🧩Out of Context: On Axioms (Definitions)

"An axiom is proposition more general than the propositions or the science in which it employed as an axiom; or, an axiom is a proposition which is true of more subjects than the subject or the science in which it is quoted as an axiom. Hence. Geometry ought to admit as axioms all Algebraic truths. The simple truths of this kind, which are commonly called axioms, ore corollaries from the definitions of such terms as equal, whole, part, sum, etc." (The Pennsylvania School Journal, 1856)

"The logical axioms are the principle of all truth." (Otto Weininger, "Sex and Character", 1903)

"An axiom is a self-evident truth, the statement of which is superfluous to the conclusiveness of the reasoning, and which only serves to show a principle involved in the reasoning. It is generally a truth of observation; such as the assertion that something is true." (Charles S Peirce, "New Elements" ["Kaina stoiceia"], 1904)

"The mathematical axioms are therefore neither synthetic nor analytic, but definitions. [...] Hence the question of whether axioms are a priori becomes pointless since they are arbitrary." (Hans Reichenbach, "The Philosophy of Space and Time", 1928)

"Axioms are instruments which are used in every department of science, and in every department there are purists who are inclined to oppose with all their might any expansion of the accepted axioms beyond the boundary of their logical application." (Max Planck, "Where Is Science Going?", 1932)

"An axiom is common to all sciences, whereas a postulate is related to a particular science; an axiom is selfevident, whereas a postulate is not; an axiom cannot be regarded as a subject for demonstration, whereas a postulate is properly such a subject; an axiom is assumed with the ready assent of the learner, whereas a postulate is assumed without, perhaps, the assent of the learner." (Howard Eves, "Foundations and Fundamental Concepts of Mathematics", 1958)

"Whenever we write an axiom, a critic can say that the axiom is true only in a certain context." (John McCarthy, "Generality in Artificial Intelligence", 1987)

🅰Saint Augustine of Hippo - Collected Quotes

"If I am given a formula, and I am ignorant of its meaning, it cannot teach me anything, but if I already know it what does the formula teach me?" (Saint Augustine of Hippo , "De Magistro", 389)

"The inclination to seek the truth is safer than the presumption which regards unknown things as known." (Augustine of Hippo, "De Trinitate", 417)

"The mind itself, its love of itself and it knowledge of itself are a kind of trinity." (Augustine of Hippo, "De Trinitate", 417)

"Then assuredly the world was made, not in time, but simultaneously with time." (Augustine of Hippo, "The City of God", 426)

"A free curiosity is more effective in learning than a rigid discipline." (Augustine of Hippo)

"Faith is to believe what you do not yet see; the reward for this faith is to see what you believe." (Augustine of Hippo)

"Free curiosity has greater power to stimulate learning than rigorous coercion. Nevertheless, the free ranging flux of curiosity is channeled by discipline under Your Law." (Augustine of Hippo)

"No one indeed believes anything unless he has first thought that it it to be believed." (Augustine of Hippo)

"Time is like a river made up of events which happen, and its current is strong; no sooner does anything appear than it is swept away." (Augustine of Hippo)

"When regard for truth has been broken down or even slightly weakened, all things will remain doubtful." (Augustine of Hippo)

🧩Out of Context: On Proofs (Definitions)

"Mathematical proofs are out of the reach of topical arguments; and are not to be attacked by the equivocal use of words or declaration, that make so great a part of other discourses, - nay, even of controversies.” (John Locke, “An Essay Concerning Human Understanding”, 1690)

"[…] it is an error to believe that rigor in the proof is the enemy of simplicity." (David Hilbert, [Paris International Congress] 1900)

"Proof is an idol before whom the pure mathematician tortures himself." (Sir Arthur S Eddington, "The Nature of the Physical World", 1928)

"A modern mathematical proof is not very different from a modern machine, or a modern test setup: the simple fundamental principles are hidden and almost invisible under a mass of technical details." (Hermann Weyl, "Unterrichtsblätter für Mathematik und Naturwissenschaften", 1932)

"A mathematical proof is demonstrative reasoning [...]" (George Pólya, "Mathematics and Plausible Reasoning", 1954)

"The result of the mathematician's creative work is demonstrative reasoning, a proof; but the proof is discovered by plausible reasoning, by guessing." (George Pólya, "Induction and Analogy in Mathematics", 1954)

"Rigorous proofs are the hallmark of mathematics, they are an essential part of mathematics’ contribution to general culture." (George Pólya, "Mathematical Discovery", 1962)

