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Philosophy of mathematics and natural science
When mathematician Hermann Weyl decided to write a book on philosophy, he faced what he referred to as "conflicts of conscience"--the objective nature of science, he felt, did not mesh easily with the incredulous, uncertain nature of philosophy. Yet the two disciplines were already intertwined. In Philosophy of Mathematics and Natural Science , Weyl examines how advances in philosophy were led by scientific discoveries--the more humankind understood about the physical world, the more curious we became. The book is divided into two parts, one on mathematics and the other on the physical sciences. Drawing on work by Descartes, Galileo, Hume, Kant, Leibniz, and Newton, Weyl provides readers with a guide to understanding science through the lens of philosophy. This is a book that no one but Weyl could have written--and, indeed, no one has written anything quite like it since.
311 pages
First published January 1, 1949
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1048 people want to read
About the author
Hermann Weyl
107 books57 followersHermann Klaus Hugo Weyl (9 November 1885 – 8 December 1955) was a German mathematician, theoretical physicist and philosopher. Although much of his working life was spent in Zürich, Switzerland and then Princeton, he is associated with the University of Göttingen tradition of mathematics, represented by David Hilbert and Hermann Minkowski. His research has had major significance for theoretical physics as well as purely mathematical disciplines including number theory. He was one of the most influential mathematicians of the twentieth century, and an important member of the Institute for Advanced Study during its early years.
Weyl published technical and some general works on space, time, matter, philosophy, logic, symmetry and the history of mathematics. He was one of the first to conceive of combining general relativity with the laws of electromagnetism. While no mathematician of his generation aspired to the 'universalism' of Henri Poincaré or Hilbert, Weyl came as close as anyone. Michael Atiyah, in particular, has commented that whenever he examined a mathematical topic, he found that Weyl had preceded him (The Mathematical Intelligencer (1984), vol.6 no.1).
Source: http://en.wikipedia.org/wiki/Hermann_...
Weyl published technical and some general works on space, time, matter, philosophy, logic, symmetry and the history of mathematics. He was one of the first to conceive of combining general relativity with the laws of electromagnetism. While no mathematician of his generation aspired to the 'universalism' of Henri Poincaré or Hilbert, Weyl came as close as anyone. Michael Atiyah, in particular, has commented that whenever he examined a mathematical topic, he found that Weyl had preceded him (The Mathematical Intelligencer (1984), vol.6 no.1).
Source: http://en.wikipedia.org/wiki/Hermann_...
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Displaying 1 - 9 of 9 reviews
December 4, 2013
Over fifty of my Goodreads friends have read Wittgenstein's Tractatus, a book which famously sets out to describe the limits of human understanding. None of them have read Weyl's Philosophy of Mathematics and Natural Science, which was written by another German-speaking author a few years later, with a related plan in mind. For the first chapter or so, I wondered why.
The reason, alas, soon becomes all too apparent; Weyl's book is much more challenging. Wittgenstein does everything from first principles (notoriously, the book does not contain a single reference). If you have no background at all in formal philosophy, it is admittedly a bit hard to understand what on earth it's about. But as long as you possess a nodding acquaintance with logic and denotational semantics, the greater part of it is reasonably straightforward. It promises a great deal, but, as the author himself admitted later in his career, delivers surprisingly little. All the same, it does what it sets out to do in a pleasingly poetic way which give an impression of Delphic wisdom, and as such it has acquired a stable fan club which it will probably keep for at least for the next century.
The nice thing about reading Wittgenstein - I admit it: I am one of those fans - is that it makes you feel that you, too, might be a great philosopher if only you could get your act together a tiny bit more. It's just a question of asking a few searching questions (what would it mean to say that it was two o'clock on the Sun?) and coming up with a striking metaphor or two (that ladder...) Weyl, alas, has the opposite effect. Dismayingly, he isn't even a member of the Philosopher's Union; when he presents his ID card for inspection, it says he's some kind of mathematician. Nevertheless, he turns out to have read all the classical Greek philosophers in Greek, all the medieval scholastics in Latin, and, needless to say, all the Germans. He has an intimate knowledge of Galileo, Newton, Descartes, Leibniz, Frege, Russell and more or less everyone else you can think of, all of whom he's also read in the original. The worst thing is that he doesn't even seem to be showing off; you get the impression that he moves in circles where people are simply expected to know this kind of stuff. When you complain, he gives you a puzzled look.
In contrast to Wittgenstein, Weyl has specific issues to discuss. What is the nature of mathematical truth? His brief but rigorous discussion of Gödel's construction and its antecendents in self-referential paradoxes going back to the Socratics is startlingly thorough, as is his contrast of Brouwer's intuitionistic logic against Russell and Whitehead. He has an unusual but appealing approach to the relationship between symbols and objects, which he develops in terms of group theory rather than mainstream semantics. He illustrates with examples from relativity and quantum mechanics, both of which he takes for granted. He has interesting interludes on the nature of the chemical bond, the logical basis of evolution and the reason why there is an arrow of time. Modern knowledge sometimes shows he's wrong, but when he is it's generally due to facts he didn't have available; who would ever have believed in 1926 that nature might violate left/right symmetry? His guesses are generally pretty good: among other things, he calls the Big Bang and DNA correctly before there was any hard evidence for either.
Looking around, I see a scattering of Nobel Prize winners, Fields medalists and similar people who hold the book in high regard. Weyl had an enormous influence on the development of twentieth century science and mathematics, but the things he did were so esoteric that it's hard to write popular books about them; the closest I've seen is Woit's Not Even Wrong, which one might briefly summarize as saying that that modern physics has screwed up by not paying enough attention to Weyl's ideas. Woit has also been largely dismissed as incomprehensible.
