Grundlagen einer allgemeinen Mannigfaltigkeitslehre [Foundations of a General Theory of Aggregates] (1883)
The Fabric of Reality (1997)
Quote (1908), # 840, in The Diaries of Paul Klee; University of California Press, 1964; as quoted by Francesco Mazzaferro, in 'The Diaries of Paul Klee - Part Three' : Klee as a Secessionist and a Neo-Impressionist Artist http://letteraturaartistica.blogspot.nl/2015/05/paul-klee-ev.html
1903 - 1910
"Polymathematics: is mathematics a single science or a set of arts?", in Mathematics: Frontiers and Perspectives (2000), edited by V. I. Arnold, M. Atiyah, P. Lax, and B. Mazur, pp. 403–416.
My Life and Confessions, for Philippine, 1786
Robert Edouard Moritz. On Mathematics and Mathematicians https://archive.org/details/onmathematicsmat00mori, 1914, 1942, 1958; p. v; Preface, lead sentence
Judea Pearl, "Trygve Haavelmo and the emergence of causal calculus." University of California Los Angeles, Computer Science Department, CA. 2012.
A History of Greek Mathematics (1921) Vol. 1. From Thales to Euclid
“Any unwillingness to learn mathematics today can greatly restrict your possibilities tomorrow.”
Methods of Mathematics Applied to Calculus, Probability, and Statistics (1985)
James Joseph Sylvester, Collected Mathematical Papers, Vol. 1 (1904), p. 91.
Univalent Foundations, Vladimir Voevodsky, IAS, March 26, 2014 http://www.math.ias.edu/vladimir/files/2014_IAS.pdf p. 13
Source: Preface to Recreations in Mathematics and Natural Philosophy. (1803), p. vi; As cited in: Tobias George Smollett. The Critical Review: Or, Annals of Literature http://books.google.com/books?id=T8APAAAAQAAJ&pg=PA412, Volume 38, (1803), p. 412
“And not very good poetry.”
Source: The Rolling Stones (1952), Chapter 14, “Flat Cats Factorial” (p. 182)
Source: Lectures on Teaching, (1906), pp. 291-292
Source: Dictionary of Burning Words of Brilliant Writers (1895), P. 580.
John Hicks (1979), quoted in: Nitasha Kaul (2007) Imagining Economics Otherwise. p. 76
Part I, Chapter 3, The Roots of Economic Orthodoxy, p. 43
The Death of Economics (1994)
The Naked Communist (1958)
Preface p. v
A History of Greek Mathematics (1921) Vol. 1. From Thales to Euclid
“My mathematics is simple: one plus one = one.”
"Mathematics," p. 29
The Shape (2000), Sequence: “Happiness of Atoms”
Justice Markandey Katju in Speech delivered on 13.10.2009 in the Indian Institute of Science Bangalore in: Sanskrit As A Language Of Science http://www.iisc.ernet.in/misc/bang_speech.html, Indian Institute of Science.
An Essay in Defence of the Female Sex. P. 54 https://archive.org/details/essayindefenceof00aste
"The Departments of Mathematics, and their Mutual Relations," Journal of Speculative Philosophy, Vol. 5, p. 170. Reported in Moritz (1914)
Journals
Source: Our Modern Idol: Mathematical Science (1984), p. 95.
“Great physics does not automatically imply complicated mathematics!”
[Martinus Veltman, Facts and mysteries in elementary particle physics, World Scientific, 2003, 981238149X, 15, https://books.google.com/books?id=CNCHDIobj0IC&pg=PA15]
Letter to James Clerk Maxwell (25 March 1857), commenting on Maxwell's paper titled "On Faraday's Lines of Force"; letter published in The Life of James Clerk Maxwell: With Selections from His Correspondence (1884), edited by Lewis Campbell and William Garnett, p. 200; also in Coming of Age in the Milky Way (2003) by Timothy Ferris, p. 186
Source: The Technological Society (1954), p. 139
Source: "Presidential Address British Association for the Advancement of Science," 1890, p. 466 : On the need of text-books on higher mathematics
Preface by Karl Pearson
The Common Sense of the Exact Sciences (1885)
Source: "Presidential Address British Association for the Advancement of Science," 1890, p. 466 : On the expansion of the field of mathematics, and on the importance of a well-chosen notation
“In mathematics the art of asking questions is more valuable than solving problems.”