"[...] the proof is a sequence of actions (applications of rules of inference) that, operating initially on the axioms, transform them into the desired theorem." (Herbert A Simon, "The Logic of Heuristic Decision Making", [in "The Logic of Decision and Action"], 1966)

"A proof is a construction that can be looked over, reviewed, verified by a rational agent." (Thomas Tymoczko, "The Four Color Problems", Journal of Philosophy , Vol. 76, 1979)

"Proof serves many purposes simultaneously […] Proof is respectability. Proof is the seal of authority. Proof, in its best instance, increases understanding by revealing the heart of the matter. Proof suggests new mathematics […] Proof is mathematical power, the electric voltage of the subject which vitalizes the static assertions of the theorems." (Reuben Hersh, "The Mathematical Experience", 1981)

"People might suppose that a mathematical proof is conceived as a logical progression, where each step follows upon the ones that have preceded it." (Roger Penrose, "The Emperor’s New Mind", 1989)

"A mathematical proof is a chain of logical deductions, all stemming from a small number of initial assumptions ('axioms') and subject to the strict rules of mathematical logic." (Eli Maor, "e: The Story of a Number", 1994)

"Mathematics rests on proof - and proof is eternal." (Saunders Mac Lane,"Reponses to …", Bulletin of the American Mathematical Society Vol. 30 (2), 1994)

"Proofs are not impersonal, they express the personality of their creator/discoverer just as much as literary efforts do. If something important is true, there will be many reasons that it is true, many proofs of that fact." (Gregory Chaitin, "Meta Math: The Quest for Omega", 2005)

"Within mathematics, a proof is an intellectual structure in which premises are conveyed to their conclusions by specific inferential steps. Assumptions in mathematics are called axioms, and conclusions theorems. This definition may be sharpened a little bit. A proof is a finite series of statements such that every statement is either an axiom or follows directly from an axiom by means of tight, narrowly defined rules." (David Berlinski, "Infinite Ascent: A short history of mathematics", 2005)

"[…] a proof is a device of communication." (Steven G Krantz, "The Proof is in the Pudding", 2007)

"A proof is a series of steps based on the (adopted) axioms and deduction rules which reaches a desired conclusion." (Cristian S Calude et al, "Proving and Programming", 2007)

"A proof is part of a situational ethic." (Steven G Krantz, "The Proof is in the Pudding", 2007)

"Heuristically, a proof is a rhetorical device for convincing someone else that a mathematical statement is true or valid." (Steven G Krantz, "The Proof is in the Pudding", 2007)

"[…] proof is central to what modern mathematics is about, and what makes it reliable and reproducible." (Steven G Krantz, "The Proof is in the Pudding", 2007)

"So the theorems and propositions are the new heights of knowledge we achieve, while the proofs are essential as they are the mortar which attaches them to the level below. Without proofs the structure would collapse." (Sidney A Morris, "Topology without Tears", 2007)

"A proof is like a piece of theatre or music, with moments of high drama where some major shift takes the audience into a new realm." (Marcus du Sautoy, "Symmetry: A Journey into the Patterns of Nature", 2008)

"[…] proof is the key ingredient of the emotional side of mathematics; proof is the ultimate explanation of why something is true, and a good proof often has a powerful emotional impact, boosting confidence and encouraging further questions ‘why’." (Alexandre V Borovik, "Mathematics under the Microscope: Notes on Cognitive Aspects of Mathematical Practice", 2009)

"A mathematical proof is a watertight argument which begins with information you are given, proceeds by logical argument, and ends with what you are asked to prove." (Sydney A Morris, "Topology without Tears", 2011)

"A proof in logic and mathematics is, traditionally, a deductive argument from some given assumptions to a conclusion. Proofs are meant to present conclusive evidence in the sense that the truth of the conclusion should follow necessarily from the truth of the assumptions. Proofs must be, in principle, communicable in every detail, so that their correctness can be checked." (Sara Negri  & Jan von Plato, "Proof Analysis", 2011)

"A proof is simply a story. The characters are the elements of the problem, and the plot is up to you." (Paul Lockhart, "Measurement", 2012)

"A mathematical proof is like a battle, or if you prefer a less warlike metaphor, a game of chess. Once a potential weak point has been identified, the mathematician’s technical grasp of the machinery of mathematics can be brought to bear to exploit it." (Ian Stewart, "Visions of Infinity", 2013)

"A proof tells us where to concentrate our doubts. […] An elegantly executed proof is a poem in all but the form in which it is written." (Morris Kline)

