hypotheses

"... there is Brouwer's , which is utterly destructive in its results. The whole theory of the \,\aleph\,'s greater than \,\aleph_1\, is rejected as meaningless (Brouwer 1907, 569). Cantor's conjecture itself receives several different meanings, all of which, though very interesting in themselves, are quite different from the original problem. They lead partly to affirmative, partly to negative answers (Brouwer, 1907, I: 9; III: 2). Not everything in this field, however, has been sufficiently clarified. The “semi-intuitionistic” standpoint along the lines of H. Poincaré and H. Weyl ... would hardly preserve substantially more of set theory."

"In the , one wholeheartedly accepts traditional mathematics at face value. All questions such as the Continuum Hypothesis are either true or false in the real world despite their independence from the various axiom systems. The Realist position is probably the one that most mathematicians would prefer to take."

"Those who argue that the concept of set is not sufficiently clear to fix the truth-value of CH have a position which is at present difficult to assail. As long as no new axiom is found which decides CH, their case will continue to grow stronger, and our assertion that the meaning of CH is clear will sound more and more empty."

"[The whole issue has been] simplified by Professor Renfrew to the ludicrous formula 7000 BCE Anatolia = farming Indo-Europeans."

"“All the migrations postulated by Renfrew ultimately stem from a single catalyst: the crossing of Anatolian farmers into Greece… For all practical purposes, Renfrew’s hypothesis disregards Tocharian and Indo-Iranian.”"

"In the 1990s, this view was briefly challenged by the British archaeologist Colin Renfrew, who posited a Homeland in Anatolia. Thence, the IE-speaking tribes would have spread as early as about 7000 BCE, on the strength of their mastery of agriculture. This new development allowed for a fast population growth, as illustrated by the more recent spread of the Bantu languages throughout Africa along with agriculture. His merit was that he tied the spectacular expansion of IE to a powerful mechanism, the demographically useful new technology of agriculture. However, this theory has been widely rejected as linguistically untenable and archaeologically unsupported. The targeted studies that sought to decide between the Anatolian and the Russian Homeland have generally favoured the latter option."

"What they do not state, however, is just how low are the proportions of Steppe ancestry in these cases. The “massive migrations” into Corded Ware northern Europe proved elusive elsewhere... 10% is no “massive migration” that might be expected to rewrite the language identity of central Anatolia (and western Anatolia, where these languages also dominated). More likely it left no major linguistic effect; the Anatolian branch remains better explained by stronger candidates."

"Anatolia is remarkable for its lack of steppe ancestry down to the Bronze Age."

"A substantial portion of Weil’s research was motivated by an effort to prove the Riemann hypothesis concerning the zeroes of the Riemann zeta function. He was continually looking for new ideas from other fields that he could bring to bear on a proof. He commented on this matter in a 1979 interview:... Asked what theorem he most wished he had proved, he responded, “In the past it sometimes occurred to me that if I could prove the Riemann hypothesis, which was formulated in 1859, I would keep it secret in order to be able to reveal it only on the occasion of its centenary in 1959. Since 1959, I have felt that I am quite far from it; I have gradually given up, not without regret.”"

"It tells us that they are very nicely distributed, about as evenly and as good as altogether possible. One cannot expect a completely even distribution, of course. But it tells us that at least in mathematics, certainly in number theory, we live in Leibniz’ “best possible of all worlds”, just as the good Candide in Voltaire’s ' is told by his teacher that he lives in the best of all possible worlds. Well, in number theory at least, one has the best relation possible among primes, even though we cannot prove it yet. It would give me great satisfaction to see a proof, because it would demonstrate that there are some things that are right in this world. There are so many other things that do not work as they should, but at least for the prime numbers, and of course also for the zeros of the zeta function, they are distributed as well as they could be."

"The dependence of so many results on Riemann's challenge is why mathematicians refer to it as a hypothesis rather than a conjecture. The word 'hypothesis' has the much stronger connotation of a necessary assumption that a mathematician makes in order to build a theory. 'Conjecture', in contrast, represents simply a prediction of how mathematicians believe their world behaves. Many have had to accept their inability to solve Riemann's riddle and have simply adopted his prediction as a working hypothesis. If someone can turn the hypothesis into a theorem, all those unproven results would be validated."

