Harvard University Faculty

"His talk was informal, often far above their heads. "Do you follow me?" asked the Professor one day. No one could say Yes. "I'm not surprised," said he; "I know of only three persons who could." At Paris, the year after, at the great Exposition, Flagg stood before a mural tablet whereon were inscribed the names of the great mathematicians of the earth for more than two thousand years. Archimedes headed, Peirce closed the list; the only American."

"Authority was nothing to Peirce. He took his own path up the mountain. … Like Pythagoras, Peirce taught that everything owes its existence and consistency to the harmony which he considered the basis of all beauty, and found music in the revolving spheres."

"Of the great mathematician as an instructor several of his pupils who ventured on the higher planes of the science have written. These were youths who, though they could follow him but a few steps in that rarefied atmosphere, had the privilege of a glimpse now and then into shining infinities wherein this giant sped rejoicing on."

"He was such a great, big ray of Light and Goodness, always so simple, cheerful and showing more than amiability, that his great power did not seem to assert itself."

"Benjamin Peirce's lectures dealt, to be sure, with the higher mathematics, but also with theories of the universe and the infinities of nature, and with man's power to deal with infinities and infinitesimals alike. His University Lectures were many a time way over the heads of his audience, but his aspect, his manner, and his whole personality held and delighted them."

"Peirce stood alone — a mountain peak whose absolute height might be hard to measure, but which towered above the surrounding country."

"He was one of the most stimulating men I have ever known."

"I always had the feeling that his attitude toward his loved science was that of a devoted worshipper, rather than a clear expounder. Although we could rarely follow him, we certainly sat up and took notice. … To him mathematics was not a humanly devised instrument of investigation, it was Philosophy itself, the divine revealer of TRUTH."

"There are many cases of these algebras which may obviously be combined into natural classes, but the consideration of this portion of the subject will be reserved to subsequent researches."

"All relations are either qualitative or quantitative. Qualitative relations can be considered by themselves without regard to quantity. The algebra of such enquiries may be called logical algebra, of which a fine example is given by Boole. Quantitative relations may also be considered by themselves without regard to quality. They belong to arithmetic, and the corresponding algebra is the common or arithmetical algebra. In all other algebras both relations must be combined, and the algebra must conform to the character of the relations."

"The branches of mathematics are as various as the sciences to which they belong, and each subject of physical enquiry has its appropriate mathematics. In every form of material manifestation, there is a corresponding form of human thought, so that the human mind is as wide in its range of thought as the physical universe in which it thinks."

"The sphere of mathematics is here extended, in accordance with the derivation of its name, to all demonstrative research, so as to include all knowledge strictly capable of dogmatic teaching. Mathematics is not the discoverer of laws, for it is not induction; neither is it the framer of theories, for it is not hypothesis; but it is the judge over both, and it is the arbiter to which each must refer its claims ; and neither law can rule nor theory explain without the sanction of mathematics. It deduces from a law all its consequences, and develops them into the suitable form for comparison with observation, and thereby measures the strength of the argument from observation in favor of a proposed law or of a proposed form of application of a law. Mathematics, under this definition, belongs to every enquiry, moral as well as physical. Even the rules of logic, by which it is rigidly bound, could not be deduced without its aid. The laws of argument admit of simple statement, but they must be curiously transposed before they can be applied to the living speech and verified by, observation."

"Mathematics is the science which draws necessary conclusions."

"I presume that to the uninitiated the formulae will appear cold and cheerless; but let it be remembered that, like other mathematical formulae, they find their origin in the divine source of all geometry. Whether I shall have the satisfaction of taking part in their exposition, or whether that will remain for some more profound expositor, will be seen in the future."

"The strongest use of the symbol is to be found in its magical power of doubling the actual universe, and placing by its side an ideal universe, its exact counterpart, with which it can be compared and contrasted, and, by means of curiously connecting fibres, form with it an organic whole, from which modern analysis has developed her surpassing geometry."

