Princeton University Faculty

"What is needed is a homely experiment which could be carried out in the basement with parts from an old sewing machine and an Ingersoll watch, with an old file of Popular Mechanics standing by for reference! This I am, alas, afraid we have not achieved, but I do believe that the following example... is adequate to expose the principles..."

"We have merely (!) to measure the volume V of a sphere of radius r or the sum \sigma of the angles of a triangle of measured are \delta, and from the results to compute the value of K."

"In all these congruence geometries, except the Euclidean, there is at hand a natural unit of length R = \frac{1}{K^\frac{1}{2}}; this length we shall, without prejudice, call the "radius of curvature" of the space."

"The value of the intrinsic approach is especially apparent in considering 3-dimensional congruence spaces... The intrinsic geometry of such a space of curvature K provides formulae for the surface area S and the volume V of a "small sphere" of radius r, whose leading terms are 3)S = 4 \pi r^2 (1 - \frac{Kr^2}{3} + ...), V = \frac{4}{3} \pi r^3 (1 - \frac{Kr^2}{5} + ...)."

"These formulae [in (1) and (2) above] may be shown to be valid for a circle or a triangle in the hyperbolic plane... for which K < 0. Accordingly here the perimeter and area of a circle are greater, and the sum of the three angles of a triangle are less, than the corresponding quantities in the Euclidean plane. It can also be shown that each full line is of infinite length, that through a given point outside a given line an infinity of full lines may be drawn which do not meet the given line (the two lines bounding the family are said to be "parallel" to the given line), and that two full lines which meet do so in but one point."

"[T]he space constant K... "" may in principle at least be determined by measurement on the surface, without recourse to its embodiment in a higher dimensional space."

"In the sum \sigma of the three angles of a triangle (whose sides are arcs of s) is greater than two right angles [180&deg;]; it can... be shown that this "spherical excess" is given by 2)\sigma - \pi = K \deltawhere \delta is the area of the spherical triangle and the angles are measured in s (in which 180&deg; = \pi [radians]). Further, each full line (great circle) is of finite length 2 \pi R, and any two full lines meet in two points—there are no parallels!"

"Measurements which may be made on the surface of the earth... is an example of a 2-dimensional congruence space of positive curvature K = \frac{1}{R^2}... [C]onsider... a "small circle" of radius r (measured on the surface!)... its perimeter L and area A... are clearly less than the corresponding measures 2\pi r and \pi r^2... in the Euclidean plane. ...for sufficiently small r (i.e., small compared with R) these quantities on the sphere are given by 1):L = 2 \pi r (1 - \frac{Kr^2}{6} + ...), A = \pi r^2 (1 - \frac{Kr^2}{12} + ...)"

"[W]e propose... to deal exclusively with properties intrinsic to the space... measured within the space itself... in terms of... inner properties."

"Euclidean geometry is only one of several congruence geometries... Each of these geometries is characterized by a real number K, which for Euclidean geometry is 0, for the hyperbolic negative, and for the spherical and elliptic geometries, positive. In the case of 2-dimensional congruence spaces... K may be interpreted as the ' of the surface into the third dimension—whence it derives its name..."

"That the existence of these motions (the "axiom of free mobility") is a desideratum, if not... a necessity, for a geometry applicable to physical space, has been forcefully argued on a priori grounds by von Helmholtz, Whitehead, Russell and others; for only in a homogeneous and isotropic space can the traditional concept of a rigid body be maintained."

"An "empty world," i.e., a homogeneous manifold at all points at which equations (1) are satisfied, has, according to the theory, a constant Riemann curvature, and any deviation from this fundamental solution is to be directly attributed to the influence of matter or energy."

"The general theory of relativity considers physical space-time as a four-dimensional manifold whose line element coefficients g_{\mu \nu} satisfy the differential equationsG_{\mu \nu} = \lambda g_{\mu \nu} \qquad .\;.\;.\;.\;.\;.\; (1)in all regions free from matter and electromagnetic field, where G_{\mu \nu} is the contracted Riemann-Christoffel tensor associated with the fundamental tensor g_{\mu \nu}, and \lambda is the ."

"We should, of course, expect that any universe which expands without limit will approach the empty de Sitter case, and that its ultimate fate is a state in which each physical unit—perhaps each nebula or intimate group of nebulae—is the only thing which exists within its own observable universe."

"Langlands spent every morning, seven days a week, for five years working on the paper he delivered in Oslo. It is written entirely in Russian and dedicated in large part to reformulating the geometric program championed by Frenkel. This new paper is an attempt to shift the field toward a more traditional approach: it proposes a new mathematical basis for the geometric theory that relates more closely to Langlands’s own conjectures by using similar tools to the ones he used in the ’60s—in the process, restoring his work back to its original arithmetic purity."