In re mathematica ars proponendi quaestionem pluris facienda est quam solvendi.
Doctoral thesis (1867); variant translation: In mathematics the art of proposing a question must be held of higher value than solving it.
George Horne, written anonymously in his A Fair, Candid, and Impartial Statement of the Case between Sir Isaac Newton and Mr. Hutchinson (1753)
“I lose faith in mathematics, logical and rigid. What with those that even zero doesn’t accept?”
"I and I," p. 30
The Shape (2000), Sequence: “Happiness of Atoms”
Letter to Dr. Bostock (June, 1798) as quoted by George Peacock, Life of Thomas Young (1855)
Source: The Philosophical and Mathematical Commentaries of Proclus on the First Book of Euclid's Elements Vol. 1 (1788), Ch. IV.
"The Mathematician", in The Works of the Mind (1947) edited by R. B. Heywood, University of Chicago Press, Chicago
Context: I think that it is a relatively good approximation to truth — which is much too complicated to allow anything but approximations — that mathematical ideas originate in empirics. But, once they are conceived, the subject begins to live a peculiar life of its own and is … governed by almost entirely aesthetical motivations. In other words, at a great distance from its empirical source, or after much "abstract" inbreeding, a mathematical subject is in danger of degeneration. Whenever this stage is reached the only remedy seems to me to be the rejuvenating return to the source: the reinjection of more or less directly empirical ideas.
Introduction
Higher Mathematics for Chemical Students (1911)
Context: The philosopher Comte has made the statement that chemistry is a non-mathematical science. He also told us that astronomy had reached a stage when further progress was impossible. These remarks, coming after Dalton's atomic theory, and just before Guldberg and Waage were to lay the foundations of chemical dynamics, Kirchhoff to discover the reversal of lines in the solar spectrum, serve but to emphasize the folly of having "recourse to farfetched and abstracted Ratiocination," and should teach us to be "very far from the litigious humour of loving to wrangle about words or terms or notions as empty".
Methods of Mathematics Applied to Calculus, Probability, and Statistics (1985)
Context: Increasingly... the application of mathematics to the real world involves discrete mathematics... the nature of the discrete is often most clearly revealed through the continuous models of both calculus and probability. Without continuous mathematics, the study of discrete mathematics soon becomes trivial and very limited.... The two topics, discrete and continuous mathematics, are both ill served by being rigidly separated.
from On the Method of Theoretical Physics, p. 183. The Herbert Spencer Lecture, delivered at Oxford (10 June 1933). Quoted in Einstein's Philosophy of Science http://plato.stanford.edu/entries/einstein-philscience/
1930s
Context: Our experience hitherto justifies us in trusting that nature is the realization of the simplest that is mathematically conceivable. I am convinced that purely mathematical construction enables us to find those concepts and those lawlike connections between them that provide the key to the understanding of natural phenomena. Useful mathematical concepts may well be suggested by experience, but in no way can they be derived from it. Experience naturally remains the sole criterion of the usefulness of a mathematical construction for physics. But the actual creative principle lies in mathematics. Thus, in a certain sense, I take it to be true that pure thought can grasp the real, as the ancients had dreamed.
“Where there is life there is a pattern, and where there is a pattern there is mathematics.”
The Artful Universe (1995)
Context: Where there is life there is a pattern, and where there is a pattern there is mathematics. Once that germ of rationality and order exists to turn a chaos into a cosmos, then so does mathematics. There could not be a non-mathematical Universe containing living observers.<!-- Ch. 5, p. 230
Part II. Ch. 2 : Mathematical Definitions and Education, p. 128
Variant translation: The chief aim of mathematics teaching is to develop certain faculties of the mind, and among these intuition is by no means the least valuable.
Science and Method (1908)
Context: The principal aim of mathematical education is to develop certain faculties of the mind, and among these intuition is not the least precious. It is through it that the mathematical world remains in touch with the real world, and even if pure mathematics could do without it, we should still have to have recourse to it to fill up the gulf that separates the symbol from reality.
