Mathematicians From The United States

"...only one man lived who could understand Gibbs's papers. That was Maxwell, and now he is dead."

"Willard Gibbs is the type of the imagination at work in the world. His story is that of an opening up which has had its effect on our lives and our thinking; and, it seems to me, it is the emblem of the naked imagination —which is called abstract and impractical, but whose discoveries can be used by anyone who is interested, in whatever "field"— an imagination which for me, more than that of any other figure in American thought, any poet, or political, or religious figure, stands for imagination at its essential points."

"Let a drop of wine fall into a glass of water; whatever be the law that governs the internal movement of the liquid, we will soon see it tint itself uniformly pink and from that moment on, however we may agitate the vessel, it appears that the wine and water can separate no more. All this, Maxwell and Boltzmann have explained, but the one who saw it in the cleanest way, in a book that is too little read because it is difficult to read, is Gibbs, in his Principles of Statistical Mechanics."

"On the one hand, the student has been informed by some writers that the only certain way lies in the use of the entropy-function and the thermodynamic potentials; on the other hand, he is told with equal authority that the method used by the original investigators has been the consideration of cyclic processes, and that the former method is nothing but a mathematical (perhaps unnecessary) refinement of the results obtained by the latter. These extreme attitudes appear to me to be unfortunate, and more especially when one observes the physical clearness introduced by the use of cyclic processes, but at the same time remembers that most of the results obtained by separate investigators using cyclic processes had, with a great many more, previously been found by J. Willard Gibbs by means of a purely analytical method."

"In the last generation, this country produced one of the most eminent men of science in the whole world. His name was quite unknown among us while he lived, and it is still unknown. Yet I may say without too great exaggeration that when I heard it mentioned in a professional assembly in the Netherlands two years ago, everybody got down under the table and touched their foreheads to the floor. His name was Josiah Willard Gibbs."

"Well, that's a subject interested me a lot because I've seen so many young people in our culture today, that emphasizes the success all the time and is always pointing to this success of researchers and developing some wonderful new thing. This pressure to succeed I found has caused many young students who I thought would make a very good researcher start doing research and then finding it they failed again and again and again and became discouraged very rapidly because the media makes it, creates the impression they should succeed right away. So they leave research and I think it's a great shame that young people have to understand research is 90% failure. It's very painful to the ego to fail again and again and again. I know I've done an awful a lot of failing in my career and I think that young students should recognize that what's research is about and if they can't stand failing then don't try to do the research. But it can be borne."

"Much of my work has come from being lazy. I didn't like writing programs, and so, when I was working on the IBM 701 (an early computer), writing programs for computing missile trajectories, I started work on a programming system to make it easier to write programs."

"Von Neumann languages do not have useful properties for reasoning about programs. Axiomatic and denotational semantics are precise tools for describing and understanding conventional programs, but they only talk about them and cannot alter their ungainly properties. Unlike von Neumann languages, the language of ordinary algebra is suitable both for stating its laws and for transforming an equation into its solution, all within the "language.""

"For twenty years programming languages have been steadily progressing toward their present condition of obesity; as a result, the study and invention of programming languages has lost much of its excitement. Instead, it is now the province of those who prefer to work with thick compendia of details rather than wrestle with new ideas. Discussions about programming languages often resemble medieval debates about the number of angels that can dance on the head of a pin instead of exciting contests between fundamentally differing concepts. Many creative computer scientists have retreated from inventing languages to inventing tools for describing them. Unfortunately, they have been largely content to apply their elegant new tools to studying the warts and moles of existing languages."

"You need the willingness to fail all the time. You have to generate many ideas and then you have to work very hard only to discover that they don’t work. And you keep doing that over and over until you find one that does work ."

