On Manifolds: Definitions

"A manifold, roughly, is a topological space in which some neighborhood of each point admits a coordinate system, consisting of real coordinate functions on the points of the neighborhood, which determine the position of points and the topology of that neighborhood; that is, the space is locally cartesian. Moreover, the passage from one coordinate system to another is smooth in the overlapping region, so that the meaning of 'differentiable' curve, function, or map is consistent when referred to either system." (Richard L Bishop & Samuel I Goldberg, "Tensor Analysis on Manifolds", 1968)

"A manifold M of dimension n, or n-manifold, is a topological space with the following properties: (i) M is Hausdorff, (ii) M is locally Euclidean of dimension n, and (iii) M has a countable basis of open sets." (William M Boothby, "An introduction to differentiable manifolds and Riemannian geometry" 2nd Ed., 1986)

"[...] a manifold is a set M on which 'nearness' is introduced (a topological space), and this nearness can be described at each point in M by using coordinates. It also requires that in an overlapping region, where two coordinate systems intersect, the coordinate transformation is given by differentiable transition functions." (Kenji Ueno & Toshikazu Sunada, "A Mathematical Gift, III: The Interplay Between Topology, Functions, Geometry, and Algebra", Mathematical World Vol. 23, 1996)

"A manifold Mn of dimension n is a Hausdorff topological space such that each point P of Mn has a neighborhood Ω homeomorphic to Rn (or equivalently to an open set of Rn." (Thierry Aubin, "A Course in Differential Geometry", 2000)

"Manifolds are a type of topological spaces we are interested in. They correspond well to the spaces we are most familiar with, the Euclidean spaces. Intuitively, a manifold is a topological space that locally looks like Rn. In other words, each point admits a coordinate system, consisting of coordinate functions on the points of the neighborhood, determining the topology of the neighborhood." (Afra J Zomorodian, "Topology for Computing", 2005)

"Roughly speaking, a manifold is essentially a space that is locally similar to the Euclidean space. This resemblance permits differentiation to be defined. On a manifold, we do not distinguish between two different local coordinate systems. Thus, the concepts considered are just those independent of the coordinates chosen. This makes more sense if we consider the situation from the physics point of view. In this interpretation, the systems of coordinates are systems of reference." (Ovidiu Calin & Der-Chen Chang,  "Geometric Mechanics on Riemannian Manifolds : Applications to partial differential equations", 2005)

"A manifold is an abstract mathematical space, which locally (i.e., in a close–up view) resembles the spaces described by Euclidean geometry, but which globally (i.e., when viewed as a whole) may have a more complicated structure." (Vladimir G Ivancevic & Tijana T Ivancevic, "Applied Differential Geometry: A Modern Introduction", 2007)

"A topological manifold of dimension k is a Hausdorff topological space M with a countable base such that for all x ∈ M, there exists an open neighborhood of x that is homeomorphic to an open set of Rk." (Stephen Lovett, "Differential Geometry of Manifolds", 2010)

"Roughly speaking, a manifold is a set whose points can be labeled by coordinates." (Gerardo F. Torres del Castillo, "Differentiable Manifolds: A Theoretical Physics Approach", 2010)

"You can very generally think of a manifold as a space which is locally Euclidian - that means that if you look closely enough at one small part of a manifold then it basically looks like Rn for some n." (Jon P Fortney, "A Visual Introduction to Differential Forms and Calculus on Manifolds", 2018)

Out of Context: On Fractals (Definitions)

 "A fractal is a mathematical set or concrete object that is irregular or fragmented at all scales [...]" (Benoît Mandelbrot, "The Fractal Geometry of Nature", 1982)

"A fractal is by definition a set for which the Hausdorff-Besicovitch dimension strictly exceeds the topological dimension." (Benoît Mandelbrot, "The Fractal Geometry of Nature", 1982)

"In the mind's eye, a fractal is a way of seeing infinity." (James Gleick, "Chaos: Making a New Science, A Geometry of Nature", 1987)

