On Art IX: Metaphysics

"Metaphysics may be, after all, only the art of being sure of something that is not so and logic only the art of going wrong with confidence." (Joseph W Krutch, "The Modern Temper", 1929)

"As to the role of emotions in art and the subconscious mechanism that serves as the integrating factor both in artistic creation and in man's response to art, they involve a psychological phenomenon which we call a sense of life. A sense of life is a pre-conceptual equivalent of metaphysics, an emotional, subconsciously integrated appraisal of man and of existence." (Ayn Rand, "The Romantic Manifesto: A Philosophy of Literature", 1969)

"For some years now the activity of the artist in our society has been trending more toward the function of the ecologist: one who deals with environmental relationships. Ecology is defined as the totality or pattern of relations between organisms and their environment. Thus the act of creation for the new artist is not so much the invention of new objects as the revelation of previously unrecognized relation- ships between existing phenomena, both physical and metaphysical. So we find that ecology is art in the most fundamental and pragmatic sense, expanding our apprehension of reality." (Gene Youngblood, "Expanded Cinema", 1970)

"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)

 "Science, as usually taught to liberal arts students, emphasizes results rather than method, and tries to teach technique rather than to give insight into and understanding of the scientific habit of thought. What is needed, however, is not a dose of metaphysics but a truly humanistic teaching of science." (Harry D Gideonse)

Out of Context: On Metaphysics (Definitions)

"Metaphysics is universal and is exclusively concerned with primary substance." (Aristotle, "Metaphysics", 340 BC

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

"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)

"[...] metaphysics is almost the last thing that one discovers." (Pierre-Simon Laplace [letter to Sylvestre F Lacroix] 1792)

"Metaphysics is the finding of bad reasons for what we believe upon instinct, but to find these reasons is no less an instinct." (Francis H Bradley, "Appearance and Reality: A Metaphysical Essay", 1893)

"Metaphysics is the science of the ultimate states of the universe." (Ian Hacking, "The Taming of Chance", 1990)

"Metaphysics is one of the main branches of philosophy, the branch that is concerned with the concept of being (that is, existence) and with several other closely related concepts. The study of concepts plays a central role in all of philosophy." (Michael Jubien, "Contemporary Metaphysics", 1997)

"Metaphysics is not about any of the things that exist, or their existence under certain limited conditions. It is purely about existence." (Earl Conee & Theodore Sider, "Riddles of Existence: A Guided Tour of Metaphysics", 2005)

"Metaphysics is rooted in the natural human desire to know, the longing to behold and participate in the beautiful, to find one's place within, and to conceive all one's activities in relation to, the whole. Metaphysics is at once about being, truth, goodness, and beauty." (Thomas Hibbs, "Aquinas, Ethics, and Philosophy of Religion: Metaphysics and Practice", 2007)

"Metaphysics, or ontology, is the study of the most basic and general problems about the universe and the mind."  (Mario Bunge, "Matter and Mind: A Philosophical Inquiry", 2010)

"Metaphysics is concerned, first and foremost, with the nature of reality" (Kitt Fine, "What is metaphysics?", Contemporary Aristotelian Metaphysics, 2012)

"Metaphysics, in whatever latitude the term be taken, is a science or complement of sciences exclusively occupied with mind." (Sir William R Hamilton)

On Conjecture (1800-1899)

"In order to supply the defects of experience, we will have recourse to the probable conjectures of analogy, conclusions which we will bequeath to our posterity to be ascertained by new observations, which, if we augur rightly, will serve to establish our theory and to carry it gradually nearer to absolute certainty." (Johann H Lambert, "The System of the World", 1800)

"In all speculations on the origin, or agents that have produced the changes on this globe, it is probable that we ought to keep within the boundaries of the probable effects resulting from the regular operations of the great laws of nature which our experience and observation have brought within the sphere of our knowledge. When we overleap those limits, and suppose a total change in nature's laws, we embark on the sea of uncertainty, where one conjecture is perhaps as probable as another; for none of them can have any support, or derive any authority from the practical facts wherewith our experience has brought us acquainted." (William Maclure, "Observations on the Geology of the United States of America", 1817)

"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)

"We know the effects of many things, but the causes of few; experience, therefore, is a surer guide than imagination, and inquiry than conjecture." (Charles C Colton, "Lacon", 1820)

"Let me be permitted to recall that the object of mathematics is not to investigate the causes that one can assign to natural phenomena. This science would lose both its character and credit if, renouncing the support of general well-confirmed facts, it sought within the realm of nebulous conjectures, a realm which has always been a fertile source of error for ways of satisfying the thirst fo rexplanation." (Sophie Germain, "Examen des principes qui peuvent conduire a la connaissance des lois de requilibre et du mouvement des solides elastiques", Annales de Chimie 38, 1828)

"Life is not the object of Science: we see a little, very little; And what is beyond we can only conjecture." (Samuel Johnson, "Causes Which Produce Diversity of Opinion", 1840)

"The entire annals of Observation probably do not elsewhere exhibit so extraordinary a verification of any theoretical conjecture adventured on by the human spirit!" (John P Nichol, "The Planet Neptune: An Exposition and History", 1848)

"The philosophical study of nature rises above the requirements of mere delineation, and does not consist in the sterile accumulation of isolated facts. The active and inquiring spirit of man may therefore be occasionally permitted to escape from the present into the domain of the past, to conjecture that which cannot yet be clearly determined, and thus to revel amid the ancient and ever-recurring myths of geology." (Alexander von Humboldt, "Views of Nature: Or Contemplation of the Sublime Phenomena of Creation", 1850)

"The rules of scientific investigation always require us, when we enter the domains of conjecture, to adopt that hypothesis by which the greatest number of known facts and phenomena may be reconciled." (Matthew F Maury, "The Physical Geography of the Sea", 1855)

"There is something fascinating about science. One gets such wholesale returns of conjecture out of such a trifling investment of fact." (Samuel L Clemens [Mark Twain], "Life on the Mississippi", 1883)

"I have been able to solve a few problems of mathematical physics on which the greatest mathematicians since Euler have struggled in vain. [...] But the pride I might have held in my conclusions was perceptibly lessened by the fact that I knew that the solution of these problems had almost always come to me as the gradual generalization of favorable examples, by a series of fortunate conjectures, after many errors." (Hermann von Helmholtz, 1891)

On Generalization (1970-1979)

"Accordingly there are two main types of science, exact science [...] and empirical science [...] seeking laws which are generalizations from particular experiences and are verifiable (or, more strictly, 'probabilities') only by observation and experiment." (Errol E Harris, "Hypothesis and Perception: The Roots of Scientific Method", 1970)

"One often hears that successive theories grow ever closer to, or approximate more and more closely to, the truth. Apparently, generalizations like that refer not to the puzzle-solutions and the concrete predictions derived from a theory but rather to its ontology, to the match, that is, between the entities with which the theory populates nature and what is ‘really there’." (Thomas S Kuhn, "The Structure of Scientific Revolutions", 1970)

"Science uses the senses but does not enjoy them; finally buries them under theory, abstraction, mathematical generalization." (Theodore Roszak, "Where the Wasteland Ends", 1972)

"A single observation that is inconsistent with some generalization points to the falsehood of the generalization, and thereby 'points to itself'." (Ian Hacking, "The Emergence Of Probability", 1975)

"The sciences have started to swell. Their philosophical basis has never been very strong. Starting as modest probing operations to unravel the works of God in the world, to follow its traces in nature, they were driven gradually to ever more gigantic generalizations. Since the pieces of the giant puzzle never seemed to fit together perfectly, subsets of smaller, more homogeneous puzzles had to be constructed, in each of which the fit was better." (Erwin Chargaff, "Voices in the Labyrinth", 1975)

"The word generalization in literature usually means covering too much territory too thinly to be persuasive, let alone convincing. In science, however, a generalization means a principle that has been found to hold true in every special case. [...] The principle of leverage is a scientific generalization." (Buckminster Fuller, "Synergetics: Explorations in the Geometry of Thinking", 1975)

"And when such claims are extraordinary, that is, revolutionary in their implications for established scientific generalizations already accumulated and verified, we must demand extraordinary proof." (Marcello Truzzi, Zetetic Scholar, Vol. 1 (1), 1976)

"If it is to be effective as a tool of thought, a notation must allow convenient expression not only of notions arising directly from a problem, but also of those arising in subsequent analysis, generalization, and specialization." (Kenneth E Iverson, "Notation as a Tool of Thought", 1979)

"Prediction can never be absolutely valid and therefore science can never prove some generalization or even test a single descriptive statement and in that way arrive at final truth." (Gregory Bateson, "Mind and Nature, A necessary unity", 1979)

William Fleming - Collected Quotes

"Science is knowledge certain and evident in itself, or by the principles from which it is deducted, or with which it is certainly connected. It is subjective, as existing in the mind; objective, as embodied in truths; speculative, as leading to do something, as in practical science." (William Fleming, "The Vocabulary of Philosophy, Mental, Moral, and Metaphysical", 1857)

