William C Dampier - Collected Quotes

"[…] we can only study Nature through our senses - that is […] we can only study the model of Nature that our senses enable our minds to construct; we cannot decide whether that model, consistent though it be, represents truly the real structure of Nature; whether, indeed, there be any Nature as an ultimate reality behind its phenomena." (William C Dampier, "The Recent Development of Physical Science", 1904)

"Confronted with the mystery of the Universe, we are driven to ask if the model our minds have framed at all corresponds with the reality; if, indeed, there be any reality behind the image." (Sir William C Dampier, "The Recent Development of Physical Science", 1904)

"The different sciences are not even parts of a whole; they are but different aspects of a whole, which essentially has nothing in it corresponding to the divisions we make; they are, so to speak, sections of our model of Nature in certain arbitrary planes, cut in directions to suit our convenience." (Sir William C Dampier, "The Recent Development of Physical Science", 1904)

"The mind of man, learning consciously and unconsciously lessons of experience, gradually constructs a mental image of its surroundings - as the mariner draws a chart of strange coasts to guide him in future voyages, and to enable those that follow after him to sail the same seas with ease and safety." (William C Dampier, "The Recent Development of Physical Science" , 1904)

"We can only study Nature through our senses – that is […] we can only study the model of Nature that our senses enable our minds to construct; we cannot decide whether that model, consistent though it be, represents truly the real structure of Nature; whether, indeed, there be any Nature as an ultimate reality behind its phenomena." (Sir William C Dampier, "The Recent Development of Physical Science", 1904)

"A false hypothesis, if it serve as a guide for further enquiry, may, at the right stage of science, be as useful as, or more useful than, a truer one for which acceptable evidence is not yet at hand." (William C Dampier, "Science and the Human Mind" , 1912)

"Sometimes the probability in favor of a generalization is enormous, but the infinite probability of certainty is never reached." (William C Dampier, "Science and the Human Mind" , 1912)

"Indeed the intellectual basis of all empirical knowledge may be said to be a matter of probability, expressible only in terms of a bet." (William C Dampier, "A History of Science and its Relations with Philosophy and Religion", 1936) 

"The fundamental concepts of physical science, it is now understood, are abstractions, framed by our mind, so as to bring order to an apparent chaos of phenomena." (Sir William C Dampier, "A History of Science and its Relations with Philosophy and Religion", 1929)

Émile Boutroux - Collected Quotes

"In spite of their relations, science and religion remain, and must remain, distinct. If there were no other way of establishing a rational order between things than that of reducing the many to the one, either by assimilation or by elimination, the destiny of religion would appear doubtful." (Émile Boutroux, "Science and Religion in Contemporary Philosophy", 1908)

"There can be nothing clearer or more convenient for the purpose of setting one's ideas in order and for conducting an abstract discussion, than precise definitions and inviolable lines of demarcation." (Émile Boutroux, "Science and Religion in Contemporary Philosophy", 1908)

"The senses afford a primary conception of the world, which they show to be a mass of facts, endless in their variety. Man may observe, analyse, and describe them with ever-increasing exactness: this very description constitutes science." (Émile Boutroux, "The Contingency of the Laws of Nature", 1911)

"Science is reduction. Mathematics is its ideal, its form par excellence, for it is in mathematics that assimilation, identification, is most perfectly realized. The universe, scientifically explained, would be a certain formula, one and eternal, regarded as the equivalent of the entire diversity and movement of things." (Émile Boutroux, "Natural law in Science and Philosophy", 1914)

"The mathematical laws presuppose a very complex elaboration. They are not known exclusively either a priori or a posteriori, but are a creation of the mind; and this creation is not an arbitrary one, but, owing to the mind’s resources, takes place with reference to experience and in view of it. Sometimes the mind starts with intuitions which it freely creates; sometimes, by a process of elimination, it gathers up the axioms it regards as most suitable for producing a harmonious development, one that is both simple and fertile. The mathematics is a voluntary and intelligent adaptation of thought to things, it represents the forms that will allow of qualitative diversity being surmounted, the moulds into which reality must enter in order to become as intelligible as possible." (Émile Boutroux, "Natural Law in Science and Philosophy", 1914)

"The philosopher asks himself whether natural law as assumed by science, wholly coincides with law as really existing in nature; whether science and reality are so alike that science may be regarded as exhausting everything intelligible and true that the real contains." (Émile Boutroux, "Natural Law in Science and Philosophy", 1914)

"The world is an endless variety of facts, linked together by necessary and immutable bonds." (Émile Boutroux, "Natural law in Science and Philosophy", 1914)

Science vs Religion IV

"Modern masters of science are much impressed with the need of beginning all inquiry with a fact. The ancient masters of religion were quite equally impressed with that necessity. They began with the fact of sin - a fact as practical as potatoes. Whether or not man could be washed in miraculous waters, there was no doubt at any rate that he wanted washing." (Gilbert K Chesterton, "Orthodoxy", 1908)

"Art and Religion are, then, two roads by which men escape from circumstance to ecstasy. Between aesthetic and religious rapture there is a family alliance. Art and Religion are means to similar states of mind." (Clive Bell, "Art", 1913)

