It also keeps alive that virile vigor for the lack of which in the average individual no possible perfection of law or of community action can ever atone. But such teaching, if not corrected by other teaching, means acquiescence in a riot of lawless business individualism which would be quite as destructive to real civilization as the lawless military individualism of the Dark Ages.

We were promised economic dynamism in exchange for inequality,” write Eric Posner and Glen Weyl in their book, Radical Markets. “We got the inequality, but dynamism is actually declining.”20

In fact, it was by demanding a local version of special relativity that Einstein got the equations for the metric field that are the core of general relativity! And it is by demanding local versions of rotations in property spaces that C.N. Yang and Robert Mills found the equations that bear their name and govern the weak and strong fluids. Yang and Mills built on the work of Herman Weyl, who showed that Maxwell's equations for the electromagnetic fluid can be derived in that way.

Weyl curvature tensor . . . but consciousness was no

When Martino was released, he wrote a book titled I Was Castro’s Prisoner, published in August, 1963. His collaborator on the book was Nathaniel Weyl, author of Red Star Over Cuba, who worked with Frank Sturgis after the Kennedy assassination pushing stories about Oswald in Miami. Weyl was also a member of the Citizens Committee to Free Cuba.

Mathematics with Brouwer gains its highest intuitive clarity. He succeeds in developing the beginnings of analysis in a natural manner, all the time preserving the contact with intuition much more closely than had been done before. It cannot be denied, however, that in advancing to higher and more general theories the inapplicability of the simple laws of classical logic eventually results in an almost unbearable awkwardness. And the mathematician watches with pain the larger part of his towering edifice which he believes to be built of concrete blocks dissolve into mist before his eyes.

Brouwer's remark is simple but deep: we have here the creation of the "continuum," which, although containing individual real numbers, does not dissolve into a set of real numbers as finished beings; we rather have a medium of free Becoming. We found ourselves in the domain of an ageold problem of thought, the problem of continuity, of change, and of Becoming.

According to Weyl 1946, 'Brouwer made it clear, as I think beyond any doubt, that there is no evidence supporting the belief in the existential character of the totality of all natural numbers ... the sequence of numbers which grows beyond any stage already reached by passing to the next number, is a manifold of possibilities open towards infinity; it remains forever in the status of creation, but is not a closed realm of things existing in themselves. That we blindly converted one into the other is the true source of our difficulties, including the antinomies – a source of more fundamental nature than Russell's vicious circle principle indicated. Brouwer opened our eyes and made us see how far classical mathematics, nourished by a belief in the 'absolute' that transcends all human possibilities of realization, goes beyond such statements as can claim real meaning and truth founded on evidence.

However the power of science rests on the combination of experiment, i.e., observation under freely chosen conditions, with symbolic construction, and the latter is its mathematical aspect. Thus if science is found guilty, mathematics cannot evade the verdict.

The introduction of numbers and coordinates is an act of violence.

Mathematics is the science of the infinite. The great achievement of the Greeks was to have made the tension between the finite and the infinite fruitful for the knowledge of reality. The feeling of the calm and unquestioning acknowledgement of the infinite belongs to the Orient, but for the East it remained a mere abstract awareness that left the concrete manifold of existence lying indifferently to one side, unshaped and impervious. Coming out of the Orient, the religious feeling of the infinite, the apeiron, took possession of the Greek soul in the Dionysian-Orphic epoch that preceded the Persian Wars. Here the Persian Wars also mark the release of the Occident from the Orient. That tension and its overcoming became for the Greeks the driving motive of knowledge. But every synthesis, as soon as it was achieved, permitted the old antithesis to break out anew in deepened form. Thus, it determined the history of theoretical knowledge into our time. Indeed, today we are compelled everywhere in the foundations of mathematics to return directly to the Greeks.

In Brouwer’s analysis, the individual place in the continuum, the real number, is to be defined not by a set but by a sequence of natural numbers, namely, by a law which correlates with every natural number n a natural number φ(n). . . How then do assertions arise which concern. . . all real numbers, i.e., all values of a real variable? Brouwer shows that frequently statements of this form in traditional analysis, when correctly interpreted, simply concern the totality of natural numbers. In cases where they do not, the notion of sequence changes its meaning: it no longer signifies a sequence determined by some law or other, but rather one that is created step by step by free acts of choice, and thus necessarily remains in statu nascendi. This “becoming” selective sequence (werdende Wahlfolge) represents the continuum, or the variable, while the sequence determined ad infinitum by a law represents the individual real number in the continuum. The continuum no longer appears, to use Leibniz’s language, as an aggregate of fixed elements but as a medium of free “becoming”. Of a selective sequence in statu nascendi, naturally only those properties can be meaningfully asserted which already admit of a yes-or-no decision (as to whether or not the property applies to the sequence) when the sequence has been carried to a certain point; while the continuation of the sequence beyond this point, no matter how it turns out, is incapable of overthrowing that decision.

Mathematics is the science of the infinite. The great achievement of the Greeks was to have made the tension between the finite and the infinite fruitful for the knowledge of reality. The feeling of the calm and unquestioning acknowledgement of the infinite belongs to the Orient, but for the East it remained a mere abstract awareness that left the concrete manifold of existence lying indifferently to one side, unshaped and impervious. Coming out of the Orient, the religious feeling of the infinite, the apeiron, took possession of the Greek soul in the DionysianOrphic epoch that preceded the Persian Wars. Here the Persian Wars also mark the release of the Occident from the Orient. That tension and its overcoming became for the Greeks the driving motive of knowledge. But every synthesis, as soon as it was achieved, permitted the old antithesis to break out anew in deepened form. Thus, it determined the history of theoretical knowledge into our time. Indeed, today we are compelled everywhere in the foundations of mathematics to return directly to the Greeks.

While Brouwer has made clear to us to what extent the intuitively certain falls short of the mathematically provable, Gödel shows conversely to what extent the intuitively certain goes beyond what (in an arbitrary but fixed formalism) is capable of mathematical proof.

The question of the ultimate foundations and the ultimate meaning of mathematics remains open; we do not know in which direction it will find its final solution nor even whether a final objective answer can be expected at all. "Mathematizing" may well be a creative activity of man, like language or music, of primary originality, whose historical decisions defy complete objective rationalization.