"SYSTEM (σύστημα, σύν ἵστημιavu, to place together) - is a full and connected view of all the truths of some department of knowledge. An organized body of truth, or truths arranged under one and the same idea, which idea is as the life or soul which assimilates all those truths. No truth is altogether isolated. Every truth has relation to some other. And we should try to unite the facts of our knowledge so as to see them in their several bearings. This we do when we frame them into a system. To do so legitimately we must begin by analysis and end with synthesis. But system applies not only to our knowledge, but to the objects of our knowledge. Thus we speak of the planetary system, the muscular system, the nervous system. We believe that the order to which we would reduce our ideas has a foundation in the nature of things. And it is this belief that encourages us to reduce our knowledge of things into systematic order. The doing so is attended with many advantages. At the same time a spirit of systematizing may be carried too far. It is only in so far as it is in accordance with the order of nature that it can be useful or sound." (William Fleming, "Vocabulary of philosophy, mental, moral, and metaphysical; with quotations and references; for the use of students", 1857)

"The principle of deduction is, that things which agree with the same thing agree with one another. The principle of induction is, that in the same circumstances and in the same substances, from the same causes the same effects will follow. The mathematical and metaphysical sciences are founded on deduction; the physical sciences rest on induction." (William Fleming, "A vocabulary of the philosophical sciences", 1857)

"The term 'intellect' includes all those powers by which we acquire, retain, and extend our knowledge; as perception, memory, imagination, judgment, and the like." (William Fleming, "A vocabulary of the philosophical sciences", 1857)

"Common sense is a phrase employed to denote that degree of intelligence, sagacity, and prudence, which is common to all men." (William Fleming)

"The difference between a parable and an apologue is, that the former, being drawn from human life, requires probability in the narration, whereas the apologue, being taken from inanimate things or the inferior animals, is not confined strictly to probability." (William Fleming) 

On Integrals I

"I see with much pleasure that you are working on a large work on the integral Calculus [...] The reconciliation of the methods which you are planning to make, serves to clarify them mutually, and what they have in common contains very often their true metaphysics; this is why that metaphysics is almost the last thing that one discovers. The spirit arrives at the results as if by instinct; it is only on reflecting upon the route that it and others have followed that it succeeds in generalising the methods and in discovering its metaphysics." (Pierre-Simon Laplace [letter to Sylvestre F Lacroix] 1792)

"Certain authors who seem to have perceived the weakness of this method assume virtually as an axiom that an equation has indeed roots, if not possible ones, then impossible roots. What they want to be understood under possible and impossible quantities, does not seem to be set forth sufficiently clearly at all. If possible quantities are to denote the same as real quantities, impossible ones the same as imaginaries: then that axiom can on no account be admitted but needs a proof necessarily." (Carl F Gauss, "New proof of the theorem that every algebraic rational integral function in one variable can be resolved into real factors of the first or the second degree", 1799)

"The integrals which we have obtained are not only general expressions which satisfy the differential equation, they represent in the most distinct manner the natural effect which is the object of the phenomenon [...] when this condition is fulfilled, the integral is, properly speaking, the equation of the phenomenon; it expresses clearly the character and progress of it, in the same manner as the finite equation of a line or curved surface makes known all the properties of those forms." (Jean-Baptiste-Joseph Fourier, "Théorie Analytique de la Chaleur", 1822)

"If one looks at the different problems of the integral calculus which arise naturally when he wishes to go deep into the different parts of physics, it is impossible not to be struck by the analogies existing. Whether it be electrostatics or electrodynamics, the propagation of heat, optics, elasticity, or hydrodynamics, we are led always to differential equations of the same family." (Henri Poincaré, "Sur les Equations aux Dérivées Partielles de la Physique Mathématique", American Journal of Mathematics Vol. 12, 1890)

"Every one who understands the subject will agree that even the basis on which the scientific explanation of nature rests, is intelligible only to those who have learned at least the elements of the differential and integral calculus, as well as of analytical geometry." (Felix Klein, Jahresbericht der Deutschen Mathematiker Vereinigung Vol. 11, 1902)

"The method of successive approximations is often applied to proving existence of solutions to various classes of functional equations; moreover, the proof of convergence of these approximations leans on the fact that the equation under study may be majorised by another equation of a simple kind. Similar proofs may be encountered in the theory of infinitely many simultaneous linear equations and in the theory of integral and differential equations. Consideration of the semiordered spaces and operations between them enables us to easily develop a complete theory of such functional equations in abstract form." (Leonid Kantorovich, "On one class of functional equations", 1936)

"The chief difficulty of modern theoretical physics resides not in the fact that it expresses itself almost exclusively in mathematical symbols, but in the psychological difficulty of supposing that complete nonsense can be seriously promulgated and transmitted by persons who have sufficient intelligence of some kind to perform operations in differential and integral calculus […]" (Celia Green, "The Decline and Fall of Science", 1976)

"But just as much as it is easy to find the differential of a given quantity, so it is difficult to find the integral of a given differential. Moreover, sometimes we cannot say with certainty whether the integral of a given quantity can be found or not." (Johann Bernoulli) [attributed to]

"Therefore one has taken everywhere the opposite road, and each time one encounters manifolds of several dimensions in geometry, as in the doctrine of definite integrals in the theory of imaginary quantities, one takes spatial intuition as an aid. It is well known how one gets thus a real overview over the subject and how only thus are precisely the essential points emphasized." (Bernhard Riemann)

On Abstraction (2000-2009)

"Abstraction is itself an abstract word and has no single meaning. […] Every word in our language is abstract, because it represents something else." (Eric Maisel, "The Creativity Book: A Year's Worth of Inspiration and Guidance", 2000)

"What cognitive capabilities underlie our fundamental human achievements? Although a complete answer remains elusive, one basic component is a special kind of symbolic activity - the ability to pick out patterns, to identify recurrences of these patterns despite variation in the elements that compose them, to form concepts that abstract and reify these patterns, and to express these concepts in language. Analogy, in its most general sense, is this ability to think about relational patterns." (Keith Holyoak et al, "Introduction: The Place of Analogy in Cognition", 2001)

"[…] we underestimate the share of randomness in about everything […]  The degree of resistance to randomness in one’s life is an abstract idea, part of its logic counterintuitive, and, to confuse matters, its realizations nonobservable." (Nassim N Taleb, "Fooled by Randomness", 2001)

"A model isolates one or a few causal connections, mechanisms, or processes, to the exclusion of other contributing or interfering factors - while in the actual world, those other factors make their effects felt in what actually happens. Models may seem true in the abstract, and are false in the concrete. The key issue is about whether there is a bridge between the two, the abstract and the concrete, such that a simple model can be relied on as a source of relevantly truthful information about the complex reality." (Uskali Mäki, "Fact and Fiction in Economics: Models, Realism and Social Construction", 2002)

"[Primes] are full of surprises and very mysterious […]. They are like things you can touch […] In mathematics most things are abstract, but I have some feeling that I can touch the primes, as if they are made of a really physical material. To me, the integers as a whole are like physical particles." (Yoichi Motohashi, "The Riemann Hypothesis: The Greatest Unsolved Problem in Mathematics", 2002)

"To criticize mathematics for its abstraction is to miss the point entirely. Abstraction is what makes mathematics work. If you concentrate too closely on too limited an application of a mathematical idea, you rob the mathematician of his most important tools: analogy, generality, and simplicity. Mathematics is the ultimate in technology transfer." (Ian Stewart, "Does God Play Dice: The New Mathematics of Chaos", 2002)

"Do not be afraid of the word 'theory'. Yes, it can sound dauntingly abstract at times, and in the hands of some writers can appear to have precious little to do with the actual, visual world around us. Good theory however, is an awesome thing. [...] But unless we actually use it, it borders on the metaphysical and might as well not be used at all." (Richard Howells,  Visual Culture, 2003)

"Group theory is a branch of mathematics that describes the properties of an abstract model of phenomena that depend on symmetry. Despite its abstract tone, group theory provides practical techniques for making quantitative and verifiable predictions about the behavior of atoms, molecules and solids." (Arthur M Lesk, "Introduction to Symmetry and Group Theory for Chemists", 2004)

"Mathematics is not about abstract entities alone but is about relation of abstract entities with real entities. […] Adequacy relations between abstract and real entities provide space or opportunity where mathematical and logical thought operates parsimoniously."  (Navjyoti Singh, "Classical Indian Mathematical Thought", 2005)

"That is, the physicist likes to learn from particular illustrations of a general abstract concept. The mathematician, on the other hand, often eschews the particular in pursuit of the most abstract and general formulation possible. Although the mathematician may think from, or through, particular concrete examples in coming to appreciate the likely truth of very general statements, he will hide all those intuitive steps when he comes to present the conclusions of his thinking to outsiders. It presents the results of research as a hierarchy of definitions, theorems and proofs after the manner of Euclid; this minimizes unnecessary words but very effectively disguises the natural train of thought that led to the original results." (John D Barrow, "New Theories of Everything", 2007)

"Abstraction is a mental process we use when trying to discern what is essential or relevant to a problem; it does not require a belief in abstract entities." (Tom G Palmer, Realizing Freedom: Libertarian Theory, History, and Practice, 2009)

"In order to deal with these phenomena, we abstract from details and attempt to concentrate on the larger picture - a particular set of features of the real world or the structure that underlies the processes that lead to the observed outcomes. Models are such abstractions of reality. Models force us to face the results of the structural and dynamic assumptions that we have made in our abstractions." (Bruce Hannon and Matthias Ruth, "Dynamic Modeling of Diseases and Pests", 2009)