"He would become fluent in French, Russian, German and Turkish, and well-versed in their literature. Frenkel, who exchanges emails with Langlands in Russian, speculates that his versatility with languages may have had something to do with his ability to see connections in disparate fields of mathematics."

"He’s clearly one of the most important living mathematicians. His legend precedes him. But the question is, ‘Do mathematicians really know what he has done?’ It’s like having a famous writer but no one has read his books."

"He’s like a modern-day Einstein. But everybody knows about Einstein and nobody knows about Langlands. Why is that?"

"He was a visionary. He pointed us into a direction where we can go and find the truth, find out what’s really going on. It’s about seeing the world in the right light."

"Langlands' life has been by no means as extravagant as Grothendieck's, but his romanticism is evident to anyone who reads his prose; the audacity of his program, one of the most elaborate syntheses of conjectures and theorems ever undertaken, has few equivalents in any field of scholarship."

"What I have achieved has been largely a matter of chance. Many problems I thought about at length with no success. With other problems, there was the inspiration—indeed, some that astound me today. Certainly the best times were when I was alone with mathematics, free of ambition and of pretense, and indifferent to the world."

"Mathematical maturity is anyhow an uncertain concept, for the mind’s natural competence seems to change with age, its purview variable."

"Although there are at present many occupations that require a good deal of skill and training in advanced mathematics, mathematics itself is still often regarded as a curious profession demanding singular talents and a singular personality."

"is a congruence geometry, or equivalently the space comprising its elements is homogeneous and isotropic; the intrinsic relations between... elements of a configuration are unaffected by the position or orientation of the configuration. ...[M]otions of are the familiar translations and rotations... made in proving the theorems of Euclid."

"The solution of (1), which represents a homogeneous manifold, may be written in the form:ds^2 = \frac{d\rho^2}{1 - \kappa^2\rho^2} - \rho^2 (d\theta^2 + sin^2 \theta \; d\phi^2) + (1 - \kappa^2 \rho^2)\; c^2 d\tau^2, \qquad (2)where \kappa = \sqrt \frac{\lambda}{3}. If we consider \rho as determining distance from the origin... and \tau as measuring the proper-time of a clock at the origin, we are led to the de Sitter spherical world..."

"In considerations involving the nature of the world as a whole the irregularities caused by the aggregation of matter into stars and stellar systems may be ignored; and if we further assume that the total matter in the world has but little effect on its macroscopic properties, we may consider them as being determined by the solution of an empty world."

"Our natures are, indeed, elusively insubstantial — notoriously less stable and less inherent than the natures of other things. And insofar as this is the case, sincerity itself is bullshit."

"It is frequently insufficient to identify the motives that guide our conduct, or that shape our attitudes and our thinking, just by observing vaguely that there are various things we want. That often leaves out too much. In numerous contexts, it is both more precise and more fully explanatory to say that there is something we care about."

"To know thyself is the ultimate form of aggression."

"Marion J. Levy Jr. [was] a scholar of sociology and international affairs who wrote about modernization theory, which seeks to explain why some societies undergo rapid economic development and corresponding social change."

"A comparative social science requires a generalized system of concepts which will enable the scientific observer to compare and contrast large bodies of concretely different social phenomena in consistent terms."

"The fundamental problem posed by modernization is whether human animals can adjust as readily to longevity, affluence, and peace as they have in the past to shortgevity, poverty, and war."

"[Leonard Jimmie Savage is] one of the few people I have met whom I would unhesitatingly call a genius."

"R. A. Fisher, J. Neyman, R. von Mises, W. Feller, and L. J. Savage denied vehemently that probability theory is an extension of logic, and accused Laplace and Jeffreys of committing metaphysical nonsense for thinking that it is."

"It was only when L. J. Savage arrived on the scene, and championed the work of Ramsey and de Finetti that the work of these two pioneers in subjective probability first received serious philosophical attention."

"For a person who wants to do original, realistic, and critical work in statistics there is no atmosphere anywhere in the world today to compare with this Department."

"My dissertation for the Ph.D. degree at the University of Michigan was on applications of vectorial methods to metric geometry (in the sense of the Menger school), especially with a view to the merging of metric geometry in that sense with differential geometry. Professor S B Myers at the University of Michigan sponsored my dissertation, but I was particularly close to R L Wilder there."