Source: Vestiges of the Natural History of Creation (1844), p. 170-171 ( 1846 edition http://books.google.com/books?id=UkoWAAAAYAAJ)
Context: This statistical regularity in moral affairs fully establishes their being under the presidency of law. Man is now seen to be an enigma only as an individual; in the mass he is a mathematical problem. It is hardly necessary to say, much less to argue, that mental action, being proved to be under law, passes at once into the category of natural things. Its old metaphysical character vanishes in a moment, and the distinction usually taken between physical and moral is annulled, as only an error in terms. This view agrees with what all observation teaches, that mental phenomena flow directly from the brain.
From the Author's Preface to First Edition (1918)
Space—Time—Matter (1952)
Context: It was my wish to present this great subject as an illustration of the itermingling of philosophical, mathematical, and physical thought, a study which is dear to my heart. This could be done only by building up the theory systematically from the foundations, and by restricting attention throughout to the principles. But I have not been able to satisfy these self-imposed requirements: the mathematician predominates at the expense of the philosopher.
Gompers v. United States, 233 U.S. 604, 610 (1914).
1910s
Curriculum Vitae (1843)
Context: What attracted me so strongly and exclusively to mathematics, apart from the actual content, was particularly the specific nature of the mental processes by which mathematical concepts are handled. This way of deducing and discovering new truths from old ones, and the extraordinary clarity and self-evidence of the theorems, the ingeniousness of the ideas... had an irresistible fascination for me. Beginning from the individual theorems, I grew accustomed to delve more deeply into their relationships and to grasp whole theories as a single entity. That is how I conceived the idea of mathematical beauty...
"Will Mathematics Survive? Report on the Zurich Congress" in The Mathematical Intelligencer, Vol. 17, no. 3 (1995), pp. 6–10.
Context: At the beginning of this century a self-destructive democratic principle was advanced in mathematics (especially by Hilbert), according to which all axiom systems have equal right to be analyzed, and the value of a mathematical achievement is determined, not by its significance and usefulness as in other sciences, but by its difficulty alone, as in mountaineering. This principle quickly led mathematicians to break from physics and to separate from all other sciences. In the eyes of all normal people, they were transformed into a sinister priestly caste... Bizarre questions like Fermat's problem or problems on sums of prime numbers were elevated to supposedly central problems of mathematics.
Science and the Unseen World (1929)
Context: 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.<!--III, p.37-38
in a review of William Herschel's A Treatise on Sound, Quarterly Review, Vol. 44, No. 88 (January-February 1831), p. 476 http://books.google.com/books?id=742veo7MzswC&printsec=titlepage#PPA476,M1.
also quoted by Brewster himself in his Treatise on Optics http://books.google.com/books?id=opYAAAAAMAAJ&printsec=frontcover#PPR4,M1 and by Dionysius Lardner as frontispiece or presentation of his works (see for example: Popular lectures on science and art http://books.google.com/books?id=uZ9LAAAAMAAJ&pg=PA3, Cabinet Cyclopaedia http://books.google.com/books?id=5T43oHhqyxUC&pg=RA1-PT7).
Context: It is not easy to devise a cure for such a state of things (the declining taste for science). The most obvious remedy is to provide the educated classes with a series of works on popular and practical science, freed from mathematical symbols and technical terms, written in simple and perspicuous language, and illustrated by facts and experiments which are level to the capacity of ordinary minds.
Archimedes or the Future of Physics (1927)
Preface.
Linear Associative Algebra (1882)
Context: I presume that to the uninitiated the formulae will appear cold and cheerless; but let it be remembered that, like other mathematical formulae, they find their origin in the divine source of all geometry. Whether I shall have the satisfaction of taking part in their exposition, or whether that will remain for some more profound expositor, will be seen in the future.
...Ut pendet continaum flexile, sic stabit contiguum rigidum, which is the Linea Catenaria.
Tr: As hangs the flexible line, so but inverted will stand the rigid arch.
