University Of California Berkeley Faculty

"A less democratic regime than czarist Russia would be hard to imagine. Starting in the 17th century, serfdom enslaved a high proportion of the country’s citizens—a system maintained by whips, chains, the threat of separating families and exiling rebels to Siberia, and the massacre of tens of thousands of serfs who staged hundreds of revolts over the years."

"In the American South, there are hundreds of Civil War battle monuments and preserved plantation manor houses for every exhibit that in any way marks the existence of slavery. And yet the world we live in—its divisions and conflicts, its widening gap between rich and poor, its seemingly inexplicable outbursts of violence—is shaped far less by what we celebrate and mythologize than by the painful events we try to forget. Leopold's Congo is but one of those silences of history."

"Although mortal, existential enemies, both Reds and Whites were united on one point: They wanted the boundaries of the Russia they hoped to control to be as wide as possible. Both sides had little but hatred for these non-Russian independence movements, especially the one in Ukraine, a land so rich in grain, iron, and coal."

"One more aspect of the Russian Civil War reverberates directly with the conflict we are now watching play out. The war was not just about who would rule Russia, but about whom Russia would rule."

"Outlying areas of the old Russian empire took advantage of the Red-White struggle to battle for independence. Poland, Finland, and the Baltic states did so successfully, Ukraine unsuccessfully. The fighting in the latter, among Reds, Whites, and several rival Ukrainian forces, convulsed cities in the headlines today: Kyiv, Odesa, Kharkiv, Kherson, Mariupol."

"Vladimir Nabokov's father, a democratically minded politician who had been arrested by a Communist Red Guard, managed to escape and flee the country, but not before the family’s cook made him caviar sandwiches for the journey."

"Before the U.S.S.R.'s collapse, in 1991, its rulers portrayed that war starkly: The Whites were evil reactionaries who tried to delay the glorious triumph of Soviet rule. But Putin, whose passion is for empire, not communism, has a different view. He would love to restore the power of both czarist Russia and the Soviet Union, which extended over territory far larger than his own shrunken Russia of today."

"Long before the civil war tore Russia apart, the challenges of holding such a huge country together, against threats without and centrifugal forces within, had been handled with widespread oppression as well as tight control from the top."

"In a nation so deeply xenophobic to begin with, the ultimately victorious Communists never forgot the foreign troops who had tried to strangle their baby in its cradle."

"The movement's other great achievement is this. Among its supporters, it kept alive a tradition, a way of seeing the world, a human capacity for outrage at pain inflicted on another human being, no matter whether that pain is inflicted on someone of another color, in another country, at another end of the earth."

"From the colonial era, the major legacy Europe left to Africa was not democracy as it is practiced today in countries like England, France, and Belgium; it was authoritarian rule and plunder."

"In the Soviet Union, for example, shooting or jailing political opponents at first helped the Communist Party and then Josef Stalin gain absolute power. But after there were no visible opponents left, seven million more people were executed, and many millions more died in the far-flung camps of the gulag."

"[T]o fight a complex, mechanized war, a disciplined army responsible to a central command is far more effective than a range of militias reporting to a crazy quilt of political parties and trade unions."

"It is impossible to watch Vladimir Putin's arrogant invasion of Ukraine without being appalled by its savagery. Dead men and women strewn on the streets of Bucha, hands bound behind their backs. Russian soldiers raping women, sometimes in front of husbands or children. Russians seizing loot of every size, from cellphones to giant John Deere wheat-harvesting combines. And, again and again, testimony about torture: beatings, electric shocks, near suffocation with plastic bags."

"One of the points I have tried to make is that mathematics is extremely useful to our society. If this is true, one would think that we as a society would vigorouly support the research that leads to new uses and that students would be at an all time high. Today that is not the case. The mathematics community has yet to effectvely demonstrate to the public and their elected representatives that our subject is dfferent from the sciences. We do not design widgets or cure diseases, yet our impact on engineering and medicine is enabling and significant. But the community has dwelled so long in splendid isolation that the public poorly understands what we do."