"The familiar proposition that all A is B, and all B is C, and therefore all A is C, is contracted in its domain by the substitution of significant words for the symbolic letters. The A, B, and C, are subject to no limitation for the purposes and validity of the proposition; they may represent not merely the actual, but also the ideal, the impossible as well as the possible. In Algebra, likewise, the letters are symbols which, passed through a machinery of argument in accordance with given laws, are developed into symbolic results under the name of formulas. When the formulas admit of intelligible interpretation, they are accessions to knowledge; but independently of their interpretation they are invaluable as symbolical expressions of thought. But the most noted instance is the symbol called the impossible or imaginary, known also as the square root of minus one, and which, from a shadow of meaning attached to it, may be more definitely distinguished as the symbol of semi-inversion. This symbol is restricted to a precise signification as the representative of perpendicularity in quaternions, and this wonderful algebra of space is intimately dependent upon the special use of the symbol for its symmetry, elegance, and power."

"Some definite interpretation of a linear algebra would, at first sight, appear indispensable to its successful application. But on the contrary, it is a singular fact, and one quite consonant with the principles of sound logic, that its first and general use is mostly to be expected from its want of significance. The interpretation is a trammel to the use. Symbols are essential to comprehensive argument."

"There is proof enough furnished by every science, but by none more than geometry, that the world to which we have been allotted is peculiarly adapted to our minds, and admirably fitted to promote our intellectual progress. There can be no reasonable doubt that it was part of the Creator's plan. How easily might the whole order have been transposed! How readily might we have been assigned to some complicated system which our feeble and finite powers could not have unravelled!"

"The very spirits of the winds, when they were sent to carry the grateful harvest to the thirsting fields of Calabria, did not forget the geometry which they had studied in the caverns of Æolus and of which the geologist is daily discovering the diagrams."

"Throughout nature the omnipresent beautiful revealed an all-pervading language spoken to the human mind, and to man's highest capacity of comprehension. By whom was it spoken? Whether by the gods of the ocean, or the land, by the ruling divinities of the sun, moon, and stars, or by the dryads of the forest and the nymphs of the fountain, it was one speech and its written cipher was cabalistic. The cabala were those of number, and even if they transcended the gemetricl skill of the Rabbi and the hieroglyphical learning of the priest of Osiris, they were, distinctly and unmistakably, expressions of thought uttered to mind by mind; they were the solutions of mathematical problems of extraordinary complexity."

"Ascend with me above the dust, above the cloud, to the realms of the higher geometry, where the heavens are never clouded; where there is no impure vapour, and no delusive or imperfect observation, where the new truths are already arisen, while they are yet dimly dawning on the world below; where the earth is a little planet; where the sun has dwindled to a star; where all the stars are lost in the Milky Way to which they belong; where the Milky Way is seen floating through space like any other nebula; where the whole great girdle of nebulae has diminished to an atom and has become as readily and completely submissive to the pen of the geometer, and the slave of his formula, as the single drop, which falls from the clouds, instinct with all the forces of the material world."

"The Key! it is of wonderful construction, with its infinity of combination, and its unlimited capacity to fit every lock. … it is the great master-key which unlocks every door of knowledge and without which no discovery which deserves the name — which is law, and not isolated fact — has been or ever can be made. Fascinated by its symmetry the geometer may at times have been too exclusively engrossed with his science, forgetful of its applications; he may have exalted it into his idol and worshipped it; he may have degraded it into his toy . . . when he should have been hard at work with it, using it for the benefit of mankind and the glory of his Creator."

"Geometry, to which I have devoted my life, is honoured with the title of the Key of Sciences; but it is the Key of an ever open door which refuses to be shut, and through which the whole world is crowding, to make free, in unrestrained license, with the precious treasures within, thoughtless both of lock and key, of the door itself, and even of Science, to which it owes such boundless possessions, the New World included. The door is wide open and all may enter, but all do not enter with equal thoughtlessness. There are a few who wonder, as they approach, at the exhaustless wealth, as the sacred shepherd wondered at the burning bush of Horeb, which was ever burning and never consumed. Casting their shoes from off their feet and the world's iron-shod doubts from their understanding, these children of the faithful take their first step upon the holy ground with reverential awe, and advance almost with timidity, fearful, as the signs of Deity break upon them, lest they be brought face to face with the Almighty."

"Gentlemen, that is surely true, it is absolutely paradoxical; we cannot understand it, and we don't know what it means. But we have proved it, and therefore we know it must be the truth."

"Ideality is preëminently the foundation of Mathematics."