Dammit, this book has completely bummed me out: I realize that it's in fact rather difficult to be a towering genius, and a great deal of study and hard work is evidently involved in getting there. I'm going to have to drown my sorrows. Bartender, give me a double Wittgenstein with a Feynman chaser. And keep 'em coming.
The reason, alas, soon becomes all too apparent; Weyl's book is much more challenging. Wittgenstein does everything from first principles (notoriously, the book does not contain a single reference). If you have no background at all in formal philosophy, it is admittedly a bit hard to understand what on earth it's about. But as long as you possess a nodding acquaintance with logic and denotational semantics, the greater part of it is reasonably straightforward. It promises a great deal, but, as the author himself admitted later in his career, delivers surprisingly little. All the same, it does what it sets out to do in a pleasingly poetic way which give an impression of Delphic wisdom, and as such it has acquired a stable fan club which it will probably keep for at least for the next century.
The nice thing about reading Wittgenstein - I admit it: I am one of those fans - is that it makes you feel that you, too, might be a great philosopher if only you could get your act together a tiny bit more. It's just a question of asking a few searching questions (what would it mean to say that it was two o'clock on the Sun?) and coming up with a striking metaphor or two (that ladder...) Weyl, alas, has the opposite effect. Dismayingly, he isn't even a member of the Philosopher's Union; when he presents his ID card for inspection, it says he's some kind of mathematician. Nevertheless, he turns out to have read all the classical Greek philosophers in Greek, all the medieval scholastics in Latin, and, needless to say, all the Germans. He has an intimate knowledge of Galileo, Newton, Descartes, Leibniz, Frege, Russell and more or less everyone else you can think of, all of whom he's also read in the original. The worst thing is that he doesn't even seem to be showing off; you get the impression that he moves in circles where people are simply expected to know this kind of stuff. When you complain, he gives you a puzzled look.
In contrast to Wittgenstein, Weyl has specific issues to discuss. What is the nature of mathematical truth? His brief but rigorous discussion of Gödel's construction and its antecendents in self-referential paradoxes going back to the Socratics is startlingly thorough, as is his contrast of Brouwer's intuitionistic logic against Russell and Whitehead. He has an unusual but appealing approach to the relationship between symbols and objects, which he develops in terms of group theory rather than mainstream semantics. He illustrates with examples from relativity and quantum mechanics, both of which he takes for granted. He has interesting interludes on the nature of the chemical bond, the logical basis of evolution and the reason why there is an arrow of time. Modern knowledge sometimes shows he's wrong, but when he is it's generally due to facts he didn't have available; who would ever have believed in 1926 that nature might violate left/right symmetry? His guesses are generally pretty good: among other things, he calls the Big Bang and DNA correctly before there was any hard evidence for either.
Looking around, I see a scattering of Nobel Prize winners, Fields medalists and similar people who hold the book in high regard. Weyl had an enormous influence on the development of twentieth century science and mathematics, but the things he did were so esoteric that it's hard to write popular books about them; the closest I've seen is Woit's Not Even Wrong, which one might briefly summarize as saying that that modern physics has screwed up by not paying enough attention to Weyl's ideas. Woit has also been largely dismissed as incomprehensible.
Dammit, this book has completely bummed me out: I realize that it's in fact rather difficult to be a towering genius, and a great deal of study and hard work is evidently involved in getting there. I'm going to have to drown my sorrows. Bartender, give me a double Wittgenstein with a Feynman chaser. And keep 'em coming.
November 28, 2013
One of my earliest memories is of hearing my great-grandma tell the story of how she met Frau Schrödinger at a rather wild party in Vienna in 1926. She asked her why she was carrying on with Weyl when she was, as she said herself, happily married.
"Well," said Great-Grandma, "she told me that one of them had a better mind and the other one had a bigger dick, but I'd drunk so much champagne that I couldn't remember the next day who had what. And I never saw her again."
I came out from under the table where I'd been hiding and asked what a couple of words meant, but was told it was past my bedtime.
"Well," said Great-Grandma, "she told me that one of them had a better mind and the other one had a bigger dick, but I'd drunk so much champagne that I couldn't remember the next day who had what. And I never saw her again."
I came out from under the table where I'd been hiding and asked what a couple of words meant, but was told it was past my bedtime.
July 22, 2017
Weyl was extraordinary in being not just a foremost mathematician but in having a broad and deep knowledge of the history of philosophy. He recalls the excitement of youth in writing his brilliant (and today insufficiently appreciated, though gaining much greater traction in recent years) work “after due literary preparations dashing off the manuscript in a few weeks.” This work not only covers geometry and foundations of mathematics, the relativity theory and mechanics of his day, but also covers a large number of traditional philosophers. These include not only the major classical philosophers whom one would expect such a survey of philosophy of science to cover (such as Plato, Aristotle, Descartes, Leibniz, and Kant) but sympathetically discusses German romantic figures such as Fichte and Schelling, rejected as murky and anti-scientific in the Anglo-Saxon world. In his 1923 survey of the foundations of mathematics, for instance, he starts with Anaxagoras, surveys contemporary logicism, intuitionism and formalism, and concludes with Parmenides and Heraclitus. Weyl’s philosophical learning was not a mere after dinner speech decoration, but was deeply embedded in his mathematical and physical research. Weyl’s works are full not only of philosophical erudition but of powerful literary allusions and metaphors that are surprising in works of professional mathematics and physics. No wonder one contemporary American physicist found that he could not stomach Weyl’s work "The Classical Groups" because it was too much like a Wagnerian opera! Ignoring such philistinism, Weyl really illuminates the mathematical and physical issues. Weyl talked elsewhere of the devil of algebra and the angel of topology struggling for the soul of mathematics, and he said that in his work he tried to combine truth and beauty, but when he had to choose, he usually chose beauty.