"The truths of religion are unprovable; the facts of science are unproved." (Gilbert K Chesterton, "Christian Science", 1920)

"Like the great temples of some religions, mathematics may be viewed only from the outside by those uninitiated into its mysteries. (Andrew M Gleason, "Evolution of an Active Mathematical Theory", Science Vol. 145 (3631), 1964)

"All religions, arts and sciences are branches of the same tree. All these aspirations are directed toward ennobling man’s life, lifting it from the sphere of mere physical existence and leading the individual towards freedom." (Albert Einstein, "Out of My Later Years", 1937)

"A religion old or new, that stressed the magnificence of the universe as revealed by modern science, might be able to draw forth reserves of reverence and awe hardly tapped by the conventional faiths. Sooner or later such a religion will emerge." (Carl Sagan, "Pale Blue Dot: a Vision of the Human Future in Space", 1994)

"Humans may crave absolute certainty; they may aspire to it; they may pretend, as partisans of certain religions do, to have attained it. But the history of science - by far the most successful claim to knowledge accessible to humans - teaches that the most we can hope for is successive improvement in our understanding, learning from our mistakes, an asymptotic approach to the Universe, but with the proviso that absolute certainty will always elude us. We will always be mired in error. The most each generation can hope for is to reduce the error bars a little, and to add to the body of data to which error bars apply." (Carl Sagan, "The Demon-Haunted World: Science as a Candle in the Dark", 1995)

"Does there truly exist an insuperable contradiction between religion and science? Can religion be superseded by science? The answers to these questions have, for centuries, given rise to considerable dispute and, indeed, bitter fighting. Yet, in my own mind there can be no doubt that in both cases a dispassionate consideration can only lead to a negative answer. What complicates the solution, however, is the fact that while most people readily agree on what is meant by ‘science,’ they are likely to differ on the meaning of ‘religion’." (Albert Einstein)

"From religion comes a man's purpose; from science, his power to achieve it. Sometimes people ask if religion and science are not opposed to one another. They are: in the sense that the thumb and fingers of my hands are opposed to one another. It is an opposition by means of which anything can be grasped." (Sir William Bragg)

"He who posseses science and art, has religion; he who possesses neither science nor art, let him get religion." (Johann Wolfgang von Goethe)

Mathematics vs Religion I

"The mathematics are the friends to religion, inasmuch as they charm the passions, restrain the impetuosity of the imagination, and purge the mind from error and prejudice. Vice is error, confusion and false reasoning; and all truth is more or less opposite to it. Besides, mathematical truth may serve for a pleasant entertainment for those hours which young men are apt to throw away upon their vices; the delightfulness of them being such as to make solitude not only easy but desirable." (John Arbuthnot, "An Essay on the Usefulness of Mathematical Learning", 1701) 

"Like the great temples of some religions, mathematics may be viewed only from the outside by those uninitiated into its mysteries." (Andrew M Gleason, "Evolution of an Active Mathematical Theory", Science Vol. 145 (3631), 1964)
 
"The laws of nature 'discovered' by science are merely mathematical or mechanical models that describe how nature behaves, not why, nor what nature 'actually' is. Science strives to find representations that accurately describe nature, not absolute truths. This fact distinguishes science from religion." (George O Abell, "Exploration of the Universe", 1969)

"In many ways, the mathematical quest to understand infinity parallels mystical attempts to understand God. Both religions and mathematics attempt to express the relationships between humans, the universe, and infinity. Both have arcane symbols and rituals, and impenetrable language. Both exercise the deep recesses of our mind and stimulate our imagination. Mathematicians, like priests, seek ‘ideal’, immutable, nonmaterial truths and then often try to apply theses truth in the real world."  (Clifford A Pickover, "The Loom of God: Mathematical Tapestries at the Edge of Time", 1997)

"Zero is powerful because it is infinity’s twin. They are equal and opposite, yin and yang. They are equally paradoxical and troubling. The biggest questions in science and religion are about nothingness and eternity, the void and the infinite, zero and infinity. The clashes over zero were the battles that shook the foundations of philosophy, of science, of mathematics, and of religion. Underneath every revolution lay a zero - and an infinity." (Charles Seife, "Zero: The Biography of a Dangerous Idea", 2000)

"Mathematics is the life supreme. The life of the gods is mathematics. All divine messengers are mathematicians. Pure mathematics is religion. Its attainment requires a theophany." (Friederich von Hardenberg [Novalis])

"‘Tis of singular use, rightly to understand, and carefully to distinguish from hypotheses or mere suppositions, the true and certain consequences of experimental and mathematical philosophy; which do, with wonderful strength and advantage, to all such as are capable of apprehending them, confirm, establish, and vindicate against all objections, those great and fundamental truths of natural religion, which the wisdom of providence has at the same time universally implanted, in some degree, in the minds of persons even of the meanest capacities, not qualified to examine demonstrative proofs." (Samuel Clarke)

Science vs Religion III

"The essence of religion is inertia; the essence of science is change. It is the function of the one to preserve, it is the function of the other to improve. If, as in Egypt, they are firmly chained together, either science will advance, in which case the religion will be altered, or the religion will preserve its purity, and science will congeal." (Winwood Reade, "The Martyrdom of Man", 1876)