"It is from this continuousness of thought and perception that the scientist, like the writer, receives the crucial flash of insight out of which a piece of work is conceived and executed. And the scientist (again like the writer) is grateful when the insight comes, because insight is the necessary catalyst through which the abstract is made concrete, intuition be given language, language provides specificity, and real work can go forward." (Vivian Gornick, "Women in Science: Then and Now", 2009)

On Simplicity X

"One should not be deceived by philosophical works that pretend to be mathematical, but are merely dubious and murky metaphysics. Just because a philosopher can recite the words lemma, theorem and corollary doesn't mean that his work has the certainty of mathematics. That certainty does not derive from big words, or even from the method used by geometers, but rather from the utter simplicity of the objects considered by mathematics." (Pierre L Maupertuis, "Les Loix du Mouvement et du Repos, déduites d'un Principe Métaphysique", 1746)

"The model of an object, process or phenomenon is some other object, process or phenomenon having certain features in common with the original. It is ordinarily assumed that the model is a simplified version of the object of study. However, it is not always easy to give precise meaning to the concept 'simpler than the original', for the simple reason that in reality all entities or phenomena are infinitely complicated and their study can be carried out with differing and constantly increasing degrees of accuracy." (Yakov Khurgin, "Did You Say Mathematics?", 1974)

"Certainty, simplicity, vividness originate in popular knowledge. That is where the expert obtains his faith in this triad as the ideal of knowledge. Therein lies the general epistemological significance of popular science." (Ludwik Fleck, "Genesis and Development of a Scientific Fact", 1979)

"Nature is capable of building complex structures by processes of self-organization; simplicity begets complexity." (Victor J Stenger, "God: The Failed Hypothesis", 2010)
 
"Simplicity in a system tends to increase that system's efficiency. Because less can go wrong with fewer parts, less will. Complexity in a system tends to increase that system's inefficiency; the greater the number of variables, the greater the probability of those variables clashing, and in turn, the greater the potential for conflict and disarray. Because more can go wrong, more will. That is why centralized systems are inclined to break down quickly and become enmeshed in greater unintended consequences." (Lawrence K Samuels, "Defense of Chaos: The Chaology of Politics, Economics and Human Action", 2013)

"Guided only by their feeling for symmetry, simplicity, and generality, and an indefinable sense of the fitness of things, creative mathematicians now, as in the past, are inspired by the art of mathematics rather than by any prospect of ultimate usefulness." (Eric T Bell)
 
"[…] if one really understood the central point and its necessity in the construction of the world, one ought to be able to state it in one clear, simple sentence. Until we see the quantum principle with this simplicity we can well believe that we do not know the first thing about the universe, about ourselves, and about our place in the universe." (John A Wheeler)

On Metaphors IV

"A man desiring to understand the world looks about for a clue to its comprehension. He pitches upon some area of commonsense fact and tries to understand other areas in terms of this one. The original area becomes his basic analogy or root metaphor." (Stephen Pepper, "World Hypotheses: A Study in Evidence", 1948)

"Every metaphor is the tip of a submerged model. […] Use of theoretical models resembles the use of metaphors in requiring analogical transfer of a vocabulary. Metaphor and model-making reveal new relationships; both are attempts to pour new content into old bottles." (Max Black," Models and Metaphors", 1962)

"Catastrophe Theory is-quite likely-the first coherent attempt (since Aristotelian logic) to give a theory on analogy. When narrow-minded scientists object to Catastrophe Theory that it gives no more than analogies, or metaphors, they do not realise that they are stating the proper aim of Catastrophe Theory, which is to classify all possible types of analogous situations." (René F Thom," La Théorie des catastrophes: État présent et perspective", 1977)

"New metaphors are capable of creating new understandings and, therefore, new realities. This should be obvious in the case of poetic metaphor, where language is the medium through which new conceptual metaphors are created." (George Lakoff and Mark Johnson, "Metaphors We Live By", 1980)

"Metaphors deny distinctions between things: problems often arise from taking structural metaphors too literally. Because unexamined metaphors lead us to assume the identity of unidentical things, conflicts can arise which can only be resolved by understanding the metaphor (which requires its recognition as such), which means reconstructing the analogy on which it is based. […] The unexplained extension of concepts can too often result in the destruction rather than the expansion of meaning." (David Pimm,"Metaphor and Analogy in Mathematics", For the Learning of Mathematics Vol. 1 (3), 1981)

"Metaphysics in philosophy is, of course, supposed to characterize what is real - literally real. The irony is that such a conception of the real depends upon unconscious metaphors." (George Lakoff,  "Philosophy in the Flesh: The Embodied Mind and its Challenge to Western Thought", 1999)

"[…] philosophical theories are structured by conceptual metaphors that constrain which inferences can be drawn within that philosophical theory. The (typically unconscious) conceptual metaphors that are constitutive of a philosophical theory have the causal effect of constraining how you can reason within that philosophical framework." (George Lakoff, "Philosophy in the Flesh: The Embodied Mind and its Challenge to Western Thought", 1999)

"The claim that scientific models are metaphors is tied to the fact that often an analogy is exploited to construct a model about a phenomenon. [...] Scientific models appear to be, contrary to past research traditions, as central in scientific practice for describing and communicating aspects of the empirical world as metaphors are in ordinary language." (Daniela M Bailer-Jones," Models, Metaphors and Analogies", 2002)
 
"In order to understand how mathematics is applied to understanding of the real world it is convenient to subdivide it into the following three modes of functioning: model, theory, metaphor. A mathematical model describes a certain range of phenomena qualitatively or quantitatively. […] A (mathematical) metaphor, when it aspires to be a cognitive tool, postulates that some complex range of phenomena might be compared to a mathematical construction." (Yuri I Manin," Mathematics as Metaphor: Selected Essays of Yuri I. Manin" , 2007)

"When the words are used without mental image or concrete objects, we label them as metaphor. […] While concepts are being internalised, language is not only appropriated but metaphorised." (Lynne Cameron, "Metaphor in the construction of a learning environment", 2008)

On Diversity (-1899)

"There never was in the world two opinions alike, no more than two hairs or two grains; the most universal quality is diversity." (Michel de Montaigne, "Essais", 1595)

"The diversity of the phenomena of Nature is so great, and the treasures hidden in the heavens so rich, precisely in order that the human mind shall never be lacking in fresh nourishment." (Johannes Kepler, "Mysterium Cosmographicum", 1596)

"Blind metaphysical necessity, which is certainly the same always and every where, could produce no variety of things. All that diversity of natural things which we find suited to different times and places could arise from nothing but the ideas and will of a Being, necessarily existing." (Isaac Newton, "Philosophiæ Naturalis Principia Mathematica", 1687)

"Entium varietates non temere esse minuendas."
"The variety of entities should not be rashly diminished" (Immanuel Kant, "Critique of Pure Reason", 1781)

"As long as men inquire, they will find opportunities to know more upon these topics than those who have gone before them, so inexhaustibly rich is nature in the innermost diversity of her treasures of beauty, order, and intelligence." (J Louis R Agassiz, "Essay on Classification", 1859)

"Unity of plan everywhere lies hidden under the mask of diversity of structure - the complex is everywhere evolved out of the simple." (Thomas H Huxley, "A Lobster; or, the Study of Zoology", 1861)

"The simplicity of nature which we at present grasp is really the result of infinite complexity; and that below the uniformity there underlies a diversity whose depths we have not yet probed, and whose secret places are still beyond our reach." (William Spottiswoode, [Report of the Forty-eighth Meeting of the British Association for the, Advancement of Science] 1878)
 
"Number is but another name for diversity." (William S Jevons, "The Principles of Science: A Treatise on Logic and Scientific Method", 1887)

"Science arises from the discovery of Identity amidst Diversity." (William S Jevons, "The Principles of Science: A Treatise on Logic and Scientific Method", 1887)

"Nature considered rationally, that is to say, submitted to the process of thought, is a unity in diversity of phenomena; a harmony, blending together all created things, however dissimilar in form and attributes; one great whole animated by the breath of life." (Alexander von Humboldt)

André Weil - Collected Quotes

"Finally, students must learn to realize that mathematics is a science with a long history behind it, and that no true insight into the mathematics of the present day can be obtained without some acquaintance with its historical background. In the first-place time gives an additional dimension to one's mental picture both of mathematics as a whole, and of each individual branch." (André Weil, "The Mathematical Curriculum", 1954)

"Rigor is to the mathematician what morality is to man. It does not consist in proving everything, but in maintaining a sharp distinction between what is assumed and what is proved, and in endeavoring to assume as little as possible at every stage." (André Weil, Mathematical Teaching in Universities", The American Mathematical Monthly Vol. 61 (1), 1954)

"As every mathematician knows, nothing is more fruitful than these obscure analogies, these indistinct reflections of one theory into another, these furtive caresses, these inexplicable disagreements; also nothing gives the researcher greater pleasure." (André Weil, "De la Métaphysique aux Mathématiques", 1960)

"Nothing is more fruitful - all mathematicians know it - than those obscure analogies, those disturbing reflections of one theory on another; those furtive caresses, those inexplicable discords; nothing also gives more pleasure to the researcher. The day comes when this illusion dissolves: the presentiment turns into certainty; the yoked theories reveal their common source before disappearing. As the Gita teaches, one achieves knowledge and indifference at the same time. Metaphysics has become Mathematics, ready to form the material of some treatise whose cold beauty has lost the power to move us." (André Weil, "De la métaphysique aux mathématiques", 1960)