Weyl published this work in 1927, the high point of the Weimar Republic and the year of publication, of the major works of Heidegger, Cassirer, (and, by the way, Whitehead in America) and others. Later, after Hitler, and his flight to the US, as well as aging, Weyl said he no longer had the enthusiasm he had when he wrote the original part of the book. He says him it took him as long to write the appendices on quantum mechanics and genetics than it had to write the original book. These appendices are, in my opinion, less illuminating than the main part of the book.
Weyl published this work in 1927, the high point of the Weimar Republic and the year of publication, of the major works of Heidegger, Cassirer, (and, by the way, Whitehead in America) and others. Later, after Hitler, and his flight to the US, as well as aging, Weyl said he no longer had the enthusiasm he had when he wrote the original part of the book. He says him it took him as long to write the appendices on quantum mechanics and genetics than it had to write the original book. These appendices are, in my opinion, less illuminating than the main part of the book.
December 20, 2013
For the record, I am not actually reading this simultaneously w/ Dahlberg and Mailer, but I reserving Weyl at my library desk at the college. I liked Manny's review much more than to click 'Like' and have been investigating Weyl. This book and several others are in stock, and I will be spending this arctic week's evenings at my little desk dabbling very slowly in this book. Pretty excited about it. My two Dahlberg books will be arriving today, tomorrow, or sometime soon, and reading Executioner's Song in long, long bursts and loving it. Autumn is my favorite season in that it marks the beginning of a decline in east-coast social obligations, and my introverted nature makes more sense to people around me whose opinion I don't care for in the first place anyway, but around whom spending the eighteen hour days at desks, occasional walks, occasional bar, wine by night, taking everything to its furthest limit possible. So yes, work five hours on final edit of script, read Mailer, eat a piece of bread, collapse, Weyl by night. Sounds good. L8er.
Update:
After several nocturnal sessions ordered for self for Christmas present. I have not read it all but I give five stars based on the sheer phantasmagorical scope of the thing. Kudos to Manny, if he reads this: Thank you, online stranger and grade-A reader, for pointing me in the Weyl direction ahead of time. One of these days will compile my 'Lifer' shelf. Till then, cheers.
Update:
After several nocturnal sessions ordered for self for Christmas present. I have not read it all but I give five stars based on the sheer phantasmagorical scope of the thing. Kudos to Manny, if he reads this: Thank you, online stranger and grade-A reader, for pointing me in the Weyl direction ahead of time. One of these days will compile my 'Lifer' shelf. Till then, cheers.
February 21, 2018
The writer attempts to convey knowledge - his impressions of how things are - sometimes he comes off rather goofy, and some of his statements may well be out of date, since the volume was first written to 1926, though later updated. Still, I found it interesting, and easy enough to understand even for a non-math/physics nerd. The stuff I couldn't possibly get - like equations (since I did not get to calculus in school) I of course did not get much out of reading. But the book was still interesting and refreshing even. The one thing you can say about great thinkers like Weyl is that they are absolutely free to think of anything and imagine explanations to problems that have dogged mankind since time immemorial. I like that. I think their field is perhaps the most exciting since it's the free-est.
The author discusses in some detail various historical trends in physics, refers to numerous thinkers who came before him, as well as well-known contemporaries that he worked with. It's a ponderous and to me at least not particularly well-written work, yet it's permeated with the enthusiasm Weyl has for physics, and the excitement of trying to convey his ideas about the cosmos etc. to others, even laypeople. It's so difficult to break complex ideas down to understandable chunks, especially concepts that many if not most of us never really think about. This is what Weyl was trying to accomplish, and makes for a memorable, if humorous at times, read.
Anyway, here are some quotes that I thought were memorable:
"[Physics] ...is the only branch of the natural sciences with which I am familiar through my own work."
"...science would perish without a supporting transcendental faith in truth and reality, and without the continuous interplay between its facts and constructions on the one hand and the imagery of ideas on the other."
"In Euclidean geometry we are concerned with three categories of objects, points, lines, and planes, which are not defined but assumed to be intuitively given, and with the basic relations of incidence (a point lies on a line, a line lies in a plane, a point lies in a plane), between-ness (a point z lies between the points x and y), and congruence (congruence of line segments and of angles). "
"...the process of definition by abstraction. A binary relation a ~ b in a domain of objects is called an equivalence (a relation of the character of equality), if the following is universally true:
(i) a~ a;
(ii) if a~b, then b ~ a (commutativity);
(iii) if a ~b and b ~ c, then a ~ c (transitivity)."
"That part of logic which operates exclusively with the logical connectives 'not,' 'and,' 'or' will be referred to as finite logic, as opposed to transfinite logic, which in addition uses the propositional operators 'some' (or 'there is') and 'all'."
"The axiomatic method consists simply in making a complete collection of the basic concepts as well as the basic facts from which all concepts and theorems of a science can be derived by definition and deduction respectively."
"...precisely speaking, what is supposed to be deducible from the axioms are the pertinent general true propositions."
"The essential character of the continuum is clearly described in this fragment due to Anaxagoras: "Among the small there is no smallest, but always something smaller. For what is cannot cease to be no matter how far it is being subdivided."