"[The George E. P. Box paper Fitting empirical data (1960) is] a mature exposition of an important branch of statistics, to which the author has made great contributions. One feature of particular interest is practical discussion of genuinely nonlinear fitting problems and their solution with the help of tact and a special, publicly available, IBM-704 program. Another is insightful comments on the role of prior distributions in statistics."

"Interpretations of the past are subject to change in response to new evidence, new questions asked of the evidence, new perspectives gained by the passage of time."

"In this study the term "abolitionist" will be applied to those Americans who before the Civil War had agitated for immediate, unconditional, and universal abolition of slavery in the United States. Contemporaries of the antislavery movement and later historians have sometimes mistakenly used the word "abolitionist" to describe adherents of the whole spectrum of antislavery sentiment. Members of the Free Soil and Republican parties have often been called abolitionists, even though these parties were pledged officially before 1861 only to the limitation of slavery, not to its extirpation. It is a moot question whether such radical anti-slavery leaders such as Charles Sumner, John Andrew, George Julian, Thaddeus Stevens, or Owen Lovejoy were genuine "abolitionists". In their hearts they probably desired an immediate end to slavery as fervently as did William Lloyd Garrison. But they were committed publicly by political affiliation and party responsibility to a set of principles that fell short of genuine abolitionism."

"The Alabama Democratic convention [instructed] its delegates to walk out of the national convention if the party refused to adopt a platform pledging a federal slave code for the territories. Other lower-South Democratic organizations followed suit. In February, Jefferson Davis presented the substance of southern demands to the Senate in resolutions affirming that neither Congress nor a territorial legislature could impair the constitutional right of any citizen of the United States to take his slave property into the common territories."

"What were these rights and liberties for which Confederates contended? The right to own slaves; the liberty to take this property into the territories."

"To a good many southerners the events of 1861–1865 have been known as 'The War of Northern Aggression'. Never mind that the South took the initiative by seceding in defiance of an election of a president by a constitutional majority. Never mind that the Confederacy started the war by firing on the American flag. These were seen as preemptive acts of defense against northern aggression."

"Slavery was at the root of what the Civil War was all about. If there had been no slavery, there would have been no war, and that ultimately what the Confederacy was fighting for was to preserve a nation based on a social system that incorporated slavery. Had that not been the case, there would have been no war. That's an issue that a lot of Southern whites today find hard to accept."

"Powerful racial prejudices? That was not true of Thomas Wentworth Higginson, or Norwood P. Hallowell, or George T. Garrison, or many other abolitionists and sons of abolitionists who became officers in black regiments. Indeed, the contrary was true. They had spent much of their lives fighting the race prejudice endemic in American society, sometimes at the risk of their careers and even their lives. That is why they jumped at the chance of help launch an experiment with black soldiers which they hoped would help African Americans."

"Rioters were mostly Irish Catholic immigrants and their children. They mainly attacked the members of New York's small black population. For a year, Democratic leaders had been telling their Irish-American constituents that the wicked Black Republicans were waging the war to free the slaves who would come north and take away the jobs of Irish workers. The use of black stevedores as scabs in a recent strike by Irish dockworkers made this charge seem plausible. The prospect of being drafted to fight to free the slaves made the Irish even more receptive to demogogic rhetoric."

"There are all kinds of myths that a people has about itself, some positive, some negative, some healthy and some not healthy. I think that one job of the historian is to try to cut through some of those myths and get closer to some kind of reality. So that people can face their current situation realistically, rather than mythically. I guess that's my sense of what a historian ought to do."

"What has changed is that I've gained a lot more sympathy for Lincoln. At the time I was doing my dissertation I tended to take the Wendell Phillips view of Lincoln. Why didn't he move more quickly? Why was he so conservative on some of these issues? Why didn't he seize this revolutionary moment? The more I've learned about it, the more I realize that Lincoln was under extraordinary pressure from all sides. In his position he could not have acted like Wendell Phillips. He would have lost the whole war."

"The great crisis facing the country was the rebellion and anybody in the North who wanted to preserve the Union now found the principal enemy to be those Southern slave owners who had broken up the country. The institution which sustained them and the institution they went to war to defend was slavery. And more and more northerners became convinced of that. As a consequence, a lot of them went the whole way over, from being conservative, pro-Southern, pro-slavery Democrats to becoming radical Republicans. Benjamin Butler is a good example, and Edwin M. Stanton is another one."

"General Sherman, who had lived in the South, liked Southerners and did not at all sympathize with Northern racial views, yet became the most hated and feared destroyer of the South and its whole civilization. And I think he did so because he saw that as necessary to win the war. And I think Lincoln made some of his decisions—issuing the Emancipation Proclamation, for example, or turning Sherman loose — because he saw that as necessary to win the war."