Mathematicians From England

"We have been pulling at the wrong end of the tangle, which has to be unravelled by a different approach. But after a general agreement with observation is established, and the tangle begins to loosen, we should always make ready for the next knot."

"There is no need for disappointment at the failure of the model to give perfect agreement with observation; it has served its purpose, for it has distinguished what are the features of the actual which require new conditions for their explanation."

"When the properties of an ideal model have been worked out by rigorous mathematics, all the underlying assumptions being clearly understood, then it becomes possible to say that such-and-such properties and laws lead precisely to such-and-such effects. If any other disregarded factors are present, they should now betray themselves when a comparison is made with Nature."

"If indeed, the subatomic energy is being freely used to maintain their great furnaces, it seems to bring a little nearer to fulfillment our dreams of controlling this latent power for the well being of the humans-or for its suicide."

"[T]he atoms of all elements are built of atoms bound together... the interior of a star seems as likely a place as any for the evolution to have occurred... a great amount of energy must have been set free; in a star a vast amount of energy is being set free which is hitherto unaccounted for. You may draw a conclusion if you like."

"I think that the suspicion has been generally entertained that the stars are the crucibles in which the lighter atoms which abound in the nebulæ are compounded into more complex elements. In the stars matter has its preliminary brewing to prepare the greater variety of elements which are needed for a world of life. The radio-active elements must have been formed at no very distant date; and their synthesis, unlike the generation of helium from hydrogen, is endothermic."

"What is the source of heat which the sun and the stars are continually squandering? The answer given is almost unanimous... the gravitational energy converted as the star steadily contracts. ...Lord Kelvin showed that this hypothesis, due to Helmholtz, necessarily dates the birth of the sun about 20,000,000 years ago; and he made strenuous efforts to induce geologists and biologists to accommodate... But... outrageous violations of this limit have prevailed. ...Sir 's theory of the earth-moon system, to the present Lord Raleigh's ...age of terrestrial rocks from occluded helium, and to all modern discussions of the statistical equilibrium of the stellar system. No one seems to have any hesitation... in carrying back the history of the earth long before the supposed date of formation of the solar system... in some cases... this appears... justified by experimental evidence... difficult to dispute. Lord Kelvin's date of the creation of the sun is treated with no more respect than Archbishop Ussher's."

"Probably the greatest need of stellar astronomy... in order to make sure that our theoretical deductions are starting on the right lines, is some means of measuring the apparent angular diameters of stars. At present we can calculate them approximately from theory, but there is no observational check. ...If the direct measurement ...could be made with any accuracy it would make a wonderfully rapid advance in our knowledge. The prospects... are now... hopeful. ...[W]ork is being carried out by interferometer methods with the 100-in. reflector at Mt. Wilson, and the results are promising. At present the method has been applied only to measuring the separation of close s... Although the great mirror is used for convenience, the interferometer method does not in principle require great apertures, but rather two small apertures widely separated, as in a range-finder."

"Our observational knowledge of the things here discussed is chiefly of a rather vague kind, and we can scarcely claim more than a general agreement of theory and observation. What we have been able to do in the way of tests is to offer the theory a considerable number of opportunities to "make a fool of itself," and so far it has not fallen into our traps. When the theory tells us that a star having the mass of the sun will at one stage in its career reach a maximum effective temperature of 9000° (the sun's effective temperature [of the ] being 6000°) we cannot do much in the way of checking it; but an erroneous theory might well have said that the maximum temperature was 20,000° (hotter than any known star), in which case we should have detected its error. If we cannot feel confident that the answers of the theory are true, it must be admitted that it has shown some discretion in lying without being found out."

"On the observational side we have fairly satisfactory knowledge of the masses and densities of the stars and of the total radiation emitted by them; this knowledge is partly individual and partly statistical. The theoretical analysis connects these observational data... with the physical properties of the material inside the star... We can thus find certain information as to the inner material, as though we had actually bored a hole. ...[W]e depend entirely on the well-tried principle of conservation of momentum and the second law of thermodynamics. If any element of speculation remains ...it is no more than is inseparable from every kind of theoretical advance."