"What is man? … What a strange union of matter and mind! A machine for converting material into spiritual force."

"There is a road that turning always Cuts off the country of Again. Archers stand there on every side And as it runs time's deer is slain And lies where it has lain."

"Long time he lay upon the sunny hill, To his father's house below securely bound."

"The world's great day is growing late, Yet strange these fields that we have planted So long with crops of love and hate."

"The curse of Scottish literature is the lack of a whole language, which finally means the lack of a whole mind."

"I have observed in foolish awe The dateless mid-days of the law And seen indifferent justice done By everyone on everyone."

"They do not live in the world, Are not in time and space. From birth to death hurled No word do they have, not one To plant a foot upon, Were never in any place."

"After fighting, scheming and murdering in pursuit of the secure tenure of absolute power, he found himself at last on a lonely pinnacle over an abyss, with no use for his power and security unattainable. His genius was such that he ended an epoch and began another - but one of unceasing war and misery, from which exhaustion produced an approach to order after two generations and peace at last under the Roman Empire. He himself never found peace. One is tempted to see him, in medieval terms, as the man who sold his soul to the Devil for power: the Devil kept his part of the bargain but ultimately claimed his own. But to the historian, prosaically such allegory, we must put it differently: to him, when he has done all the work - work that must be done, and done carefully - of analysing the play of faction and the system of government, Alexander illustrates with startling clarity the ultimate loneliness of supreme power."

"Philip II, at least from the time of his victory over Phocis, Athens, and their allies in 346, prepared to proclaim himself the champion of a United Greece against the barbarians."

"We are all Danes now. We are all Jews now. We are all Pope Benedict."

"Cultural and civilizational diversity challenges the Western and particularly American belief in the universal relevance of Western culture. This belief is expressed both descriptively and normatively. Descriptively it holds that peoples in all societies want to adopt Western values, institutions, and practices. If they seem not to have that desire and to be committed to their own traditional cultures, they are victims of a “false consciousness” comparable to that which Marxists found among proletarians who supported capitalism. Normatively the Western universalist belief posits that people throughout the world should embrace Western values, institutions, and culture because they embody the highest, most enlightened, most liberal, most rational, most modern, and most civilized thinking of humankind. In the emerging world of ethnic conflict and civilizational clash, Western belief in the universality of Western culture suffers three problems: it is false; it is immoral; and it is dangerous. … The belief that non-Western peoples should adopt Western values, institutions, and culture is immoral because of what would be necessary to bring it about. The almost-universal reach of European power in the late nineteenth century and the global dominance of the United States in the late twentieth century spread much of Western civilization across the world. European globalism, however, is no more. American hegemony is receding if only because it is no longer needed to protect the United States against a Cold War-style Soviet military threat. Culture, as we have argued, follows power. If non-Western societies are once again to be shaped by Western culture, it will happen only as a result of the expansion, deployment, and impact of Western power. Imperialism is the necessary logical consequence of universalism. In addition, as a maturing civilization, the West no longer has the economic or demographic dynamism required to impose its will on other societies and any effort to do so is also contrary to the Western values of self-determination and democracy. As Asian and Muslim civilizations begin more and more to assert the universal relevance of their cultures, Westerners will come to appreciate more and more the connection between universalism and imperialism."

"Although Europeans universally acknowledge the fundamental significance of the dividing line between Western Christendom, on the one hand, and Orthodoxy and Islam, on the other, the United States, its secretary of state said, would "not recognize any fundamental divide among the Catholic, Orthodox, and Islamic parts of Europe." Those who do not recognize fundamental divides, however, are doomed to be frustrated by them."

"A world in which cultural identities — ethnic, national, religious, civilizational — are central, and cultural affinities and differences shape the alliances, antagonisms, and policies of states has three broad implications for the West generally and for the United States in particular. First, statesmen can constructively alter reality only if they recognize and understand it. The emerging politics of culture, the rising power of non-Western civilizations, and the increasing cultural assertiveness of these societies have been widely recognized in the non-Western world. European leaders have pointed to the cultural forces drawing people together and driving them apart. American elites, in contrast, have been slow to accept and to come to grips with these emerging realities."

