Mathematicians From England

"Objections to religious mysticism lose their force if they can equally be turned against natural mysticism."

"Not only is the universe stranger than we imagine, it is stranger than we can imagine."

"With regard to the Newtonian concept of absolute rotation, Eddington admitted that Einstein's plenum does in fact provide a world-wide inertial frame, with respect to which it can be measured. Nevertheless, Eddington believed that Einstein attributed too important a role to matter, for in his universe it appears that not only the metrical properties, as in General Relativity, but the very existence of space depends on the existence of matter. Eddington preferred to regard matter as a manifestation of the "structure" of space-time."

"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."

"The best presentation of the general theory [of relativity] is still Eddington's book of 1923, The Mathematical Theory of Relativity. … The mathematical foundation, the calculus of tensors, is given very completely in Eddington's book. For an exhaustive treatment see: Levi-Cevita, The Absolute Differential Calculus, translated by Dr. E. Perisco (1927)."

"Our sense experience has to be supplemented by all kinds of images and formulas to tell us much about the outer world, just as we learn something about the nonphysical one from mythical images tested out in experience. Arthur Eddington has discussed at length the strange way that man receives his knowledge of the physical world in Science and the Unseen World. Mythological thinking is not as strange as some people think once they begin to see the total universe as it is."

"The equations of gravitation … signify that whenever we recognise the existence of one of these physical magnitudes it is always accompanied by corresponding curvatures of space-time. It is usual to assume that the curvatures are produced by those concrete somethings which we call mass, momentum, energy, pressure. In this way, we must concede a duality to nature; there would exist both matter and space-time, or, better still, matter and the metrical field of space-time. Einstein … attempted to remove this duality by proving that it was possible to attribute the entire existence of the metrical field, hence of space-time, to the presence of matter. This attitude led to a matter-moulding conception of the universe … And … only when this attitude was adhered to could Mach's belief in the relativity of all motion be accepted. Eddington's attitude is just the reverse. He prefers to assume that the equations of gravitation are not equations in the ordinary sense of something being equal to something else. In his opinion they are identities. They merely tell us how our senses will recognize the existence of certain curvatures of space-time by interpreting them as matter, motion, and so on. In other words, there is no matter; there is nothing but a variable curvature of space-time. Matter, momentum, vis viva, are the names we give to those curvatures on account of the varying ways they affect our senses."

"One of the foremost scientists, not only of the 20th Century, but of the entire history of the human species. His achievements include, but are not limited to, the integration of modern astrophysics, major advances in the early history of relativistic cosmology, major additions to modern astronomical advancements, a superior exposition of the philosophy of physical science, and authorship of many books and articles making science more meaningful, not only to scientists, but to all rational and curious individuals. His achievements will radiate for the duration of the human species!"

"The mathematics is not there till we put it there."

"Let us suppose that an ichthyologist is exploring the life of the ocean. He casts a net into the water and brings up a fishy assortment. Surveying his catch, he proceeds in the usual manner of a scientist to systematise what it reveals. He arrives at two generalisations: No sea-creature is less than two inches long. (2) All sea-creatures have gills. These are both true of his catch, and he assumes tentatively that they will remain true however often he repeats it. In applying this analogy, the catch stands for the body of knowledge which constitutes physical science, and the net for the sensory and intellectual equipment which we use in obtaining it. The casting of the net corresponds to observation; for knowledge which has not been or could not be obtained by observation is not admitted into physical science. An onlooker may object that the first generalisation is wrong. "There are plenty of sea-creatures under two inches long, only your net is not adapted to catch them." The icthyologist dismisses this objection contemptuously. "Anything uncatchable by my net is ipso facto outside the scope of icthyological knowledge. In short, what my net can't catch isn't fish." Or — to translate the analogy — "If you are not simply guessing, you are claiming a knowledge of the physical universe discovered in some other way than by the methods of physical science, and admittedly unverifiable by such methods. You are a metaphysician. Bah!""

"For the truth of the conclusions of physical science, observation is the supreme Court of Appeal. It does not follow that every item which we confidently accept as physical knowledge has actually been certified by the Court; our confidence is that it would be certified by the Court if it were submitted. But it does follow that every item of physical knowledge is of a form which might be submitted to the Court. It must be such that we can specify (although it may be impracticable to carry out) an observational procedure which would decide whether it is true or not. Clearly a statement cannot be tested by observation unless it is an assertion about the results of observation. Every item of physical knowledge must therefore be an assertion of what has been or would be the result of carrying out a specified observational procedure."