"Fractals are geometric shapes that are equally complex in their details as in their overall form. That is, if a piece of a fractal is suitably magnified to become of the same size as the whole, it should look like the whole, either exactly, or perhaps after a slight limited deformation." (Benoît B Mandelbrot, "Fractals and an Art for the Sake of Science", 1989)

"Fractals are patterns which occur on many levels." (Györgi Ligeti, [interview] 1999)

"Mathematical fractals are generated by repeating the same simple steps at ever decreasing scales. In this way an apparently complex shape, containing endless detail, can be generated by the repeated application of a simple algorithm." (F David Peat, "From Certainty to Uncertainty", 2002)

[fractal:] "A fragmented geometric shape that can be split up into secondary pieces, each of which is approximately a smaller replica of the whole, the phenomenon commonly known as self similarity." (Khondekar et al, "Soft Computing Based Statistical Time Series Analysis, Characterization of Chaos Theory, and Theory of Fractals", 2013)

"Fractals are generally self-similar (each section looks at all) and are not subordinated to a specific scale. They are used especially in the digital modeling of irregular patterns and structures in nature." (Mauro Chiarella, "Folds and Refolds: Space Generation, Shapes, and Complex Components", 2016)

"A fractal is a mathematical set or concrete object that is irregular or fragmented at all scales […]" (Benoît B Mandelbrot)

Out of Context: On Bifurcation (Definitions)

"[…] bifurcations - the abrupt changes that can take place in the behavior, and often in the complexity, of a system when the value of a constant is altered slightly." (Edward N Lorenz, "The Essence of Chaos", 1993)

"A bifurcation is an event that occurs in the evolution of a dynamic system in which the characteristic behavior of the system is transformed." (Courtney Brown, "Chaos and Catastrophe Theories", 1995)

"The concept of bifurcation, present in the context of non-linear dynamic systems and theory of chaos, refers to the transition between two dynamic modalities qualitatively distinct; both of them are exhibited by the same dynamic system, and the transition (bifurcation) is promoted by the change in value of a relevant numeric parameter of such system." (Emilio Del-Moral-Hernandez, "Chaotic Neural Networks", Encyclopedia of Artificial Intelligence, 2009)

"In mathematical models, a bifurcation occurs when a small change made to a parameter value of a system causes a sudden qualitative or topological change in its behavior." (Dmitriy Laschov & Michael Margaliot, "Mathematical Modeling of the λ Switch: A Fuzzy Logic Approach", 2010)

"In dynamical systems, a bifurcation occurs when a small smooth change made to the parameter values (the bifurcation parameters) of a system causes a sudden 'qualitative' or topological change in its behaviour. Generally, at a bifurcation, the local stability properties of equilibria, periodic orbits or other invariant sets changes." (Gregory Faye, "An introduction to bifurcation theory",  2011)

"Most commonly applied to the mathematical study of dynamical systems, a bifurcation occurs when a small smooth change made to the parameter values (the bifurcation parameters) of a system causes a sudden 'qualitative' or topological change in its behavior. Bifurcations can occur in both continuous systems (described by ODEs, DDEs, or PDEs) and discrete systems (described by maps)." (Tianshou Zhou, "Bifurcation", 2013)

"The qualitative structure of the flow can change as parameters are varied. In particular, fixed points can be created or destroyed, or their stability can change. These qualitative changes in the dynamics are called bifurcations, and the parameter values at which they occur are called bifurcation points." (Steven H Strogatz, "Nonlinear Dynamics and Chaos: With Applications to Physics, Biology, Chemistry, and Engineering", 2015)

"[…] what exactly do we mean by a bifurcation? The usual definition involves the concept of 'topological equivalence': if the phase portrait changes its topological structure as a parameter is varied, we say that a bifurcation has occurred." (Steven H Strogatz, "Nonlinear Dynamics and Chaos: With Applications to Physics, Biology, Chemistry, and Engineering", 2015)

Out of Context: On Entropy (Just the Quotes)