"The history of thought should warn us against concluding that because the scientific theory of the world is the best that has yet to be formulated, it is necessarily complete and final. We must remember that at bottom the generalizations of science, or, in common parlance, the laws of nature are merely hypotheses devised to explain that ever-shifting phantasmagoria of thought which we dignify with the high-sounding names of the world and the universe. In the last analysis, magic, religion, and science are nothing but theories of thought." (Sir James G Frazer, "The Golden Bough", 1890)

"Both religion and natural science require a belief in God for their activities, to the former He is the starting point, and to the latter the goal of every thought process. To the former He is the foundation, to the latter, the crown of the edifice of every generalized world view." (Max Planck, "Religion and Natural Science", 1937)

"All men seek to be enlightened. Religion is but the most ancient and honorable way in which men have striven to make sense out of God's universe. Scientists seek the lawfulness of events. It is the task of Religion to fit man into this lawfulness." (Frank Herbert, "Dune", 1965)

"Both religion and science must preserve their autonomy and their distinctiveness. Religion is not founded on science nor is science an extension of religion. Each should possess its own principles, its pattern of procedures, its diversities of interpretation and its own conclusions." (Pope John Paul II, [Letter to Father George V Coyne], 1988)

"Religion is the antithesis of science; science is competent to illuminate all the deep questions of existence, and does so in a manner that makes full use of, and respects the human intellect. I see neither need nor sign of any future reconciliation." (Peter W Atkins, "Religion - The Antithesis to Science", 1997)

"As knowledge advances, science ceases to scoff at religion; and religion ceases to frown on science. The hour of mockery by the one, and of reproof by the other, is passing away. Henceforth, they will dwell together in unity and goodwill. They will mutually illustrate the wisdom, power, and grace of God. Science will adorn and enrich religion; and religion will ennoble and sanctify science." (Oliver W Holmes)

"Science is opposed to theological dogmas because science is founded on fact. To me, the universe is simply a great machine which never came into being and never will end. The human being is no exception to the natural order. Man, like the universe, is a machine." (Nikola Tesla)

"Science without religion is lame, religion without science is blind." (Albert Einstein)

"When confronted with the order and beauty of the universe and the strange coincidences of nature, it's very tempting to take the leap of faith from science into religion. I am sure many physicists want to. I only wish they would admit it." (Tony Rothman)

Science vs Religion II

"The belief in science has replaced in large measure, the belief in God. Even where religion was regarded as compatible with science, it was modified by the mentality of the believer in scientific truth." (Hans Reichenbach, "The Rise of Scientific Philosophy", 1951)

"Religion and science are the two conjugated faces of phases of one and the same act of complete knowledge - the only one which can embrace the past and future of evolution so as to contemplate, measure and fulfill them." (Pierre T de Chardin, "The Phenomenon of Man", 1959)

"Science, philosophy and religion are bound to converge as they draw nearer to the whole." (Pierre T de Chardin, "The Phenomenon of Man", 1959)

"Scientists seek the lawfulness of events. It is the task of Religion to fit man into this lawfulness." (Frank Herbert, "Dune", 1965)

"The laws of nature 'discovered' by science are merely mathematical or mechanical models that describe how nature behaves, not why, nor what nature 'actually' is. Science strives to find representations that accurately describe nature, not absolute truths. This fact distinguishes science from religion." (George O Abell, "Exploration of the Universe", 1969)

"The mystery of sound is mysticism; the harmony of life is religion. The knowledge of vibrations is metaphysics, the analysis of atoms is science, and their harmonious grouping is art. The rhythm of form is poetry, and the rhythm of sound is music. This shows that music is the art of arts and the science of all sciences; and it contains the fountain of all knowledge within itself." (Inayat Khan, "The Mysticism of Sound and Music", 1996)

"To look at the development of physics since Newton is to observe a struggle to define the limits of science. Part of this process has been the intrusion of scientific methods and ideas into domains that have traditionally been the province of metaphysics or religion. In this conflict, Hawking’s phrase ‘to know the Mind of God’ is just one example of a border infringement. But by playing the God card, Hawking has cleverly fanned the flames of his own publicity, appealing directly to the popular allure of the scientist-as-priest."  (Peter Coles, "Hawking and the Mind of God", 2000)

"Zero is powerful because it is infinity’s twin. They are equal and opposite, yin and yang. They are equally paradoxical and troubling. The biggest questions in science and religion are about nothingness and eternity, the void and the infinite, zero and infinity. The clashes over zero were the battles that shook the foundations of philosophy, of science, of mathematics, and of religion. Underneath every revolution lay a zero - and an infinity." (Charles Seife, "Zero: The Biography of a Dangerous Idea", 2000)

"Religion claims to help us understand things about the universe, but, unlike science has no way to test or verify its claims. Both science and religion compete to understand reality, but only science has the method to verify its findings, while religion merely buttresses emotional and epistemic commitments made in advance, commitments impervious to evidence." (Jerry Coyne, 2015)

"Much of the history of science, like the history of religion, is a history of struggles driven by power and money. And yet, this is not the whole story. Genuine saints occasionally play an important role, both in religion and science. For many scientists, the reward for being a scientist is not the power and the money but the chance of catching a glimpse of the transcendent beauty of nature." (Freeman J Dyson)

