Mathematicians From England

"In the application of the ordinary principles of geometry and trigonometry to such Astronomical measures... it may sometimes be expedient to resolve the process into several successive steps, and these steps may perhaps require different kinds of treatment. But... all are simple and within ordinary comprehension, and the only complexity arises from the circumstance that the student may find it necessary to have a clear view of several such steps at once..."

"In conversing with persons who are not officially attached to Observatories or in other ways professionally cognizant of the technicalities of practical Astronomy but who nevertheless display great interest... these persons appear to regard the determination of measures like those of the distance of the Sun and Moon as mysteries beyond ordinary comprehension... [and] when persons well acquainted with the general facts of Astronomy are introduced into an Observatory, they are for the most part utterly unable to understand anything which they see... The measure of the Moon's distance involves no principle more abstruse than the measure of the distance of a tree on the opposite bank of a river. The principles of construction of the best Astronomical instruments are as simple and as closely referred to matters of common school-education and familiar experience, as are those of the common globes, the steam engine, or the turning-lathe; the details are usually less complicated."

"[T]his Fifth Edition is required to meet the demand of a somewhat wider class of students than those for whom the Lectures were originally intended. ...Mr. Stirling has been at liberty to prepare the modifications and additions ..."

"In the hands of Science and indomitable energy, results the most gigantic and absorbing may be wrought out by skilful combinations of acknowledged data and the simplest means."

""On the Diffraction of an Object-glass with Circular Aperture" read (Nov. 24, 1834) Transactions of the Cambridge Philosophical Society (1835) Vol. 5, p. 283."

"The investigation of the form and brightness of the rings or rays surrounding the image of a star as seen in a good telescope, when a diaphragm bounded by a rectilinear contour is placed upon the object-glass, though sometimes tedious is never difficult. The expressions which it is necessary to integrate are always sines and cosines of multiples of the independent variable, and the only trouble consists in taking properly the limits of integration. Several cases of this problem have been completely worked out, and the result, in every instance, has been entirely in accordance with observation. These experiments... have seldom been made except by those whose immediate object was to illustrate the undulatory theory of light. There is however a case of a somewhat different kind; which in practice recurs perpetually, and which in theory requires for its complete investigation the value of a more difficult integral; I mean the usual case of an object-glass with a circular . The desire of submitting to mathematical investigation every optical phænomenon of frequent occurrence has induced me to procure the computation of the numerical values of the integral that presents itself in this inquiry: and I now beg leave to lay before the Society the calculated table, with a few remarks upon its application."

"In the early days of the computer revolution computer designers and numerical analysts worked closely together and indeed were often the same people. Now there is a regrettable tendency for numerical analysts to opt out of any responsibility for the design of the arithmetic facilities and a failure to influence the more basic features of software. It is often said that the use of computers for scientific work represents a small part of the market and numerical analysts have resigned themselves to accepting facilities "designed" for other purposes and making the best of them. [...] One of the main virtues of an electronic computer from the point of view of the numerical analyst is its ability to "do arithmetic fast." Need the arithmetic be so bad!"

"Of course everything in computerology is new; that is at once its attraction, and its weakness. Only recently I learned that computers are revolutionizing astrology. Horoscopes by computer!"

"Numerical analysis has begun to look a little square in the computer science setting, and numerical analysts are beginning to show signs of losing faith in themselves. Their sense of isolation is accentuated by the present trend towards abstraction in mathematics departments which makes for an uneasy relationship. How different things might have been if the computer revolution had taken place in the 19th century! [...] In any case "numerical analysts" may be likened to "The Establishment" in computer science and in all spheres it is fashionable to diagnose "rigor morris" in the Establishment."

"He [Turing] was particularly fond of little programming tricks (some people would say that he was too fond of them to be a "good" programmer) and would chuckle with boyish good humor at any little tricks I may have used."

"Turing had a strong predeliction for working things out from first principles, usually in the first instance without consulting any previous work on the subject, and no doubt it was this habit which gave his work that characteristically original flavor. I was reminded of a remark which Beethoven is reputed to have made when he was asked if he had heard a certain work of Mozart which was attracting much attention. He replied that he had not, and added "neither shall I do so, lest I forfeit some of my own originality.""