"If for the entire universe we conceive the same magnitude to be determined, consistently and with due regard to all circumstances, which for a single body I have called entropy, and if at the same time we introduce the other and simpler conception of energy, we may express in the following manner the fundamental laws of the universe which correspond to the two fundamental theorems of the mechanical theory of heat: (1) The energy of the universe is constant. (2) The entropy of the universe tends to a maximum." (Rudolf Clausius, "The Mechanical Theory of Heat - With its Applications to the Steam Engine and to Physical Properties of Bodies", 1867)

"The Entropy of a system is the mechanical work it can perform without communication of heat, or alteration of its total volume, all transference of heat being performed by reversible engines. When the pressure and temperature of the system have become uniform the entropy is exhausted." (James C Maxwell, "Theory of Heat", 1899)

"The second law of thermodynamics appears solely as a law of probability, entropy as a measure of the probability, and the increase of entropy is equivalent to a statement that more probable events follow less probable ones." (Max Planck, "A Survey of Physics", 1923)

"Entropy is the measure of randomness." (Lincoln Barnett, "The Universe and Dr. Einstein", 1948)

"Just as entropy is a measure of disorganization, the information carried by a set of messages is a measure of organization." (Norbert Wiener, "The Human Use of Human Beings", 1950)

"Entropy is a measure of the heat energy in a substance that has been lost and is no longer available for work. It is a measure of the deterioration of a system." (William B. Sill & Norman Hoss (Eds.), "Popular Science Encyclopedia of the Sciences", 1963)

"You should call it entropy, for two reasons. In the first place your uncertainty function has been used in statistical mechanics under that name, so it already has a name. In the second place, and more important, no one really knows what entropy really is, so in a debate you will always have the advantage." (John von Neumann) [Suggesting to Claude Shannon a name for his new uncertainty function, see Scientific American Vol. 225 (3), 1971]

"Thus, in physics, entropy is associated with the possibility of converting thermal energy into mechanical energy. If the entropy does not change during a process, the process is reversible. If the entropy increases, the available energy decreases. Statistical mechanics interprets an increase of entropy as a decrease in order or, if we wish, as a decrease in our knowledge." (John R Pierce, "An Introduction to Information Theory: Symbols, Signals & Noise" 2nd Ed., 1980)

"Entropy [...] is the amount of disorder or randomness present in any system. All non-living systems tend toward disorder; left alone they will eventually lose all motion and degenerate into an inert mass." (Lars Skyttner, "General Systems Theory: Ideas and Applications", 2001)

"Entropy is not about speeds or positions of particles, the way temperature and pressure and volume are, but about our lack of information." (Hans C von Baeyer," Information, The New Language of Science", 2003)

"Heat is the energy of random chaotic motion, and entropy is the amount of hidden microscopic information." (Leonard Susskind, "The Black Hole War", 2008)

"Entropy is the crisp scientific name for waste, chaos, and disorder." (Kevin Kelly, "What Technology Wants", 2010)

"Entropy is a measure of amount of uncertainty or disorder present in the system within the possible probability distribution. The entropy and amount of unpredictability are directly proportional to each other." (G Suseela & Y Asnath V Phamila, "Security Framework for Smart Visual Sensor Networks", 2019)

"In the physics [entropy is the] rate of system's messiness or disorder in a physical system. In the social systems theory - social entropy is a sociological theory that evaluates social behaviors using a method based on the second law of thermodynamics." (Justína Mikulášková et al, "Spiral Management: New Concept of the Social Systems Management", 2020)

Out of Context: On Emergence (Definitions)

"[...] emergence is an integral part of the dynamics of open systems [...] " (Fritjof  Capra, "The Hidden Connections", 2002)

"Emergence is not really mysterious, although it may be complex. Emergence is brought about by the interactions between the parts of a system. 