Science vs Religion I

"Therefore science always goes abreast with the just elevation of the man, keeping step with religion and metaphysics; or, the state of science is an index of our self-knowledge." (Ralph W Emerson, "The Poet", 1844)

"A mere inference or theory must give way to a truth revealed; but a scientific truth must be maintained, however contradictory it may appear to the most cherished doctrines of religion." (David Brewster, "More Worlds Than One: The Creed of the Philosopher and the Hope of the Christian", 1856)

"Without poetry our science will appear incomplete, and most of what now passes with us for religion and philosophy will be replaced by poetry." (Matthew Arnold, "The Study of Poetry", 1880)

"Science boasts of the distance of its stars; of the terrific remoteness of the things of which it has to speak. But poetry and religion always insist upon the proximity, the almost menacing closeness of the things with which they are concerned." (Gilbert K Chesterton, 'A Glimpse of My Country', 1909)

"For a long time it has been known that the first systems of representations with which men have pictured to themselves the world and themselves were of religious origin. There is no religion that is not a cosmology at the same time that it is a speculation upon divine things. If philosophy and the sciences were born of religion, it is because religion began by taking the place of the sciences and philosophy." (Émile Durkheim, "The Elementary Forms of the Religious Life", 1912)

"The mysteries of religion are of a different order from those of science; they are parts of an arbitrary system of man’s own creation; they contradict our reason and our experience, while the mysteries of science are revealed by our reason, and transcend our experience." (John Burroughs, "Scientific Faith", The Atlantic Monthly, 1915)

"Progress in truth - truth of science and truth of religion - is mainly a progress in the framing of concepts, in discarding artificial abstractions or partial metaphors, and in evolving notions which strike more deeply into the root of reality." (Alfred N Whitehead, "Religion in the Making", 1926)

"It is impossible to make a clear cut between science, religion, and art. The whole is never equal simply to the sum of its various parts." (Max Planck, "The Philosophy of Physics", 1936)

"Since science’s competence extends to observable and measurable phenomena, not to the inner being of things, and to the means, not to the ends of human life, it would be nonsense to expect that the progress of science will provide men with a new type of metaphysics, ethics, or religion."
(Jacques Maritain, "Science and Ontology", Bulletin of the Atomic Scientists Vol. 5, 1944)

"Science always goes abreast with the just elevation of the man, keeping step with religion and metaphysics; or, the state of science is an index of our self-knowledge." (Ralph W Emerson)

On Metaphysics (1850-1874)

“Metaphysics. The science to which ignorance goes to learn its knowledge, and knowledge to learn its ignorance. On which all men agree that it is the key, but no two upon how it is to be put into the lock.” (Augustus De Morgan, [letter to Dr. Whewell] 1850)

“Religion is the metaphysics of the masses […] Just as they have popular poetry, and the popular wisdom of proverbs, so they must have popular metaphysics too: for mankind absolutely needs an interpretation of life; and this, again, must be suited to popular comprehension.” (Arthur Schopenhauer, “The Horrors and Absurdities of Religion”, 1851)

“Some think to avoid the influence of metaphysical errors, by paying no attention to metaphysics; but experience shows that these men beyond all others are held in an iron vice of metaphysical theory, because by theories that they have never called in question.” (Charles S Peirce, 1867)

“Nor do I know any study which can compete with mathematics in general in furnishing matter for severe and continued thought. Metaphysical problems may be even more difficult; but then they are far less definite, and, as they rarely lead to any precise conclusion, we miss the power of checking our own operations, and of discovering whether we are thinking and reasoning or merely fancying and dreaming.” (Isaac Todhunter, “Conflict of Studies and Other Essays”, 1873)

"The history of civilization proves beyond doubt just how sterile the repeated attempts of metaphysics to guess at nature’s laws have been. Instead, there is every reason to believe that when the human intellect ignores reality and concentrates within, it can no longer explain the simplest inner workings of life’s machinery or of the world around us." (Santiago Ramón y Cajal, "Reglas y Consejos sobre Investigacíon Cientifica: Los tónicos de la voluntad", 1897)

On Metaphysics (1940-1949)

“Since science’s competence extends to observable and measurable phenomena, not to the inner being of things, and to the means, not to the ends of human life, it would be nonsense to expect that the progress of science will provide men with a new type of metaphysics, ethics, or religion.”  (Jacques Maritain, “Science and Ontology”, Bulletin of the Atomic Scientists Vol. 5, 1944)

“The complete use of pure reason brings us finally from physical to metaphysical knowledge. But the concepts of metaphysical knowledge do not in themselves fully satisfy the demand of our integral being. They are indeed entirely satisfactory to the pure reason itself, because they are the very stuff of its own existence. But our nature sees things through two eyes always, for it views them doubly as idea and as fact and therefore every concept is incomplete for us and to a part of our nature almost unreal until it becomes an experience.” (Sri Aurobindo, “The Life Divine”, 1944)

"In various ways, methods of approaching the mathematician's ideal were sought, and the resulting suggestions were the source of much that was mistaken in metaphysics and theory of knowledge.” (Bertrand Russell, “A History of Western Philosophy”, 1945)