"Alternative models are neither right nor wrong, just more or less useful in allowing us to operate in the world and discover more and better options for solving problems." (Andrew Weil," The Natural Mind: A Revolutionary Approach to the Drug Problem", 2004)

"A tediously laborious proof may be a sign that the writer has been less than felicitous in expressing himself; but more often than not, as we know, it indicates that he has been laboring under limitations which prevented him from translating directly into words or formulas some very simple ideas." (André Weil)

"In the future, as in the past, the great ideas [of mathematics] must be simplifying ideas, the creator must always be one who clarifies, for himself, and for others, the most complicated issues of formulas and concepts." (André Weil)

"Strategy means the art of recognizing the main problems, attacking them at their weak points, setting up future lines of advance. Mathematical strategy is concerned with long-range objectives; it requires a deep understanding of broad trends and of the evolution of ideas over long periods." (André Weil)

On Certainty (1925-1949)

"The certainty which science aims to bring about is not a psychologic feeling about a given proposition but a logical ground on which its claim to truth can be founded." (Morris R Cohen, "Reason and Nature", 1931)

"In every experiment on living organisms, there must remain an uncertainty as regards the physical conditions to which they are subjected, and the idea suggests itself that the minimal freedom we must allow the organism in this respect is just large enough to permit it, so to say, to hide its ultimate secrets from us." (Niels H D Bohr, "Light and Life", Nature Vol. 131 (3309), 1933)

"All the theories and hypotheses of empirical science share this provisional character of being established and accepted ‘until further notice’, whereas a mathematical theorem, once proved, is established once and for all; it holds with that particular certainty which no subsequent empirical discoveries, however unexpected and extraordinary, can ever affect to the slightest extent." (Carl G Hempel, Geometry and Empirical Science", 1935)

"There is no certainty vouchsafed us in the vast testimony of Nature that the universe was designed for man, nor yet for any purpose, even the bleak purpose of symmetry. The courageous thinker must look the inimical aspects of his environment in the face, and accept the stern fact that the universe is hostile and deadly to him save for a very narrow zone where it permits him, for a few eons, to exist." (Donald C Peattie, "An Almanac for Moderns", 1935)

"In every writer on philosophy there is a concealed metaphysic, usually unconscious; even if his subject is metaphysics, he is almost certain to have an uncritically believed system which underlies his specific arguments." (Bertrand Russell, "Dewey’s New Logic" [in "The Philosophy of John Dewey", ed. by Paul A Schilpp & Lewis E Hahn, 1939])

"For a logician of a certain sort only complete proofs exist. What intends to be a proof must leave no gaps, no loopholes, no uncertainty whatever, or else it is no proof." (George Pólya, "How to solve it", 1945)

"Heuristic reasoning is reasoning not regarded as final and strict but as provisional and plausible only, whose purpose is to discover the solution of the present problem. We are often obliged to use heuristic reasoning. We shall attain complete certainty when we shall have obtained the complete solution, but before obtaining certainty we must often be satisfied with a more or less plausible guess. We may need the provisional before we attain the final. We need heuristic reasoning when we construct a strict proof as we need scaffolding when we erect a building." (George Pólya, "How to solve it", 1945)

"The most distinctive characteristic which differentiates mathematics from the various branches of empirical science, and which accounts for its fame as the queen of the sciences, is no doubt the peculiar certainty and necessity of its results." (Carl G Hempel, "Geometry and Empirical Science", 1945)

"The study of mathematics cultivates the reason; that of the languages, at the same time, the reason and the taste. The former gives the grasp and power to the mind; the latter both power and flexibility. The former by itself, would prepare us for a state of certainties, which nowhere exists; the later, for a state of probabilities which is that of common life. Each, by itself, does but an imperfect work: in the union of both, is the best discipline for the mind, and the best mental training for the world as it is." (Tyron Edwards, "The New Dictionary of Thoughts", 1948)

On Certainty (1700-1799)

"Probability is a degree of certainty and it differs from certainty as a part from a whole." (Jacob Bernoulli, "Ars Conjectandi", 1713)

"We define the art of conjecture, or stochastic art, as the art of evaluating as exactly as possible the probabilities of things, so that in our judgments and actions we can always base ourselves on what has been found to be the best, the most appropriate, the most certain, the best advised; this is the only object of the wisdom of the philosopher and the prudence of the statesman." (Jacob Bernoulli, "Ars Conjectandi", 1713)

"[…] our reasonings about the wonderful and intricate operations of Nature are so full of uncertainty, that, as the wise-man truly observes, hardly do we guess aright at the things that are upon earth, and with labour do we find the things that are before us." (Stephen Hales, "Vegetable Staticks", 1727)

"As arithmetic and algebra are sciences of great clearness, certainty, and extent, which are immediately conversant about signs, upon the skillful use whereof they entirely depend, so a little attention to them may possibly help us to judge of the progress of the mind in other sciences, which, though differing in nature, design, and object, may yet agree in the general methods of proof and inquiry." (George Berkeley, "Alciphorn: or, the Minute Philosopher", 1732)

"It may be observed of mathematicians that they only meddle with such things as are certain, passing by those that are doubtful and unknown. They profess not to know all things, neither do they affect to speak of all things. What they know to be true, and can make good by invincible arguments, that they publish and insert among their theorems. Of other things they are silent and pass no judgment at all, choosing rather to acknowledge their ignorance, than affirm anything rashly." (Isaac Barrow, "Mathematical Lecture", 1734)

"Do not expect to arrive at certainty in every subject which you pursue. There are a hundred things wherein we mortals […] must be content with probability, where our best light and reasoning will reach no farther." (Isaac Watts," The Improvement of the Mind", 1741) 

 

"There is nothing more pleasant for man than the certainty of knowledge; whoever has once tasted of it is repelled by everything in which he perceives nothing but uncertainty. This is why the mathematicians who always deal with certain knowledge have been repelled by philosophy and other things, and have found nothing more pleasant than to spend their time with lines and letters." (Christian Wolff, 1741)

"One should not be deceived by philosophical works that pretend to be mathematical, but are merely dubious and murky metaphysics. Just because a philosopher can recite the words lemma, theorem and corollary doesn't mean that his work has the certainty of mathematics. That certainty does not derive from big words, or even from the method used by geometers, but rather from the utter simplicity of the objects considered by mathematics." (Pierre L Maupertuis, "Les Loix du Mouvement et du Repos, déduites d'un Principe Métaphysique", 1746)

"All effects follow not with like certainty from their supposed causes." (David Hume, "An Enquiry Concerning Human Understanding", 1748)

"If an inquiry thus carefully conducted should fail at last of discovering the truth, it may answer an end perhaps as useful, in discovering to us the weakness of our own understanding. If it does not make us knowing, it may make us modest. If it does not preserve us from error, it may at least from the spirit of error; and may make us cautious of pronouncing with positiveness or with haste, when so much labour may end in so much uncertainty." (Edmund Burke, "Essay on the Sublime and Beautiful", 1756)

"He who has not made the experiment, or who is not accustomed to require rigorous accuracy from himself, will scarcely believe how much a few hours take from certainty of knowledge, and distinctness of imagery; how the succession of objects will be broken, how separate parts will be confused, and how many particular features and discriminations will be compressed and conglobated into one gross and general idea." (Samuel Johnson, "A Journey to the Western Islands of Scotland", 1775)

"Metaphysical truths can only be established by producing effects from corresponding causes; and though we may confront such demonstrative evidence with the immutable laws of mathematical decision, we must be sensible that there will still remain some pretense for doubt; thus the basis of that knowledge, which on these principles we have been long labouring to accomplish, will become an endless toil, an endless force for controversy: and having the passions and the prejudices of mankind to combat, which mathematical certainty can alone effectually suppress, we must content ourselves only with making converts of those who have minds sufficiently expansive without the shackles of Euclid, and the vanity of displaying their own learning and pedantry." (James Douglas, "A Dissertation on the Antiquity of the Earth", 1785)

Martin Heidegger - Collected Quotes

"The average, vague understanding of being can be permeated by traditional theories and opinions about being in such a way that these theories, as the sources of the prevailing understanding, remain hidden." (Martin Heidegger, "Being and Time", 1927)

"Philosophy is metaphysics. Metaphysics thinks beings as a whole - the world, man, God - with respect to Being, with respect to the belonging together of beings in Being. Metaphysics thinks beings as being in the manner of representational thinking which gives reasons." (Martin Heidegger, "The End of Philosophy and the Task of Thinking", 1964)

"All great insights and discoveries are not only usually thought by several people at the same time, they must also be re-thought in that unique effort to truly say the same thing about the same thing." (Martin Heidegger, "What Is A Thing", 1967)

"Our expression 'the mathematical' always has two meanings. It means, first, what can be learned in the manner we have indicated, and only in that way, and, second, the manner of learning and the process itself. The mathematical is that evident aspect of things within which we are always already moving and according to which we experience them as things at all, and as such things. The mathematical is this fundamental position we take toward things by which we take up things as already given to us, and as they must and should be given. Therefore, the mathematical is the fundamental presupposition of the knowledge of things." (Martin Heidegger, "What Is A Thing", 1967)