Cypher at the end of his A Description of Helioscopes, and Some Other Instruments https://books.google.com/books?id=KQtPAAAAcAAJ (1676) p. 31, as quoted in "The Life of Dr. Robert Hooke," The Posthumous Works of Robert Hooke https://books.google.com/books?id=6xVTAAAAcAAJ, Richard Waller (1705) English translation in Ted Ruddock, Arch Bridges and Their Builders 1735-1835 (1979) p. 46 https://books.google.com/books?id=amQ9AAAAIAAJ&pg=PA46
“All mathematical laws which we find in Nature are always suspect to me, in spite of their beauty.”
As quoted in Lichtenberg : A Doctrine of Scattered Occasions (1959) by Joseph Peter Stern, p. 84
Context: All mathematical laws which we find in Nature are always suspect to me, in spite of their beauty. They give me no pleasure. They are merely auxiliaries. At close range it is all not true.
The close of his Nobel lecture: "The Statistical Interpretations of Quantum Mechanics" (11 December 1954) http://nobelprize.org/nobel_prizes/physics/laureates/1954/born-lecture.html
Context: Can we call something with which the concepts of position and motion cannot be associated in the usual way, a thing, or a particle? And if not, what is the reality which our theory has been invented to describe?
The answer to this is no longer physics, but philosophy. … Here I will only say that I am emphatically in favour of the retention of the particle idea. Naturally, it is necessary to redefine what is meant. For this, well-developed concepts are available which appear in mathematics under the name of invariants in transformations. Every object that we perceive appears in innumerable aspects. The concept of the object is the invariant of all these aspects. From this point of view, the present universally used system of concepts in which particles and waves appear simultaneously, can be completely justified. The latest research on nuclei and elementary particles has led us, however, to limits beyond which this system of concepts itself does not appear to suffice. The lesson to be learned from what I have told of the origin of quantum mechanics is that probable refinements of mathematical methods will not suffice to produce a satisfactory theory, but that somewhere in our doctrine is hidden a concept, unjustified by experience, which we must eliminate to open up the road.
Essay on Atomism: From Democritus to 1960 (1961)
Context: It is widely believed that only those who can master the latest quantum mathematics can understand anything of what is happening. That is not so, provided one takes the long view, for no one can see far ahead. Against a historical background, the layman can understand what is involved, for example, in the fascinating challenge of continuity and discontinuity expressed in the antithesis of field and particle.<!--p.4
“Mathematics alone will blind you and let others rob you.”
Source: The Poker Face of Wall Street (2006), Chapter 9, Who Got Game, p. 279
Context: Mathematical theory, tested in practice and constantly retested, is a valuable aid to play. Mathematics alone will blind you and let others rob you.
in La formation scientifique, Une communication du Prix Nobel d’économie, Maurice Allais http://www.canalacademie.com/Maurice-Allais-la-formation.html, address to the Académie des Sciences Morales et Politiques (1997).
Context: Any author who uses mathematics should always express in ordinary language the meaning of the assumptions he admits, as well as the significance of the results obtained. The more abstract his theory, the more imperative this obligation.
In fact, mathematics are and can only be a tool to explore reality. In this exploration, mathematics do not constitute an end in itself, they are and can only be a means.
“Only by a study of the development of mathematics can its contemporary significance be understood.”
100 Years of Mathematics: a Personal Viewpoint (1981)
Context: The professional mathematician can scarcely avoid specialization and needs to transcend his private interests and take a wide synoptic view of the whole landscape of contemporary mathematics. His scientific colleagues are continually seeking enlightenment on the relevance of mathematical abstractions. The undergraduate needs a guidebook to the topography of the immense and expanding world of mathematics. There seems to be only one way to satisfy these varied interests... a concise historical account of the main currents... Only by a study of the development of mathematics can its contemporary significance be understood.
Surviving the Future, (2016), p. 180, Epilogue http://www.flemingpolicycentre.org.uk/lean-logic-surviving-the-future/
Introduction
Cosmic Imagery: Key Images in the History of Science (2008)
Context: Mathematics became an experimental subject. Individuals could follow previously intractable problems by simply watching what happened when they were programmed into a personal computer.... The PC revolution has made science more visual and more immediate.... by creating films of imaginary experiences of mathematical worlds.... Words are no longer enough.
