Mathematicians From England

"Yet almost all the Astronomers differ'd from this Opinion of Seneca; neither did Seneca himself think fit to set down those Phænomena of the Motion, by which he was enabled to maintain his Opinion: Nor the Times of those Appearances, which might be of use to Posterity, in order to the Determining these Things. And indeed, upon the Turning over very many Histories of Comets, I find nothing at all that can be of Service in this Affair, before, A.D. 1337, at which time ', a Constantinopolitan Historian and Astronomer, did pretty accurately describe the Path of a Comet amongst the Fix'd Stars, but was too laxe as to the Account of the Time; so that this most doubtful and uncertain Comet, only deserves to be inserted in our Catalogue, for the sake of its appearing near 400 Years ago."

"But in the Year 1577, (Ticho seriously pursuing the Study of the Stars, and having gotten large Instruments for the Performing Cœlestial Mensurations, with far greater Care and Certainty, than the Ancients cou'd ever hope for) there appear'd a very remarkable Comet; to the Observation of which, Ticho vigorously applied himself; and found by many just and faithful Trials, that it had not a Diurnal Parallax that was at all perceptible: And consequently was not only no Aireal Vapour, but also much higher than the Moon; nay, might be plac'd amongst the Planets for any thing that appear'd to the Contrary; the cavilling Opposition made by some of the School-men in the mean time, being to no Purpose."

"Wherefore, if, according to what we have already said, it should return again about the year 1758, candid posterity will not refuse to acknowledge that this was first discovered by an Englishman."

"So that 'tis to the Greeks themselves as the Inventors (and especially to the Great Hipparchus) that we owe this Astronomy, which is now improv'd to such a Heigth. But yet, amongst these, the Opinion of Aristotle (who wou'd have Comets to be nothing else, but Sublunary Vapours, or Airy Meteors) prevailed so far, that this most difficult Part of the Astronomical Science lay altogether neglected; for no Body thought it worth while to take Notice of, or write about, the Wandring uncertain Motions of what they esteemed Vapours floating in the Æther; whence it came to pass, that nothing certain, concerning the Motion of Comets, can be found transmitted from them to us."

"Next to Ticho, came the Sagacious Kepler. He having the Advantage of Tichos Labours and Observations, found out the true Physical System of the World, and vastly improv'd the Astronomical Science. For he demonstrated that all the Planets perform their Revolutions in Elliptick Orbits, whose 'Plains pass thro' the Center of the Sun, observing this Law, That the Area's (of the Elliptick Sectors, taken at the Center of the Sun, which he proved to be in the common Focus of these Ellipses) are always proportional to the Times, in which the correspendent Elliptical Arches are describ'd. He discover'd also, That the Distances of the Planets from the Sun are in the Ratio [3:2] of the Periodical Times, or (which is all one) That the Cubes of the Distances are as the Squares of the Times. This great Astronomer had the Opportunity of observing Two Comets, one of which was a very remarkable one. And from the Observations of these (which afforded sufficient Indications of an Annual Parallax) he concluded, That the Comets mov'd freely thro' the Planetary Orbs, with a Motion not much different from a Rectilinear one; but of what Kind, he cou'd not then precisely determine."

"But Seneca the Philosopher, having consider'd the Phænomena of Two remarkable Comets of his Time, made no Scruple to place them amongst the Cœlestial Bodies; believing them to be Stars of equal Duration with the World, tho' he owns their Motions to be govern'd by Laws not as then known or found out. And at last (which was no untrue or vain Prediction) he foretells, that there should be Ages sometime hereafter, to whom Time and Diligence shou'd unfold all these Mysteries, and who shou'd wonder that the Ancients cou'd be ignorant of them, after some lucky Interpreter of Nature had shewn, in what Parts of the Heavens the Comets wander'd, and how great they were."

"Much of Dr. Whitrow's work is concerned with problems in cosmology and relativity. ...He has been editor of The Observatory Magazine and the Monthly Notices of the Royal Astronomical Society."

"Perhaps the most important of Whitrow's books was The Natural Philosophy of Time. He showed that time can be studied independently of its magnitude. ...Whitrow's historical work included a paper on Robert Hooke."