"The leaders of the United States have not only permitted that but assiduously promoted the diversity rather than the unity of the people they govern. The leaders of other countries have, as we have seen, at times attempted to disavow their cultural heritage and shift the identity of their country from one civilization to another. In no case to date have they succeeded and they have instead created schizophrenic torn countries. The American multiculturalists similarly reject their country’s cultural heritage. Instead of attempting to identify the United States with another civilization, however, they wish to create a country of many civilizations, which is to say a country not belonging to any civilization and lacking a cultural core. History shows that no country so constituted can long endure as a coherent society. A multicivilizational United States will not be the United States; it will be the United Nations."

"In the early 1990s Muslims were engaged in more intergroup violence than were non-Muslims, and two-thirds to three-quarters of intercivilizational wars were between Muslims and non-Muslims. Islam's borders are bloody and so are its innards."

"Muslim governments, even the bunker governments friendly to and dependent on the West, have been strikingly reticent when it comes to condemning terrorist acts against the West. On the other side, European governments and publics have largely supported and rarely criticized actions the United States has taken against its Muslim opponents, in striking contrast to the strenuous opposition they often expressed to American actions against the Soviet Union and communism during the Cold War. In civilizational conflicts, unlike ideological ones, kin stand by their kin. The underlying problem for the West is not Islamic fundamentalism. It is Islam, a different civilization whose people are convinced of the superiority of their culture and are obsessed with the inferiority of their power. The problem for Islam is not the CIA or the US department of Defense. It is the West, a different civilization whose people are convinced of the universality of their culture and believe that their superior, if declining, power imposes on them the obligation to extend that culture throughout the world. These are the basic ingredients that fuel conflict between Islam and the West."

"In Western Europe, anti-Semitism directed against Arabs has largely replaced Anti-Semitism against Jews..."

"Undoubtedly many more people in the world are concerned with sports than with human rights."

"Hypocrisy, double standards, and "but nots" are the price of universalist pretensions. Democracy is promoted, but not if it brings Islamic fundamentalists to power; nonproliferation is preached for Iran and Iraq, but not for Israel; free trade is the elixir of economic growth, but not for agriculture; human rights are an issue for China, but not with Saudi Arabia; aggression against oil-owning Kuwaitis is massively repulsed, but not against non-oil-owning Bosnians. Double standards in practice are the unavoidable price of universal standards of principle."

"Having achieved political independence, non-Western societies wish to free themselves from Western economic, military, and cultural domination. East Asian societies are well on their way to equaling the West economically. Asian and Islamic countries are looking for shortcuts to balance the West militarily. The universal aspirations of Western civilization, the declining relative power of the West, and the increasing cultural assertiveness of other civilizations ensure generally difficult relations between the West and the rest."

"The dangerous clashes of the future are likely to arise from the interaction of Western arrogance, Islamic intolerance, and Sinic assertiveness."

"Asians, in short, are determined to exclude Australia from their club for the same reason Europeans do Turkey: they are different from us. Prime Minister Keating liked to say that he was going to change Australia from "the odd man out to the odd man in" in Asia. That, however is an oxymoron: odd men don't get in."

"The West won the world not by the superiority of its ideas or values or religion (to which few members of other civilizations were converted) but rather by its superiority in applying organized violence. Westerners often forget this fact; non-Westerners never do."

"China under the Tang, Sung, and Ming dynasties, the Islamic world from the eighth to the twelfth centuries, and Byzantium from the eighth to the eleventh centuries far surpassed Europe in wealth, territory, military power, and artistic, literary and scientific achievements. Between the eleventh and thirteen centuries, European culture began to develop, facilitated by the "eager and systematic appropriation of suitable elements from the higher civilization of Islam and Bizantium, together with adaptation of this inheritance to the special conditions and interests of the West"."

"One widely articulated paradigm was based on the assumption that the end of the Cold War meant the end of significant conflict in global politics and the emergence of one relatively harmonious world. The most widely discussed formulation of this model was "the "end of history" thesis advanced by Francis Fukuyama. "We may be witnessing, Fukuyama argued, "… the end of history as such: that is, the end point of mankind's ideological evolution and the universalization of Western liberal democracy as the final form of human government." … The future will be devoted not to great exhilarating struggles over ideas but rather to resolving mundane economic and technical problems. And, he concluded rather sadly, it will all be rather boring."