 “[…] science, properly interpreted, is not dependent on any sort of metaphysics. It merely attempts to cover a maximum of facts by a minimum of laws.” (Herbert Feigl, “Naturalism and Humanism”, American Quarterly, Vol. 1, No. 2, 1949)

“When a scientific theory is firmly established and confirmed, it changes its character and becomes a part of the metaphysical background of the age: a doctrine is transformed into a dogma.” (Max Born, “Natural Philosophy of Cause and Chance”, 1949)

On Metaphysics (1800-1849)

"With the synthesis of every new concept in the aggregation of coordinate characteristics the extensive or complex distinctness is increased; with the further analysis of concepts in the series of subordinate characteristics the intensive or deep distinctness is increased. The latter kind of distinctness, as it necessarily serves the thoroughness and conclusiveness of cognition, is therefore mainly the business of philosophy and is carried farthest especially in metaphysical investigations." (Immanuel Kant, “Logic”, 1800)

“Metaphysical inquiry attempts to trace things to the very first stage in which they can, even to the most penetrating intelligences, be the subjects of a thought, a doubt, or a proposition; that profoundest abstraction, where they stand on the first step of distinction from nonentity, and where that one question might be put concerning them, the answer to which would leave no further question possible. And having thus abstracted and penetrated to the state of pure entity, the speculation would come back, tracing it into all its modes and relations; till at last metaphysical truth, approaching nearer and nearer to the sphere of our immediate knowledge, terminates on the confines of distinct sciences and obvious realities. Now, it would seem evident that this inquiry into primary truth must surpass, in point of dignity, all other speculations. If any man could carry his discoveries as far, and make his proofs as strong, in the metaphysical world, as Newton did in the physical, he would be an incomparably greater man than even Newton.” (John Foster, “Essays”, cca. 1805)

“Physics is unable to stand on its own feet, but needs a metaphysics on which to support itself, whatever fine airs it may assume towards the latter.” (Arthur Schopenhauer, “The World as Will and Representation”, 1819)

“The science of the mathematics performs more than it promises, but the science of metaphysics promises more than it performs. The study of the mathematics, like the Nile, begins in minuteness but ends in magnificence; but the study of metaphysics begins with a torrent of tropes, and a copious current of words, yet loses itself at last in obscurity and conjecture, like the Niger in his barren deserts of sand.” (Charles C Colton, “Lacon”, 1820)

"Logic and metaphysics make use of more tools than all the rest of the sciences put together, and do the least work." (Charles C Colton, "Remarks on the Talents of Lord Byron and the Tendencies of Don Juan" 1823)

"All parts of knowledge have their origin in metaphysics, and finally, perhaps, revolve into it. Mathematics has not a foot to stand upon which is not purely metaphysical." (Thomas De Quincey, "Kant in His Miscellaneous Essays", Blackwood's Edinburgh Magazine Vol. 28, 1830) 

"Metaphysics is the attempt of the mind to rise above the mind." (Thomas Carlyle, "Critical and Miscellaneous: Collected and Republished", 1839)

“Therefore science always goes abreast with the just elevation of the man, keeping step with religion and metaphysics; or, the state of science is an index of our self-knowledge.” (Ralph W Emerson, “The Poet”, 1844)

On Numbers: Perfect Numbers I

“A perfect number is that which is equal to the sum of its own parts.” (Euclid, “Elements”, cca. 300 BC)

“If as many numbers as we please beginning from a unit be set out continuously in double proportion, until the sum of all becomes a prime, and if the sum multiplied into the last make some number, the product will be perfect.” (Euclid, “Elements”, cca 300 BC)

“Among simple even numbers, some are superabundant, others are deficient: these two classes are as two extremes opposed to one another; as for those that occupy the middle position between the two, they are said to be perfect. And those which are said to be opposite to each other, the superabundant and the deficient, are divided in their condition, which is inequality, into the too much and the too little.” (Nicomachus of Gerasa, “Introductio Arithmetica”, cca. 100 AD)

"There exists an elegant and sure method of generating these numbers, which does not leave out any perfect numbers and which does not include any that are not; and which is done in the following way. First set out in order the powers of two in a line, starting from unity, and proceeding as far as you wish: 1, 2, 4, 8, 16, 32, 64, 128, 256, 512, 1024, 2048, 4096; and then they must be totalled each time there is a new term, and at each totaling examine the result, if you find that it is prime and non-composite, you must multiply it by the quantity of the last term that you added to the line, and the product will always be perfect. If, otherwise, it is composite and not prime, do not multiply it, but add on the next term, and again examine the result, and if it is composite leave it aside, without multiplying it, and add on the next term. If, on the other hand, it is prime, and non-composite, you must multiply it by the last term taken for its composition, and the number that results will be perfect, and so on as far as infinity." (Nicomachus of Gerasa, “Introductio Arithmetica”, cca. 100 AD)

"Six is a number perfect in itself, and not because God created all things in six days; rather, the converse is true. God created all things in six days because the number is perfect." (Saint Augustine, "The City of God", 426 AD)