"The golden age of mathematics - that was not the age of Euclid, it is ours. (p. 268)"

"How pitiable is it to reflect, that although you were so fully convinced of the benevolence of the Father of Mankind, and of his equal and impartial distribution of these rights and privileges, which he hath conferred upon them, that you should at the same time counteract his mercies, in detaining by fraud and violence so numerous a part of my brethren, under groaning captivity and cruel oppression, that you should at the same time be found guilty of that most criminal act, which you professedly detested in others, with respect to yourselves"

"I thank you, sincerely, for your letter of the 19th instant, and for the Almanac it contained. No body wishes more than I do, to see such proofs as you exhibit, that nature has given to our black brethren talents equal to those of the other colors of men ; and that the appearance of the want of them, is owing merely to the degraded condition of their existence, both in Africa and America. I can add with truth, that no body wishes more ardently to see a good system commenced, for raising the condition, both of their body and mind, to what it ought to be, as far as the imbecility of their present existence, and other circumstances, which cannot be neglected, will admit. I have taken the liberty of sending your Almanac to Monsieur de Condozett, Secretary of the Academy of Sciences at Paris, and Member of the Philanthropic Society, because I considered it as a document, to which your whole color had a right for their justification, against the doubts which have been entertained of them. I am with great esteem, Sir, Your most obedient Humble Servant…"

"Now let us look too at Benjamin Banneker. Look at his correspondence with Thomas Jefferson. Here's a man, some end part in the absolute center of slavery who laid out the plans for Washington, D.C. Here was a man who had been a slave, and his parents were slaves. One must perceive this and see that he was not caught up by that same system which will smother in an attempt to protect. And one walks down the same century and sees the list of thousands of Black people who said "I've got to break away. Thank you for your caring but I must break away." You cannot blame the family for its attempts to say, "Stay, I can hold you. As long as I can touch you I feel more secure." No matter how it says it. The family may say "You'll never make it." Whatever it says it means stay with me so I can keep my eyes on you. The family is not to be blamed for that."

"Presumption should never make us neglect that which appears easy to us, nor despair make us lose courage at the sight of difficulties."

"Evil communication corrupts good manners. I hope to live to hear that good communication corrects bad manners."

"Sir, I am fully sensible of the greatness of that freedom, which I take with you on the present occasion; a liberty which seemed to me scarcely allowable, when I reflected on that distinguished and dignified station in which you stand, and the almost general prejudice and prepossession, which is so prevalent in the world against those of my complexion. I suppose it is a truth too well attested to you, to need a proof here, that we are a race of beings, who have long labored under the abuse and censure of the world; that we have long been looked upon with an eye of contempt; and that we have long been considered rather as brutish than human, and scarcely capable of mental endowments. Sir, I hope I may safely admit, in consequence of that report which hath reached me, that you are a man far less inflexible in sentiments of this nature, than many others; that you are measurably friendly, and well disposed towards us; and that you are willing and ready to lend your aid and assistance to our relief, from those many distresses, and numerous calamities, to which we are reduced. Now Sir, if this is founded in truth, I apprehend you will embrace every opportunity, to eradicate that train of absurd and false ideas and opinions, which so generally prevails with respect to us ; and that your sentiments are concurrent with mine, which are, that one universal Father hath given being to us all ; and that he hath not only made us all of one flesh, but that he hath also, without partiality, afforded us all the same sensations and endowed us all with the same faculties ; and that however variable we may be in society or religion, however diversified in situation or color, we are all of the same family, and stand in the same relation to him. Sir, if these are sentiments of which you are fully persuaded, I hope you cannot but acknowledge, that it is the indispensible duty of those, who maintain for themselves the rights of human nature, and who possess the obligations of Christianity, to extend their power and influence to the relief of every part of the human race, from whatever burden or oppression they may unjustly labor under ; and this, I apprehend, a full conviction of the truth and obligation of these principles should lead all to."