"[Leibniz:] "From the fact that a mathematical solid cannot be resolved into primal elements it follows immediately that it is nothing real but merely an ideal construct designating only a possibility of parts."
"Whenever the possibility of a construction is being asserted, we have no meaningful proposition; only by virtue of an effective construction, an executed proof, does n existential statement acquire meaning. In any of the numerous existential theorems in mathematics, what is valuable in each case is not the theorem as such but the construction carried out in its proof; without it the theorem is an empty shadow."
"...Kant...emphasized the symbolic construction with concrete tokens in algebra...-- that "mathematics possesses a content that is secure independently of all logic and therefore can never be based upon logic alone."
"...a theoretical desire, incomprehensible from the merely phenomenal point of view, is alive in us which urges toward totality."
"Nowhere do mathematics, natural sciences,and philosophy permeate one another so intimately as in the problem of space."
"...the essential fact that objectivity is an issue decidable on the ground of experience only. It also accounts for the two main sources of the error so often committed in the history of knowledge, that of mistaking a statement for objective that is not: (1) one overlooked certain relevant circumstantial factors on which the meaning of the statement depends although they are not mentioned explicitly in its elliptic form, (2) though these factors were recognized, one did not investigate carefully enough whether or not the truth of the statement is affected by their variation."
"All knowledge, while it starts with intuitive description, tends toward symbolic construction."
"To fulfill the demand of objectivity we construct an image of the world in symbols."
"Only in the infinitely small may we expect to encounter the elementary and uniform laws, hence the world must be comprehended though its behavior in the infinitely small."
"How can we, ... we must ask, ascertain objectively the equi-locality or the simultaneity of events, the equality of time intervals and the congruence of spatial configurations?"
"By shooting bullets from O in all possible directions with all possible velocities I can only hit those world points which are later than O; I cannot shoot into the past. Likewise any event happening at O has influence only upon the events at later world points; the past cannot be changed. That is to say, the stratification has a causal meaning; it determines the causal connection of the world."
"If I am at O, then O will divide my life line, that is the world line of my body, into two parts, past and future..."
"...once the inertial structure of the world is accepted as the cause for the dynamical inequivalence of motions, we recognize clearly why the situation appeared so unsatisfactory. We were asked to believe that something producing such enormous effects as inertia .... is a rigid geometrical property of the world, fixed once and for all. Leibniz...emphatically stressed the dynamic character of inertia as a tendency to resist deflecting forces..."
"...Aristotelian space (the interior of the crystal sphere) differs only in its metrical relations, not topologically, from the infinite one. ...infinite Euclidean space leds to absurdities if we assume that the masses are on the whole uniformly distributed throughout the universe and that Newton's law of attraction is valid. Even though the gravitational force of a constant mass decreases with the inverse square of the distance, the far-off masses would then be so predominant in the entire gravitational effect that the total force exerted upon any one star would remain completely indeterminate. It is possible, however, that space is finite and yet unbounded; indeed it may be a closed manifold, like the two-dimensional surface of a sphere."
"The doctrine of the subjectivity of sense qualities has been intimately connected with the progress of science ever since Democritus laid down the principle, "Sweet and bitter, cold and warm, as well as the colors, all these things exist but in opinion and not in reality..."; what really exists are unchangeable particles, atoms, which move in empty space."
"No less than the sense qualities must the intuitions of space and time be reliquinshed as its building material; they must be replaced by a four-dimensional continuum in the abstract arithmetical sense."
"The objective world simply is, it does not happen. Only to the gaze of my consciousness, crawling upward along the life line of my body, does a section of this world come to life as a fleeting image in space which continuously changes in time."
"Hume attempted to uphold with inexorable consistency the viewpoint that the given is the whole of reality. Since it became apparent through him that this viewpoint fails completely in the explanation of those cognitive positions which play a basic role in everyday life and in science, he was indeed the first to reveal the problem of reality in its full difficulty. Reason it its function of constituting reality is described by him as the faculty of imagination. With complete sincerity he confesses the irreconcilable conflict between thought and life, into which he finds himself thrown."
"The fact that both constituents, extension and quality, are bound to each other is the root of Aristotle's thesis of the impossibility of empty space."
"Under the influence of modern mathematical axiomatic investigations one has come to distinguish the 'mathematical space,' whose laws are logical consequences of arbitrarily assumed axioms, from the 'physical space,' the ordering scheme of the real things, which enters as an integral component into the theoretical construction of the world. With regard to this distinction Einstein says... "As far as the propositions of mathematics refer to reality they are not certain, and in so far as they are certain they do not refer to reality."
"Thus it can be argued that the middle case of 1 negative dimension is realized by the metrical field of the real world because of the necessity of a causal structure by virtue of which an ego may be actively and passively connected with the world in such a manner as to separate past and future, which is known from what is planned."
""Thus," says Maxwell... "all the mathematical sciences are founded on relations between physical laws and laws of numbers."
"Dilthey, in hs essay on the autonomy of thought in the 17th century...describes the development of mechanics up to Galileo. "Galileo came, and with him there followed an actual analysis of nature, after more than two thousand years of mere description and consideration of form in nature, that had culminated in Copernicus's picture of the world."
"What is decisive is this: the farther the analysis progresses, the more detailed the observations become and the finer the elements into which we dissect the phenomena, the simpler -- and not the more complicated, as might be expected -- become the basic laws, and the more completely and accurately do they explain the factual course of events."