“We should not leave unmentioned the principal numbers […] those which are called ‘perfect numbers’. These have parts which are neither larger nor smaller than the number itself, such as the number six, whose parts, three, two, and one, add up to exactly the same sum as the number itself. For the same reason twenty-eight, four hundred ninety-six, and eight thousand one hundred twenty-eight are called perfect numbers.” (Hrotsvit of Gandersheim, “Sapientia”, 10th century)

"[…] I think I am able to prove that there are no even numbers which are perfect apart from those of Euclid; and that there are no odd perfect numbers, unless they are composed of a single prime number, multiplied by a square whose root is composed of several other prime number. But I can see nothing which would prevent one from finding numbers of this sort. For example, if 22021 were prime, in multiplying it by 9018009 which is a square whose root is composed of the prime numbers 3, 7, 11, 13, one would have 198585576189, which would be a perfect number. But, whatever method one might use, it would require a great deal of time to look for these numbers […]" (René Descartes, [a letter to Marin Mersenne] 1638)

“The existence of an odd perfect number – its escape, so to say, from the complex web of conditions which hem it in on all sides – would be little short of a miracle.” (James J Sylvester)

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Resources:
Wikipedia (2018) List of perfect numbers [Online] Available from: https://en.wikipedia.org/wiki/List_of_perfect_numbers

On Complex Numbers V

“It is generally true, that wherever an imaginary expression occurs the same results will follow from the application of these expressions in any process as would have followed had the proposed problem been possible and its solution real.” (Augustus de Morgan, “On the Study and Difficulties of Mathematics”, 1898) 

“What could be more beautiful than a deep, satisfying relation between whole numbers. How high they rank, in the realms of pure thought and aesthetics, above their lesser brethren: the real and complex numbers.” (Manfred Schroeder, “Number Theory in Science and Communication”, 1984)

“The attitudes of mathematicians can be found not only in what they wrote, but in what they did not write. It is possible to divide mathematicians into those who gave complex numbers some kind of coverage, and those who sometimes or always ignored them.” (Diana Willment, “Complex Numbers from 1600 to 1840” [Masters thesis], 1985)

“The lack of a visual representation for √-1 had a profound influence on attitudes to it, and complex numbers were not widely accented until after the invention of the Argand diagram.” (Diana Willment, “Complex Numbers from 1600 to 1840” [Masters thesis], 1985)

“The square roots of negative numbers! If negative numbers were false, absurd or fictitious, it is hardly to be wondered at that their square roots were described as 'imaginary'.” (David Wells, “The Penguin Dictionary of Curious and Interesting Numbers”, 1986)

“The original purpose and immediate objective in introducing complex numbers into mathematics is to express laws of dependence between variables by simpler operations on the quantities involved. If one applies these laws of dependence in an extended context, by giving the variables to which they relate complex values, there emerges a regularity and harmony which would otherwise have remained concealed.” (Heinz-Dieter Ebbinghaus et al, “Numbers”, 1990)

"The number ‘i’ is evidence that much real progress can result from the positing of imaginary entities. Theologians who have built elaborate systems on much flimsier analogies should perhaps take heart." (John A Paulos, “Beyond Numeracy”, 1991)

“If we think of square roots in the geometric manner, as we have just done, to ask for the square root of a negative quantity is like asking: ‘What is the length of the side of a square whose area is less than zero?’ This has more the ring of a Zen koan than of a question amenable to a quantitative answer.” (Barry Mazur, “Imagining Numbers”, 2003)

“To have the courage to think outside the square, we need to be intrigued by a problem. This intrigue will encourage us to use our imaginations to find solutions which are beyond our current view of the world. This was the challenge that faced mathematicians as they searched for a solution to the problem of finding meaning for the square root of a negative number, in particular √-1.” (Les Evans, “Complex Numbers and Vectors”, 2006)

“Unfortunately, if we were to use geometry to explore the concept of the square root of a negative number, we would be setting a boundary to our imagination that would be difficult to cross. To represent -1 using geometry would require us to draw a square with each side length being less than zero. To be asked to draw a square with side length less than zero sounds similar to the Zen Buddhists asking ‘What is the sound of one hand clapping?’” (Les Evans, “Complex Numbers and Vectors”, 2006)

See also:
5 Books 10 Quotes: Complex Numbers V
Complex Numbers III
Complex Numbers II
Complex Numbers I  

On Numbers: Zero

"When sunya [zero] is added to a number or subtracted from a number, the number remains unchanged; and a number multiplied by sunya becomes sunya." (Brahmagupta, 628)

"Every number arises from One, and this in turn from the Zero. In this lies a great and sacred mystery - in hoc magnum latet sacramentum: HE is symbolized by that which has neither beginning nor end; and just as the zero neither increases nor diminishes another number to which it is added or from which it is subtracted so does HE neither wax nor wane. And as the zero multiplies by ten the number behind which it is placed so does HE increase not tenfold, but a thousand fold - nay, to speak more correctly, HE creates all out of nothing, preserves and rules it  - omnia ex nichillo creat, conservat atque gubernat." ("Salem Codex", 12th century)