" (Derek Hitchins, "Advanced Systems Thinking, Engineering and Management", 2003)

"Emergence is the phenomenon of properties, capabilities and behaviours evident in the whole system that are not exclusively ascribable to any of its parts." (Derek Hitchins, "Advanced Systems Thinking, Engineering and Management", 2003)

"Emergence refers to the relationship between the details of a system and the larger view." (Yaneer Bar-Yam, "Making Things Work: Solving Complex Problems in a Complex World", 2004)

[emergence:] "The process of complex pattern formation from simpler rules; emergent properties are neither properties had by any parts of the system taken in isolation nor a resultant of a mere summation of properties of parts of the system." (Ani Calinescu & Janet Efstathiou, "Measures of Network Structure", Encyclopedia of Networked and Virtual Organizations, 2008) 

"Emergence is defined as the occurrence of new processes operating at a higher level of abstraction then is the level at which the local rules operate." (Jirí Kroc & Peter M A Sloot, "Complex Systems Modeling by Cellular Automata", Encyclopedia of Artificial Intelligence, 2009)

"This spontaneous emergence of order at critical points of instability, which is often referred to simply as 'emergence', is one of the hallmarks of life." (Fritjof Capra, "The Systems View of Life: A Unifying Vision", 2014)

"Emergence is a nontrivial relationship between the properties of a system at microscopic and macroscopic scales. Macroscopic properties are called emergent when it is hard to explain them simply from microscopic properties." (Hiroki Sayama, "Introduction to the Modeling and Analysis of Complex Systems", 2015)

[emergence:] "A feature in a complex system that is generated through the dynamic interactions between the parts of a system at one level, and is realized at the next level of organization without intentionality or causality." (A Faye Bres, "Integral Post-Analysis of Design-Based Research of an Organizational Learning Process for Strategic Renewal of Environmental Management", Integral Theory and Transdisciplinary Action Research in Education, 2019)

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)

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)

Out of Context: On Laws (Definitions)

"Laws are generally found to be nets of such a texture, as the little creep through, the great break through, and the middle-sized are alone entangled in." (William Shenstone, "Works in Verse and Prose", 1764)

"The laws of nature are the rules according to which the effects are produced; but there must be a cause which operates according to these rules." (Thomas Reid, "Essays on the Intellectual Powers of Man", 1785)

"[...] law is the expression of the will of God." (Agnes M Clerke, "Problems in Astrophysics", 1903)

"[...] laws are only approximations." (William James, "Pragmatism: A New Name for Some Old Ways of Thinking", 1914)

"Scientific laws, when we have reason to think them accurate, are different in form from the common-sense rules which have exceptions: they are always, at least in physics, either differential equations, or statistical averages." (Bertrand A Russell, "The Analysis of Matter", 1927)

"Natural law is not applicable to the unseen world behind the symbols, because it is unadapted to anything except symbols, and its perfection is a perfection of symbolic linkage." (Arthur S Eddington, "Science and the Unseen World", 1929)

"Being built on concepts, hypotheses, and experiments, laws are no more accurate or trustworthy than the wording of the definitions and the accuracy and extent of the supporting experiments." (Gerald Holton, "Introduction to Concepts and Theories in Physical Science", 1952)

"It seems to be one of the fundamental features of nature that fundamental physical laws are described in terms of a mathematical theory of great beauty and power, needing quite a high standard of mathematics for one to understand it." (Paul Dirac, "The Evolution of the Physicist's Picture of Nature", 1963)

"A law is a permanent cause-and-effect relation of phenomena or processes. A law is a necessary, essential, stable and repetitive relation among phenomena." (Dmitry A Novikov, "Cybernetics: From Past to Future", 2016)

Out of Context: On Artificial Intelligence (Definitions)

"Artificial intelligence is the science of making machines do things that would require intelligence if done by men." (Marvin Minsky, 1968)

"Artificial intelligence is based on the assumption that the mind can be described as some kind of formal system manipulating symbols that stand for things in the world." (George Johnson, Machinery of the Mind: Inside the New Science of Artificial Intelligence, 1986)

"Artificial intelligence is the mimicking of human thought and cognitive processes to solve complex problems automatically. AI uses techniques for writing computer code to represent and manipulate knowledge." (Radian Belu, "Artificial Intelligence Techniques for Solar Energy and Photovoltaic Applications", 2013)