“Mathematics is written for mathematicians”
Translation as quoted in The Gradual Acceptance of the Copernican Theory of the Universe (1917) by Dorothy Stimson, p. 115
Context: If perchance there should be foolish speakers who, together with those ignorant of all mathematics, will take it upon themselves to decide concerning these things, and because of some place in the Scriptures wickedly distorted to their purpose, should dare to assail this my work, they are of no importance to me, to such an extent do I despise their judgment as rash. For it is not unknown that Lactantius, the writer celebrated in other ways but very little in mathematics, spoke somewhat childishly of the shape of the earth when he derided those who declared the earth had the shape of a ball. So it ought not to surprise students if such should laugh at us also. Mathematics is written for mathematicians to whom these our labors, if I am not mistaken, will appear to contribute something even to the ecclesiastical state the headship of which your Holiness now occupies. (Author's preface to de revolutionibus) http://la.wikisource.org/wiki/Pagina:Nicolai_Copernici_torinensis_De_revolutionibus_orbium_coelestium.djvu/8
Vol. I, p. 238
Joannis Kepleri Astronomi Opera Omnia, ed. Christian Frisch (1858)
Physics and Philosophy (1958)
Context: The words "position" and "velocity" of an electron... seemed perfectly well defined... and in fact they were clearly defined concepts within the mathematical framework of Newtonian mechanics. But actually they were not well defined, as seen from the relations of uncertainty. One may say that regarding their position in Newtonian mechanics they were well defined, but in their relation to nature, they were not. This shows that we can never know beforehand which limitations will be put on the applicability of certain concepts by the extension of our knowledge into the remote parts of nature, into which we can only penetrate with the most elaborate tools. Therefore, in the process of penetration we are bound sometimes to use our concepts in a way which is not justified and which carries no meaning. Insistence on the postulate of complete logical clarification would make science impossible. We are reminded... of the old wisdom that one who insists on never uttering an error must remain silent.
Methods of Mathematics Applied to Calculus, Probability, and Statistics (1985)
Context: It is easy to measure your mastery of the results via a conventional examination; it is less easy to measure your mastery of doing mathematics, of creating new (to you) results, and of your ability to surmount the almost infinite details to see the general situation.
“The very possibility of the science of mathematics seems an insoluble contradiction.”
Source: Science and Hypothesis (1901), Ch. I: On the Nature of Mathematical Reasoning (1905) Tr. https://books.google.com/books?id=5nQSAAAAYAAJ George Bruce Halstead
Context: The very possibility of the science of mathematics seems an insoluble contradiction. If this science is deductive only in appearance, whence does it derive that perfect rigor no one dreams of doubting? If, on the contrary, all the propositions it enunciates can be deduced one from another by the rules of formal logic, why is not mathematics reduced to an immense tautology? The syllogism can teach us nothing essentially new, and, if everything is to spring from the principle of identity, everything should be capable of being reduced to it. Shall we then admit that the enunciations of all those theorems which fill so many volumes are nothing but devious ways of saying A is A!... Does the mathematical method proceed from particular to the general, and, if so, how can it be called deductive?... If we refuse to admit these consequences, it must be conceded that mathematical reasoning has of itself a sort of creative virtue and consequently differs from a syllogism.<!--pp.5-6
“The great advances in mathematics have not been made by logic but by creative imagination.”
100 Years of Mathematics: a Personal Viewpoint (1981)
Context: Logical analysis is indispensable for an examination of the strength of a mathematical structure, but it is useless for its conception and design. The great advances in mathematics have not been made by logic but by creative imagination.
"On teaching mathematics", as translated by A. V. Goryunov, in Russian Mathematical Surveys Vol. 53, no. 1 (1998), p. 229–236.
Context: In the middle of the twentieth century it was attempted to divide physics and mathematics. The consequences turned out to be catastrophic. Whole generations of mathematicians grew up without knowing half of their science and, of course, in total ignorance of any other sciences. They first began teaching their ugly scholastic pseudo-mathematics to their students, then to schoolchildren (forgetting Hardy's warning that ugly mathematics has no permanent place under the Sun).