"The mathematical is that evident aspect of things within which we are always already moving and according to which we experience them as things at all, and as such things. The mathematical is this fundamental position we take toward things by which we take up things as already given to us, and as they must and should be given. Therefore, the mathematical is the fundamental presupposition of the knowledge of things." (Martin Heidegger, "Modern Science, Metaphysics and Mathematics", 1967)

"We take cognizance of all this and learn it without regard for the things. Numbers are the most familiar form of the mathematical because, in our usual dealing with things, when we calculate or count, numbers are the closest to that which we recognize in things without deriving it from them. For this reason numbers are the most familiar form of the mathematical. In this way, this most familiar mathematical becomes mathematics. But the essence of the mathematical does not lie in number as purely delimiting the pure ‘how much’, but vice versa. Because number has such a nature, therefore, it belongs to the learnable in the sense of mathesis." (Martin Heidegger, "Modern Science, Metaphysics and Mathematics", 1967)

"Teaching is more difficult than learning because what teaching calls for is this: to let learn. The real teacher, in fact, let nothing else be learned than learning. His conduct, therefore, often produces the impression that we properly learn nothing from him, if by ‘learning’ we now suddenly understand merely the procurement of useful information." (Martin Heidegger, "What is called thinking?", 1968)

On Discovery (1950-1974)

"Scientific discovery consists in the interpretation for our own convenience of a system of existence which has been made with no eye to our convenience at all." (Norbert Wiener, "Human Use of Human Beings: Cybernetics and Society", 1950)

"The scientist who discovers a theory is usually guided to his discovery by guesses; he cannot name a method by means of which he found the theory and can only say that it appeared plausible to him, that he had the right hunch or that he saw intuitively which assumption would fit the facts." (Hans Reichenbach, "The Rise of Scientific Philosophy", 1951)

"A discovery in science, or a new theory, even when it appears most unitary and most all-embracing, deals with some immediate element of novelty or paradox within the framework of far vaster, unanalysed, unarticulated reserves of knowledge, experience, faith, and presupposition. Our progress is narrow; it takes a vast world unchallenged and for granted. This is one reason why, however great the novelty or scope of new discovery, we neither can, nor need, rebuild the house of the mind very rapidly. This is one reason why science, for all its revolutions, is conservative. This is why we will have to accept the fact that no one of us really will ever know very much. This is why we shall have to find comfort in the fact that, taken together, we know more and more." (J. Robert Oppenheimer, Science and the Common Understanding, 1954)

"It is important for him who wants to discover not to confine himself to one chapter of science, but to keep in touch with various others." (Jacques S Hadamard, "An Essay on the Psychology of Invention in the Mathematical Field", 1954)

"The true aim of science is to discover a simple theory which is necessary and sufficient to cover the facts, when they have been purified of traditional prejudices." (Lancelot L Whyte, "Accent on Form", 1954)

"At bottom, the society of scientists is more important than their discoveries. What science has to teach us here is not its techniques but its spirit: the irresistible need to explore." (Jacob Bronowski, "Science and Human Values", 1956)

"A change in science, whether novelty or discovery, when properly understood, when the linguistic problem is adequately solved, will even then provide only a hunch, a starting point for looking at an area of experience other than the science in which it was nourished and born." (J Robert Oppenheimer, "The Growth of Science and the Structure of Culture", Daedalus, 1958) 

"Creating a new theory is not like destroying an old barn and erecting a skyscraper in its place. It is rather like climbing a mountain, gaining new and wider views, discovering unexpected connections between our starting point and its rich environment. But the point from which we started out still exists and can be seen, although it appears smaller and forms a tiny part of our broad view gained by the mastery of the obstacles on our adventurous way up." (Leopold Infeld, "The Evolution of Physics", 1961)

"First, I should be clear about what the act of discovery entails. It is rarely, on the frontier of knowledge or elsewhere, that new facts are 'discovered' in the sense of being encountered, as Newton suggested, in the form of islands of truth in an uncharted sea of ignorance. Or if they appear to be discovered in this way, it is almost always thanks to some happy hypothesis about where to navigate. Discovery, like surprise, favors the well-prepared mind." (Jerome S Bruner, "On Knowing: Essays for the Left Hand", 1962)

"We must now ask how changes of this sort can come about, considering first discoveries, or novelties of fact, and then inventions, or novelties of theory. That distinction between discovery and invention or between fact and theory will, however, immediately prove to be exceedingly artificial." (Thomas Kuhn, "The Structure of Scientific Revolutions", 1962)

“Discovery follows discovery, each both raising and answering questions, each ending a long search, and each providing the new instruments for a new search.” (J Robert Oppenheimer, "Prospects in the Arts and Sciences", 1964)

"The central task of a natural science is to make the wonderful commonplace: to show that complexity, correctly viewed, is only a mask for simplicity; to find pattern hidden in apparent chaos. […] This is the task of natural science: to show that the wonderful is not incomprehensible, to show how it can be comprehended - but not to destroy wonder. For when we have explained the wonderful, unmasked the hidden pattern, a new wonder arises at how complexity was woven out of simplicity. The aesthetics of natural science and mathematics is at one with the aesthetics of music and painting - both inhere in the discovery of a partially concealed pattern." (Herbert A Simon, "The Sciences of the Artificial", 1968)

"Discovery always carries an honorific connotation. It is the stamp of approval on a finding of lasting value. Many laws and theories have come and gone in the history of science, but they are not spoken of as discoveries. […] Theories are especially precarious, as this century profoundly testifies. World views can and do often change. Despite these difficulties, it is still true that to count as a discovery a finding must be of at least relatively permanent value, as shown by its inclusion in the generally accepted body of scientific knowledge." (Richard J. Blackwell, "Discovery in the Physical Sciences", 1969)

"A discovery must be, by definition, at variance with existing knowledge." (Albert Szent-Gyorgyi, "Dionysians and Apollonians", Science 176, 1972)

"Taken as a story of human achievement, and human blindness, the discoveries in the sciences are among the great epics." (J Robert Oppenheimer, "Reflections on the resonances of physics history" , 1972) 

"Discoveries are made by pursuing possibilities suggested by existing knowledge." (Michael Polanyi, "Meaning", 1975)

"Metaphysics attempts to discover the ultimate nature of reality, and in this sense, the innerspace of science fiction is metaphysical fiction." (Kate Wilhelm, 1974)

On Discovery (1700-1799)

"Wit, you know, is the unexpected copulation of ideas, the discovery of some occult relation between images in appearance remote from each other." (Samuel Johnson, "The Rambler", 1750)

"[…] chance, that is, an infinite number of events, with respect to which our ignorance will not permit us to perceive their causes, and the chain that connects them together. Now, this chance has a greater share in our education than is imagined. It is this that places certain objects before us and, in consequence of this, occasions more happy ideas, and sometimes leads us to the greatest discoveries […]" (Claude Adrien Helvetius, "On Mind", 1751)

"If an inquiry thus carefully conducted should fail at last of discovering the truth, it may answer an end perhaps as useful, in discovering to us the weakness of our own understanding. If it does not make us knowing, it may make us modest. If it does not preserve us from error, it may at least from the spirit of error; and may make us cautious of pronouncing with positiveness or with haste, when so much labour may end in so much uncertainty." (Edmund Burke, "Essay on the Sublime and Beautiful", 1756)

"To endeavor at discovering the connections that subsist in nature, is no way inconsistent with prudence; but it is downright folly to push these researches too far; as it is the lot only of superior Beings to see the dependence of events, from one end to the other, of the chain which supports them." (Pierre Louis Maupertuis, "An Essay Towards a History of the Principal Comets Since 1742", 1769)

"Cultivate that kind of knowledge which enables us to discover for ourselves in case of need that which others have to read or be told of." (Georg C Lichtenberg, Notebook D, 1773-1775)

"It falls into this difficulty without any fault of its own. It begins with principles, which cannot be dispensed with in the field of experience, and the truth and sufficiency of which are, at the same time, insured by experience. With these principles it rises, in obedience to the laws of its own nature, to ever higher and more remote conditions. But it quickly discovers that, in this way, its labours must remain ever incomplete, because new questions never cease to present themselves; and thus it finds itself compelled to have recourse to principles which transcend the region of experience, while they are regarded by common sense without distrust. It thus falls into confusion and contradictions, from which it conjectures the presence of latent errors, which, however, it is unable to discover, because the principles it employs, transcending the limits of experience, cannot be tested by that criterion. The arena of these endless contests is called Metaphysic." (Immanuel Kant, "The Critique of Pure Reason", 1781)

"This schematism of our understanding, in its application to appearances and their mere form, is an art concealed in the depths of the human soul, whose real modes of activity nature is hardly likely ever to allow us to discover, and to have open to our gaze." (Immanuel Kant, "Critique of Pure Reason", 1781)

"A good method of discovery is to imagine certain members of a system removed and then see how what is left would behave: for example, where would we be if iron were absent from the world: this is an old example." (Georg C Lichtenberg, Notebook J, 1789-1793)

"Every science has for its basis a system of principles as fixed and unalterable as those by which the universe is regulated and governed. Man cannot make principles; he can only discover them." (Thomas Paine, "The Age of Reason", 1794)

Map vs. Territory II

"We say the map is different from the territory. But what is the territory? Operationally, somebody went out with a retina or a measuring stick and made representations which were then put on paper. What is on the paper map is a representation of what was in the retinal representation of the man who made the map; and as you push the question back, what you find is an infinite regress, an infinite series of maps. The territory never gets in at all. […] Always, the process of representation will filter it out so that the mental world is only maps of maps, ad infinitum." (Gregory Bateson, "Steps to an Ecology of Mind", 1972)