"Artificial intelligence is defined as the branch of science and technology that is concerned with the study of software and hardware to provide machines the ability to learn insights from data and the environment, and the ability to adapt in changing situations with high precision, accuracy and speed." (Amit Ray, "Compassionate Artificial Intelligence", 2018)

"Artificial Intelligence is not just learning patterns from data, but understanding human emotions and its evolution from its depth and not just fulfilling the surface level human requirements, but sensitivity towards human pain, happiness, mistakes, sufferings and well-being of the society are the parts of the evolving new AI systems." (Amit Ray, "Compassionate Artificial Intelligence", 2018)

"Artificial intelligence is the elucidation of the human learning process, the quantification of the human thinking process, the explication of human behavior, and the understanding of what makes intelligence possible." (Kai-Fu Lee, "AI Superpowers: China, Silicon Valley, and the New World Order", 2018)

"AI is a simulation of human intelligence through the progress of intelligent machines that think and work like humans carrying out such human activities as speech recognition, problem-solving, learning, and planning." (Hari K Kondaveeti et al, "Deep Learning Applications in Agriculture: The Role of Deep Learning in Smart Agriculture", 2021)

Out of Context: On Cybernetics (Definitions)

"Cybernetics is a word invented to define a new field in science. It combines under one heading the study of what in a human context is sometimes loosely described as thinking and in engineering is known as control and communication. In other words, cybernetics attempts to find the common elements in the functioning of automatic machines and of the human nervous system, and to develop a theory which will cover the entire field of control and communication in machines and in living organisms." (Norbert Wiener, "Cybernetics", 1948)

"Cybernetics is similar in its relation to the actual machine. It takes as its subject-matter the domain of 'all possible machines', and is only secondarily interested if informed that some of them have not yet been made, either by Man or by Nature."(W Ross Ashby, "An Introduction to Cybernetics", 1956)

"[Cybernetics is] the art of ensuring the efficacy of action." (Louis Couffignal, 1958)

"Cybernetics is the science of the process of transmission, processing and storage of information." (Sergei Sobolew, Woprosy Psychology, 1958)

"Cybernetics is the general science of communication. But to refer to communication is consciously or otherwise to refer to distinguishable states of information inputs and outputs and /or to information being processed within some relatively isolated system." (Henryk Greniewski, "Cybernetics without Mathematics", 1960)

"Cybernetics is concerned primarily with the construction of theories and models in science, without making a hard and fast distinction between the physical and the biological sciences." (Frank H George, "The Brain As A Computer", 1962)

"Cybernetics is the science or the art of manipulating defensible metaphors; showing how they may be constructed and what can be inferred as a result of their existence." (Gordon Pask, "The Cybernetics of Human Performance and Learning", 1966)

"For cybernetics is an interdisciplinary science, owing as much to biology as to physics, as much to the study of the brain as to the study of computers, and owing also a great deal to the formal languages of science for providing tools with which the behaviour of all these systems can be objectively described." (A Stafford Beer, 1966)

"Cybernetics is a homogenous and coherent scientific complex, a science resulting from the blending of at least two sciences - psychology and technology; it is a general and integrative science, a crossroads of sciences, involving both animal and car psychology. It is not just a discipline, circumscribed in a narrow and strictly defined field, but a complex of disciplines born of psychology and centered on it, branched out as branches of a tree in its stem. It is a stepwise synthesis, a suite of multiple, often reciprocal, modeling; syntheses and modeling in which, as a priority, and as a great importance, the modeling of psychology on the technique and then the modeling of the technique on psychology. Cybernetics is an intellectual symphony, a symphony of ideas and sciences." (Stefan Odobleja, 1978)

"Cybernetics is concerned with scientific investigation of systemic processes of a highly varied nature, including such phenomena as regulation, information processing, information storage, adaptation, self-organization, self-reproduction, and strategic behavior." (Fritz B Simon et al, "Language of Family Therapy: A Systemic Vocabulary and Source Book", 1985)