On the Social State of Marxism (1978)
New Theories of Everything (2007)
Context: Scanning the past millennia of human achievement reveals just how much has been achieved during the last three hundred years since Newton set in motion the effective mathematization of Nature. We found that the world is curiously adapted to a simple mathematical description. It is enigma enough that the world is described by mathematics; but by simple mathematics, of the sort that a few years energetic study now produces familiarity with, this is an enigma within an enigma.<!--Ch. 1, p. 2
Source: Language, Truth, and Logic (1936), p. 77.
Context: The principles of logic and mathematics are true simply because we never allow them to be anything else. And the reason for this is that we cannot abandon them without contradicting ourselves, without sinning against the rules which govern the use of language, and so making our utterances self-stultifying. In other words, the truths of logic and mathematics are analytic propositions or tautologies.
“Contrary to popular opinion, mathematics is about simplifying life, not complicating it.”
Source: The (Mis)Behavior of Markets (2004, 2008), Ch. 7, p. 125
Context: Contrary to popular opinion, mathematics is about simplifying life, not complicating it. A child learns a bag of candies can be shared fairly by counting them out: That is numeracy. She abstracts that notion to dividing a candy bar into equal pieces: arithmetic. Then, she learns how to calculate how much cocoa and sugar she will need to make enough chocolate for fifteen friends: algebra.
Ideas and Opinions (1954), pp. 238–239; quoted in "Einstein's Philosophy of Science" http://plato.stanford.edu/entries/einstein-philscience/
1950s
Context: The theory of relativity is a beautiful example of the basic character of the modern development of theory. That is to say, the hypotheses from which one starts become ever more abstract and more remote from experience. But in return one comes closer to the preeminent goal of science, that of encompassing a maximum of empirical contents through logical deduction with a minimum of hypotheses or axioms. The intellectual path from the axioms to the empirical contents or to the testable consequences becomes, thereby, ever longer and more subtle. The theoretician is forced, ever more, to allow himself to be directed by purely mathematical, formal points of view in the search for theories, because the physical experience of the experimenter is not capable of leading us up to the regions of the highest abstraction. Tentative deduction takes the place of the predominantly inductive methods appropriate to the youthful state of science. Such a theoretical structure must be quite thoroughly elaborated in order for it to lead to consequences that can be compared with experience. It is certainly the case that here, as well, the empirical fact is the all-powerful judge. But its judgment can be handed down only on the basis of great and difficult intellectual effort that first bridges the wide space between the axioms and the testable consequences. The theorist must accomplish this Herculean task with the clear understanding that this effort may only be destined to prepare the way for a death sentence for his theory. One should not reproach the theorist who undertakes such a task by calling him a fantast; instead, one must allow him his fantasizing, since for him there is no other way to his goal whatsoever. Indeed, it is no planless fantasizing, but rather a search for the logically simplest possibilities and their consequences.
Source: Flatland: A Romance of Many Dimensions (1884), PART II: OTHER WORLDS, Chapter 16. How the Stranger Vainly Endeavoured to Reveal to Me in Words the Mysteries of Spaceland
Context: You are living on a Plane. What you style Flatland is the vast level surface of what I may call a fluid, on, or in, the top of which you and your countrymen move about, without rising above it or falling below it.I am not a plane Figure, but a Solid. You call me a Circle; but in reality I am not a Circle, but an infinite number of Circles, of size varying from a Point to a Circle of thirteen inches in diameter, one placed on the top of the other. When I cut through your plane as I am now doing, I make in your plane a section which you, very rightly, call a Circle. For even a Sphere — which is my proper name in my own country — if he manifest himself at all to an inhabitant of Flatland — must needs manifest himself as a Circle.Do you not remember — for I, who see all things, discerned last night the phantasmal vision of Lineland written upon your brain — do you not remember, I say, how, when you entered the realm of Lineland, you were compelled to manifest yourself to the King, not as a Square, but as a Line, because that Linear Realm had not Dimensions enough to represent the whole of you, but only a slice or section of you? In precisely the same way, your country of Two Dimensions is not spacious enough to represent me, a being of Three, but can only exhibit a slice or section of me, which is what you call a Circle.The diminished brightness of your eye indicates incredulity. But now prepare to receive proof positive of the truth of my assertions. You cannot indeed see more than one of my sections, or Circles, at a time; for you have no power to raise your eye out of the plane of Flatland; but you can at least see that, as I rise in Space, so my sections become smaller. See now, I will rise; and the effect upon your eye will be that my Circle will become smaller and smaller till it dwindles to a point and finally vanishes.There was no "rising" that I could see; but he diminished and finally vanished. I winked once or twice to make sure that I was not dreaming. But it was no dream. For from the depths of nowhere came forth a hollow voice — close to my heart it seemed — "Am I quite gone? Are you convinced now? Well, now I will gradually return to Flatland and you shall see my section become larger and larger."Every reader in Spaceland will easily understand that my mysterious Guest was speaking the language of truth and even of simplicity. But to me, proficient though I was in Flatland Mathematics, it was by no means a simple matter.