"In the spring of 1997, Connes went to Princeton to explain his new ideas to the big guns: Bombieri, Selberg and Sarnak. Princeton was still the undisputed Mecca of the Riemann Hypothesis despite Paris's push to reclaim its dominance. Selberg had become godfather to the problem - nothing could pass muster before being vetted by a man who had spent half a century doing battle with the primes. Sarnak was the young gun whose rapier-like intellect would cut through anything that was found slightly wanting. He'd recently joined forces with Nick Katz, also at Princeton, one of the undisputed masters of the mathematics developed by Weil and Grothendieck. Together they had proved that the strange statistics of random drums that we believe describe the zeros in Riemann's landscape are definitely present in the landscapes considered by Weil and Grothendieck. Katz's eyes were particularly sharp, and little escaped his penetrating stare. It was Katz who, some years before, had found the mistake in Wiles's first erroneous proof of Fermat's Last Theorem. And finally there was Bombieri, sitting in state as the undisputed master of the Riemann Hypothesis. He had earned his Fields Medal for the most significant result to date about the error between the true number of primes and Gauss's guess - a proof of something mathematicians call the 'Riemann Hypothesis on average'. In the quiet of his office overlooking the woods that surround the Institute, Bombieri has been marshalling all his insights of previous years for a final push for the complete solution. Bombieri, like Katz, has a fine eye for detail. A keen philatelist, he once had the chance to purchase a very rare stamp to add to his collection. After scrutinising it carefully he discovered three flaws. He returned the stamp to the dealer, pointing out two of them. The third subtle flaw he kept to himself - in case he is offered an improved forgery at a future date. Any aspiring proof of the Riemann Hypothesis is subjected to an equally painstaking examination."

"I believe that the vast majority of statements about the integers are totally and permanently beyond proof in any reasonable system. Here I am using proof in the sense that mathematicians use that word. Can statistical evidence be regarded as proof ? I would like to have an open mind, and say ‘Why not?’. If the first ten billion zeros of the zeta function lie on the line whose real part is 1/2, what conclusion shall we draw? I feel incompetent even to speculate on how future generations will regard numerical evidence of this kind."

"The Riemann hypothesis is, and will hopefully remain for a long time, a great motivation to uncover and explore new parts of the mathematical world. After reviewing its impact on the development of we discuss three strategies, working concretely at the level of the explicit formulas. The first strategy is “analytic” and is based on Riemannian spaces and and its comparison with the explicit formulas. The second is based on algebraic geometry and the . We establish a framework in which one can transpose many of the ingredients of the Weil proof as reformulated by , Tate and Grothendieck. This framework is elaborate and involves noncommutative geometry, es and . We point out the remaining difficulties and show that RH gives a strong motivation to develop algebraic geometry in the emerging world of characteristic one. Finally we briefly discuss a third strategy based on the development of a suitable “”, the role of Segal’s Γ-rings and of topological cyclic homology as a model for “absolute algebra” and as a cohomological tool."

"Hilbert, in his 1900 address to the Paris , listed the Riemann Hypothesis as one of his 23 problems for mathematicians of the twentieth century to work on. Now we find it is up to twenty-first century mathematicians! The Riemann Hypothesis (RH) has been around for more than 140 years, and yet now is arguably the most exciting time in its history to be working on RH. Recent years have seen an explosion of research stemming from the confluence of several areas of mathematics and physics."

"At the beginning of the new millennium the most famous unsolved problem in complex analysis, if not in all of mathematics, is to determine whether the Riemann hypothesis holds."

"The truth is, that these writings of mine were meant to protect the arguments of Parmenides against those who make fun of him and seek to show the many ridiculous and contradictory results which they suppose to follow from the affirmation of the one. My answer is addressed to the partisans of the many, whose attack I return with interest by retorting upon them that their hypothesis of the being of many, if carried out, appears to be still more ridiculous than the hypothesis of the being of one. Zeal for my master led me to write the book in the days of my youth, but some one stole the copy; and therefore I had no choice whether it should be published or not; the motive, however, of writing, was not the ambition of an elder man, but the pugnacity of a young one."

"Prof. Michael Foster has somewhere said that ‘hypothesis is the salt of science.’ Without hypothesis there is no possibility of fruitful investigation. But it is equally true that where the desire to prove a particular hypothesis is dominant, hypothesis becomes the poison of science. The Aryan race theory of Western scholars is as good an illustration of how hypothesis can be the poison of science as one can think of."

"I do not yet want to form a hypothesis to test, because as soon as you make a hypothesis, you become prejudiced. Your mind slides into a groove, and once it is in that groove, has difficulty noticing anything outside of it. During this time, my sense must be sharp; that is the main thing — to be sharp, yet open."

"It is human nature to avoid being consumed by hypotheticals until they are staring us squarely in the face."

"Now all hypotheses, by definition, involve make-believe. Many of them...involve sort-crossing, and are therefore metaphors. The conclusion ... is to try to adopt the actual technique of Plato...Then, whether we suppose that man is a state, or that the world is a machine, or that man is a wolf, the risk of confusing the facts of one sort with those of the other will be lessened."

"There are two possible outcomes: if the result confirms the hypothesis, then you've made a measurement. If the result is contrary to the hypothesis, then you've made a discovery."

"One thinker no less brilliant than the heresiarch himself, but in the orthodox tradition, advanced a most daring hypothesis. This felicitous supposition declared that there is only one Individual, and that this indivisible Individual is every one of the separate beings in the universe, and that these beings are the instruments and masks of divinity itself."

"The functional validity of a working hypothesis is not a priori certain, because often it is initially based on intuition. However, logical deductions from such a hypothesis provide expectations (so called prognoses) as to the circumstances under which certain phenomena will appear in nature. Such a postulate or working hypothesis can then be substantiated by additional observations or by experiments especially arranged to test details. The value of the hypothesis is strengthened if the observed facts fit the expectation within the limits of permissible error."