"It seems that cybernetics is many different things to many different people. But this is because of the richness of its conceptual base; and I believe that this is very good, otherwise cybernetics would become a somewhat boring exercise. However, all of those perspectives arise from one central theme; that of circularity." (Heinz von Foerster, "Ethics and Second-Order Cybernetics", 1991)

"Cybernetics is a science of purposeful behavior. It helps us explain behavior as the continuous action of someone (or thing) in the process, as we see it, of maintaining certain conditions near a goal state, or purpose." (Jeff Dooley, "Thoughts on the Question: What is Cybernetics", 1995)

"Cybernetics is the science of effective organization, of control and communication in animals and machines. It is the art of steersmanship, of regulation and stability." (Chris Lucas, "Cybernetics and Stochastic Systems", 1999)

"The science of cybernetics is not about thermostats or machines; that characterization is a caricature. Cybernetics is about purposiveness, goals, information flows, decision-making control processes and feedback (properly defined) at all levels of living systems." (Peter Corning, "Synergy, Cybernetics, and the Evolution of Politics", 2005)

"Cybernetics is the study of systems and processes that interact with themselves and produce themselves from themselves." (Louis Kauffman, 2007)

"Cybernetics is the art of creating equilibrium in a world of possibilities and constraints. This is not just a romantic description, it portrays the new way of thinking quite accurately." (Ernst von Glasersfeld, "Partial Memories: Sketches from an Improbable Life", 2010)

"Cybernetics is the study of systems which can be mapped using loops (or more complicated looping structures) in the network defining the flow of information." (Alan Scrivener, "A Curriculum for Cybernetics and Systems Theory", 2012)

Out of Context: On Architecture (Definitions)

"Architecture is frozen music." (Friedrich Schelling, "Philosophie der Kunst", cca 1805)

"[...] architecture is a kind of oratory in forms, sometimes persuading or even flattering, sometimes simply commanding." (Friedrich Nietzsche, "Twilight of the Idols", 1889)

"Architecture is geometry made visible in the same sense that music is number made audible." (Claude F Bragdon, "The Beautiful Necessity: Seven Essays on Theosophy and Architecture", 1910)

"Architecture is the first manifestation of man creating his own universe, creating it in the image of nature, submitting to the laws of nature, the laws which govern our own nature, our universe." (Charles-Edouard Jeanneret [Le Corbusier], Towards a New Architecture, 1923)

"Architecture is the masterly, correct and magnificent play of masses brought together in light." (Charles-Edouard Jeanneret [Le Corbusier], "Towards a New Architecture", 1923)

"Architecture is preeminently the art of significant forms in space - that is, forms significant of their functions." (Claude Bragdon, "Wake Up and Dream", Outlook, 1931)

"Architecture, of all the arts, is the one which acts the most slowly, but the most surely, on the soul." (Ernest Dimnet, "What We Live By", 1932)

"Among the planets of the arts, architecture is the dark side of the moon." (Bruno Zevi, "Architecture as Space: How to Look at Architecture", 1951)

"Architecture [...] is like a great hollowed-out sculpture which man enters and apprehends by moving about within it." (Bruno Zevi, "Architecture as Space: How to Look at Architecture", 1951)

"Architecture is not art alone, it is not merely a reflection of conceptions of life or a portrait of systems of living. Architecture is environment, the stage on which our lives unfold." (Bruno Zevi, "Architecture as Space: How to Look at Architecture", 1951)

"Architecture is the art of how to waste space." (Philip Johnson, "Ideas and Men" New York Times, 1964)

"Architecture is the art and science of designing buildings and the spaces between them." (Robert J Piper, "Opportunities in an Architecture Career", 1970)

"Architecture is defined as the art and science of creating buildings." (Derek Hitchins, "Advanced Systems Thinking, Engineering and Management", 2003)

"Architecture is akin to music in that both should be based on the symmetry of mathematics." (Frank L Wright)

"Architecture is the triumph of human imagination over materials, methods and men, to put man into possession of his own earth." (Frank L Wright)