"For our purposes, a simple way to understand paradigms is to see them as maps. We all know that ‘the map is not the territory’. A map is simply an explanation of certain aspects of the territory. That’s exactly what a paradigm is. It is a theory, an explanation, or model of something else." (Stephen R Covey, "The 7 Habits of Highly Effective People", 1989)

"Principles are the territory. Values are maps. When we value correct principles, we have truth - a knowledge of things as they are." (Stephen R Covey, "The 7 Habits of Highly Effective People", 1989)

"Today abstraction is no longer that of the map, the double, the mirror, or the concept. Simulation is no longer that of a territory, a referential being or substance. It is the generation by models of a real without origin or reality: A hyperreal. The territory no longer precedes the map, nor does it survive it. It is nevertheless the map that precedes the territory - precession of simulacra - that engenders the territory." (Baudrillard Jean, "Simulacra and Simulation", 1981)

"The ultimate metaphysical secret, if we dare state it so simply, is that there are no boundaries in the universe. Boundaries are illusions, products not of reality but of the way we map and edit reality. And while it is fine to map out the territory, it is fatal to confuse the two." (Ken Wilber, "No Boundary: Eastern and Western Approaches to Personal Growth", 1979)

"Our view of reality is like a map with which to negotiate the terrain of life. If the map is true and accurate, we will generally know how to get there. If the map is false and inaccurate, we generally will be lost." (M Scott Peck, "Wisdom from the Road Less Traveled", 2001)

"A map does not just replicate the shape of a territory; rather, it actively inflects and works over that territory." (Steven Shaviro, "Post Cinematic Affect", 2010)

"An all-inclusive model would be like the map in the famous story by Borges - perfect and inclusive because it was identical to the territory it was mapping." (Reuben Hersh,” Mathematics as an Empirical Phenomenon, Subject to Modeling”, 2017) 

Pierre L Maupertuis - Collected Quotes

"Nature always uses the simplest means to accomplish its effects." (Pierre L Maupertuis, "Accord between different laws of Nature that seemed incompatible", Mémoires de l'académie royale des sciences, 1744)

"A true philosopher does not engage in vain disputes about the nature of motion; rather, he wishes to know the laws by which it is distributed, conserved or destroyed, knowing that such laws is the basis for all natural philosophy." (Pierre L Maupertuis, "Les Loix du Mouvement et du Repos, déduites d'un Principe Métaphysique", 1746)

"Despite the disorder observed in Nature, one finds enough traces of the wisdom and power of its Author that one cannot fail to recognize Him." (Pierre L Maupertuis, "Les Loix du Mouvement et du Repos, déduites d'un Principe Métaphysique", 1746)

"Everything is so arranged that the blind logic of mathematics executes the will of the most enlightened and free Mind." (Pierre L Maupertuis, "Les Loix du Mouvement et du Repos, déduites d'un Principe Métaphysique", 1746)

"One should not be deceived by philosophical works that pretend to be mathematical, but are merely dubious and murky metaphysics. Just because a philosopher can recite the words lemma, theorem and corollary doesn't mean that his work has the certainty of mathematics. That certainty does not derive from big words, or even from the method used by geometers, but rather from the utter simplicity of the objects considered by mathematics." (Pierre L Maupertuis, "Les Loix du Mouvement et du Repos, déduites d'un Principe Métaphysique", 1746)

"The laws of movement and of rest deduced from this principle being precisely the same as those observed in nature, we can admire the application of it to all phenomena. The movement of animals, the vegetative growth of plants [...] are only its consequences; and the spectacle of the universe becomes so much the grander, so much more beautiful, the worthier of its Author, when one knows that a small number of laws, most wisely established, suffice for all movements." (Pierre L M Maupertuis, "Les Loix du mouvement et du repos déduites d'un principe metaphysique", Histoire de l'Académie Royale des Sciences et des Belles Lettres, 1746)

"The supreme Being is everywhere; but He is not equally visible everywhere. Let us seek Him in the simplest things, in the most fundamental laws of Nature, in the universal rules by which movement is conserved, distributed or destroyed; and let us not seek Him in phenomena that are merely complex consequences of these laws." (Pierre L Maupertuis, "Les Loix du Mouvement et du Repos, déduites d'un Principe Métaphysique", 1746) 

"When a change occurs in Nature, the quantity of action necessary for that change is as small as possible." (Pierre L Maupertuis, "Les Loix du Mouvement et du Repos, déduites d'un Principe Métaphysique", 1746)

"To endeavor at discovering the connections that subsist in nature, is no way inconsistent with prudence; but it is downright folly to push these researches too far; as it is the lot only of superior Beings to see the dependence of events, from one end to the other, of the chain which supports them." (Pierre Louis Maupertuis, "An Essay Towards a History of the Principal Comets Since 1742", 1769)

Aristotle - Collected Quotes

"A deduction is a demonstration when the premisses from which the deduction starts are true and primitive, or are such that our knowledge of them has originally come through premisses which are primitive and true." (Aristotle, "Topics", cca. 350 BC)

"A problem that is suitable for argumentative reasoning is one for which there are numerous and good arguments." (Aristotle, "Topics", cca. 350 BC)

"A quantity is infinite if it is such that we can always take a part outside what has been already taken." (Aristotle, "Physics")

"All human actions have one or more of these seven causes: chance, nature, compulsions, habit, reason, passion, desire." (Aristotle)

"But Nature flies from the infinite, for the infinite is unending or imperfect, and Nature ever seeks an end." (Aristotle, "Generation of Animals")

"Educating the mind without educating the heart is no education at all." (Aristotle)

"Education and morals will be found almost the whole that goes to make a good man." (Aristotle)

"Education is the best provision for old age." (Aristotle)

"For all who argue per impossibile infer by syllogism a false conclusion, and prove the original conclusion hypothetically when something impossible follows from a contradictory assumption, as, for example, that the diagonal [of a square] is incommensurable [with the side] because odd numbers are equal to even if it is assumed to be commensurate. It is inferred by syllogism that odd numbers are equal to even, and proved hypothetically that the diagonal is incommensurate, since a false conclusion follows from the contradictory assumption." (Aristotle, "Laws")

"For generally the infinite has this mode of existence: one thing is always being taken after another, and each thing that is taken is always finite […]." (Aristotle, "Physics")

"For that which is probable is that which generally happens." (Aristotle, "The Art of Rhetoric", 4th century BC)

"For the things we have to learn before we can do, we learn by doing." (Aristotle, "Nicomachean Ethics", Book II, 349 BC)

"[Imagination is] that in virtue of which we say that an image occurs to us and not as we speak of it metaphorically."  (Aristotle, "De Anima" III, cca. 350 BC)

"If 'bounded by a surface' is the definition of body there cannot be an infinite body either intelligible or sensible." ((Aristotle, "De Caelo" ["On the Heavens"], cca. 350 BC)

"In all disciplines in which there is systematic knowledge of things with principles, causes, or elements, it arises from a grasp of those: we think we have knowledge of a thing when we have found its primary causes and principles, and followed it back to its elements." (Aristotle, "Physics", cca. 350 BC)

"In all things which have a plurality of parts, and which are not a total aggregate but a whole of some sort distinct from the parts, there is some cause." (Aristotle, "Metaphysics", cca. 335-323 BC)

"Intuition is the source of scientific knowledge." (Aristotle)

"It is impossible that the existence of something [one thing] should necessitate both the existence and the non-existence of something else [i.e., one (same) other thing]." (Aristotle, "Posterior Analytics", cca. 350 BC)

"It is obvious then, that memory belongs to that part of the soul to which imagination belongs. […] Just as the picture painted on the panel is at once a picture and a portrait, and though one and the same, is both, yet the essence of the two is not the same, and it is possible to think of it both as a picture and as a portrait, so in the same way we must regard the mental picture within us both as an object of contemplation in itself and as a mental picture of something else […]. Insofar as we consider it in relation to something else, e.g. as a likeness, it is also an aid to memory." (Aristotle, "De Memoria et Reminiscentia" [On Memory and Recollection], 4th century BC)

"It is the mark of an educated mind to rest satisfied with the degree of precision which the nature of the subject admits and not to seek exactness where only an approximation is possible."  (Aristotle, "Nicomachean Ethics", Book II, 349 BC)

"Knowledge, then, is a state of capacity to demonstrate, and has the other limiting characteristics which we specify in the Analytics; for it is when one believes in a certain way and the principles are known to him that he has knowledge, since if they are not better known to him than the conclusion, he will have his knowledge only on the basis of some concomitant." (Aristotle," Nicomachean Ethics", cca. 340 BC)

"Metaphors do somehow make known what they mean by the comparison they involve. Whenever we use them, we transfer meaning according to some similarity. But statements such as the aforesaid do not make anything known. There is no inherent resemblance to justify calling law ‘a measure’ or ‘an image’ , nor is it customary to refer to it as such. If one says that law is literally ‘a measure’ or ‘an 

image’, he is either deceiving or being deceived. An image is something fashioned in the likeness of something else, but such is not an inherent characteristic of law. If, on the other hand, the statement is not made in a literal sense, it is evident that it is obscure, and worse than any metaphorical expression." (Aristotle)