"Very belatedly in 1947, Darwin [Sir Charles Darwin, great-grandson of the famous Charles Darwin] agreed to set up a very small electronics group [...] It was not easy to have the imagination to foresee that computers were to become one of the most important developments of the century."

"It is not simply that a clear understanding is acquired of the movements of the great bodies which we regard as the system of the world, but it is that we are introduced to a perception of laws governing the motion of all matter, from the finest particle of dust to the largest planet or sun, with a degree of uniformity and constancy, which otherwise we could hardly have conceived. Astronomy is pre-eminently the science of order."

"[P]erhaps one of the most valuable results to be derived from a truly intellectual study of Astronomy is, the habit of keeping up a sustained attention to all the successive steps of a long series of reasonings. Power, and with it dignity, are gained to the mind by this noble exercise."

"Complete knowledge of every theoretical and instrumental detail can only be obtained by those who will devote... a large portion of their lives; but sound knowledge of the principles... can be obtained by the reasonable efforts of persons possessing common opportunities for general knowledge."

"The elucidation of the theory of centripetal and disturbing forces is necessarily less complete. Still... a general conception of the nature of the action of those forces... sufficient to preserve the student from the gross errors... may be obtained from explanations like those here offered. The methods of ascertaining the weight of the Earth and other bodies are... more difficult of explanation; yet... something may be done even in these."

"[T]he methods used for measuring Astronomical distances are in some applications absolutely the same as the methods of ordinary -surveying, and are in other applications equivalent to them..."

"[T]he instrumental conceptions derived from the use of a common globe are sufficient, in almost every case, for the understanding of the instruments in an Observatory..."

"[H]ow much of the fundamentals of Astronomy may be obtained with the coarsest observation with the unaided eye. ...the science which is thus obtained by personal observations is vastly superior (as far as it goes) to that which is obtained by any other method. ...The knowledge ...inferred from actual personal observation carries with it a degree of reality and certainty, as the veritable science of external objects, which nothing else can give."

"The properties of bodies were investigated by several distinguished French mathematicians on the hypothesis that they are systems of molecules in equilibrium. The somewhat unsatisfactory nature of the results... produced... a reaction in favour of the opposite method of treating bodies as if they were... continuous. This method, in the hands of Green, Stokes, and others, has led to results the value of which does not at all depend on what theory we adopt as to the ultimate constitution of bodies."

"Although many of the artifices employed in the works before mentioned are remarkable for their elegance, it is easy to see they are adapted only to particular objects, and that some general method, capable of being employed in every case, is still wanting."

"Game theory concepts were first explicitly applied in evolutionary biology by Lewontin (1961). His approach, however, was to picture a species as playing a game against nature, and to seek strategies which minimised the probability of extinction. A similar line has been taken by Slobodkin & Rapoport (1974). In contrast, here we picture members of a population as playing games against each other, and consider the population dynamics and equilibria which can arise."

"The theory of games was first formalised by Von Neumann & Morgenstern (1953) in reference to human economic behaviour. Since that time, the theory has undergone extensive development... Sensibly enough, a central assumption of classical game theory is that the players will behave rationally, and according to some criterion of self-interest. Such an assumption would clearly be out of place in an evolutionary context. Instead, the criterion of rationality is replaced by that of population dynamics and stability, and the criterion of self-interest by Darwinian fitness."

"Evolutionary game theory is a way of thinking about evolution at the phenotypic level when the fitnesses of particular phenotypes depend on their frequencies in the population."

"Paradoxically, it has turned out that game theory is more readily applied to biology than to the field of economic behavior for which it was originally designed"

"Тhе сепtгаl роіпt геmаіпs that Darwin provided a theory which predicts that organisms should have parts adapted to ensure their survival and . This has led to the suggestion that life should be defined by the possession of those properties which are needed to ensure evolution by natural selection. That is, entities with the properties of multiplication, variation, and are alive, and entities lacking one or more of those properties are not."

"Natural selection is the only workable explanation for the beautiful and compelling illusion of 'design' that pervades every living body and every organ. Knowledge of evolution may not be strictly useful in everyday commerce. You can live some sort of life and die without ever hearing the name of Darwin. But if, before you die, you want to understand why you lived in the first place, Darwinism is the one subject that you must study."