"The theory of s in approximately one century old, although its origin may be traced back much further. As originally formulated by and subsequently used throughout his work, the theory was intended as a tool to be used in the study of geometric problems. After two periods of theoretical development, one in the 1930s and the other in the 1960s, there has recently been a renewed interest in exterior differential systems as providing a systematic framework for the study of geometric problems. It is my opinion that this development is just beginning, and that exterior differential systems should become a standard tool for geometers, especially for questions where the differential equations expressing the problem are overdetermined systems, and for global questions. When used properly, the theory has a marvelous ability to reveal the underlying geometry in a complicated problem."

"for a smooth algebraic curve includes both the Hodge structure (period matrix) on cohomology and the use of that Hodge structure to study the geometry of the curve, via the . extended the theory of the period matrix to smooth algebraic varieties of any dimension, defining in general a Hodge structure on the cohomology of the variety. He gave a few applications to the geometry of the variety, but these did not attain the richness of the Jacobian variety. In recent years, Hodge theory has been successfully extended to arbitrary varieties, and to families of varieties."

"We clap when our infants don't spill their food. We can afford to let go of clapping when exotic folks don't, when in our times, celebrating diversity is a shibboleth of moral legitimacy among thinking First World people, and considerably, if not comprehensibly, beyond."

"If your opinions never evolve, you’re either not paying attention or not genuinely interested."

"[T]here comes a point when a usage is so common that we must consider it not slovenliness but change."

"Unfortunately, for a short period it became fashionable to read this epic in cultural terms – Aryan vs. Dravidian. This, in my view, is a misreading of the fundamental premise of the epic: the opposition between two views of life, one epitomized by Rama, the other by Ravana. What makes Kamban so great is that he presents both views in extremely convincing and beautiful terms – Ravana is the greatest of all kings and symbolizes this world, Rama symbolizes another dimension. And don't forget, Ravana is a Brahmin."

"Freud also changed the vocabulary with which we understand ourselves and others. […] While Freud had an enormous impact on 20th century culture, he has been a dead weight on 20th century psychology . . . At best, Freud is a figure of only historical interest for psychologists. He is better studied as a writer, in departments of [Western] language and literature, than as a scientist, in departments of psychology. Psychologists can get along without him […] Of course, Freud lived at a particular period of time, and it might be argued that his theories were valid when applied to European culture at the turn of the last century, even if they are no longer apropos today. However, recent historical analyses show that Freud’s construal of his case material was systematically distorted and biased by his theories of unconscious conflict and infantile sexuality, and that he misinterpreted and misrepresented the scientific evidence available to him. Freud’s theories were not just a product of his time: they were misleading and incorrect even when he published them."

"Freud’s cultural influence [on the West] is based, at least implicitly, on the premise that his theory is scientifically valid. But from a scientific point of view, classical Freudian psychoanalysis is dead as both a theory of the mind and a mode of therapy (Crews, 1998; Macmillan, 1996). No empirical evidence supports any specific proposition of psychoanalytic theory, such as the idea that development proceeds through oral, anal, phallic, and genital stages, or that little boys lust after their mothers and hate and fear their fathers. […] It is one thing to say that unconscious motives play a role in behavior. It is something quite different to say that our every thought and deed is driven by repressed sexual and aggressive urges; that children harbor erotic feelings toward the parent of the opposite sex; and that young boys are hostile toward their fathers, who they regard as rivals for their mothers’ affections. This is what Freud believed, and so far as we can tell Freud was wrong in every respect. For example, the unconscious mind revealed in laboratory studies of automaticity and implicit memory bears no resemblance to the unconscious mind of psychoanalytic theory"

"Just your weekly reminder that this won't end with abortion.... protections for same-sex marriage, contraception, and individual autonomy that are not explicit in the Constitution are up for grabs, too."