"The whole science of mathematics depends upon zero. Zero alone determines the value in mathematics. Zero is in itself nothing. Mathematics is based upon nothing, and, consequently, arises out of nothing." (Lorenz Oken, "Elements of Physiophilosophy", 1847)

"A great deal of misunderstanding is avoided if it be remembered that the terms infinity, infinite, zero, infinitesimal must be interpreted in connexion with their context, and admit a variety of meanings according to the way in which they are defined." (George B Mathews, "Theory of Numbers", 1892)

"The point about zero is that we do not need to use it in the operations of daily life. No one goes out to buy zero fish. It is in a way the most civilized of all the cardinals, and its use is only forced on us by the needs of cultivated modes of thought." (Alfred N Whitehead, "An Introduction to Mathematics", 1911)

"In the history of culture the discovery of zero will always stand out as one of the greatest achievements of the human race." (Tobias Danzig, "Number: The Language of Science", 1930)

"The zero is the most important digit. It is a stroke of genius, to make something out of noting by giving it a name and inventing a symbol for it." (B L van der Waerden, "Science Awakening", 1962)

"[…] it took men about five thousand years, counting from the beginning of number symbols, to think of a symbol for nothing." (Isaac Asimov, "Of Time and Space and Other Things", 1965)

"[zero is] A mysterious number, which started life as a space on a counting board, turned into a written notice that a space was present, that is to say that something was absent, then confused medieval mathematicians who could not decide whether it was really a number or not, and achieved its highest status in modern abstract mathematics in which numbers are defined anyway only by their properties, and the properties of zero are at least as clear, and rather more substantial, than those of many other numbers." (David Wells, "The Penguin Dictionary of Curious and Interesting Numbers", 1986)

"Clearly, however, a zero probability is not the same thing as an impossibility; […] In systems that are now called chaotic, most initial states are followed by nonperiodic behavior, and only a special few lead to periodicity. […] In limited chaos, encountering nonperiodic behavior is analogous to striking a point on the diagonal of the square; although it is possible, its probability is zero. In full chaos, the probability of encountering periodic behavior is zero." (Edward N Lorenz, "The Essence of Chaos", 1993)

"Zero is behind all of the big puzzles in physics. The infinite density of the black hole is a division by zero. The big bang creation from the void is a division by zero. The infinite energy of the vacuum is a division by zero. Yet dividing by zero destroys the fabric of mathematics and the framework of logic - and threatens to undermine the very basis of science. […] The universe begins and ends with zero." (Charles Seife ."Zero, the Biography of a Dangerous Idea", 2000)

"Mathematics is an activity about activity. It hasn't ended - has hardly in fact begun, although the polish of its works might give them the look of monuments, and a history of zero mark it as complete. But zero stands not for the closing of a ring: it is rather a gateway." (Robert Kaplan, "The Nothing that Is: A Natural History of Zero", 2000)

"Zero was at the heart of the battle between East and West. Zero was at the center of the struggle between religion and science. Zero became the language of nature and the most important tool in mathematics. And the most profound problems in physics - the dark core of a black hole and the brilliant flash of the big bang - are struggles to defeat zero." (Charles Seife ."Zero, the Biography of a Dangerous Idea", 2000)

"In one of the largest calculations done to date, it was checked that the first ten trillion of these zeros lie on the correct line. So there are ten trillion pieces of evidence indicating that the Riemann hypothesis is true and not a single piece of evidence indicating that it is false. A physicist might be overwhelmingly pleased with this much evidence in favour of the hypothesis, but to some mathematicians this is hardly evidence at all. However, it is interesting ancillary information." (Peter Borwein et al, "The Riemann Hypothesis: A Resource for the Afficionado and Virtuoso Alike", 2007)

"One of the current ideas regarding the Riemann hypothesis is that the zeros of the zeta function can be interpreted as eigenvalues of certain matrices. This line of thinking is attractive and is potentially a good way to attack the hypothesis, since it gives a possible connection to physical phenomena. [...] Empirical results indicate that the zeros of the Riemann zeta function are indeed distributed like the eigenvalues of certain matrix ensembles, in particular the Gaussian unitary ensemble. This suggests that random matrix theory might provide an avenue for the proof of the Riemann hypothesis." (Peter Borwein et al, "The Riemann Hypothesis: A Resource for the Afficionado and Virtuoso Alike", 2007)

"The concept of zero is so familiar that it takes a great deal of effort to recapture how mysterious, subtle, and contradictory the idea really is." (William Byers, "How Mathematicians Think", 2007)

"Zero is the mathematically defined numerical function of nothingness. It is used not for an evasion but for an apprehension of reality. Zero is by far the most interesting number among all the others: It is a symbol for what is not there. It is an emptiness that increases any number it's added to. Zero is an inexhaustible and indispensable paradox. Zero is the only number which can be divided by every other number. Zero is also only number which can divide no other number. It seems zero is also the most debated number in mathematics. We know that mathematicians are involved in heated philosophical and logical discussions around the issues of zero: Can we divide a number by zero? Is the result of this division infinity or not? Is zero a positive or a negative number? Is it even or is it odd?" (Fahri Karakas, "Reflections on zero and zero-centered spirituality in organizations", 2008)

