Mathematicians From The United States

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

"‘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]."

"[…] 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."

"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."

"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."

"‘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)"

"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."

"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.”"

"‘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)"

"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?"

"The Earth is an irregular oblate spheroid (a sphere flattened at the poles). Measurements of its dimensions and the amount of its flattening are subjects of geodesy. However, for most navigational purposes, assuming a spherical Earth introduces insignificant error. The Earth’s axis of rotation is the line connecting the north and south geographic poles. A great circle is the line of intersection of a sphere and a plane through its center. This is the largest circle that can be drawn on a sphere. The shortest line on the surface of a sphere between two points on the surface is part of a great circle. On the spheroidal Earth the shortest line is called a geodesic. A great circle is a near enough approximation to a geodesic for most problems of navigation."

"Marine navigation blends both science and art. A good navigator constantly thinks strategically, operationally, and tactically. He plans each voyage carefully. As it proceeds, he gathers navigational information from a variety of sources, evaluates this information, and determines his ship’s position. He then compares that position with his voyage plan, his operational commitments, and his predetermined “dead reckoning” position. A good navigator anticipates dangerous situations well before they arise, and always stays “ahead of the vessel.” He is ready for navigational emergencies at any time. He is increasingly a manager of a variety of resources--electronic, mechanical, and human. Navigation methods and techniques vary with the type of vessel, the conditions, and the navigator’s experience. The navigator uses the methods and techniques best suited to the vessel, its equipment, and conditions at hand. Some important elements of successful navigation cannot be acquired from any book or instructor. The science of navigation can be taught, but the art of navigation must be developed from experience."

"I think in other things that I've done, I can look back and see: "This idea developed from something else." Sometimes it would lead back to a previous idea of mine, very often it would lead to something somebody else had done. But the Bakery algorithm just seemed to come out of thin air to me. There was nothing like it that preceded it, so perhaps that's why I'm proudest of it."

"A distributed system is one in which the failure of a computer you didn't even know existed can render your own computer unusable."

"The first thing is deciding what the program should do. If you don’t think carefully about that, it’s going to wind up not doing something that it probably should do or you wind up with an ‘absolute bug-free program’. Because to have a bug, you have to have some notion of what it means for the program to be operating correctly. And there is no precise definition of what it means for the program to be correctly operating — so, well, no bugs. Not a very good situation."

"With so many people doing so much writing, great writing is hard to find ... If you succeed in attaining a position that allows you to do something great, if you do something that really is great, and if you realize that it’s great, there’s still one more hurdle: You have to convince others that it’s great. This will require writing."

"Thinking doesn't guarantee that we won't make mistakes. But not thinking guarantees that we will."

"Though there are important differences among species, there is reason to be optimistic that these methods can provide useful interventions for humans...This paper and related studies have the potential for huge clinical impact in Alzheimer's disease and others involving brain inflammation."

"What is the question, to which this is the answer?"

"When scientific research is increasingly funded by private corporations that have a financial interest in particular outcomes of that research—is the drug effective or not?—scientific objectivity is undermined. When universities are more interested in patent royalties than in the open sharing of scientific information, the public suffers. There are hundreds of important political and economic issues surrounding science and technology. Sociology of science, at its best, has done much to clarify these issues. But sloppy sociology, like sloppy science, is useless or even counterproductive."

"But why did I do it? I confess that I'm an unabashed Old Leftist who never quite understood how deconstruction was supposed to help the working class. And I'm a stodgy old scientist who believes, naively, that there exists an external world, that there exist objective truths about that world, and that my job is to discover some of them."

"I was really amazed by my first encounters with serious mathematics textbooks...I could appreciate that the mathematics was an impressive intellectual edifice, and I could follow the steps of proofs. I assumed that such an elaborate buildup must be leading to a fantastic denouement, which I eagerly awaited -- and waited, and waited. It was only much later, after much of the mathematics I had studied had come alive for me that I came to appreciate how ineffective and denatured the standard ((definition theorem proof)n remark)m style is for communicating mathematics. When I reread some of these early texts, I was stunned by how well their formalism and indirection hid the motivation, the intuition and the multiple ways to think about their subjects: they were unwelcoming to the full human mind."

"Anyone who believes that the laws of physics are mere social conventions is invited to try transgressing those conventions from the windows of my apartment. (I live on the twenty-first floor.)"

"The results of my little experiment [Sokal affair] demonstrate, at the very least, that some fashionable sectors of the American academic Left have been getting intellectually lazy. The editors of Social Text liked my article because they liked its conclusion: that "the content and methodology of postmodern science provide powerful intellectual support for the progressive political project" [sec. 6]. They apparently felt no need to analyze the quality of the evidence, the cogency of the arguments, or even the relevance of the arguments to the purported conclusion."

"The term `geometry'...refers to a pattern of processing within our brains related to our spatial and visual senses, more than it refers to a separate content area of mathematics."

"Mathematics is a process of staring hard enough with enough perseverence at at the fog of muddle and confusion to eventually break through to improved clarity. I'm happy when I can admit, at least to myself, that my thinking is muddled, and I try to overcome the embarrassment that I might reveal ignorance or confusion.Over the years, this has helped me develop clarity in some things, but I remain muddled in many others."

"Mathematics is primarily a tool for human thought."

"The most important thing about mathematics is how it resides in the human brain."

"People can be fooled into thinking of mathematics as logical, formal, symbolic reasoning. But this is far from reality...computers are far better at formal computation and formal reasoning, but humans are far better mathematicians."

"When mathematics is explained, formalized and written down, there is a strong tendency to favor symbolic modes of thought at the expense of everything else, because symbols are easier to write and more standardized than other modes of reasoning. But when mathematics loses its connection to our minds, it dissolves into a haze."