"We planted our crops in the spring, and they came up, and were looking nicely, and we were cheered with the hopes of having a very abundant harvest. But alas! it very soon appeared as if our crops were going to be swallowed up by a vast horde of crickets, that came down from these mountains-crickets very different to what I used to be acquainted with in the State of New York. They were crickets nearly as large as a man's thumb. They came in immense droves, so that men and women with brush could make no headway against them; but we cried unto the Lord in our afflictions, and the Lord heard us, and sent thousands and tens of thousands of a small white bird. I have not seen any of them lately. Many called them gulls, although they were different from the seagulls that live on the Atlantic coast. And what did they do for us? They went to work, and by thousands and tens of thousands, began to devour them up, and still we thought that even they could not prevail against so large and mighty an army. But we noticed, that when they had apparently filled themselves with these crickets, they would go and vomit them up, and again go to work and fill themselves, and so they continued to do, until the land was cleared of crickets, and our crops were saved. There are those who will say that this was one of the natural courses of events, that there was no miracle in it. Let that be as it may, we esteemed it as a blessing from the hand of God; miracle or no miracle, we believe that God had a hand in it, and it does not matter particularly whether strangers believe or not."

"The Universalists think they are very charitable. Why? Because they send all to heaven, whether they are good or evil, saints or sinners. Murderers, drunkards, and all classes of society are to dwell together in heaven. And what a heaven it would be!—Methodists contending against Baptists, and Baptists against Methodists, Presbyterians against Quakers, Roman Catholics against Protestants, and Nothingarians against Sectarians, and Sectarians against Nothingarians; and then add to the whole catalogue of contending sects drunkards, blasphemers, whoremongers, murderers, and every species of wicked beings, all jumbled up together. Oh, what a happy place! Brother Kimball says—"And all of them with a revolver and bowie knife at their sides.""

"Another great change happened nearly two thousand years after the earth was made. It was baptized by water. A great flow of water come, the great deep was broken up, the windows of heaven were opened from on high, and the waters prevailed upon the face of the earth, sweeping away all wickedness and transgression-a similitude of baptism for the remission of sins. God requires the children of men to be baptized. What for? For the remission of sins. So he required our globe to be baptized by a flow of waters, and all of its sins were washed away, not one sin remaining."

"But by and by the time came when the Christian Church apostatized and turned away, and began to follow after their own wisdom, and the Prophets and Apostles ceased, so far as the affairs of the Christian Church on the earth were concerned. Revelations, and visions, and the various gifts of the spirit were also taken away, according to their unbelief and apostacy; but in the latter days God intends to again raise up a Christian Church upon the earth. Do not be startled, you who think that God will no more have a Church on the earth, for he has promised that he would again have one, and that he would set up his kingdom, and when he does you may look out for a great many Prophets and inspired men; and if you ever see a Church arise, calling itself a Christian Church, and it has not inspired Apostles like those in ancient times, you may know that it is a spurious church, and that it makes pretensions to something that it does not enjoy. If you ever find a church called a Christian Church that has no men to foretell future events, you may know, at once, that it is not a Christian Church. If you find a Christian Church that has not the ancient gifts, for instance the gift of healing, opening the eyes of the blind, unstopping the ears of the deaf, causing the tongue of the dumb to speak and the lame to walk; if you ever find a people calling themselves a Christian Church and they have not these gifts among them, you may know with a perfect knowledge that they do not agree with the pattern given in the New Testament. The Christian Church is always characterized with inspired men, whose revelations are just as sacred as any contained in the Bible; and, if written and published, just as binding upon the human family. The Christian Church will always lay hands upon the sick in the name of Jesus, in order that the sick may be healed. The Christian Church will always have those among its members who have heavenly visions, the ministration of angels, and the various gifts that are promised according to the Gospel."

"The Apostles were put to death; they were hunted from nation to nation; they wandered about in sheep skins and goat skins in the dens and caves of the earth, of whom the world was unworthy. Their followers were put to death by hundreds, by thousands, by tens of thousands; and after a while there sprang up a people that pretended to be Christians--followers of the meek and lowly Jesus, having no apostles, no inspired men, no revelation, no ministration of angels, none of the characteristics, except a few forms, of the Christian Church as it existed in the first century of the Christian era. This class of men, calling themselves Christian, uniting with the various forms of the pagan religion, adopting many of their ceremonies and institutions, became very popular, and finally some of the pagans embraced Christianity and were placed, as it were, upon the throne, and what they termed Christianity became very popular indeed."