"Proofs are to mathematics what spelling (or even calligraphy) is to poetry. Mathematical works do consist of proofs, just as poems do consist of words." (Vladimir Arnold)

🧩Out of Context: On Curiosity (Definiions)

"Curiosity is only vanity. We usually only want to know something so that we can talk about it." (Blaise Pascal, "Pensées", 1669)

"Curiosity is, in great and generous minds, the first passion and the last." (Samuel Johnson, 1751)

"Curiosity is one of the permanent and certain characteristics of a vigorous intellect." (Samuel Johnson, 1751) 

"Curiosity is the most superficial of all the affections; it changes its object perpetually; it has an appetite which is very sharp, but very easily satisfied, and it has always an appearance of giddiness, restlessness and anxiety." (Edmund Burke, "A Philosophical Enquiry into the Origin of our Ideas of the Sublime and Beautiful", 1757)

"Curiosity is one of those insatiable passions that grow by gratification." (Sarah Scott, "A Description of Millenium Hall", 1762)

"Curiosity is lying in wait for every secret." (Ralph W Emerson, "Letters and Social Aims, Progress of Culture", 1884)

"Curiosity is the aspect of the universe seeking to realise itself, and the fruit of such activity is new reality, stimulating to new research." (Cassius J Keyser, "The Human Worth of Rigorous Thinking: Essays and Addresses", 1916)

"Curiosity and the urge to solve problems are the emotional hallmarks of our species [...]" (Carl E Sagan, "The Dragons of Eden", 1977)

"Curiosity, especially intellectual inquisitiveness, is what separates the truly alive from those who are merely going through the motions." (Tom Robbins, "Villa Incognito", 2003)

"Curiosity is the greatest source of ideas, retail revolutions, and insights." (Michael J Silverstein, "Rocket: Eight Lessons to Secure Infinite Growth", 2015)

"Curiosity is a delicate little plant which, aside from stimulation, stands mainly in need of freedom." (Albert Einstein)

"Curiosity is the one thing invincible in Nature." (Freya Stark)

"Curiosity is the wick in the candle of learning." (William A Ward)

🧩Out of Context: On Teaching (Definitions)

"Teaching is just pointing out what you cannot see without help." (Issai Chozan, "The Mysterious Skills of the Old Cat" ["Neko No Myoujutsu"], 1727)

"Teaching is not a lost art, but the regard for it is a lost tradition." (Jacques Barzun, "Teacher in America", 1945)

"Teaching is more difficult than learning because what teaching calls for is this: to let learn." (Martin Heidegger, "What is called thinking?", 1968)

"Learning is finding out what you already know. Doing is demonstrating that you know it. Teaching is reminding others that they know just as well as you. You are all learners, doers, teachers." (Richard Bach, "Illusions: The Adventures of a Reluctant Messiah", 1977)

"Teaching is an instinctual art, mindful of potential, craving of realizations, a pausing, seamless process." (A Bartlett Giamatti, "The University and the Public Interest", 1981)

"Teaching is only demonstrating that it is possible. Learning is making it possible for yourself." (Paulo Coelho, "The Pilgrimage", 1987)

"Teaching is mostly listening, and learning is mostly telling." (Deborah Meier, "The Power of Their Ideas", 1995)

"Teaching is not about how we see things, it is about how children see things." (Kavita B Ghosh)

"Teaching is not about information. It's about having an honest intellectual relationship with your students." (Paul Lockhart)

🧩Out of Context: On Learning (Definitions)

"Learning is, in too many cases, but a foil to common sense; a substitute for true knowledge." (William Hazlitt, "Table Talk; or, Original Essays", 1821-1822)

"Learning is its own exceeding great reward; and at the period of which we speak, it bore other fruits, not unworthy of it." (William Hazlitt, "The Plain Speaker", 1826)

"Learning is motivated by intent and understanding by visualization." (A B Garrett, "Visualization: A Step to Understanding", Journal of Chemical Education Vol. 25 (10), 1948)

"Learning is a property of all living organisms." (Winfred B. Hirschmann, "Profit from the Learning Curve", Harvard Business Review, 1964)

"Learning is any change in a system that produces a more or less permanent change in its capacity for adapting to its environment." (Herbert A Simon, "The Sciences of the Artificial", 1968)

"Learning is finding out what you already know." (Richard Bach, "Illusions: The Adventures of a Reluctant Messiah", 1977)

"Teaching is only demonstrating that it is possible. Learning is making it possible for yourself." (Paulo Coelho, "The Pilgrimage", 1987)