"For a long time the laws of Euclidean geometry were held to be sacrosanct. The principles of the conservation of energy and momentum are of comparable if not higher, stability."
"H.A. Lorenz arrived at the fundamental electromagnetic laws of the theory of electrons by taking the phenomenological Maxwell equations, which had been derived from observation and with which the electrical engineer works, and crossing out all quantities in which the influence of matter manifests itself in the form of material constants, such as conductivity, electrical polarization and magnetization. Under the assumption that the true 'microscopic' electromagnetic field obeys these simplified harmonic laws, in conjunction with certain ideas on the atomic structure of matter, he was able to obtain once more the old phenomenological laws for the macroscopic field by identifying the macroscopic field quantities with certain average values of the microscopic field quantities."
"Mach speaks of a progressive "adaptation of thoughts to facts." The justification for the formation of theories he sees in the ensuing economy of comprehending and communicating facts and procedures... Others have adhered to the belief that reason is here at work, reason which strives according to immanent principles to construct symbolically its correlate, transcendent reality. Without this belief science, to them, seems an empty shell. But all are of one opinion as to the ultimate goal, the prediction of events."
"Kant, in his transcendental logic, made the attempt to ascertain by a systematic procedure the aprioristic principles for the construction of empirical reality. ... Kant's problem....remains open for the future, presumably as an infinite task. Kant...considered metaphysics...as "the only one of all sciences which, through a small but united effort, may count on such completion in a short time, so that nothing will remain to posterity."
"Just as in intuitive space extension and quality are tied to each other, so, in the field theory, the state quantities of the field or the field structure on the one hand, and its spatio-temporal medium, the structureless four-dimensional continuum on the other, depend on one another."
"While according to Democritus the distinction of full and empty forms the basis of substance theory, any field theory is founded on certain state quantities spread out in the four-dimensional space-time continuum."
"Proceeding in this way we reestablish the duality of field and matter. Their connection is a dynamic one; matter excites the field, the field acts upon matter. If less attention is paid to the connecting medium of the field, then matter and force appear as the interdependent constituents of the world."
"[Leibniz:] "For not all truths relating to the world of bodies can be derived from merely arithmetical and geometrical axioms, that is, from axioms of larger and smaller, of shape and position," he says in criticism of Descartes ... "but others must be added concerning cause and effect, activity and passivity, in order to give an account of the order of things." The ultimate element is the monad, an indecomposable unit without extension, from which the force busts forth as a transcendental power."
"...as the optical phenomena subordinate themselves as a partial domain to the electromagnetic ones and the conception of electromagnetic field is developed by Faraday and Maxwell, the ether is divested of its substatio-physical character, and nothing remains but the absolute space as the medium of electromagnetic field states."
"Aristotelian philosophy considers matter ... as the determinable, in contrast to the determining form..."
"[Leibniz:] "However much I stand on the side of scholasticism in the general and, so to speak, metaphysical explanation of the principles of the physical world, I am, on the other hand, the most radical adherent of the corpuscular philosophy with regard to the explanation of particular phenomena..." "My opinion thus is to the effect that bodies, which are commonly considered substances, are nothing but real phenomena and are as unsubstantial a a mock sun or a rainbow... The monad alone is substance..."
"...the description of the world according to the field theory consists of the here-thus relations -- the here being represented by the space-time coordinates, the thus by the state quantities. If the latter are given as functions of the former, then the course of the world is completely known."
".. now physics entered into its golden era of atomic research. During the last half century it has provided a through and brilliant corroboration for the basic tenets of atomism and penetrated into ever deeper layers of the stange atomic world."
"...it seems clear hat quantum physics cannot posit matter and ether as the basic polarity underlying all phenomena, as the ether theory had done. Light is not only ether wave but also corpuscle, an electron is not only a corpuscle but also a wave."
"The causal relation does not hold between individual events but between classes of events."
"The abandonment of the metaphysical quest for the cause in favor of the scientific quest for the law is preached by all great scientists."
"...in nature cause and effect lie within the same plane."
"This leads to the formulation that the content of the past determines by law the content of the future. However this is not the causal law itself but only a special form of it which fits well into field physics."
"Subjectively that part of the light cone which opens towrad the furure plays an entirely different role frothat which is directed rtoward the past."
"And yet one must demand that the basic fact of consciousness, the one-way direction of the flow of time, have a physical foundation. Phenomenological thermodynamics finds it in the law of entropy, according to which the natural processes are irreversible and take place in the sense of increasing entropy. Since the atomic theory reduces phenomenological thermodynamics to the statistics of atomic motion and thereby introduces an element of chance, we postpone a discussion of the problem and turn first to an analysis of chance."
"...most of the usual physical 'laws,' especially those concerning matter, must not be construed as strictly valid laws of nature but as statistical regularities. Statics (which treats of the laws valid for thermodynamic equilibrium) and dynamics (which treats of the laws regulating the transition from a disturbed state to a state of equilibrium) are supplemented by a theory of fluctuations, which investigates the fluctuations about the statistical mean values in equilibrium and the physical effects connected therewith."
"Our consciousness does not reflect the molecular chaos of the phenomena but exerts and integrating function with respect to both space and time, from which results the apparent homogeneity and continuity of the phenomena."
"Of the two laws which are of universal significance for all physical phenomena, the law of conservation of energy and the law of continuously increasing entropy, one is the prototype of a strict law, the other of a statistical law. The latter ...states that heat flows from the warmer to the colder body... that in general, an improbable state in the course of time changes into that 'most probable' state from which the overwhelming majority of possible states differs but little; or, otherwise expressed, that order gradually changes to disorder."