"The shift toward a linear time conception, a confutation against (instead of a development from) the age-old cyclic time conception, did not occur suddenly and lasted well into the nineteenth century. ...Attention had clearly drifted away from seeking an eternally valid order toward a focus on change; truth had now ceased to lie in an unchanging order of things—rather, it tended to be regarded as dependent on process. Gerald Whitrow has put the matter succinctly, that in the nineteenth century " interest was transferred from the 'thing completed' to the genetic process, that is, from 'being' to 'becoming.' ""

"By combining prodigious scholarship from the ancient Greeks to modern physicists, he argued persuasively in more than 100 academic papers and a string of books that an integrated, interdisciplinary understanding of time should be possible."

"Next, Hevelius (a Noble Emulator of Ticho Brahe) following in Keplers Steps, embraced the same Hypothesis of the Rectilinear Motion of Comets, himself accurately observing many of them. Yet, he complain'd, that his Calculations did not perfectly agree to the Matter of Fact in the Heavens: And was aware, that the Path of a Comet was bent into a Curve Line towards the Sun."

"Professor Gerald Whitrow, who has died aged 87, wrote The Natural Philosophy of Time (1960) a tour de force that examined the subject from every side—mathematical, cosmological, historical, biological and psychological. ...As an opening for his talks, he would sometimes recount one of his favourite stories on time. It concerned the Russian poet Samuel Marshak, on a visit to London before 1914. Marshak's English was not too good and when he asked a man in the street "Please, what is time?", he received the surprised response, "But that's a big question. Why ask me?""

"As the degree of observational accuracy at which general relativity becomes significantly different from Newtonian theory is far from being achieved in this field, and as stellar velocities are small compared with light, there is no sign yet that any non-Newtonian theory is required."

"Einstein's pioneer application in 1917 of his newly developed general relativity to the problem of world-structure ushered in a new phase in the theoretical approach to the subject. Then, some seven years later, Hubble's discovery of Cepheid variables in the Andromeda nebula finally settled the long-debated question concerning this and similar nebulae in the Milky Way."

"The models of Einstein and de Sitter are static solutions of Einstein's modified gravitational equations for a world-wide homogeneous system. They both involve a positive cosmological constant λ, determining the curvature of space. If this constant is zero, we obtain a third model in classical infinite Euclidean space. This model is empty, the space-time being that of Special Relativity. It has been shown that these are the only possible static world models based on Einstein's theory. In 1922, Friedmann... broke new ground by investigating non-static solutions to Einstein's field equations, in which the radius of curvature of space varies with time. This Possibility had already been envisaged, in a general sense, by Clifford in the eighties."

"By the time of Comte, scientists unanimously rejected the idea that there was any essential difference between celestial and terrestrial matter, but they still had no empirical evidence to support their view any more than had Aristotle to support his, and to the positivist philosopher it seemed that none could ever be obtained. ...The possibility of a solution to this problem appeared shortly after Comte's pronouncement with the rise of the science of astronomical spectroscopy..."

"From a careful determination of the amount of solar heat that which would fall per minute on an area of one square centimetre placed perpendicular to the radiation as it falls on Earth's surface and from a knowledge of the Earth's distance, we deduce that each square centimetre of the solar surface radiates on the average of about the rate of a nine horse-power engine."

"Another interesting feature of the Einstein universe is that in principle it could be circumnavigated by a ray of light... it is unlikely that the rays would converge with sufficient accuracy. Nevertheless it is interesting to consider the possibility that some of the stars and nebulae which we see may after all be only optical ghosts."

"Although the Special Theory of Relativity does not account for electromagnetic phenomena, it explains many of their properties. General Relativity, however, tells us nothing about electromagnetism. In Einstein's space-time continuum gravitational forces are absorbed in the geometry, but the electromagnetic forces are quite unaffected. Various attempts have been made to generate the geometry of space-time so as to produce a unified field theory incorporating both gravitational and electromagnetic forces."

"The philosophical consequences of the General Theory of Relativity are perhaps more striking than the experimental tests. As Bishop Barnes has reminded us, "The astonishing thing about Einstein's equations is that they appear to have come out of nothing." We have assumed that the laws of nature must be capable of expression in a form which is invariant for all possible transformations of the space-time co-ordinates and also that the geometry of space-time is Riemannian. From this exiguous basis, formulae of gravitation more accurate than those of Newton have been derived. As Barnes points out..."

"Space-time is curved in the neighborhood of material masses, but it is not clear whether the presence of matter causes the curvature of space-time or whether this curvature is itself responsible for the existence of matter."