"[T]heoretical researches of Einstein and Weyl make it probable that the space which remains beyond is not illimitable; not merely the material universe, but also space itself, is perhaps finite..."

"We used to think that if we knew one, we knew two, because one and one are two. We are finding that we must learn a great deal more about "and"."

"I am continually trying to find out why people find the procedure obscure. But I would point out that even Einstein was considered obscure, and hundreds of people have thought it necessary to explain him. I cannot seriously believe that I ever attain the obscurity that Dirac does. But in the case of Einstein and Dirac people have thought it worthwhile to penetrate the obscurity. I believe they will understand me all right when they realize they have got to do so--and when it becomes the fashion "to explain Eddington"."

"Observation and theory get on best when they are mixed together, both helping one another in the pursuit of truth. It is a good rule not to put overmuch confidence in a theory until it has been confirmed by observation. I hope I shall not shock the experimental physicists too much if I add that it is also a good rule not to put overmuch confidence in the observational results that are put forward until they have been confirmed by theory."

"It is impossible to trap modern physics into predicting anything with perfect determinism because it deals with probabilities from the outset."

"There once was a brainy baboon, Who always breathed down a bassoon, For he said, "It appears That in billions of years I shall certainly hit on a tune"."

"To the pure geometer the radius of curvature is an incidental characteristic — like the grin of the Cheshire cat. To the physicist it is an indispensable characteristic. It would be going too far to say that to the physicist the cat is merely incidental to the grin. Physics is concerned with interrelatedness such as the interrelatedness of cats and grins. In this case the "cat without a grin" and the "grin without a cat" are equally set aside as purely mathematical phantasies."

"I think that science would never have achieved much progress if it had always imagined unknown obstacles hidden round every corner. At least we may peer gingerly round the corner, and perhaps we shall find there is nothing very formidable after all."

"A star is drawing on some vast reservoir of energy by means unknown to us. This reservoir can scarcely be other than the sub-atomic energy which, it is known, exists abundantly in all matter; we sometimes dream that man will one day learn to release it and use it for his service. The store is well-nigh inexhaustible, if only it could be tapped."

"We do not argue with the critic who urges that the stars are not hot enough for this process; we tell him to go and find a hotter place."

"At terrestrial temperatures matter has complex properties which are likely to prove most difficult to unravel; but it is reasonable to hope that in the not too distant future we shall be competent to understand so simple a thing as a star."

"The present revolution of scientific thought follows in natural sequence on the great revolutions at earlier epochs in the history of science. Einstein's special theory of relativity, which explains the indeterminateness of the frame of space and time, crowns the work of Copernicus who first led us to give up our insistence on a geocentric outlook on nature; Einstein's general theory of relativity, which reveals the curvature or non-Euclidean geometry of space and time, carries forward the rudimentary thought of those earlier astronomers who first contemplated the possibility that their existence lay on something which was not flat. These earlier revolutions are still a source of perplexity in childhood, which we soon outgrow; and a time will come when Einstein's amazing revelations have likewise sunk into the commonplaces of educated thought."

"Physics has in the main contented itself with studying the abridged edition of the book of nature."

"Our environment may and should mean something towards us which is not to be measured with the tools of the physicist or described by the metrical symbols of the mathematician."

"Our system of philosophy is itself on trial; it must stand or fall according as it is broad enough to find room for this experience as an element of life."

"Consciousness is not wholly, nor even primarily a device for receiving sense-impressions. … there is another outlook than the scientific one, because in practice a more transcendental outlook is almost universally admitted. … who does not prize these moments that reveal to us the poetry of existence?"

"The scientific answer is relevant so far as concerns the sense-impressions … For the rest the human spirit must turn to the unseen world to which it itself belongs."

"Study of the scientific world cannot prescribe the orientation of something which is excluded from the scientific world."

"The desire for truth so prominent in the quest of science, a reaching out of the spirit from its isolation to something beyond, a response to beauty in nature and art, an Inner Light of conviction and guidance — are these as much a part of our being as our sensitivity to sense impressions?"

"As truly as the mystic, the scientist is following a light; and it is not a false or an inferior light."