"Darwin's theory of evolution by natural selection is the only workable explanation that has ever been proposed for the remarkable fact of our own existence, indeed the existence of all life wherever it may turn up in the universe."

"It is an occupational risk of biologists to claim, towards the end of their careers, that the problems which they have not solved are insoluble."

"It is in the nature of science that once a position becomes orthodox it should be subjected to criticism.... It does not follow that, because a position is orthodox, it is wrong."

"The last decade has seen a steady increase in the application of concepts from the theory of games to the study of evolution. Fields as diverse as sex ratio theory, animal distribution, contest behaviour and reciprocal altruism have contributed to what is now emerging as a universal way of thinking about phenotypic evolution."

"John Maynard Smith, an engineer by training, knows much about his biology secondhand. He seldom deals with live organisms. He computes and he reads. I suspect that it's very hard for him to have insight into any group of organisms when he does not deal with them directly. Biologists, especially, need direct sensory communication with the live beings they study and about which they write."

"Does life in some way make use of the potentiality for vast quantum superpositions, as would be required for serious quantum computation? How important are the quantum aspects of DNA molecules? Are cellular microtubules performing some essential quantum roles? Are the subtleties of quantum field theory important to biology? Shall we gain needed insights from the study of quantum toy models? Do we really need to move forward to radical new theories of physical reality, as I myself believe, before the more subtle issues of biology — most importantly conscious mentality — can be understood in physical terms? How relevant, indeed, is our present lack of understanding of physics at the quantum/classical boundary? Or is consciousness really “no big deal,” as has sometimes been expressed? It would be too optimistic to expect to find definitive answers to all these questions, at our present state of knowledge, but there is much scope for healthy debate..."

"[I]n the June, 1960 issue of... the ', there appeared a paper by... Roger Penrose with the esoteric title "A Approach to General Relativity." Although the paper was highly mathematical, it outlined a very elegant and streamlined technique for solving certain problems in general relativity. ...this new method made some computations almost easy."

"Although I'm regarded as a dangerous radical by particle physicists for proposing that there may be loss of quantum coherence, I'm definitely a conservative compared to Roger. I take the positivist viewpoint that a is just a and that it is meaningless to ask whether it corresponds to reality. All that one can ask is that its predictions should be in agreement with observation. I think Roger is a Platonist at heart but he must answer for himself."

"Reading about the lives of geniuses can be a powerful antidote against any envy we might feel for not being among their number. Almost invariably they turn out not only to have had the same flaws, trials and heartbreaks as the rest of us but to be deeply weird to boot. The Nobel Prize-winning British physicist Roger Penrose certainly fits the template."

"Whereas originally the hopes for string theory, and its descendants, were that some kind of uniqueness would be arrived at, whereby the theory would supply mathematical explanations for the measured numbers of experimental physics, the string theorists were driven to find refuge in the strong anthropic argument in an attempt to narrow down an absolutely vast number of alternatives. In my own view, this a very sad and unhelpful place for a theory to find itself."

"What the anthropic principle depends upon is the idea that whatever is the nature of the universe, or universe portion that we see about us, being subject to whatever dynamical laws govern its actions, this must be strongly favourable to our very existence."

"One is left with the uneasy feeling that even if supersymmetry is actually false, as a feature of nature, and that accordingly no supersymmetry partners are ever found by the LHC or by any later more powerful accelerator, then the conclusion that some supersymmetry proponents might come to would not be that supersymmetry is false for the actual particles of nature, but merely that the level of supersymmetry breaking must be greater even that the level reached at that moment, and that a new even more powerful machine would be required to observe it!"

"The idea of having an ambient space-time of some specific dimension seems to play less of a role of string theory than in conventional physics, and certainly less than the kind of role that I would myself feel comfortable with. It is particularly difficult to assess the functional freedom that is involved in a physical theory unless one has a clear idea of its actual space-time dimensionality."

"General relativity is certainly a very beautiful theory, but how does one judge the elegance of physical theories generally?"