"The solution... was found only after the rise of nuclear physics, and, strange to relate, was not known to Eddington when he developed his celebrated theory of stellar structure between 1916 and 1924. Indeed, it is one of the most intriguing facts in the history of science that the two most influential theories concerning the stars—Newton's theory of gravitation and Eddington's theory of stellar construction—were each developed so successfully although Newton was ignorant of the origin of gravitation and Eddington of the origin of stellar energy."

"[Time is not] a mysterious illusion of the intellect. ..It is an essential feature of the universe."

"At length, came that prodigious Comet of the Year 1680, which descending (as it were) from an infinite Distance Perpendicularly towards the Sun, arose from him again with as great a Velocity. This Comet, (which was Seen for Four Months continually) by the very remarkable and peculiar Curvity of its Orbit (above all others) gave the fittest Occasion for investigating the Theory of the Motion. And the Royal Observatories at Paris and Greenwich having been for some time founded, and committed to the Care of most excellent Astronomers, the apparent Motion of this Comet was most accurately (perhaps as far as Humane Skill cou'd go) observ'd by Mrs. Cassini and Flamsteed."

"Not until the pioneer work of Rutherford and his colleagues was the possibility of nuclear reactions and transformations as sources of stellar energy envisaged."

"Newton's laws of motion and gravitation achieved their original success when applied to the solar system. The first definite evidence that they were applicable on a larger scale came from the study of binary stars towards the eighteenth century. In recent times the limitations of Newton's theory have become apparent. Even on the scale of the solar system, it has been challenged by Einstein's."

"According to the Special Theory of Relativity, the velocity of a moving body is always less than the velocity of light. Since the energy of motion of a body depends on its inertial mass and its velocity, it follows that if the energy of a body is increased indefinitely by the continual application of a force, the inertial mass of the body must be increased too; for, if not, the velocity would ultimately increase indefinitely and exceed the velocity of light. Einstein found that, corresponding to any increase in the energy content of a body, there is an equivalent increase in its inertial mass. Mass and energy thus appeared to be different names for the same thing, the energy associated with a mass M being Mc2, where c is the velocity of light; and the mass M of a body moving with velocity v he found to be given by the following formulaM = \frac {m}{\sqrt{(1 - \frac {v^2}{c^2}}}"

"Cosmology is peculiar among the sciences for it is both the oldest and the youngest. From the dawn of civilization man has speculated about the nature of the starry heavens and the origin of the world, but only in the present century has physical cosmology split away from general philosophy to become an independent discipline."

"The point at issue between the two theories [A and B theory] is whether 'time' really is, in some deep ontological sense, differentiated into past, present and future. ...Reichenbach and Whitrow propose that there is indeed such a type of event and this is the 'becoming', or 'coming into being' of factual states-of-affairs in the physical world. ...Whitrow expressed ..."The past is the determined, the present is the moment of 'becoming' when events become determined, and the future is as-yet undetermined. Although neither Reichenbach nor Whitrow developed their thesis at any length, the general purport of what they meant is clear: there is a basic chance element in nature, at least at the micro-level, and the moment of 'becoming', which they identify with 'the present', is marked by a tranisition from what is merely possible to what is factual. However... this important attempt to provide a physical basis for the A-theory is by no means immune from criticism."

"Most of the illustrations in this book were scanned... from the volumes in which they originally appeared. One of these deserves special mention. It is... from the library of the late Gerald James Whitrow... I was honoured to receive this small book as a gift in 2001 from Professor Whitrow's widow, Magda... I wished to include a copy of this historic image in my own book both as a tribute to Pofessor Whitrow's memory and to express my sincere gratitude to all responsible for the gift."

"While at Oxford, he was much influenced by cosmologist, E. A. Milne. ...Whitrow is... remembered for his very loud voice which could be heard 'from miles away.'"

"Whitrow... proposed an anthropic resolution of the venerable philosophical question Why physical space has three dimensions? (arguing that with a space of different dimensionality there would be no living being to pose the question) and, similarly to [Grigory Moiseevich] Idlis, alluded around 1955 to an anthropic explanation of the size of the observable universe. Anyway, he never published these last ideas, which were developed years later by Wheeler. The only reference to Whitrow’s argument that appeared in print during the 1950s seems to be that due to the philosopher of religion Eric Lionel Mascall, who attributed to the English’s mathematician thatit may be necessary for the universe to have the enormous size and complexity which modern astronomy has revealed, in order for the earth to be a possible habitation for living beings."