"Jesus appeared unto the ancient Nephites, in the northern part of what we call South America…"

"God prior to this time, had given revelation to this youth, on many occasions. The first one that he gave to him was in the spring of 1820, before Joseph Smith was of the age of fifteen. Then a wonderful revelation was given to him, the first one he ever received. In a great and glorious open vision, in answer to his prayers, there was the manifestation of two of the great personages in the heavens-not angels, not messengers, but two persons that hold the keys of authority over all the creations of the universe. Who were they? God the Eternal Father and his Son Jesus Christ, through whom God the Father made the worlds!"

"In the year 1823 the angel first came to him, telling him about the plates, also telling him that he would return one year afterwards, when he would give him further instruction. Why this delay? why not commit to his charge the plates at once? It was because of his want of experience. It is true, he had previously received a heavenly vision some four years before, in which he had seen the face of God, the Father."

"Finally the light rested down upon and overwhelmed him in the midst of it, and his mind at the same time seemed to be caught away from surrounding objects, and he saw nothing excepting the light and two glorious personages standing before him in the midst of this light. One of these personages, pointing to the other, said-"Behold my beloved Son, hear ye him.""

"[T]he heavens were opened, and two personages clothed in light or fire descended and stood before him. As soon as this light surrounded him, and he was enclosed or enveloped in it, his mind was caught away from earthly objects and things, and he saw these two glorious personages, their countenances shining with exceeding great brilliancy. One of them, while pointing to the other, addressed him in this language, "Behold my beloved son, hear ye him.""

"He saw in this light two glorious personages, one of whom spoke to him, pointing to the other, saying, "This is my beloved Son, hear ye him." This was a glorious vision given to this boy."

"By and by an obscure individual, a young man, rose up, and, in the midst of all Christendom, proclaimed the startling news that God had sent an angel to him;... This young man, some four years afterwards, was visited again by a holy angel."

"By and by an obscure individual, a young man, rose up, and, in the midst of all Christendom, proclaimed the startling news that God had sent an angel to him; that through his faith, prayers, and sincere repentance he had beheld a supernatural vision, that he had seen a pillar of fire descend from Heaven, and saw two glorious personages clothed upon with this pillar of fire, whose countenance shone like the sun at noonday; that he heard one of these personages say, pointing to the other, 'This is my beloved Son, hear ye him.' This occurred before this young man was fifteen years of age; and it was a startling announcement to make in the midst of a generation so completely given up to the traditions of their fathers; and when this was proclaimed by this young, unlettered boy to the priests and the religious societies in the State of New York, they laughed him to scorn. 'What!' said they, "visions and revelations in our day! God speaking to men in our day!" They looked upon him as deluded; they pointed the finger of scorn at him and warned their congregations against him. 'The canon of Scripture is closed up; no more communications are to be expected from Heaven. The ancients saw heavenly visions and personages; they heard the voice of the Lord; they were inspired by the Holy Ghost to receive revelations, but behold no such thing is to be given to man in our day, neither has there been for many generations past.' This was the style of the remarks made by religionists forty years ago. This young man, some four years afterwards, was visited again by a holy angel."

"He saw the light gradually approaching him until it rested upon the tops of the trees. He beheld that the leaves of the trees were not consumed by it, although its brightness, apparently, was sufficient, as he at first thought, to consume everything before it. But the trees were not consumed by it, and it continued to descend until it rested upon him and enveloped him in its glorious rays. When he was thus encircled about with this pillar of fire his mind was caught away from every object that surrounded him, and he was filled with the visions of the Almighty, and he saw, in the midst of this glorious pillar of fire, two glorious personages, whose countenances shone with an exceeding great lustre. One of them spoke to him, saying, while pointing in the other, "This is my beloved Son in whom I am well pleased, hear ye him.""