"One of the things that we need to take from this moment is that this isn’t just about abortion and it’s not going to end with abortion. But if we are going to register any kind of objections, we need a functioning, healthy democracy. And that’s the first thing that they have disrupted."

"[ Gay rights, contraceptives, certain fertility treatments and even interracial marriage ] are imperiled because they’re all rooted in that right to privacy. All of this has been implied because they’re understood to be core, basic human rights. You don’t need the state to recognize them because they are vested in you by virtue of being a human."

"Marriage has played a critical role in the operation of the criminal justice system, including serving as a defense to crime and as a form of punishment"

"I want to denaturalize the present."

"It was a commonplace in medieval accounts of kings that in fact the Muslims are incarnations of the demons, while the Hindu king who wages war against them is an incarnation of one of the Hindu Gods, whose function it is to descend to earth in human form and extirpate the demons, thus winning one more round in the on-going battle between the Gods and the demons (§). The comparison, then, that the Muslims do no more against the divine images than the demons have always done against the Gods themselves, suggests this deeper identification."

"The Ekalingamahdatmya is clearly sufficiently preoccupied with the Muslim invasions of North India to raise directly the issue of how and why the Muslims could have overrun the land. Indeed as we have just seen it asks forthright how the images of the Hindu Gods, which it regards as the Gods themselves, could have been broken and battered by the unbelievers. The central stories of the text, the stories designed to explain the origins of the holy site and its images, I believe should be understood in this broader context of the text. Doing so reveals the consummate artistry that is at work in the reshaping of the underlying puranic stories to fit its special circumstances: the curse of Parvati, making the hard-hearted Gods into stones, is aptly suited to a period in history when building of more precious materials was a clear invitation to disaster. The Ekalingamabatmya may use the understatement of story telling, but it is nonetheless a valuable testimony to fears and attitudes in this perilous period of Indian religious history."

"Seidenberg (1983), "regard[s] it as certain that knowledge of Pythagoras' Theorem was known to the Satapatha Brahmana, which mentions calculations connected with the purusa bird altar, and to the Taittiriya Samhita, which showed similar geometrical awareness" (106). Since these texts are generally dated to around 1000-800 B.C.E., "Greek geometry did not somehow make its way into Vedic geometry, as Greek geometry is only supposed to have started about 600 B.C." (108). Scholars no longer consider Vedic geometry to have been borrowed from the Greeks, so Seidenberg's more controversial claim in the modern context is his rejection of the possibility that the algebra either of the Indians or the Greeks was derived from Babylonia, since the former were aware of aspects of the theorem to which the Babylonians make no reference. He also rejects the possibility that these aspects could have been discovered by the Indians after receiving the basic theorem from Babylonia and then transforming it, and concludes that either "Old Babylonia got the theorem of Pythagoras from India or that Old Babylonia and India got it from a third source" (Seidenberg 1983, 121)."

"Seidenberg's next steps involve a series of assumptions that need to be laid out. He first states that since the Babylonian sources are dated to 1700 B.C.E., the mathematical knowledge in the Sulvasutras must predate that. His next statement is significant: "Now the Sanskrit scholars do not give a date for the geometric rituals in question as early as 1700 B.C. Therefore I postulate a pre-Babylonian (i.e., pre-1700 B.C.) source for the kind of geometric rituals we see preserved in the Sulvasutras, or at least for the mathematics involved in these rituals" (1983, 121). Seidenberg assigns a date of 2200 B.C.E. for the common source, but upon being told by his Indological colleagues that the Aryans were not even in India in 1700 B.C.E. (1978, 324), let alone 2200 B.C.E., he is forced to postulate that the Aryans and Greeks inherited their mathematics from the joint Indo- European period, and the Indo-Aryans brought the knowledge into the subcontinent with them."

"‘a number of points in Greek geometry are illuminated by the hypothesis that it started from a tradition of peg-and-cord constructions of the kind we find with the Vedic ritualists’ [45]."