"Metaphysics is universal and is exclusively concerned with primary substance. […] And here we will have the science to study that which is, both in its essence and in the properties which, just as a thing that is, it has. […] That among entities there must be some cause which moves and combines things. […] There must then be a principle of such a kind that its substance is activity." (Aristotle, "Metaphysics", cca. 335-323 BC)

"Of the figures, the first is especially scientific. The mathematical sciences carry out their demonstrations through it – e.g. arithmetic and geometry and optics – and so do almost all those sciences which inquire into the reason why. For deductions giving the reason why are carried out, either in general or for the most part and in most cases, through this figure. For this reason, then, it is especially scientific; for study of the reason why has most importance for knowledge." (Aristotle, "Posterior Analytics", cca. 350 BC) 

"Our account does not rob mathematicians of their science, by disproving the actual existence of the infinite in the direction of increase, in the sense of the untraceable. In point of fact they do not need the infinite and do not use it. They postulate any that the finite straight line may be produced as far as they wish." (Aristotle, "Physics")

"Sense perception is a prerequisite for memory; the memory of frequently repeated sense perceptions results in experimental proof; experimental proofs provide the materials for a science or an art.” (Aristotle)

"So poetry is something more philosophical and more worthy of serious attention than history, for while poetry is concerned with universal truth, history treats of particular facts [...]" (Aristotle, "Poetics", cca. 350 BC)

"The chief forms of beauty are order and symmetry and definiteness, which the mathematical sciences demonstrate in a special degree. And since these (e.g. order and definiteness) are obviously causes of many things, evidently these sciences must treat this sort of causative principle also (i.e. the beautiful) as in some sense a cause." (Aristotle, "Metaphysica", cca. 350 BC)

"The infinite is imperfect, unfinished and therefore, unthinkable; it is formless and confused."  (Aristotle)

"The intellectual capacity thinks the forms in the phantasmata (mental images) […] And for the following reason, as without having perceptual awareness no one could either learn or understand anything, so when one engages in intellectual activity one must at that time do so by means of a phantasma. For, phantasmata are just as perceptual states (aisthemata) are [in actual external perception] but without matter." (Aristotle, "De Anima" III, cca. 350 BC)

"The mathematical sciences particularly exhibit order, symmetry, and limitation; and these are the greatest forms of the beautiful." (Aristotle, "Metaphysics", cca. 335-323 BC)

"The only possible way to conceive universal is by induction, since we come to know abstractions by induction. But unless we have sense experience, we cannot make inductions. Even though sense perception relates to particular things, scientific knowledge concerning such can only be constructed by the successive steps of sense perception, induction, and formulation of universals." (Aristotle, "Posterior Analytics", cca. 350 BC)

"The point is prior to, and in itself more evident than the line. The same may be said of the line relative to the plane surface, and of the plane surface with reference to the solid. It is likewise true of unity in relation to plurality, for which unity is the principle. This also holds in regard to the letter relative to the syllable. [...] The reverse, however, sometimes occurs in the case of our knowledge. Generally the average mind more readily perceives what is posterior, whereas the comprehension of what is prior is reserved to the more profound and learned intellect." (Aristotle, "Topics", cca. 350 BC)

"The same ideas, one must believe, recur in men’s minds not once or twice but again and again." (Aristotle, "De Caelo" ["On the Heavens"], cca. 350 BC)

"The soul never thinks without a picture." (Aristotle)

"The whole is more than the sum of its parts." (Aristotle, "Metaphysics", cca. 335-323 BC)

"Those who assert that the mathematical sciences make no affirmation about what is fair or good make a false assertion; for they do speak of these and frame demonstrations of them in the most eminent sense of the word. […] Of what is fair, however, the most important species are order and symmetry, and that which is definite, which the mathematical sciences make manifest in a most eminent degree." (Aristotle, "Metaphysics", cca. 335-323 BC)

"Time and space are divided into the same and equal divisions. Wherefore also, Zeno’s argument, that it is impossible to go through an infinite collection or to touch an infinite collection one by one in a finite time, is fallacious. For there are two senses in which the term ‘infinte’ is applied both to length and to time and in fact to all continuous things: either in regard to divisibility or in regard to number. Now it is not possible to touch things infinite as to number in a finite time, but it is possible to touch things infinite in regard to divisibility; for time itself is also infinite in this sense." (Aristotle)

"What is one is indivisible whatever it may be, e.g. a man is one man, not many. Number on the other hand is a plurality of 'ones' and a certain quantity of them. Hence number must stop at the indivisible: for 'two' and 'three' are merely derivative terms, and so with each of the other numbers." (Aristotle, "Physics", cca. 4th-century BC)

"What we know is not capable of being otherwise; of things capable of being otherwise we do not know, when they have passed outsideour observation, whether they exist or not. Therefore the object of knowledge is of necessity. Therefore it is eternal; for things that are of necessity in the unqualified sense are all eternal; and things that are eternal are ungenerated and imperishable. " (Aristotle, "Nicomachean Ethics", cca. 340 BC)

"When the consequences of either assumption are the same, we should always assume that things are finite rather than infinite in number, since in things constituted by nature that which is infinite and that which is better ought, if possible, to be present rather than the reverse […]" (Aristotle)

"While those whom devotion to abstract discussions has rendered unobservant of the facts are too ready to dogmatize on the basis of a few observations." (Aristotle, "De Caelo" ["On the Heavens"], cca. 350 BC)

Gottfried W Leibniz - Collected Quotes

"For any number there exists a corresponding even number which is its double. Hence the number of all numbers is not greater than the number of even numbers, that is, the whole is not greater than the part." (Gottfried W Leibniz, "De Arte Combinatoria", 1666)

"Reality cannot be found except in One single source, because of the interconnection of all things with one another. […] It is a good thing to proceed in order and to establish propositions (principles). This is the way to gain ground and to progress with certainty." (Gottfried W Leibniz, 1670)

"I believe that we need another analysis properly geometric or linear, which treats PLACE directly the way that algebra treats MAGNITUDE." (Gottfried W Leibniz, 1670s)

"But if now a simple, that is, a linear equation, is multiplied by a quadratic, a cubic equation will result, which will have  real roots if the quadratic is possible, or two imaginary roots and only one real one if the quadratic is impossible. […] How can it be, that a real quantity, a root of the proposed equation, is expressed by the intervention of an imaginary? For this is the remarkable thing, that, as calculation shows, such an imaginary quantity is only observed to enter those cubic equations that have no imaginary root, all their roots being real or possible, as has been shown by trisection of an angle, by Albert Girard and others. […] This difficulty has been too much for all writers on algebra up to the present, and they have all said they that in this case Cardano’s rules fail." (Gottfried W Leibniz, cca. 1675)

"For this evil I have found a remedy and obtained a method, by which without experimentation the roots of such binomials can be extracted, imaginaries being no hindrance, and not only in the case of cubics but also in higher equations. This invention rests upon a certain peculiarity which I will explain later. Now I will add certain rules derived from the consideration of irrationals (although no mention is made of irrationals), by which a rational root can easily be extracted from them." (Gottfried W Leibniz, cca. 1675)

"Only geometry can hand us the thread [which will lead us through] the labyrinth of the continuum's composition, the maximum and the minimum, the infinitesimal and the infinite; and no one will arrive at a truly solid metaphysics except he who has passed through this [labyrinth]." (Gottfried W Leibniz, "Dissertatio Exoterica De Statu Praesenti et Incrementis Novissimis Deque Usu Geometriae", 1676)

"Probability is a degree of possibility." (Gottfried W Leibniz, "On estimating the uncertain", 1676)

"It is obvious that if we could find characters or signs suited for expressing all our thoughts as clearly and as exactly as arithmetic expresses numbers or geometry expresses lines, we could do in all matters, insofar as they are subject to reasoning, all that we can do in arithmetic and geometry." (Gottfried W Leibniz, 1677)

"After all the progress I have made in these matters, I am still not happy with Algebra, because it provides neither the shortest ways nor the most beautiful constructions of Geometry. This is why when it comes to that, I think that we need another analysis which is properly geometric or linear, which expresses to us directly situm, in the same way as algebra expresses magnitudinem. And I think that I have the tools for that, and that we might represent figures and even engines and motion in character, in the same way as algebra represents numbers in magnitude." (Gottfried W Leibniz, [letter to Christiaan Huygens] 1679)

"I found the elements of a new characteristic, completely different from Algebra and which will have great advantages for the exact and natural mental representation, although without figures, of everything that depends on the imagination. Algebra is nothing but the characteristic of undetermined numbers or magnitudes. But it does not directly express the place, angles and motions, from which it follows that it is often difficult to reduce, in a computation, what is in a figure, and that it is even more difficult to find geometrical proofs and constructions which are enough practical even when the Algebraic calculus is all done." (Gottfried W Leibniz, [letter to Christiaan Huygens] 1679)

"Probability, however, is not something absolute, [it is] drawn from certain information which, although it does not suffice to resolve the problem, nevertheless ensures that we judge correctly which of the two opposites is the easiest (facilius) given the conditions known to us." (Gottfried W Leibniz, "Forethoughts for an encyclopaedia or universal science", cca. 1679)

"But the most powerful proof of the reality of phenomena (a proof which is, indeed, sufficient by itself) is success in predicting future phenomena from those which are past and present, whether the prediction be founded upon the success, so far, of a reason or hypothesis, or upon custom so far observed." (Gottfried W Leibniz, "De Modo Distinguendi phenomena realia ab imaginariis" ["On the Method of Distinguishing Real from Imaginary Phenomena"], cca. 1684)