"Since December 2015, there have 10 incidents that killed 10 or more people. That’s more than there was in 30 years between 1982 and 2011. And five of the 10 deadliest mass shootings in modern American history have all happened in the last five years."

"GOP are so all-in on an "all base mobilization, all the time" strategy that people just take for granted it must be smart politics when it's far from obvious that's the case."

"There are entire disciplines in which predictions have been failing, often at great cost to society. Consider something like biomedical research. In 2005, an Athens-raised medical researcher named John P. Ioannidis published a controversial paper titled "Why Most Published Research Findings Are False." ... The paper studied positive findings documented in peer-reviewed journals: descriptions of successful predictions of medical hypotheses carried out in laboratory experiments. It concluded that most of these findings were likely to fail when applied in the real world. Bayer Laboratories recently confirmed Ioannidis's hypothesis. They could not replicate about two-thirds of the positive findings in medical journals when they attempted the experiments themselves. ..."

"Bernie Sanders proudly describes himself as a “socialist” (or more commonly, as a “democratic socialist”). To Americans of a certain age, this is a potential liability. I’m just old enough (38) to have grown up during the Cold War, a time when “socialist” did not just mean “far left” but also implied something vaguely un-American. If you’re older than me, you may have even more acutely negative associations with “socialism” and may see it as a step on the road to communism. If you’re a few years younger than me, however, you may instead associate “socialism” with the social democracies of Northern Europe, which have high taxes and large welfare states. Sweden may not be your cup of tea, but it isn’t scary in the way the USSR was to people a generation ago."

"I'd say I am somewhere in between being a libertarian and a liberal. So if I were to vote it would be kind of a Gary Johnson versus Mitt Romney decision, I suppose."

"I've always felt like something of an outsider. I've always had friends, but I've always come from an outside point of view. I think that's important. If you grow up gay, or in a household that's agnostic, when most people are religious, then from the get-go, you are saying that there are things that the majority of society believes that I don't believe."

"In February William Perry let slip that if this reform falls through in Russia, then the West should be ready to strengthen its collective security. What's the connection? Since when can the defense secretary of the U.S. determine the type of economic reform Russia needs? And why are we being threatened with external pressure if Russia changes the direction and tempo of reform?"

"Any further spread of nuclear weapons multiplies the possibilities of nuclear confrontation and magnifies the danger of diversion. Thus, if proliferation of weapons of mass destruction continues into Iran and remains in North Korea in the face of all ongoing negotiations, the incentives for other countries to follow the same path will become overwhelming. Considerations as these have induced former Senator Sam Nunn, former Secretary of Defense William Perry, former Secretary of State George Shultz, and I -- two Democrats and two Republicans -- to publish recommendations for systematically reducing and eventually eliminating reliance on nuclear weapons. We have a record of strong commitment to national defense and security. We continue to affirm the importance of adequate deterrent forces, and we do not want our recommendations to diminish essentials for the defense of free peoples while a process of adaptation to new realities is going on. At the same time, we reaffirm the objective of a world without nuclear weapons that has been proclaimed by every American President since President Eisenhower."

"Our first action that really set us off in a bad direction was when Nato started to expand, bringing in eastern European nations, some of them bordering Russia. At that time we were working closely with Russia and they were beginning to get used to the idea that Nato could be a friend rather than an enemy ... but they were very uncomfortable about having Nato right up on their border and they made a strong appeal for us not to go ahead with that."

"We stand today, I believe, in greater danger of a nuclear catastrophe than we faced during the Cold War. And I can explain why I believe that, but the first point I want to make, though, is that hardly anybody understands that. And because we don’t understand it, our policies are not responsive to those dangers. During the Cold War, at least we had, we understood the danger and were taking action to try to deal with it; we had very serious arms control discussions, for example. Today, it’s so far in the background that people don’t understand it at all, particularly the new generation of people who didn’t live through the Cold War. But it’s a very dangerous situation today. And the headlines, which are about North Korea, just emphasize the danger; but the greater danger, really, is–the ultimate danger–is some sort of an exchange between the United States and Russia today. That would lead not just to a great catastrophe, it would basically lead to the end of civilization. So that’s what’s at stake here: ending our civilization, and the nuclear weapons have the power to do that."

"It’s always healthy to recognize facts."

"While not a panacea, candidly recognizing the absence of any good logical arguments for God’s existence, giving up on divine allies and advocates as well as taskmasters and tormentors, and prizing a humane, reasonable, and brave outlook just might help move this world a bit closer to a heaven on earth."

"Confirmation of a person’s unreliable statement by another unreliable person makes the statement even less reliable."

"Claiming that a holy book’s claims are undeniable because the book itself claims them to be is convincing only to the convinced."

"The whole weight of science is the prima facie evidence against a miracle having occurred."

"For the record, natural selection is a highly nonrandom process that acts on the genetic variation produced by random mutation and genetic drift and results in those organisms with more adaptive traits differentially surviving and reproducing."

"Define God in a sufficiently nebulous way as beauty, love, mysterious complexity, or the ethereal taste of strawberry shortcake, and most atheists become theists. Still, although one can pose as Humpty Dumpty and aver, “When I use a word, it means just what I choose it to mean, neither more nor less,” others needn’t play along."

"You would think that the obvious irreligious objection would come to almost anyone’s mind when reading a religious tome or holy book. What if you don’t believe the holy book’s presuppositions and narrative claims and simply ask for independent argument or evidence for God’s existence? What if you’re not persuaded by the argument that God exists because His assertion that He exists and discussion of His various exploits appear in this book about Him that believers say He inspired?"