"Learning is the process of obtaining new knowledge. It results in a better reaction to the same inputs at the next session of operation. It means improvement. It is a step toward adaptation. Learning is a major characteristic of intelligent systems." (Nikola K Kasabov, "Foundations of Neural Networks, Fuzzy Systems, and Knowledge Engineering", 1996)

"Learning is a multi-faceted, integrated process where changes with any one element alters the larger network. Knowledge is subject to the nuances of complex, adaptive systems." (George Siemens, "Knowing Knowledge", 2006)

"Learning is the process of creating networks." (George Siemens, "Knowing Knowledge", 2006)

"Learning is a process of modifying or completely changing our mental models based on new experiences or evidence." (Edward D Hess, "Learn or Die: Using Science to Build a Leading-Edge Learning Organization", 2014)

"Learning is a dynamic event and so the belief that learning is primarily about the acquisition of facts is fundamentally flawed - the acquisition and manipulation of data is at best a prerequisite to learning. Real learning involves acquiring knowledge and understanding." (William Byers, "Deep Thinking: What Mathematics Can Teach Us About the Mind", 2015)

"A little learning is a dangerous thing." (Alexander Pope)

"Learning is a treasure which accompanies its owner everywhere." (proverb)

"Learning is the only thing the mind never exhausts, never fears, and never regrets." (Leonardo da Vinci)

"Learning is not attained by chance. It must be sought for with ardor and attended to with diligence." (Abigail Adams)

🧩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)

🧩Out of Context: On Scientific method (Definitions)

"[...] scientific method is simply the attempt to acquire knowledge of general laws directly or indirectly by experience, by the use of our five senses. The only limitations that can be assigned to the applicability of this process are those due to the character of experience. Anything that is logically related to experience by discoverable laws and is capable of description in general terms can be dealt with by the scientific method." (Arthur D Ritchie, "Scientific Method: An Inquiry Into the Character and Validity of Natural Laws", 1923)

"Scientific method is what working scientists do, not what other people or even they themselves may say about it." (Percy W Bridgman, "Reflections of a Physicist", 1950)

"Scientific method is the way to truth, but it affords, even in principle, no unique definition of truth. Any so-called pragmatic definition of truth is doomed to failure equally." (Willard v O Quine, "Word and Object", 1960)

"The scientific method is a potentiation of common sense, exercised with a specially firm determination not to persist in error if any exertion of hand or mind can deliver us from it." (Sir Peter B Medawar, "Pluto’s Republic: Incorporating the Art of the Soluble and Induction Intuition in Scientific Thought", 1982)

"Scientific method is concerned with efficient ways of generating knowledge." (George E P Box, "Total Quality: Its Origins and its Future", 1995)

"Scientific method is not much different from our day-to-day ways of learning about the world. Without really thinking about the steps or the standards, common sense invokes the same process of evidence and reasoning as scientists more explicitly follow." (Peter Kosso, "A Summary of Scientific Method, "2011")

"Scientific method is the gateway into scientific discoveries that in turn prompt technological advances and cultural influences." (Hugh G Gauch Jr., "Scientific Method in Brief", 2012)

"The scientific method is the foundation of modern research. It’s how we prove a theory. It’s how we demonstrate cause and effect. It’s how we discover, innovate, and invent." (Kristin H Jarman, "The Art of Data Analysis: How to answer almost any question using basic statistics", 2013)

The traditional scientific method is hypothesis driven. The researcher formulates a theory of how the world works, and then seeks to support or reject this hypothesis based on data." (Steven S Skiena, "The Data Science Design Manual", 2017)

On Noise VI

"In a white-noise process, every value of the process (e.g., the successive frequencies of a melody) is completely independent of its past - it is a total surprise. By contrast, in 'brown music' (a term derived from Brownian motion), only the increments are independent of the past, giving rise to a rather boring tune." (Manfred Schroeder, "Fractals, Chaos, Power Laws Minutes from an Infinite Paradise", 1990)

"In contrast to gravitation, interatomic forces are typically modeled as inhomogeneous power laws with at least two different exponents. Such laws (and exponential laws, too) are not scale-free; they necessarily introduce a characteristic length, related to the size of the atoms. Power laws also govern the power spectra of all kinds of noises, most intriguing among them the ubiquitous (but sometimes difficult to explain)." (Manfred Schroeder, "Fractals, Chaos, Power Laws Minutes from an Infinite Paradise", 1990)