"The problem of experiences is not limited to the interpretation of sense-impressions."

"Our story of evolution ended with a stirring in the brain-organ of the latest of Nature's experiments; but that stirring of consciousness transmutes the whole story and gives meaning to its symbolism. Symbolically it is the end, but looking behind the symbolism it is the beginning."

"It remains a real world if there is a background to the symbols — an unknown quantity which the mathematical symbol x stands for. We think we are not wholly cut off from this background. It is to this background that our own personality and consciousness belong, and those spiritual aspects of our nature not to be described by any symbolism … to which mathematical physics has hitherto restricted itself."

"Mind is the first and most direct thing in our experience; all else is remote inference."

"Physics most strongly insists that its methods do not penetrate behind the symbolism."

"The chairs and tables around us which broadcast to us incessantly those signals which affect our sight and touch cannot in their nature be like unto the signals or to the sensations which the signals awake at the end of their journey."

"It is an astonishing feat of deciphering that we should have been able to infer an orderly scheme of natural knowledge from such indirect communication."

"Matter and all else that is in the physical world have been reduced to a shadowy symbolism."

"We have travelled far from the standpoint which identifies the real with the concrete. Even the older philosophy found it necessary to admit exceptions; for example, time must be admitted to be real, although no one could attribute to it a concrete nature."

"We are no longer tempted to condemn the spiritual aspects of our nature as illusory because of their lack of concreteness."

"If to-day you ask a physicist what he has finally made out the æther or the electron to be, the answer will not be a description in terms of billiard balls or fly-wheels or anything concrete; he will point instead to a number of symbols and a set of mathematical equations which they satisfy. What do the symbols stand for? The mysterious reply is given that physics is indifferent to that; it has no means of probing beneath the symbolism. To understand the phenomena of the physical world it is necessary to know the equations which the symbols obey but not the nature of that which is being symbolised. … this newer outlook has modified the challenge from the material to the spiritual world."

"Wind, earthquake, fire — meteorology, seismology, physics — pass in review, as we have been reviewing the natural forces of evolution; the Lord was not in them. Afterwards, a stirring, an awakening in the organ of the brain, a voice which asks "What doest thou here?""

"There is another passage from the Old Testament that comes nearer to my own sympathies — "And behold the Lord passed by, and a great and strong wind rent the mountains, and brake in pieces the rocks before the Lord; but the Lord was not in the wind: and after the wind an earthquake; but the Lord was not in the earthquake: and after the earthquake a fire; but the Lord was not in the fire: and after the fire a still small voice. … And behold there came a voice unto him, and said. What doest thou here, Elijah?""

"I think it is not irreligion but a tidiness of mind, which rebels against the idea of permeating scientific research with a religious implication."

"Proof is the idol before whom the pure mathematician tortures himself."

"It is difficult for the matter-of-fact physicist to accept the view that the substratum of everything is of mental character. But no one can deny that mind is the first and most direct thing in our experience, and all else is remote inference — inference either intuitive or deliberate."

"Consciousness is not sharply defined, but fades into sub-consciousness; and beyond that we must postulate something indefinite but yet continuous with our mental nature. This I take it be the world-stuff."

"The mind-stuff is not spread in space and time. But we must presume that in some other way or aspect it can be differentiated into parts. Only here and there does it arise to the level of consciousness, but from such islands proceeds all knowledge. The latter includes our knowledge of the physical world."

"The universe is of the nature of a thought or sensation in a universal Mind … To put the conclusion crudely — the stuff of the world is mind-stuff. As is often the way with crude statements, I shall have to explain that by "mind" I do not exactly mean mind and by "stuff" I do not at all mean stuff. Still that is about as near as we can get to the idea in a simple phrase. The mind-stuff of the world is something more general than our individual conscious minds; but we may think of its nature as not altogether foreign to feelings in our consciousness … Having granted this, the mental activity of the part of world constituting ourselves occasions no great surprise; it is known to us by direct self-knowledge, and we do not explain it away as something other than we know it to be — or rather, it knows itself to be."

"The whole subject-matter of exact science consists of pointer readings and similar indications."