"It's a very plausible thing now that the entropy should increase all the time... [T]hese volumes... are... enormously different in scale... I can't convey to you in the picture the absolute stupendous difference in the sizes of these volumes. So if you happen to find yourself in one of them, and you wiggle around, the next one you find yourself in will be overwhelmingly likely to be much much larger, and the entropy therefor goes up."

"[S]ome of these regions may be... indistinguishable, for example the air in the room. We might have molecules in some other places. You might like to say we don't care where the individual molecules are. We just care about overall parameters, and so we lump together the systems which look very much the same. ...[L]et's say with regard to macroscopic parameters we lump them together, and so we have these things called course graining cells in the phase space... [Y]ou then say, well let's measure the volume of these regions... V... and the logarithm of that volume is the entropy. This is a marvelous formula due to Boltzmann. This [k] is Boltzmann's constant, the only thing in the formula that wasn't due to Boltzmann... This was named afterwards. I don't think he was particularly interested in constants..."

"[T]he randomness is measured... by... entropy, and it's telling us that this entropy is increasing with time. ...[I]t can be given a clearer definition ...the idea due to Boltzmann ...we imagine... a ... a space... of a very large number of dimensions, where each point in the space represents a state of the system at one moment. In fact it contains both the positions of all the particles and the momenta (or velocities) of all the particles. So if you know where the point is in this large dimensional space at any moment that describes a particular thing... then the dynamics will tell you where that point moves. So that there will be a unique path through that point, wiggling around somewhere through this phase space."

"The 2nd law of thermodynamics... tells you that randomness increases with time. It's a sort of depressing law... It depends on how you look at it, really..."

"In its simplest form, the 2nd law of thermodynamics... You imagine... a glass of wine sitting on a table... it falls off and wine splashes out onto the carpet...[etc.] If you just think of this as a Newtonian situation, as the system evolves the thing proceeds according to Newtonian laws, but Newtonian laws are reversible in time... What's not so agreeable [about the reverse] is that it violates the 2nd law..."

"If you want fantasy... first of all, you have to believe in string theory... these extra dimensions and the D brains... and these D brains are supposed to have collided in the period before the Big Bang and there they come together and produced our Big Bang... and that expands... [T]he trouble... is a strong element of fantasy. We really haven't the remotest idea... what kind of physics is supposed to go on here, but there's a more serious problem... [T]his... has different forms, one... is... in terms of the 2nd law of thermodynamics... and it's related to a geometrical issue... [T]hese pictures are hard to draw.... because the singularity in the black hole doesn't really fit on the Big Bang singularity... It's a stretch of geometrical imagination... [I]t doesn't make them wrong, because... you really do need some fantasy, and this is an example of this possible kind of fantasy that you might need, but I want to give you a different kind which... has some greater plausibility..."

"Somewhat more exotic is the idea... by Lee Smolin in his book... [T]hese pictures are a little hard to draw... The difficulty seems... a... drawback. It may mean... something... troublesome about the geometry. ...[W]e have black holes forming ...You must imagine each one of these forming ...take this funnel ...that's supposed to represent the universe ...which expands from the Big Bang and ...its expansion accelerates because of ... or, if you're more boring like me, the cosmological constant ...and according to Smolin, all these black holes, which form at various places, could be the origins of new universes, and you see them sprouting off at various places... [Y]ou can adopt the Wheeler idea of maybe having the constants of nature changing to reach one of these phases."

"[T]here's a version of this a version of this idea which John Wheeler has promoted, which is that in each of these cycles, since nobody really knows what goes on at the crunch, bang stage... you can... invent any physics you like, and one idea... is to suggest that the... fundamental constants of nature might get changed every time you go through one of these cycles... [T]his might help to explain... puzzles that... the constants have to be just such and such in order that life should exist...[etc.] I always have trouble with many of these arguments. It's not at all clear whether you need them or not. They might be true but we don't know. It may be that these numbers are fixed and they might change through each cycle...[etc.] but our current physics... doesn't allow this kind of thing. These are singular states according to classical theory. Maybe if we had quantum gravity... one could imagine such a scheme..."

"I'm not sure what Friedmann actually said, but he... produced a model in which the universe... started in a Big Bang... expanded to a maximum size... then would shrink down to a crunch, and then start all over again. ...There would be several Big Bangs and before each one, would be a collapsing phase of the universe..."