"We might expect that the noise will 'smear out' each data point and make it difficult for the network to fit individual data points precisely and hence will reduce over-fitting. In practice it has been demonstrated that training with noise can indeed lead to improvements in network generalization." (Christopher M Bishop, "Neural Networks for Pattern Recognition", 1995)

"A moderate amount of noise leads to enhanced order in excitable systems, manifesting itself in a nearly periodic spiking of single excitable systems, enhancement of synchronized oscillations in coupled systems, and noise-induced stability of spatial pattens in reaction-diffusion systems." (Benjamin Lindner et al, "Effects of Noise in Excitable Systems", Physical Reports. vol. 392, 2004)

"Linear systems do not benefit from noise because the output of a linear system is just a simple scaled version of the input [...] Put noise in a linear system and you get out noise. Sometimes you get out a lot more noise than you put in. This can produce explosive effects in feedback systems that take their own outputs as inputs." (Bart Kosko, "Noise", 2006)

"One person’s signal is another person’s noise and vice versa. We call this relative role reversal the noise-signal duality." (Bart Kosko, "Noise", 2006)

"Many scientists who work not just with noise but with probability make a common mistake: They assume that a bell curve is automatically Gauss's bell curve. Empirical tests with real data can often show that such an assumption is false. The result can be a noise model that grossly misrepresents the real noise pattern. It also favors a limited view of what counts as normal versus non-normal or abnormal behavior. This assumption is especially troubling when applied to human behavior. It can also lead one to dismiss extreme data as error when in fact the data is part of a pattern." (Bart Kosko, "Noise", 2006)

"Noise is an unwanted signal. A signal is anything that conveys information or ultimately anything that has energy. The universe consists of a great deal of energy. Indeed a working definition of the universe is all energy anywhere ever. So the answer turns on how one defines what it means to be wanted and by whom." (Bart Kosko, "Noise", 2006)

"The noise takes its toll on the message as it randomly turns some of the 1 bits into 0 bits and randomly turns some of the 0 bits into 1 bits: Noise randomly flips bits. [...] But noise can be subtler in a digital system. Noise can disturb the timing of when a bit value arrives at a receiver as well as randomly flipping that bit value." (Bart Kosko, "Noise", 2006)

"[...] in the statistical world, what we see and measure around us can be considered as the sum of a systematic mathematical idealized form plus some random contribution that cannot yet be explained. This is the classic idea of the signal and the noise." (David Spiegelhalter, "The Art of Statistics: Learning from Data", 2019)

🅷Ian Hacking - Collected Quotes

"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)

"Many modern philosophers claim that probability is relation between an hypothesis and the evidence for it." (Ian Hacking, "The Emergence of Probability", 1975)

"Determinism was eroded during the nineteenth century and a space was cleared for autonomous laws of chance. The idea of human nature was displaced by a model of normal people with laws of dispersion. These two transformations were parallel and fed into each other. Chance made the world seem less capricious; it was legitimated because it brought order out of chaos. The greater the level of indeterminism in our conception of the world and of people, the higher the expected level of control." (Ian Hacking, "The Taming of Chance", 1990)

"Epistemology is the theory of knowledge and belief." (Ian Hacking, "The Taming of Chance", 1990)

"Logic is the theory of inference and argument. For this purpose we use the deductive and often tautological unravelling of axioms provided by pure mathematics, but also, and for most practical affairs, we now employ- sometimes precisely, sometimes informally - the logic of statistical inference." (Ian Hacking, "The Taming of Chance", 1990)

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

"The systematic collection of data about people has affected not only the ways in which we conceive of a society, but also the ways in which we describe our neighbour. It has profoundly transformed what we choose to do, who we try to be, and what we think of ourselves." (Ian Hacking, "The Taming of Chance", 1990)

"There is a seeming paradox: the more the indeterminism, the more the control. This is obvious in the physical sciences. Quantum physics takes for granted that nature is at bottom irreducibly stochastic. Precisely that discovery has immeasurably enhanced our ability to interfere with and alter the course of nature." (Ian Hacking, "The Taming of Chance", 1990)

"I write of the taming of chance, that is, of the way in which apparently chance or irregular events have been brought under the control of natural or social law. The world became not more chancy, but far less so. Chance, which was once the superstition of the vulgar, became the centrepiece of natural and social science, or so genteel and rational people are led to believe." (Ian Hacking, "The Taming of Chance", 1990)

"The best reaction to a paradox is to invent a genuinely new and deep idea." (Ian Hacking, "An Introduction to Probability and Inductive Logic", 2001)