"Perhaps space and time...merely have the function of a probability field for physics."
"...more important, perhaps, than the openness toward the infinitely distant fringes of the world is the inner infinitude toward the atoms."
"In a sweeping way and in conformity with the experience in all branches of physics, the law of entropy accounts for 'time's arrow,' for the different roles of past and future, the irreversibility of natural processes."
"We thus arrive at the conviction that the distinctiveness of the direction from past to future finds its expression not in the elementary laws but in the probability judgment."
"The contrast of right and left, however much it occupies mythical thinking, does not pose as basic a problem to scientific thought as that of past and future."
"Order, organization, is the characteristic of life. Hence the impression that life, in its evolution on earth, resists the plunge into the abyss of 'thermal death,' which the entropy law imposes upon inorganic matter."
"The real riddle, if I am not mistaken, lies in the double position of the ego: it is not merely an existing individual which carries out real psychic acts but also 'vision,' a self-penetrating light (sense-giving consciousness, knowledge, image, or however you may call it); as an individual capable of positing reality, its vision open to reason; "a force into which an eye has been put," as Fichte says, or "an organization turned toward two worlds at once," in the words of Schelling... But this secret, by its very nature, lies beyond the cognitive means of natural science."
The author discusses in some detail various historical trends in physics, refers to numerous thinkers who came before him, as well as well-known contemporaries that he worked with. It's a ponderous and to me at least not particularly well-written work, yet it's permeated with the enthusiasm Weyl has for physics, and the excitement of trying to convey his ideas about the cosmos etc. to others, even laypeople. It's so difficult to break complex ideas down to understandable chunks, especially concepts that many if not most of us never really think about. This is what Weyl was trying to accomplish, and makes for a memorable, if humorous at times, read.
Anyway, here are some quotes that I thought were memorable:
"[Physics] ...is the only branch of the natural sciences with which I am familiar through my own work."
"...science would perish without a supporting transcendental faith in truth and reality, and without the continuous interplay between its facts and constructions on the one hand and the imagery of ideas on the other."
"In Euclidean geometry we are concerned with three categories of objects, points, lines, and planes, which are not defined but assumed to be intuitively given, and with the basic relations of incidence (a point lies on a line, a line lies in a plane, a point lies in a plane), between-ness (a point z lies between the points x and y), and congruence (congruence of line segments and of angles). "
"...the process of definition by abstraction. A binary relation a ~ b in a domain of objects is called an equivalence (a relation of the character of equality), if the following is universally true:
(i) a~ a;
(ii) if a~b, then b ~ a (commutativity);
(iii) if a ~b and b ~ c, then a ~ c (transitivity)."
"That part of logic which operates exclusively with the logical connectives 'not,' 'and,' 'or' will be referred to as finite logic, as opposed to transfinite logic, which in addition uses the propositional operators 'some' (or 'there is') and 'all'."
"The axiomatic method consists simply in making a complete collection of the basic concepts as well as the basic facts from which all concepts and theorems of a science can be derived by definition and deduction respectively."
"...precisely speaking, what is supposed to be deducible from the axioms are the pertinent general true propositions."
"The essential character of the continuum is clearly described in this fragment due to Anaxagoras: "Among the small there is no smallest, but always something smaller. For what is cannot cease to be no matter how far it is being subdivided."
"[Leibniz:] "From the fact that a mathematical solid cannot be resolved into primal elements it follows immediately that it is nothing real but merely an ideal construct designating only a possibility of parts."
"Whenever the possibility of a construction is being asserted, we have no meaningful proposition; only by virtue of an effective construction, an executed proof, does n existential statement acquire meaning. In any of the numerous existential theorems in mathematics, what is valuable in each case is not the theorem as such but the construction carried out in its proof; without it the theorem is an empty shadow."
"...Kant...emphasized the symbolic construction with concrete tokens in algebra...-- that "mathematics possesses a content that is secure independently of all logic and therefore can never be based upon logic alone."
"...a theoretical desire, incomprehensible from the merely phenomenal point of view, is alive in us which urges toward totality."
"Nowhere do mathematics, natural sciences,and philosophy permeate one another so intimately as in the problem of space."
"...the essential fact that objectivity is an issue decidable on the ground of experience only. It also accounts for the two main sources of the error so often committed in the history of knowledge, that of mistaking a statement for objective that is not: (1) one overlooked certain relevant circumstantial factors on which the meaning of the statement depends although they are not mentioned explicitly in its elliptic form, (2) though these factors were recognized, one did not investigate carefully enough whether or not the truth of the statement is affected by their variation."
"All knowledge, while it starts with intuitive description, tends toward symbolic construction."
"To fulfill the demand of objectivity we construct an image of the world in symbols."
"Only in the infinitely small may we expect to encounter the elementary and uniform laws, hence the world must be comprehended though its behavior in the infinitely small."
"How can we, ... we must ask, ascertain objectively the equi-locality or the simultaneity of events, the equality of time intervals and the congruence of spatial configurations?"
"By shooting bullets from O in all possible directions with all possible velocities I can only hit those world points which are later than O; I cannot shoot into the past. Likewise any event happening at O has influence only upon the events at later world points; the past cannot be changed. That is to say, the stratification has a causal meaning; it determines the causal connection of the world."
"If I am at O, then O will divide my life line, that is the world line of my body, into two parts, past and future..."