"All change is relative. The universe is expanding relatively to our common material standards; our material standards are shrinking relatively to the size of the universe. The theory of the "expanding universe" might also be called the theory of the "shrinking atom". […] Let us then take the whole universe as our standard of constancy, and adopt the view of a cosmic being whose body is composed of intergalactic spaces and swells as they swell. Or rather we must now say it keeps the same size, for he will not admit that it is he who has changed. Watching us for a few thousand million years, he sees us shrinking; atoms, animals, planets, even the galaxies, all shrink alike; only the intergalactic spaces remain the same. The earth spirals round the sun in an ever‑decreasing orbit. It would be absurd to treat its changing revolution as a constant unit of time. The cosmic being will naturally relate his units of length and time so that the velocity of light remains constant. Our years will then decrease in geometrical progression in the cosmic scale of time. On that scale man's life is becoming briefer; his threescore years and ten are an ever‑decreasing allowance. Owing to the property of geometrical progressions an infinite number of our years will add up to a finite cosmic time; so that what we should call the end of eternity is an ordinary finite date in the cosmic calendar. But on that date the universe has expanded to infinity in our reckoning, and we have shrunk to nothing in the reckoning of the cosmic being. We walk the stage of life, performers of a drama for the benefit of the cosmic spectator. As the scenes proceed he notices that the actors are growing smaller and the action quicker. When the last act opens the curtain rises on midget actors rushing through their parts at frantic speed. Smaller and smaller. Faster and faster. One last microscopic blurr of intense agitation. And then nothing."

"Accordingly the primordial state of things which I picture is an even distribution of protons and electrons, extremely diffuse and filling all (spherical) space, remaining nearly balanced for an exceedingly long time until its inherent instability prevails. We shall see later that the density of this distribution can be calculated; it was about one proton and electron per litre. There is no hurry for anything to begin to happen. But at last small irregular tendencies accumulate, and evolution gets under way. The first stage is the formation of condensations ultimately to become the galaxies; this, as we have seen, started off an expansion, which then automatically increased in speed until it is now manifested to us in the recession of the spiral nebulae."

"If our so-called facts are changing shadows, they are shadows cast by the light of constant truth. So too in religion we are repelled by that confident theological doctrine … but we need not turn aside from the measure of light that comes into our experience showing us a Way through the unseen world."

"You will understand the true spirit neither of science nor of religion unless seeking is placed in the forefront."

"It is perhaps difficult sufficiently to emphasise Seeking without disparaging its correlative Finding. But I must risk this, for Finding has a clamorous voice that proclaims its own importance; it is definite and assured, something that we can take hold of — that is what we all want, or think we want. Yet how transitory it proves. The finding of one generation will not serve for the next. It tarnishes rapidly except it be reserved with an ever-renewed spirit of seeking."

"If the kind of controversy which so often springs up between modernism and traditionalism in religion were applied to more commonplace affairs of life we might see some strange results. … It arises, let us say, from a passage in an obituary notice which mentions that the deceased had loved to watch the sunsets from his peaceful country home.. … it is forgotten that what the deceased man looked out for each evening was an experience and not a creed."

"Science has an important part to play in our everyday existence, and there is far too much neglect of science; but its intention is to supplement not to supplant the familiar outlook."

"We have to build the spiritual world out of symbols taken from our own personality, as we build the scientific world out of the symbols of the mathematician."

"If in a community of the blind one man suddenly received the gift of sight, he would have much to tell which would not be at all scientific."

"Physical science comes nearest to that complete system of exact knowledge which all sciences have before them as an ideal. Some fall far short of it. The physicist who inveighs against the lack of coherence and the indefiniteness of theological theories, will probably speak not much less harshly of the theories of biology and psychology. They also fail to come up to his standard of methodology. On the other side of him stands an even superior being — the pure mathematician — who has no high opinion of the methods of deduction used in physics, and does not hide his disapproval of the laxity of what is accepted as proof in physical science. And yet somehow knowledge grows in all these branches. Wherever a way opens we are impelled to seek by the only methods that can be devised for that particular opening, not over-rating the security of our finding, but conscious that in this activity of mind we are obeying the light that is in our nature."