"When, where, and how were you, Joseph Smith, first called? How old were you? and what were you qualifications? I was between fourteen and fifteen years of age. Had you been to college? No. Had you studied in any seminary of learning? No. Did you know how to read? Yes. How to write? Yes. Did you understand much about arithmetic? No. About grammar? No. Did you understand all the branches of education which are generally taught in our common schools? No. But yet you say the Lord called you when you were but fourteen or fifteen years of age? How did he call you? I will give you a brief history as it came from his own mouth. I have often heard him relate it. He was wrought upon by the Spirit of God, and felt the necessity of repenting of his sins and serving God. He retired from his father's house a little way, and bowed himself down in the wilderness, and called upon the name of the Lord. He was inexperienced, and in great anxiety and trouble of mind in regard to what church he should join. He had been solicited by many churches to join with them, and he was in great anxiety to know which was right. He pleaded with the Lord to give him wisdom on the subject; and while he was thus praying, he beheld a vision, and saw a light approaching him from the heavens; and as it came down and rested on the tops of the trees, it became more glorious; and as it surrounded him, his mind was immediately caught away from beholding surrounding objects. In this cloud of light he saw two glorious personages; and one, pointing to the other, said, "Behold my beloved son! hear ye him.""

"Every moment we're making decisions. If you want to understand the universe as a whole you'd have to include all those decisions including our own lives so that even in the future our presence is going to be felt one way or the other — so I think … our death as being — it's a death to a micro-phase self or a small self … it's a death into our larger self in that … this whole vast adventure is our larger identity."

"What's evil for the hawk is the mouse because, you know, the mouse is quick and gets away but I did this one realization every scientist goes through at one point... if you gave the hawk power, the power of God, the first thing the hawk might do is to slow down the mouse. But then the hawk would lose it's speed. And then if you slow the mouse all the way down, so it can just barely move, the hawk would lose its flight. So that in a weird way the tension between those two say natural enemies is what gives birth to their beauty. So I definitely feel that the tension we have right now within the human community in particular — that those are ultimately going to be resolved with a deeper harmony and a deeper appreciation for one another."

"I have a sense that something amazing is at work … I think our planet is actually moving into a time of profound harmony and fecundity and peace but whether that's going to take 600 years or 6 days I don't know. I mean, I think that as humans begin to take seriously... the planetary dimension of conscious self-awareness, then we will become homonized versions of natural selection — so that we will begin to make decision with the large scale dynamic of the planet in mind. So I see that we're actually entering into a transformation of the human species out of the modern period into this new era... It may take centuries... but like the past and it's catastrophes I think that's... what's taking place in the midst of so many hardships."

"The break-through moments are unimaginable until they happen."

"I think the discovery of nonlocality is touching in on the whole. So that these these seemingly separate events are somehow connected through the whole. … you have this larger enveloping field and we're, you know, just beginning to understand something about that... so I love that discovery although I don't think we're anywhere near really knowing what we've come upon."

"The more I learn about light the more I realize, man, we don't know anything about light... It's just bizarre... a particle has its own proper time which slows down as you speed up. But at the speed of light... there's no time. That's bizarre … that we can, right now, as you know, see — interact with the light that has come from the birth of the universe. So … from our point of view, that light traveled for 14 billion years but from the point of view of the light it's the moment of creation."

"There's a great phrase from Eric Jantsch … and he says, "these self-organizing dynamics are in every place in the universe, waiting at their marks". I love that phrase because you get that … the power for making water exists everywhere in the universe but the conditions have to be right. But if the conditions are right, then these self-organizing dynamics leap to it. So I think it's something like that, that the possibility for sentience has always been there but has been waiting for a chance to really display."

"If you if you take Buddhism and Christianity and so forth there's a kind of battle — a subtle sort of struggle taking place because they're not standing in a common ground but … take the Earth or ecology then suddenly they can begin to explore what they have to offer. So I do think I do think absolutely that … there will be a flourishing of religions, not a withering away. And they will flourish to the degree that they will move into the context of planet and universe. I even think that as a matter of fact that … some of the central insights of the religions are more powerfully presented by what we know about the universe now then when they were first formulated."

