Unification in science and mathematics

“Whatever its source, mathematics has come down to the present by the two main streams of number and form. The first carried along arithmetic and algebra, the second, geometry. In the seventeenth century these two united, forming the ever-broadening river of mathematical analysis.”

“People are always asking for the latest developments in the unification of this theory with that theory, and they don't give us a chance to tell them anything about what we know pretty well. They always want to know the things we don't know.”

“The decisive steps toward a clear understanding of non-Euclidean geometry were taken by Riemann, Helmholtz, and Poincaré, who recognized the essential unity of geometry and physics. However, the understanding did not come into its own until Einstein showed that such a combination of geometry and physics was really necessary for the derivation of phenomena which had actually been observed.”

“All knowledge... is unification of the multiple.”

“Every kind of science, if it has only reached a certain degree of maturity, automatically becomes a part of mathematics.”

“Reduced to their most pregnant difference, empiricism means the habit of explaining wholes by parts, and rationalism means the habit of explaining parts by wholes. Rationalism thus preserves affinities with monism, since wholeness goes with union, while empiricism inclines to pluralistic views. No philosophy can ever be anything but a summary sketch, a picture of the world in abridgment, a foreshortened bird's-eye view of the perspective of events.”

“Klein showed that the Reimannian species of non-Euclidean geometry can be developed in a fashion completely analogous to the Lobachevskian type by choosing an "imaginary" absolute, that is, an "imaginary" point pair or conic, and an imaginary value of the constant k. Euclidean geometry can also be treated in the same way by choosing a "degenerate" point pair or conic.”

“As long as algebra and geometry travelled separate paths their advance was slow and their applications limited. But when these two sciences joined company, they drew from each other fresh vitality and thenceforward marched on at a rapid pace towards perfection. It is to Descartes that we owe the application of algebra to geometry,—an application which has furnished the key to the greatest discoveries in all branches of mathematics.”

“While Descartes' theory of vortices was spectacularly wrong (as Newton ruthlessly pointed out later), it was still interesting, being the first serious attempt to formulate a theory of the universe as a whole based upon the same laws that apply on the Earth's surface. In other words, to Descartes there was no difference between terrestrial and celestial phenomena—the Earth was part of a universe that obeyed uniform physical laws.”

“Nature does not begin with elements, as we are obliged to begin with them. It is certainly fortunate... that we can... turn aside our eyes from the over powering unity of the All, and allow them to rest on individual details. But we should not omit, ultimately, to complete and correct our views by a thorough consideration of the things which for the time being we left out of account.”

“Velocity of transverse undulations in our hypothetical medium, calculated from the electromagnetic experiments of 'MM'. Kohlrausch and Weber, agrees so exactly with the velocity of light calculated from the optical experiments of M. Fizeau, that we can scarcely avoid the conclusion that light consists in the transverse undulations of the same medium which is the cause of electric and magnetic phenomena.”

“The agreement of the results seems to show that light and magnetism are affections of the same substance, and that light is an electromagnetic disturbance propagated through the field according to electromagnetic laws.”

“The object of the book is philosophical, in the sense now accepted by many and by divergent schools—i.e., it desires to contribute something towards a unification of thought.”

“In whatever they focus on, physicists seek the simplicity in complexity and the unity in diversity. Like philosophers, their intellectual siblings, they are driven by the conviction that the universe is within the human power to understand and that if you look beneath its variety and intricacy, you will find comprehensible rules.”

“In early physical systems we have optics dealing with phenomena perceived by the eye; acoustics treating of auditory percepts, and so on. The subjective concepts of "tone" and "colour" have now been replaced by the objectified concepts of frequency of vibration; and wave-length. The object of this process of elimination is, according to Planck, the striving towards a unification of the whole theoretical system, so that it shall be equally significant for all intelligent beings.”

“The unification of knowledge is the natural consequence of the intellectual and moral development of the race.”

“The Pythagoreans were the first who attempted a complete classification of the facts of the universe. Their effort, though feeble, was in the right direction; for the first principle of perception is analysis, or classification; and knowledge can never be unified until an ultimate or complete analysis has been performed.”

“It is the opinion of all competent authorities that the fundamental principles of Kant's philosophy declare against the possibility of a unification of knowledge.”

“Now what is science? ...it is before all a classification, a manner of bringing together facts which appearances separate, though they are bound together by some natural and hidden kinship.”

“Maxwell got a huge bonus for understanding the unification of electricity and magnetism. He understood the nature of light! When I first heard about this in high school I thought this was the coolest thing, and I still do. It's what we're all trying to do.”

“There is obviously only one alternative, namely the unification of minds or consciousnesses. Their multiplicity is only apparent, in truth there is only one mind.”

“In Newton's system of mechanics... there is an absolute space and an absolute time. In Einstein's theory time and space are interwoven, and the way in which they are interwoven depends on the observer. Instead of three plus one we have four dimensions.”

“In both quantum theory and general relativity, we encounter predictions of physically sensible quantities becoming infinite. This is likely the way that nature punishes impudent theorists who dare to break her unity. ...If infinities are signs of missing unification, a unified theory will have none. It will be what we call a finite theory.”

“The theories that we describe here provide the basis of progress toward a unification of macroeconomics and microeconomics.”

“We only call an elephant or a bacterium an 'organism' because, by analogy we attribute to those beings a similar unification of sensation and of consciousness to that we are conscious of in ourselves; but in human societies and in humanity this essential indication is lacking, and therefore, however many other indications we may detect that are common to humanity and to an organism, in the absence of that essential indication, the acknowledgement of humanity as an organism is incorrect.”

“From Pappus it appears, however, that the early Mathematicians had at first some reluctance in admitting either the Conic Sections or superior curves in the solution of problems, considering them as not strictly geometrical; but afterwards these lines became objects of much curious investigation, even among the ancients; and in modern times ultimately were of the most extensive utility, both in abstract and in physical science.”

“To see what is general in what is particular and what is permanent in what is transitory is the aim of scientific thought. ...[W]e ...endeavour to imagine the world as one connected set of things which underlies all the perceptions of all people.”