"The exploration of the external world by the methods of physical science leads not to a concrete reality but to a shadow world of symbols, beneath which those methods are unadapted for penetrating."

"Theological or anti-theological argument to prove or disprove the existence of a deity seems to me to occupy itself largely with skating among the difficulties caused by our making a fetish of this word [existence]."

"Ninety-nine people out of a hundred have not seriously considered what they mean by the term "exist" nor how a thing qualifies itself to be labelled real."

"We do not want a religion that deceives us for our own good."

"If God is as real as the shadow of the Great War on Armistice Day, need we seek further reason for making a place for God in our thoughts and lives? We shall not be concerned if the scientific explorer reports that he is perfectly satisfied that he has got to the bottom of things without having come across either."

"However closely we may associate thought with the physical machinery of the brain, the connection is dropped as irrelevant as soon as we consider the fundamental property of thought — that it may be correct or incorrect. … that involves recognising a domain of the other type of law — laws which ought to be kept, but may be broken."

"The mind has an outlook which transcends the natural law by which it functions."

"The laws of logic do not prescribe the way our minds think; they prescribe the way our minds ought to think."

"Even if religion and morality are dismissed as illusion, the word "Ought" still has sway."

"The essential difference, which we meet in entering the realm of spirit and mind, seems to hang round the word "Ought.""

"To those who have any intimate acquaintance with the laws of chemistry and physics the suggestion that the spiritual world could be ruled by laws of allied character is as preposterous as the suggestion that a nation could be ruled by laws like the laws of grammar."

"Natural law is not applicable to the unseen world behind the symbols, because it is unadapted to anything except symbols, and its perfection is a perfection of symbolic linkage. You cannot apply such a scheme to the parts of our personality which are not measurable by symbols any more than you can extract the square root of a sonnet."

"Materialism in its literal sense is long since dead. … It is … belief in the universal dominance of scientific law which is nowadays generally meant by materialism."

"What I may attempt is to dispel the feeling that in using the eye of the body or the eye of the soul, and incorporating what is thereby revealed in our conception of reality, we are doing something irrational and disobeying the leading of truth which as scientists we are pledged to serve."

"We should not argue with the blind man who maintained that sight was an illusion to which some abnormal people were subject. Therefore in speaking of religious experience I do not attempt to prove the existence of religious experience …"

"We have found a strange foot-print on the shores of the unknown. We have devised profound theories, one after another, to account for its origins. At last, we have succeeded in reconstructing the creature that made the footprint. And lo! It is our own."

"We can see that, the constant in the law of gravitation being fixed, there may be some upper limit to the amount of matter possible; as more and more matter is added in the distant parts, space curves round and ultimately closes; the process of adding more matter must stop, because there is no more space, and we can only return to the region already dealt with. But there seems nothing to prevent a defect of matter, leaving space unclosed. Some mechanism seems to be needed, whereby either gravitation creates matter, or all the matter in the universe conspires to define a law of gravitation."

"Spherical space is not very easy to imagine. We have to think of the properties of the surface of a sphere — the two-dimensional case — and try to conceive something similar applied to three-dimensional space. Stationing ourselves at a point let us draw a series of spheres of successively greater radii. The surface of a sphere of radius r should be proportional to r2; but in spherical space the areas of the more distant spheres begin to fall below the proper proportion. There is not so much room out there as we expected to find. Ultimately we reach a sphere of biggest possible area, and beyond it the areas begin to decrease. The last sphere of all shrinks to a point — our antipodes. Is there nothing beyond this? Is there a kind of boundary there? There is nothing beyond and yet there is no boundary. On the earth's surface there is nothing beyond our own antipodes but there is no boundary there."

"It is of interest to inquire what happens when the aviator's speed … approximates to the velocity of light. Lengths in the direction of flight become smaller and smaller, until for the speed of light they shrink to zero. The aviator and the objects accompanying him shrink to two dimensions. We are saved the difficulty of imagining how the processes of life can go on in two dimensions, because nothing goes on. Time is arrested altogether. This is the description according to the terrestrial observer. The aviator himself detects nothing unusual; he does not perceive that he has stopped moving. He is merely waiting for the next instant to come before making the next movement; and the mere fact that time is arrested means that he does not perceive that the next instant is a long time coming."