Out of Context: On Observation (Definitions)

"[...] observation is the mind's support in reasoning [...]" (Claude Bernard, "An Introduction to the Study of Experimental Medicine", 1865)

"Observation is a passive science, experimentation an active science." (Claude Bernard, "An Introduction to the Study of Experimental Medicine", 1865)

"Observation is a putting together of several results of sensation which are or are supposed to be connected with each other according to the law of causality, so that some represent causes and others their effects." (Thorvald N Thiele, "Theory of Observations", 1903)

"For the truth of the conclusions of physical science, observation is the supreme Court of Appeal." (Sir Arthur Eddington, "The Philosophy of Physical Science", 1939)

"[…] observation is not enough, and it seems to me that in science, as in the arts, there is very little worth having that does not require the exercise of intuition as well as of intelligence, the use of imagination as well as of information." (Kathleen Lonsdale, "Facts About Crystals", American Scientist Vol. 39 (4), 1951)

"Scientific observation is always a viewing of things through the refracting medium of a symbol system, and technological praxis is always handling of things in ways that some symbol system has dictated. Education in science and technology is essentially education on the symbol level." (Aldous L Huxley, "Essay", Daedalus, 1962)

"Innocent, unbiased observation is a myth." (Sir Peter B Medawar, Induction and Intuition in Scientific Thought, 1969)

"All perceiving is also thinking, all reasoning is also intuition, all observation is also invention." (Rudolf Arnheim, "Entropy and Art: An Essay on Disorder and Order", 1974)

"For scientists, observation is the highest authority [...]." (Raymond S Nickerson, "Mathematical Reasoning: Patterns, Problems, Conjectures and Proofs", 2009)

Out of Context: Experiment is... (Definitions)

"Experiment is fundamentally only induced observation." (Claude Bernard, "An Introduction to the Study of Experimental Medicine", 1865)

"Observation, then, is what shows facts; experiment is what teaches about facts and gives experience in relation to anything." (Claude Bernard, "An Introduction to the Study of Experimental Medicine", 1865)

"Experiment is the sole source of truth. It alone can teach us something new; it alone can give us certainty." (Henri Poincaré, "Science and Hypothesis", 1902)

"An experiment is an observation that can be repeated, isolated and varied." (Edward B Titchener, "A Text-Book of Psychology", 1909)

"Experiments are like cross-questioning a witness who will tell the truth but not the whole truth." (Alan Gregg, "The Furtherance of Medical Research", 1941)

"An experiment is a question which science poses to Nature, and a measurement is the recording of Nature’s answer." (Max Plank, "The Meaning and Limits of Exact Science", Science, 1949)

"Experimenters are the shocktroops of science." (Max Planck, "Scientific Autobiography, and Other Papers", 1949)

"An experiment is a question which man asks of nature; one result of the observation is an answer which nature yields to man." (Ferdinand Gonseth, "The Primeval Atom", 1950)

"Experiment is the sole judge of scientific ‘truth’." (Richard Feynman, "Six Easy Pieces", 1994)

"One good experiment is worth a thousand models […]; but one good model can make a thousand experiments unnecessary." (David Lloyd & Evgenii I Volkov, "The Ultradian Clock: Timekeeping for Intracelular Dynamics"  2013)

"An Experiment, like every other event which takes place, is a natural phenomenon; but in a Scientific Experiment the circumstances are so arranged that the relations between a particular set of phenomena may be studied to the best advantage." (James C Maxwell)

Out of Context: On Scientists (Definitions)

"The scientist is a lover of truth for the very love of truth itself, wherever it may lead." (Luther Burbank, "Why I Am An Infidel", 1926)

"The scientist is a practical man and his are practical aims. He does not seek the ultimate but the proximate. He does not speak of the last analysis but rather of the next approximation. […] On the whole, he is satisfied with his work, for while science may never be wholly right it certainly is never wholly wrong; and it seems to be improving from decade to decade." (Gilbert N Lewis, "The Anatomy of Science", 1926)