"One of the few authors to have explicitly connected the physical issue of the expansion of the universe with the philosophical topic of the metaphysical status of space is Gerald James Whitrow."

"Remarks on the concept of simultaneity may mislead the reader to believing that only modern physicists and philosophers recognize the crucial importance of this notion. ...this concept has occupied the attention of philosophers and scientists throughout the whole history of human thought and played an important role in the writings of such intellectual giants as Aristotle, St. Augustine, Leibniz, and Kant. It would be a serious mistake to associate the concept of simultaneity exclusively with philosophic or scientific reasoning. In fact it was at the level of prescientific apprehension, a fundamental ingredient in the process of human apperception and conception of time. As Gerald Whitrow rightly pointed out, "our conscious appreciation of the fact that one event follows another is of a different kind from our awareness of either event separately. If two events are to be represented as occurring in succession, then—paradoxically—they must also be thought of simultaneously.""

"Whitrow's stance... is probably the first attempt to introduce such a philosophical approach in modern cosmology... it could be the a stimulus for new insights and a better comprehension of the physical foundations of cosmology itself."

"Language itself inevitably introduced an element of permanence into the world. For, although speech itself is transitory, the conventionalized sound symbols of language transcended time."

"To obtain a greater degree of permanence the time symbols of oral speech had to be converted into the space symbols of written speech. ...The crucial stage in the evolution of writing occurred when ideographs became phonograms..."

"Our idea of the universe as a whole remains a product of the imagination."

"The development of rational thought actually seems to have impeded man's appreciation for the significance of time. ...Belief that the ultimate reality is timeless is deeply rooted in human thinking, and the origin of rational investigation of the world was the search for permanent factors that lie behind the ever-changing pattern of events."

"Man must have been conscious of memories and purposes long before he made any explicit distinction between past, present, and future."

"The famous palaeolithic paintings found in caves such as that at Lascaux in the Dordogne have been interpreted as evidence that, at least implicitly people were operating 20,000 or more years ago with teleological intent in terms of past, present, and future. It may well be that those responsible for the so-called 'Dancing Sorcerer' ...may have felt that the actual performance of the dance was insufficient, since they were concerned with the magical efficacy of the dance after it ended."

"The history of natural philosophy is characterized by the interplay of two rival philosophies of time — one aiming at its "elimination" and the other based on the belief that it is fundamental and irreducible."

"It must have required enormous effort for man to overcome his natural tendency to live like the animals in a continual present."

"It became clear that our Galaxy is only one system among many, and that the universe is far vaster than the particular stellar system to which the Sun and planets belong. Since then developments have been more rapid than at any time since the days of Copernicus, Digges and Bruno when the geocentric hypothesis of the cosmos received its death-blow."

"The basic objection to attempts to deduce the unidirectional nature of time from concepts such as entropy is that they are attempts to reduce a more fundamental concept to a less fundamental one."

"FIN [from finite speed] is the axiom... from relativity theory that information... can't travel faster than the speed of light."

"SPIN and TWIN are operationally definable... Do the operation that's called, measuring... as many times as you like, and see that they always give the same answers. That's what is meant by saying that those things are operationally definable..."

"SPIN... is a... curious axiom. If you take one of these particles and ask it what... it's squared component of spin is, in three... mutually perpendicular directions, it always happens that two of the answers are 1, and one of them is 0. That's most mysterious... and... it's not possible to solve this puzzle. ...[W]e have these 33 directions, and it's not possible to assign 0s and 1s to them, subject to that condition... the 1-0-1 rule. ...[T]he particle is acting somewhat like a little boy ...making up its mind as it goes along. It doesn't stop it from giving answers, but it does stop the answers from being determined ahead of time, and that's the guts of it."

"[T]he strangest contribution of quantum mechanics to this discussion is the EPR paradox. ...That's an essential contribution to our theorem too. ...Despite the fact that information can't be transmitted faster than the speed of light, ...remotely separated events can be correlated ...and this is the content of our TWIN axiom, you can put two particles into a... singleton state... the angular momentum of the pair of particles is zero... [B]y the conservation of angular momentum... if you measure the angular momentum of this in any direction, then for the angular momentum of the other you get the negative answer, but... we're going to square it, that means... the squared component of spin is the same... [T]hese particles have been sort of hypnotized. If you ask... they will give the same answer... like I and my twin brother... [T]he funny thing is, even though the proves that the answers do not exist ahead of time, the equality of the answers can exist..."

"[R]elativity is an important part of the game."