"However, in contrast to one, which is singularly straightforward, zero is secretly peculiar. If you pierce the obscuring haze of familiarity around it, you’ll see that it is a quantitative entity that, curiously, is really the absence of quantity. It took people a long time to get their minds around that." (David Stipp, "A Most Elegant Equation: Euler's Formula and the Beauty of Mathematics", 2017)

"Zero is not a point of non-existence. Zero is always a balance point of existents. The human understanding of 'zero' must undergo the most radical of all transformations. Most people, especially scientists, associate it with absolute nothingness, with non-existence. This is absolutely untrue. Or, to put it another way, we can define it in two ways: 1) nothing as non-existence, in which case it has absolutely no consequences but leads to all manner of abstract paradoxes and contradictions, or 2) nothing as existence, in which case it is always a mathematical balance point for somethings. It is purely mathematical, not scientific, or religious, or spiritual, or emotional, or sensory, or mystical. It is analytic nothing and whenever you encounter it you have to establish the exact means by which it is maintaining its balance of zero." (Thomas Stark, "God Is Mathematics: The Proofs of the Eternal Existence of Mathematics", 2018)

On Numbers: From Zero to Infinity

“A quantity divided by zero becomes a fraction the denominator of which is zero. This fraction is termed an infinite quantity. In this quantity consisting of that which has zero for its divisor, there is no alteration, though many may be inserted or extracted; as no change takes place in the infinite and immutable God when worlds are created or destroyed, though numerous orders of beings are absorbed or put forth.” (Bhaskara II, “Bijaganita”, 12th century)

 “The Infinite is often confounded with the Indefinite, but the two conceptions are diametrically opposed. Instead of being a quantity with unassigned yet assignable limits, the Infinite is not a quantity at all, since it neither admits of augmentation nor diminution, having no assignable limits; it is the operation of continuously withdrawing any limits that may have been assigned: the endless addition of new quantities to the old: the flux of continuity. The Infinite is no more a quantity than Zero is a quantity. If Zero is the sign of a vanished quantity, the Infinite is a sign of that continuity of Existence which has been ideally divided into discrete parts in the affixing of limits.” (George H. Lewes, “Problems of Life and Mind”, 1873)

“One microscopic glittering point; then another; and another, and still another; they are scarcely perceptible, yet they are enormous. This light is a focus; this focus, a star; this star, a sun; this sun, a universe; this universe, nothing. Every number is zero in the presence of the infinite.” (Victor Hugo, “The Toilers of the Sea”, 1874)

"A great deal of misunderstanding is avoided if it be remembered that the terms infinity, infinite, zero, infinitesimal must be interpreted in connexion with their context, and admit a variety of meanings according to the way in which they are defined." (George B Mathews, "Theory of Numbers", 1892)

“Infinity is the land of mathematical hocus pocus. There Zero the magician is king. When Zero divides any number he changes it without regard to its magnitude into the infinitely small [great?], and inversely, when divided by any number he begets the infinitely great [small?]. In this domain the circumference of the circle becomes a straight line, and then the circle can be squared. Here all ranks are abolished, for Zero reduces everything to the same level one way or another. Happy is the kingdom where Zero rules!” (Paul Carus, “The Nature of Logical and Mathematical Thought”; Monist Vol 20, 1910)

“I do not say that the notion of infinity should be banished; I only call attention to its exceptional nature, and this so far as I can see, is due to the part which zero plays in it, and we must never forget that like the irrational it represents a function which possesses a definite character but can never be executed to the finish If we bear in mind the imaginary nature of these functions, their oddities will not disturb us, but if we misunderstand their origin and significance we are confronted by impossibilities.” (Paul Carus, “The Nature of Logical and Mathematical Thought”; Monist Vol 20, 1910)

"Each act of creation could be symbolized as a particular product of infinity and zero. From each such product could emerge a particular entity of which the appropriate symbol was a particular number." (Srinivasa Ramanujan)

“If you look at zero you see nothing; but look through it and you will see the world. For zero brings into focus the great, organic sprawl of mathematics, and mathematics in turn the complex nature of things.” (Robert Kaplan, “The Nothing that Is: A Natural History of Zero”, 2000)

“Zero is powerful because it is infinity’s twin. They are equal and opposite, yin and yang. They are equally paradoxical and troubling. The biggest questions in science and religion are about nothingness and eternity, the void and the infinite, zero and infinity. The clashes over zero were the battles that shook the foundations of philosophy, of science, of mathematics, and of religion. Underneath every revolution lay a zero - and an infinity.” (Charles Seife, “Zero: The Biography of a Dangerous Idea”, 2000)

“Zero seems as diaphanous as a fairy’s wing, yet it is as powerful as a black hole. The obverse of infinity, it’s enthroned at the center of the number line - at least as the line is usually drawn - making it a natural center of attention. It has no effect when added to other numbers, as if it were no more substantial than a fleeting thought. But when multiplied times other numbers it seems to exert uncanny power, inexorably sucking them in and making them vanish into itself at the center of things. If you’re into stark simplicity, you can express any number (that is, any number that’s capable of being written out) with the use of zero and just one other number, one.” (David Stipp, “A Most Elegant Equation: Euler's Formula and the Beauty of Mathematics”, 2017)

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