"[...] scientists are not a select few intelligent enough to think in terms of 'broad sweeping theoretical laws and principles'. Instead, scientists are people specifically trained to build models that incorporate theoretical assumptions and empirical evidence." (Peter Imhof, Science Vol. 287, 1935–1936)

"A good scientist is a person with original ideas." (Freeman Dyson, "Disturbing the Universe", 1979)

"Scientists are generally reluctant to accept the existence of a phenomenon when they do not know how to explain it. On the other hand, they will often accept a theory that is especially plausible before there exists any data to support it.” (Richard Morris, 1983) 

"A scientist is a person who knows more and more about less and less, until he knows everything about nothing." (John M Ziman, "Knowing Everything about Nothing: Specialization and Change in Scientific Careers", 1987)

"A scientist is no more a collector and classifier of facts than a historian is a man who complies and classifies a chronology of the dates of great battles and major discoveries." (Sir Peter B Medawar, "Aristotle to Zoos: A Philosophical Dictionary of Biology", 1983)

"It seems that scientists are often attracted to beautiful theories in the way that insects are attracted to flowers - not by logical deduction, but by something like a sense of smell." (Steven Weinberg, "Physics Today", 2005)

"The best scientists aren't the ones who know the most data; they're the ones who know what they're looking for." (Noam Chomsky, [Guardian] 2005)

"Under normal conditions the research scientist is not an innovator but a solver of puzzles, and the puzzles upon which he concentrates are just those which he believes can be both stated and solved within the existing scientific tradition." (Thomas S Kuhn, "The Essential Tension: Selected Studies in Scientific Tradition and Change", 2011)

"Great scientists are virtuosi of the art of discovering the meaning of what otherwise might seem barren observations." (Theodosius Dobzhansky)

"Scientists are not dependent on the ideas of a single man, but on the combined wisdom of thousands of men, all thinking of the same problem, and each doing his little bit to add to the great structure of knowledge which is gradually being erected." (Ernest Rutherford)

"The scientist is not content to stop at the obvious." (Charles H Mayo)

Out of Context: On Information (Definitions)

"Information is a set of marks that have meaning." (Edmund C Berkeley & Lawrence Wainwright, Computers: Their Operation and Applications", 1956)

"Information is carried by physical entities, such as books or sound waves or brains, but it is not itself material. Information in a living system is a feature of the order and arrangement of its parts, which arrangement provides the signs that constitute a ‘code’ or ‘language’." (John Z Young, "Programs of the Brain", 1978)

"Information is recorded in vast interconnecting networks. Each idea or image has hundreds, perhaps thousands, of associations and is connected to numerous other points in the mental network." (Peter Russell, "The Brain Book: Know Your Own Mind and How to Use it", 1979)

"Neither noise nor information is predictable." (Ray Kurzweil, "The Age of Spiritual Machines: When Computers Exceed Human Intelligence", 1999)

"Most of the information is fuzzy and linguistic in form." (Timothy J Ross & W Jerry Parkinson, "Fuzzy Set Theory, Fuzzy Logic, and Fuzzy Systems", 2002)

"Information is assimilated to the slots of a mental model in the form of ‘frames’ which are understood here as ‘chunks’ of knowledge with a well-defined meaning anchored in a given body of shared knowledge." (Jürgen Renn, "Before the Riemann Tensor: The Emergence of Einstein’s Double Strategy", "The Universe of General Relativity" Ed. by A.J. Kox & Jean Eisenstaedt, 2005)

"One advantage of the use of fuzzy models is the fact that their complexity can be gradually increased as more information is gathered. This increase in complexity can be done automatically or manually by a careful commission of the new operating point." (Jairo Espinosa et al, "Fuzzy Logic, Identification and Predictive Control", 2005)

"In a physical system, information is the opposite of entropy, as it involves uncommon and highly correlated configurations that are difficult to arrive at." (César A Hidalgo, "Why Information Grows: The Evolution of Order, from Atoms to Economies", 2015)