"It is the reciprocity of these appearances — that each party should think the other has contracted — that is so difficult to realise. Here is a paradox beyond even the imagination of Dean Swift. Gulliver regarded the Lilliputians as a race of dwarfs; and the Lilliputians regarded Gulliver as a giant. That is natural. If the Lilliputians had appeared dwarfs to Gulliver, and Gulliver had appeared a dwarf to the Lilliputians — but no! that is too absurd for fiction, and is an idea only to be found in the sober pages of science. … It is not only in space but in time that these strange variations occur. If we observed the aviator carefully we should infer that he was unusually slow in his movements; and events in the conveyance moving with him would be similarly retarded — as though time had forgotten to go on. His cigar lasts twice as long as one of ours. … But here again reciprocity comes in, because in the aviator's opinion it is we who are travelling at 161,000 miles a second past him; and when he has made all allowances, he finds that it is we who are sluggish. Our cigar lasts twice as long as his."

"Phys. I know that it is often a help to represent pressure and volume as height and width on paper; and so geometry may have applications to the theory of gases. But is it not going rather far to say that geometry can deal directly with these things and is not necessarily concerned with lengths in space? Math. No. Geometry is nowadays largely analytical, so that in form as well as in effect, it deals with variables of an unknown nature. … It is literally true that I do not want to know the significance of the variables x, y, z, t that I am discussing. … Phys. Yours is a strange subject. You told us at the beginning that you are not concerned as to whether your propositions are true, and now you tell us you do not even care to know what you are talking about. Math. That is an excellent description of Pure Mathematics, which has already been given by an eminent mathematician [Bertrand Russell]."

"[Experimental Physicist] Phys. I cannot imagine myself perceiving non-Euclidean space! Math. Look at the reflection of the room in a polished doorknob, and imagine yourself one of the actors in what you see going on there."

"[Relativist] Rel. There is a well-known proposition of Euclid which states that "Any two sides of a triangle are together greater than the third side." Can either of you tell me whether nowadays there is good reason to believe that this proposition is true? [Pure Mathematician] Math. For my part, I am quite unable to say whether the proposition is true or not. I can deduce it by trustworthy reasoning from certain other propositions or axioms, which are supposed to be still more elementary. If these axioms are true, the proposition is true; if the axioms are not true, the proposition is not true universally. Whether the axioms are true or not I cannot say, and it is outside my province to consider."

"The writings of Einstein, Minkowski, Hilbert, Lorentz, Weyl, Robb, and others, have provided the groundwork; in the give and take of debate with friends and correspondents, the extensive ramifications have gradually appeared."

"A non-mathematical presentation has necessary limitations; and the reader who wishes to learn how certain exact results follow from Einstein's, or even Newton's, law of gravitation is bound to seek the reasons in a mathematical treatise. ...[T]he geometry of relativity in its perfect harmony expresses a truth... which my bowdlerised version misses. But the mind is not content to leave scientific Truth in a dry husk of mathematical symbols, and demands that it shall be alloyed with familiar images."

"By his theory of relativity Albert Einstein has provoked a revolution of thought in physical science. ...Physical space and time are found to be closely bound... with... motion of the observer; and only an amorphous combination of the two is left... It is my aim to give an account of this work without introducing anything very technical in... mathematics, physics, or philosophy. ...[T]he task is one of interpreting a clear-cut theory... although perhaps not everyone would accept the author's views of its meaning."

"[P]erhaps there is something to be said for . The classic authorities tell us that he was only "doing a stunt," but I... think of him as a man who... brought to light a constructional defect in the flying-machines of his day. So too, in science, cautious Dædalus will apply his theories where he feels most confident... but by... excess caution their hidden weaknesses cannot be brought to light. Icarus will strain his theories to the breaking-point until the weak joints gape. ...But if he is not yet destined to reach the sun and solve for all time the riddle of its constitution, yet he may hope to learn from his journey some hints to build a better machine."

"Our model of Nature should not be like a building... for the populace to admire... until the edifice comes tumbling down. It should be like an engine with movable parts. ...[A]ny one lever ...is to be adjusted from time to time as the latest observations indicate. The aim of the theorist is to know the train of wheels which the lever sets in motion—that binding of the parts which is the soul of the engine."

"I suppose that the applied mathematician whose theory had just passed one still more stringent test by observation ought not to feel satisfaction, but rather disappointment—"Foiled again! This time I had hoped to find a discordance which would throw light on the points where my model could be improved." ...I own that I have never felt very keenly a disappointment of this kind."