Source: The Philosophical and Mathematical Commentaries of Proclus on the First Book of Euclid's Elements Vol. 1 (1788), Ch. IV.
Source: Infinite in All Directions (1988), Ch. 3 : Manchester and Athens
Context: Fifty years ago Kurt Gödel... proved that the world of pure mathematics is inexhaustible. … I hope that the notion of a final statement of the laws of physics will prove as illusory as the notion of a formal decision process for all mathematics. If it should turn out that the whole of physical reality can be described by a finite set of equations, I would be disappointed, I would feel that the Creator had been uncharacteristically lacking in imagination.
Knowledge and Power : The Information Theory of Capitalism and How it is Revolutionizing our World (2013), Ch. 10: Romer's Recipes and Their Limits <!-- Regnery Publishing -->
Context: Academic scientists of any sort expect to be struck by lightning if they celebrate real creation de novo in the world. One does not expect modern scientists to address creation by God. They have a right to their professional figments such as infinite multiple parallel universes. But it is a strange testimony to our academic life that they also feel it necessary of entrepreneurship to chemistry and cuisine, Romer finally succumbs to the materialist supersition: the idea that human beings and their ideas are ultimately material. Out of the scientistic fog there emerged in the middle of the last century the countervailling ideas if information theory and computer science. The progenitor of information theory, and perhaps the pivotal figure in the recent history of human thought, was Kurt Gödel, the eccentric Austrian genius and intimate of Einstein who drove determinism from its strongest and most indispensable redoubt; the coherence, consistency, and self-sufficiency of mathematics.
Gödel demonstrated that every logical scheme, including mathematics, is dependent upon axioms that it cannot prove and that cannot be reduced to the scheme itself. In an elegant mathematical proof, introduced to the world by the great mathematician and computer scientist John von Neumann in September 1930, Gödel demonstrated that mathematics was intrinsically incomplete. Gödel was reportedly concerned that he might have inadvertently proved the existence of God, a faux pas in his Viennese and Princeton circle. It was one of the famously paranoid Gödel's more reasonable fears. As the economist Steven Landsberg, an academic atheist, put it, "Mathematics is the only faith-based science that can prove it."
The Rights of Conscience Inalienable (1791)
Context: Government has no more to do with the religions opinions of men, than it has with the principles of mathematics. Let every man speak freely without fear, maintain the principles that he believes, worship according to his own faith, either one God, three Gods, no God, or twenty Gods; and let government protect him in so doing, i. e., see that he meets with no personal abuse, or loss of property, from his religious opinions. (p. 184)
The Days of My Life : An Autobiography (1989), p. 212
Context: My first TV series on demonstrations in physics — titled Why Is It So? were now seen and heard over the land. The mail was massive. The academics were a special triumph for me. They charged me with being superficial and trivial. If I had done what they wanted my programs would be as dull as their classes! I knew my purpose well and clear: to show how Nature behaves without cluttering its beauty with abtruse mathematics. Why cloud the charm of a Chladni plate with a Bessel function?
On the Social State of Marxism (1978)
As We May Think (1945)
Context: If scientific reasoning were limited to the logical processes of arithmetic, we should not get far in our understanding of the physical world. One might as well attempt to grasp the game of poker entirely by the use of the mathematics of probability. The abacus, with its beads strung on parallel wires, led the Arabs to positional numeration and the concept of zero many centuries before the rest of the world; and it was a useful tool — so useful that it still exists.