"With the appearance of the human, the coding process of life burst beyond the DNA molecule and began carving its information in stone!"

"This is the greatest discovery of the scientific enterprise: You take hydrogen gas, and you leave it alone, and it turns into rosebushes, giraffes, and humans."

"Earth was once molten rock, and now it makes spaceships."

"The primary challenge of this cosmological transformation of consciousness is the awareness that each being in the universe is an origin of the universe. "The center of the cosmos" refers to that place where the great birth of the universe happened at the beginning of time, but it also refers to the upwelling of the universe as river, as star, as raven, as you, the universe surging into existence anew. The consciousness that learns it is at the origin point of the universe is itself an origin of the universe. The awareness that bubbles up each moment that we identify as ourselves is rooted in the originating activity of the universe. We are all of us arising together at the center of the cosmos."

"Genetics is the study of similarity between relatives, and the problem of human genetics is that, in a species with a family and social structure and a taboo against manipulating individual life histories experimentally, there is a confounding between the similarity of relatives that arises from biological causes and the similarity that arise from social causes."

"An important consequence of the unique interaction between internal and external forces … is that knowledge of genetic differences contains no information at all about whether a characteristic can be changed by environmental and social arrangements. The most elementary error about genetics and development is to suppose that “genetic” is the opposite of “changeable” and that an answer to the question “how much can a trait be changed by social, historical, and individual circumstances” is given by an answer to the question “how important are genes.”"

"[T]he psychic and physical characteristics of human beings, and the differences between individuals, are the consequence of an interaction between the genes that are present in the fertilized egg and the sequence of environmental circumstances that the developing organism experiences during its entire life history. With a few exceptions, like cystic fibrosis where possession of the defective genotype leads ineluctably to the disease, or language acquisition where the language spoken depends only on experience and not at all on genotype, human characteristics are all subject to this interaction of forces. There are, moreover, random events in cell growth and differentiation that are neither genetic nor environmental in the usual sense, and which play an extremely important part in development, especially in behavioral traits."

"The division between those who try to learn about the world by manipulating it and those who can only observe it had led, in natural science, to a struggle for legitimacy. The experimentalists look down on the observers as merely telling uncheckable just-so stories, while the observers scorn the experimentalists for their cheap victories over excessively simple phenomena. In biology the two camps are now generally segregated in separate academic departments where they can go about their business unhassled by their unbelievers. But the battle is unequal because the observers' consciousness of what it is to do "real" science has been formed in a world dominated by the manipulators of nature. The observers then pretend to an exactness that they cannot achieve and they attempt to objectify a part of nature that is completely accessible only with the air of subjective tools."

"It is characteristic of the design of scientific research that exquisite attention is devoted to methodological problems that can be solved, while the pretense is made that the ones that cannot be solved are really nothing to worry about. On the one hand, biologists will apply the most critical and demanding canons of evidence in the design of measuring instruments or in the procedure for taking an unbiased samples of organisms to be tested, but when asked whether the conditions in the laboratory are likely to be relevant to the situation in nature, they will provide a hand-waving intuitive argument filled with unsubstantiated guesses and prejudices because, in the end, that is all they can do."

"The social scientist is in a difficult, if not impossible position. On the one hand there is the temptation to see all of society as one's autobiography writ large, surely not the path to general truth. On the other, there is the attempt to be general and objective by pretending that one knows nothing about the experience of being human, forcing the investigator to pretend that people usually know and tell the truth about important issues, when we all know from our lives how impossible that is. How, then, can there be a "social science"? The answer, surely, is to be less ambitious and stop trying to make sociology into a natural science although it is, indeed, the study of natural objects. There are some things in the world that we will never know and many that we will never know exactly. Each domain of phenomena has its characteristic grain of knowability. Biology is not physics, because organisms are such complex physical objects, and sociology is not biology because human societies are made by self-conscious organisms. By pretending to a kind of knowledge that it cannot achieve, social science can only engender the scorn of natural scientists and the cynicism of the humanists."