"...once the inertial structure of the world is accepted as the cause for the dynamical inequivalence of motions, we recognize clearly why the situation appeared so unsatisfactory. We were asked to believe that something producing such enormous effects as inertia .... is a rigid geometrical property of the world, fixed once and for all. Leibniz...emphatically stressed the dynamic character of inertia as a tendency to resist deflecting forces..."
"...Aristotelian space (the interior of the crystal sphere) differs only in its metrical relations, not topologically, from the infinite one. ...infinite Euclidean space leds to absurdities if we assume that the masses are on the whole uniformly distributed throughout the universe and that Newton's law of attraction is valid. Even though the gravitational force of a constant mass decreases with the inverse square of the distance, the far-off masses would then be so predominant in the entire gravitational effect that the total force exerted upon any one star would remain completely indeterminate. It is possible, however, that space is finite and yet unbounded; indeed it may be a closed manifold, like the two-dimensional surface of a sphere."
"The doctrine of the subjectivity of sense qualities has been intimately connected with the progress of science ever since Democritus laid down the principle, "Sweet and bitter, cold and warm, as well as the colors, all these things exist but in opinion and not in reality..."; what really exists are unchangeable particles, atoms, which move in empty space."
"No less than the sense qualities must the intuitions of space and time be reliquinshed as its building material; they must be replaced by a four-dimensional continuum in the abstract arithmetical sense."
"The objective world simply is, it does not happen. Only to the gaze of my consciousness, crawling upward along the life line of my body, does a section of this world come to life as a fleeting image in space which continuously changes in time."
"Hume attempted to uphold with inexorable consistency the viewpoint that the given is the whole of reality. Since it became apparent through him that this viewpoint fails completely in the explanation of those cognitive positions which play a basic role in everyday life and in science, he was indeed the first to reveal the problem of reality in its full difficulty. Reason it its function of constituting reality is described by him as the faculty of imagination. With complete sincerity he confesses the irreconcilable conflict between thought and life, into which he finds himself thrown."
"The fact that both constituents, extension and quality, are bound to each other is the root of Aristotle's thesis of the impossibility of empty space."
"Under the influence of modern mathematical axiomatic investigations one has come to distinguish the 'mathematical space,' whose laws are logical consequences of arbitrarily assumed axioms, from the 'physical space,' the ordering scheme of the real things, which enters as an integral component into the theoretical construction of the world. With regard to this distinction Einstein says... "As far as the propositions of mathematics refer to reality they are not certain, and in so far as they are certain they do not refer to reality."
"Thus it can be argued that the middle case of 1 negative dimension is realized by the metrical field of the real world because of the necessity of a causal structure by virtue of which an ego may be actively and passively connected with the world in such a manner as to separate past and future, which is known from what is planned."
""Thus," says Maxwell... "all the mathematical sciences are founded on relations between physical laws and laws of numbers."
"Dilthey, in hs essay on the autonomy of thought in the 17th century...describes the development of mechanics up to Galileo. "Galileo came, and with him there followed an actual analysis of nature, after more than two thousand years of mere description and consideration of form in nature, that had culminated in Copernicus's picture of the world."
"What is decisive is this: the farther the analysis progresses, the more detailed the observations become and the finer the elements into which we dissect the phenomena, the simpler -- and not the more complicated, as might be expected -- become the basic laws, and the more completely and accurately do they explain the factual course of events."
"For a long time the laws of Euclidean geometry were held to be sacrosanct. The principles of the conservation of energy and momentum are of comparable if not higher, stability."
"H.A. Lorenz arrived at the fundamental electromagnetic laws of the theory of electrons by taking the phenomenological Maxwell equations, which had been derived from observation and with which the electrical engineer works, and crossing out all quantities in which the influence of matter manifests itself in the form of material constants, such as conductivity, electrical polarization and magnetization. Under the assumption that the true 'microscopic' electromagnetic field obeys these simplified harmonic laws, in conjunction with certain ideas on the atomic structure of matter, he was able to obtain once more the old phenomenological laws for the macroscopic field by identifying the macroscopic field quantities with certain average values of the microscopic field quantities."
"Mach speaks of a progressive "adaptation of thoughts to facts." The justification for the formation of theories he sees in the ensuing economy of comprehending and communicating facts and procedures... Others have adhered to the belief that reason is here at work, reason which strives according to immanent principles to construct symbolically its correlate, transcendent reality. Without this belief science, to them, seems an empty shell. But all are of one opinion as to the ultimate goal, the prediction of events."
"Kant, in his transcendental logic, made the attempt to ascertain by a systematic procedure the aprioristic principles for the construction of empirical reality. ... Kant's problem....remains open for the future, presumably as an infinite task. Kant...considered metaphysics...as "the only one of all sciences which, through a small but united effort, may count on such completion in a short time, so that nothing will remain to posterity."
"Just as in intuitive space extension and quality are tied to each other, so, in the field theory, the state quantities of the field or the field structure on the one hand, and its spatio-temporal medium, the structureless four-dimensional continuum on the other, depend on one another."
"While according to Democritus the distinction of full and empty forms the basis of substance theory, any field theory is founded on certain state quantities spread out in the four-dimensional space-time continuum."
"Proceeding in this way we reestablish the duality of field and matter. Their connection is a dynamic one; matter excites the field, the field acts upon matter. If less attention is paid to the connecting medium of the field, then matter and force appear as the interdependent constituents of the world."
"[Leibniz:] "For not all truths relating to the world of bodies can be derived from merely arithmetical and geometrical axioms, that is, from axioms of larger and smaller, of shape and position," he says in criticism of Descartes ... "but others must be added concerning cause and effect, activity and passivity, in order to give an account of the order of things." The ultimate element is the monad, an indecomposable unit without extension, from which the force busts forth as a transcendental power."