Source: Vestiges of the Natural History of Creation (1844), p. 35
Context: It is remarkable of the simple substances that they are generally in some compound form. Thus oxygen and nitrogen, though in union they form the aerial envelope of the globe, are never found separate in nature. Carbon is pure only in the diamond. And the metallic bases of the earths, though the chemist can disengage them, may well be supposed unlikely to remain long uncombined, seeing that contact with moisture makes them burn. Combination and re-combination are principles largely pervading nature. There are few rocks, for example, that are not composed of at least two varieties of matter, each of which is again a compound of elementary substances. What is still more wonderful with respect to this principle of combination, all the elementary substances observe certain mathematical proportions in their unions. It is hence supposed that matter is composed of infinitely minute particles or atoms, each of which belonging to any one substance, can only (through the operation of some as yet hidden law) associate with a certain number of the atoms of any other.
This work is also noteworthy because it contains the first of an effort to represent the imaginary number graphically by the method now used. The effort stopped short of success but was an ingenious beginning.
History of Mathematics (1923) Vol.1
The Common Law (1881), p. 1.
1880s
100 Years of Mathematics: a Personal Viewpoint (1981)
Source: The Value of Science (1905), Ch. 5: Analysis and Physics
Context: All laws are... deduced from experiment; but to enunciate them, a special language is needful... ordinary language is too poor...
This... is one reason why the physicist can not do without mathematics; it furnishes him the only language he can speak. And a well-made language is no indifferent thing;
... the analyst, who pursues a purely esthetic aim, helps create, just by that, a language more fit to satisfy the physicist.
... law springs from experiment, but not immediately. Experiment is individual, the law deduced from it is general; experiment is only approximate, the law is precise...
In a word, to get the law from experiment, it is necessary to generalize... But how generalize?... in this choice what shall guide us?
It can only be analogy.... What has taught us to know the true profound analogies, those the eyes do not see but reason divines?
It is the mathematical spirit, which disdains matter to cling only to pure form.<!--pp.76-77
Letter to Abigail Adams (12 May 1780)
1780s
Context: The science of government it is my duty to study, more than all other sciences; the arts of legislation and administration and negotiation ought to take the place of, indeed exclude, in a manner, all other arts. I must study politics and war, that our sons may have liberty to study mathematics and philosophy. Our sons ought to study mathematics and philosophy, geography, natural history and naval architecture, navigation, commerce and agriculture in order to give their children a right to study painting, poetry, music, architecture, statuary, tapestry and porcelain.
“The assumptions and definitions of mathematics and science come from our intuition”
Methods of Mathematics Applied to Calculus, Probability, and Statistics (1985)
Context: The assumptions and definitions of mathematics and science come from our intuition, which is based ultimately on experience. They then get shaped by further experience in using them and are occasionally revised. They are not fixed for all eternity.
from "Doron Zeilberger's 126th Opinion", Dr. Z's Homepage http://www.math.rutgers.edu/~zeilberg/Opinion126.html
Mathematical Methods in Science (1977)
Context: We wish to see... the typical attitude of the scientist who uses mathematics to understand the world around us.... In the solution of a problem... there are typically three phases. The first phase is entirely or almost entirely a matter of physics; the third, a matter of mathematics; and the intermediate phase, a transition from physics to mathematics. The first phase is the formulation of the physical hypothesis or conjecture; the second, its translation into equations; the third, the solution of the equations. Each phase calls for a different kind of work and demands a different attitude.<!--p.164
“Mathematics… is the set of all possible self-consistent structures”
Source: Hyperspace (1995), Ch.15 Conclusion<!--p.328-->
Context: Mathematics... is the set of all possible self-consistent structures, and there are vastly more logical structures than physical principles.
Gauss-Schumacher Briefwechsel (1862)
Context: It may be true, that men, who are mere mathematicians, have certain specific shortcomings, but that is not the fault of mathematics, for it is equally true of every other exclusive occupation. So there are mere philologists, mere jurists, mere soldiers, mere merchants, etc. To such idle talk it might further be added: that whenever a certain exclusive occupation is coupled with specific shortcomings, it is likewise almost certainly divorced from certain other shortcomings.
Reported in Eugene Gerhart, America's Advocate: Robert H. Jackson (1958), p. 289