"A distinction must be made between true and false ideas, and that too much rein must not be given to a man's imagination under pretext of its being a clear and distinct intellection." (Gottfried W Leibniz, "Discours de métaphysique", 1686)

"Every substance is as a world apart, independent of everything else except God." (Gottfried W Leibniz, "Discours de métaphysique", 1686)

"In whatever manner God created the world, it would always have been regular and in a certain general order. God, however, has chosen the most perfect, that is to say, the one which is at the same time the simplest in hypothesis and the richest in phenomena." (Gottfried W Leibniz, "Discours de métaphysique", 1686)

"It is also only by virtue of the continual action of God upon us that we have in our soul the ideas of all things; that is to say, since every effect expresses its cause, the essence of our soul is a certain expression, imitation or image of the divine essence, thought, and will and of all the ideas which are comprised in God." (Gottfried W Leibniz, "Discourse on Metaphysics", 1686)

"When a rule is extremely complex, that which conforms to it passes for irregular (random)." (Gottfried Leibniz, "Discourse on Metaphysics", 1686)

"Consider however (imitating Mathematicians) certainty or truth to be like the whole; and probabilities [to be like] parts, such that probabilities would be to truths what an acute angle [is] to a right [angle]." (Gottfried W Leibniz, [to Vincent Placcius] 1687)

"Infinities and infinitely small quantities could be taken as fictions, similar to imaginary roots, except that it would make our calculations wrong, these fictions being useful and based in reality." (Gottfried W Leibniz, [letter to Johann Bernoulli] 1689)

"God makes nothing without order, and everything that forms itself develops imperceptibly out of small parts." (Gottfried W Leibniz, "Protogaea", 1693/1759)

"Imaginary numbers are a fine and wonderful refuge of the divine spirit almost an amphibian between being and non-being." (Gottfried Leibniz, 1702)

"[...] nature has established patterns originating in the return of events, but only for the most part." (Gottfried W Leibniz [letter to Jacob Bernoulli], 1703)

"The calculation of probabilities is of the utmost value, […] but in statistical inquiries there is need not so much of mathematical subtlety as of a precise statement of all the circumstances. The possible contingencies are too numerous to be covered by a finite number of experiments, and exact calculation is, therefore, out of the question. Although nature has her habits, due to the recurrence of causes, they are general, not invariable. Yet empirical calculation, although it is inexact, may be adequate in affairs of practice." (Gottfried W Leibniz [letter to Bernoulli], 1703)

"Two things are identical if one can be substituted for the other without affecting the truth." (Gottfried W Leibniz, "Table de definitions", 1704)

"Nothing is accomplished all at once, and it is one of my great maxims, and one of the most completely verified, that Nature makes no leaps: a maxim which I have called the law of continuity." (Gottfried W Leibniz, "New Essays Concerning Human Understanding", 1704/1765)

"Progress depends on our sagacity in finding intermediate ideas. Those who are ignorant of algebra cannot imagine the wonderful things that may be done in this field by means of this science. And I do not see that it is easy to determine what new means of perfecting other parts of our knowledge may yet be found out by a penetrating mind." (Gottfried W Leibniz, "New Essays Concerning Human Understanding", 1704/1765)

"The art of discovering the causes of phenomena, or true hypothesis, is like the art of decyphering, in which an ingenious conjecture greatly shortens the road." (Gottfried W Leibniz, "New Essays Concerning Human Understanding", 1704/1765)

"As in mathematics, when there is no maximum nor minimum, in short nothing distinguished, everything is done equally, or when that is not nothing at all is done: so it may be said likewise in respect of perfect wisdom, which is no less orderly than mathematics, that if there were not the best (optimum) among all possible worlds, God would not have produced any." (Gottfried W Leibniz, "Theodicy: Essays on the Goodness of God and Freedom of Man and the Origin of Evil", 1710)

"There are two famous labyrinths where our reason very often goes astray. One concerns the great question of the free and the necessary, above all in the production and the origin of Evil. The other consists in the discussion of continuity, and of the indivisibles which appear to be the elements thereof, and where the consideration of the infinite must enter in." (Gottfried W Leibniz, "Theodicy: Essays on the Goodness of God and Freedom of Man and the Origin of Evil", 1710) 

"Moreover, it must be confessed that perception and that which depends upon it are inexplicable on mechanical grounds, that is to say, by means of figures and motions. And supposing there were a machine, so constructed as to think, feel, and have perception, it might be conceived as increased in size, while keeping the same proportions, so that one might go into it as into a mill. That being so, we should, on examining its interior, find only parts which work one upon another, and never anything by which to explain a perception. Thus it is in a simple substance, and not in a compound or in a machine, that perception must be sought for." (Gottfried W Leibniz,  "Monadology", 1714)

"There are two kinds of truths: those of reasoning and those of fact. The truths of reasoning are necessary and their opposite is impossible; the truths of fact are contingent and their opposites are possible." (Gottfried W Leibniz,  "Monadology", 1714)

"This interconnection or accommodation of all created things to each other, and each to all the others, brings it about that each simple substance has relations that express all the others, and consequently, that each simple substance is a perpetual, living mirror of the universe." (Gottfried W Leibniz,  "Monadology", 1714)

"If a plurality of states of things is assumed to exist which involves no opposition to each other, they are said to exist simultaneously. Thus we deny that what occurred last year and this year are simultaneous, for they involve incompatible states of the same thing.  If one of two states which are not simultaneous involves a reason for the other, the former is held to be prior, the latter posterior. My earlier state involves a reason for the existence of my later state. And since my prior state, by reason of the connection between all things, involves the prior state of other things as well, it also involves a reason for the later state of these other things and is thus prior to them. Therefore whatever exists is either simultaneous with other existences or prior or posterior." (Gottfried W Leibniz, "Initium rerum Mathematicarum metaphysica", 1715)

"Time is the order of existence of those things which are not simultaneous. Thus time is the universal order of changes when we do not take into consideration the particular kinds of change. Duration is magnitude of time. If the magnitude of time is diminished uniformly and continuously, time disappears into moment, whose magnitude is zero. Space is the order of coexisting things, or the order of existence for things which are simultaneous." (Gottfried W Leibniz, "Initium rerum Mathematicarum metaphysica", 1715)

"We should like Nature to go no further; we should like it to be finite, like our mind; but this is to ignore the greatness and majesty of the Author of things." (Gottfried W Leibniz, [letter to S Clarke] 1715)

"It is easily seen from a consideration of the nature of demonstration and analysis that there can and must be truths which cannot be reduced by any analysis to identities or to the principle of contradiction but which involve an infinite series of reasons which only God can see through." (Gottfried W Leibniz, "Nouvelles lettres et opuscules inédits", 1857)

"All things in the whole wide world happen mathematically." (Gottfried W Leibniz [attributed])

"Atoms are the effect of the weakness of our imagination, for it likes to rest and therefore hurries to arrive at a conclusion in subdivisions or analyses; this is not the case in Nature, which comes from the infinite and goes to the infinite. Atoms satisfy only the imagination, but they shock the higher reason." (Gottfried W Leibniz)

"I am so in favor of the actual infinite that instead of admitting that Nature abhors it, as is commonly said, I hold that Nature makes frequent use of it everywhere, in order to show more effectively the perfections of its Author." (Gottfried W Leibniz)

"I did not understand how such a quantity could be real, when imaginary or impossible numbers were used to express it." (Gottfried W Leibniz)

"In signs, one sees an advantage for discovery that is greatest when they express the exact nature of a thing briefly and, as it were, picture it; then indeed, the labor of thought is wonderfully diminished" (Gottfried W Leibniz)

"If it were as easy in other matters to verify reasonings by experiments, there would not be such differing opinions. But the trouble is that experiments in physics are difficult and cost a great deal; and in metaphysics they are impossible, unless God out of love for us perform a miracle in order to acquaint us with remote immaterial things." (Gottfried W Leibniz)

"It is a good thing to proceed in order and to establish propositions. This is the way to gain ground and to progress with certainty." (Gottfried W Leibniz)

"Music is the hidden arithmetical exercise of a soul unconscious that it is calculating." (Gottfried W Leibniz)

"Music is the pleasure the human soul experiences from counting without being aware it is counting." (Gottfried W Leibniz)

"Nothing is more important than to see the sources of invention which are, in my opinion, more interesting than the inventions themselves." (Gottfried W Leibniz)

"The pleasure we obtain from music comes from counting, but counting unconsciously. Music is nothing but unconscious arithmetic." (Gottfried W Leibniz)

"There is nothing waste, nothing sterile, nothing dead in the universe; no chaos, no confusions, save in appearance." (Gottfried W Leibniz)

"This is why the ultimate reason of things must lie in a necessary substance, in which the differentiation of the changes only exists eminently as in their source; and this is what we call God." (Gottfried W Leibniz)

"Those few things having been considered, the whole matter is reduced to pure geometry, which is the one aim of physics and mechanics." (Gottfried W Leibniz)

"Without mathematics we cannot penetrate deeply into philosophy.

Without philosophy we cannot penetrate deeply into mathematics.

Without both we cannot penetrate deeply into anything." (Gottfried W Leibniz)