"...as the optical phenomena subordinate themselves as a partial domain to the electromagnetic ones and the conception of electromagnetic field is developed by Faraday and Maxwell, the ether is divested of its substatio-physical character, and nothing remains but the absolute space as the medium of electromagnetic field states."
"Aristotelian philosophy considers matter ... as the determinable, in contrast to the determining form..."
"[Leibniz:] "However much I stand on the side of scholasticism in the general and, so to speak, metaphysical explanation of the principles of the physical world, I am, on the other hand, the most radical adherent of the corpuscular philosophy with regard to the explanation of particular phenomena..." "My opinion thus is to the effect that bodies, which are commonly considered substances, are nothing but real phenomena and are as unsubstantial a a mock sun or a rainbow... The monad alone is substance..."
"...the description of the world according to the field theory consists of the here-thus relations -- the here being represented by the space-time coordinates, the thus by the state quantities. If the latter are given as functions of the former, then the course of the world is completely known."
".. now physics entered into its golden era of atomic research. During the last half century it has provided a through and brilliant corroboration for the basic tenets of atomism and penetrated into ever deeper layers of the stange atomic world."
"...it seems clear hat quantum physics cannot posit matter and ether as the basic polarity underlying all phenomena, as the ether theory had done. Light is not only ether wave but also corpuscle, an electron is not only a corpuscle but also a wave."
"The causal relation does not hold between individual events but between classes of events."
"The abandonment of the metaphysical quest for the cause in favor of the scientific quest for the law is preached by all great scientists."
"...in nature cause and effect lie within the same plane."
"This leads to the formulation that the content of the past determines by law the content of the future. However this is not the causal law itself but only a special form of it which fits well into field physics."
"Subjectively that part of the light cone which opens towrad the furure plays an entirely different role frothat which is directed rtoward the past."
"And yet one must demand that the basic fact of consciousness, the one-way direction of the flow of time, have a physical foundation. Phenomenological thermodynamics finds it in the law of entropy, according to which the natural processes are irreversible and take place in the sense of increasing entropy. Since the atomic theory reduces phenomenological thermodynamics to the statistics of atomic motion and thereby introduces an element of chance, we postpone a discussion of the problem and turn first to an analysis of chance."
"...most of the usual physical 'laws,' especially those concerning matter, must not be construed as strictly valid laws of nature but as statistical regularities. Statics (which treats of the laws valid for thermodynamic equilibrium) and dynamics (which treats of the laws regulating the transition from a disturbed state to a state of equilibrium) are supplemented by a theory of fluctuations, which investigates the fluctuations about the statistical mean values in equilibrium and the physical effects connected therewith."
"Our consciousness does not reflect the molecular chaos of the phenomena but exerts and integrating function with respect to both space and time, from which results the apparent homogeneity and continuity of the phenomena."
"Of the two laws which are of universal significance for all physical phenomena, the law of conservation of energy and the law of continuously increasing entropy, one is the prototype of a strict law, the other of a statistical law. The latter ...states that heat flows from the warmer to the colder body... that in general, an improbable state in the course of time changes into that 'most probable' state from which the overwhelming majority of possible states differs but little; or, otherwise expressed, that order gradually changes to disorder."
"Perhaps space and time...merely have the function of a probability field for physics."
"...more important, perhaps, than the openness toward the infinitely distant fringes of the world is the inner infinitude toward the atoms."
"In a sweeping way and in conformity with the experience in all branches of physics, the law of entropy accounts for 'time's arrow,' for the different roles of past and future, the irreversibility of natural processes."
"We thus arrive at the conviction that the distinctiveness of the direction from past to future finds its expression not in the elementary laws but in the probability judgment."
"The contrast of right and left, however much it occupies mythical thinking, does not pose as basic a problem to scientific thought as that of past and future."
"Order, organization, is the characteristic of life. Hence the impression that life, in its evolution on earth, resists the plunge into the abyss of 'thermal death,' which the entropy law imposes upon inorganic matter."
"The real riddle, if I am not mistaken, lies in the double position of the ego: it is not merely an existing individual which carries out real psychic acts but also 'vision,' a self-penetrating light (sense-giving consciousness, knowledge, image, or however you may call it); as an individual capable of positing reality, its vision open to reason; "a force into which an eye has been put," as Fichte says, or "an organization turned toward two worlds at once," in the words of Schelling... But this secret, by its very nature, lies beyond the cognitive means of natural science."
July 29, 2018
This book is meant for smarter people than I.
August 22, 2018
As Feynman said, this lecture, in this case this Book, requires Infinite Intelligence(II),whilst having Zero Knowledge(ZK). Perhaps I should say II maping to ZK...
August 14, 2014
One of those books that I took longest to read, that too partially with a lot of skimming and I would come back to those parts if I ever familiarize myself with pre-requisites, which among other things, include some minimum basic essentials of Reimannian geometry and topology and the the underlying philosophies. If you have not done so, you can skim the chapter about Geometry and also parts of the last chapter on physical picture of the universe. For complete laymen, Burtt's Metaphysical Foundation of Modern Science and some decent introduction to number theory and real analysis is kind of mandatory to get going. I think I would now move to Weyl's essays and lectures, that is Levels of Infinity and then come back here to re-read some of the parts.
September 1, 2010
You could only give this four stars if you use it like an encyclopaedia or a reference. If you do this, the historical significance of some of the ideas and philosophical points is not withered with the passage of time.
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