"If the Indians invented plane geometry, what was to become of Greek ‘genius’ or of the Greek ‘miracle’?"

"As N.S. Rajaram has rightly observed, Seidenberg traces Babylonian mathematics and astronomy to Indian models. He suggests the Kassite dynasty (18th-16th century) as the channel of transmission, as the Kassite language has an Indo-Aryan substrate. This is eminently reasonable. Thus, Babylonian astronomy divided the ecliptic in 18, yet by the first millennium it had adopted a division in 12, the same as existed in Vedic culture, where a nightly division into 28 lunar houses was complemented by a daily division of the ecliptic in 12 half-seasons (Madhu, Madhava etc.), and where the rishi Dirghatamas introduced the first-ever division of the circle into 12 and 360. Till today, the division into 360 is explained in textbooks as a Babylonian invention, but the earliest mention is Indian."

"Seidenberg (1983), a historian of science, described the reaction to Thibaut's claim: "Thibaut himself never belabored or elaborated these views, nor did he formulate the obvious conclusion, namely, that it was not the Greeks who invented plane geometry, it was the Indians. At least this was the message that the Greek scholars saw in Thibaut's paper. And they didn't like it" (103). Eventually, Thibaut was pressured into proposing a date for the sutras that would disarm the Greek scholars, but he would not forgo the possibility that the two different peoples could at least have developed the same knowledge independently. The date he offered was the fourth or third century B.C.E. Seidenberg (1978) comments: "A terrible statement! I cannot help thinking that it shows battle- weariness rather than a considered opinion. . . . Anyway, the damage had been done and the Sulvasutras have never taken the position in the history of mathematics that they deserve" (306)"

"[…] nor did he [Thibaut] formulate the obvious conclusion, namely, that the Greeks were not the inventors of plane geometry, rather it was the Indians. At least this was the message that the Greek scholars saw in Thibaut’s paper. And they didn’t like it."

"‘Greece and India have a common heritage that cannot have derived from Old-Babylonia, i.e., the Old-Babylonia of about 1700 B.C.’ (Seidenberg 1978)"

"Its mathematics was very much like what we see in the Sulvasutras [szulbasu utras]. In the first place, it was associated with ritual. Second, there was no dichotomy between number and magnitude … In geometry it knew the Theorem of Pythagoras and how to convert a rectangle into a square. It knew the isosceles trapezoid and how to compute its area … [and] some number theory centered on the existence of Pythagorean triplets … [and how] to compute a square root. …The arithmetical tendencies here encountered [ie in the SZulbasuutras] were expanded and in connection with observations on the rectangle led to Babylonian mathematics. A contrary tendency, namely, a concern for exactness of thought … together with a recognition that arithmetic methods are not exact, led to Pythagorean mathematics."

"‘A comparison of Pythagorean and Vedic mathematics together with some chronological consideration showed that the current view [of Greek influence on Vedic thought] is not tenable. A common source for the Pythagorean and Vedic mathematics is to be sought either in the Vedic mathematics [i.e. the Sulbasutras] or in an older mathematics very much like it.’ (Seidenberg 1978)"

"However, Seidenberg was told by the Indologists that these Sutras, or any Vedic text for that matter, were definitely written later than 1700 BC. But mathematical data cannot be manipulated just like that, and Seidenberg remained convinced of his case: “Whatever the difficulty there may be [concerning chronology], it is small in comparison with the difficulty of deriving the Vedic ritual application of the theorem from Babylonia. (The reverse derivation is easy)… the application involves geometric algebra, and there is no evidence of geometric algebra from Babylonia. And the geometry of Babylonia is already secondary whereas in India it is primary.” [To satisfy the indologists, he said that the Shulba Sutra had conserved an older tradition, and that it is from this one that the Babylonians had learned their mathematics:] “Hence we do not hesitate to place the Vedic (…) rituals, or more exactly, rituals exactly like them, far back of 1700 BC. (…) elements of geometry found in Egypt and Babylonia stem from a ritual system of the kind described in the Sulvasutras.”"

"By examining the evidence in the Shatapatha Brahmana, we now know that Indian geometry predates Greek geometry by centuries. For example, the earth was represented by a circular altar and the heavens were represented by a square altar and the ritual consisted of converting the circle into a square of an identical area. There we see the beginnings of geometry! Two aspects of the 'Pythagoras' theorem are described in the Vedic literature. One aspect is purely algebraic that presents numbers a, b, c for which the sum of the squares of the first two equals the square of the third. The second is the geometric, according to which the sum of the areas of two square areas of different size is equal to another square. The Babylonians knew the algebraic aspect of this theorem as early as 1700 BCE, but they did not seem to know the geometric aspect. The Shatapatha Brahmana, which precedes the age of Pythagoras, knows both aspects. Therefore, the Indians could not have learnt it from the Old-Babylonians or the Greeks, who claim to have rediscovered the result only with Pythagoras. India is thus the cradle of the knowledge of geometry and mathematics."

"I think its mathematics [of the common source] was very much like what we see in the Sulbasutras. (Seidenberg 1978)"

"I cannot help thinking that it shows battle weariness rather than a considered opinion."

"Yet long before 1937 people had suggested a non-Greek origin to Greek mathematics; and long before 1943 people had pointed out that sacred books of the East contain the ‘Pythagorean numbers’ [and indeed the ‘theorem’]. Such numbers are mentioned in the Sulvasutras; ancient Indian works on altar constructions. […] Neugebauer does not mention the Sulvasutras in his book Vorlesungen über Geschichte der Antiken Mathematischen Wissenschaften, nor does B. L. van der Waerden them in his book Science Awakening. Why this omission?"

"You are all computer scientists. You know what FINITE AUTOMATA can do. You know what TURING MACHINES can do. For example, Finite Automata can add but not multiply. Turing Machines can compute any computable function. Turing machines are incredibly more powerful than Finite Automata. Yet the only difference between a FA and a TM is that the TM, unlike the FA, has paper and pencil. Think about it. It tells you something about the power of writing. Without writing, you are reduced to a finite automaton. With writing you have the extraordinary power of a Turing machine."

"Transforming hereditary privilege into ‘merit,’ the existing system of educational selection, with the Big Three [Harvard, Princeton, Yale] as its capstone, provides the appearance if not the substance of equality of opportunity. In so doing, it legitimates the established order as one that rewards ability over the prerogatives of birth. The problem with a ‘meritocracy,’ then, is not only that its ideals are routinely violated (though that is true), but also that it veils the power relations beneath it. For the definition of ‘merit,’ including the one that now prevails in America’s leading universities, always bears the imprint of the distribution of power in the larger society. Those who are able to define ‘merit’ will almost invariably possess more of it, and those with greater resources—cultural, economic and social—will generally be able to ensure that the educational system will deem their children more meritorious."

"By the end of 2006 it was generally believed that Perelman’s proof was correct. That year, the journal Science named Perelman’s proof the “Breakthrough of the Year.” Like Smale and Freedman before him, the forty-year old Perelman was tapped to be a Fields Medals recipient for his contributions to the Poincaré conjecture (in fact, Thurston also received a Fields Medal for his work that indirectly led to the final proof). The countdown for the $1 million prize had begun (some wonder if Perelman and Hamilton will be offered the prize jointly)."

"If the proof is correct then no other recognition is needed."

"Revolutions in mathematics are quiet affairs. No clashing armies and no guns. Brief news stories far from the front page. Unprepossessing. Just like the raw damp Monday afternoon of April 7, 2003, in Cambridge, Massachusetts. Young and old crowded the lecture theater at the Massachusetts Institute of Technology (MIT). They sat on the floor and in the aisles, and stood at the back. The speaker, Russian mathematician Grigory Perelman, wore a rumpled dark suit and sneakers, and paced while he was introduced."