Out of Context: On Music (Definitions)

"Music is the discipline which treats of numbers in their relation to those things which are found in sound." (Cassiodorus, cca. 6th century)

"Music is fashioned wholly in the likeness of numbers." (Anon, "Scholia Enchiriadis", cca. 900)

"There are certain pleasures which only fill the outward senses, and there are others also which pertain only to the mind or reason; but music is a delectation so put in the midst that both by the sweetness of the sounds it moveth the senses, and by the artificiousness of the number and proportions it delighteth reason itself." (John Northbrooke , "Against Dicing", 1577)

"Music is a science which should have definite rules; these rules should be drawn from an evident principle; and this principle cannot really be known to us without the aid of mathematics." (Jean-Philippe Rameau, "Treatise on Harmony reduced to its natural principles", 1722)

"Music is like geometric figures and numbers, which are the universal forms of all possible objects of experience." (Friedrich Nietzsche, "Birth of Tragedy", 1872)

"Music is an order of mystic, sensuous mathematics." (James Huneker, "Chopin: The Man and His Music", 1900)

"Music is mathematics, the mathematics of listening, mathematics for the ears." (Karlheinz Stockhausen, "Conversations with Stockhausen", 1987)

"Music is a science which should have definite rules; these rules should be drawn from an evident principle; and this principle cannot really be known to us without the aid of mathematics." (Jean-Philippe Rameau, Treatise on Harmony, 2012)

"Mathematics is music for the mind; Music is mathematics for the soul." (Stanley Gudder)

"Music is architecture translated or transposed from space into time; for in music, besides the deepest feeling, there reigns also a rigorous mathematical intelligence." (Georg W F Hegel)

"Music is the arithmetic of sounds as optics is the geometry of light." (Claude Debussy)

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

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

Daniel Levitin - Collected Quotes

"When a musical piece is too simple we tend not to like it, finding it trivial. When it is too complex, we tend not to like it, finding it unpredictable - we don't perceive it to be grounded in anything familiar. Music, or any art form […] has to strike the right balance between simplicity and complexity […]" (Daniel Levitin, "This is Your Brain on Music", 2006)

"A well-designed graph clearly shows you the relevant end points of a continuum. This is especially important if you’re documenting some actual or projected change in a quantity, and you want your readers to draw the right conclusions. […]" (Daniel J Levitin, "Weaponized Lies", 2017)

"Collecting data through sampling therefore becomes a never-ending battle to avoid sources of bias. [...] While trying to obtain a random sample, researchers sometimes make errors in judgment about whether every person or thing is equally likely to be sampled." (Daniel J Levitin, "Weaponized Lies", 2017)

"GIGO is a famous saying coined by early computer scientists: garbage in, garbage out. At the time, people would blindly put their trust into anything a computer output indicated because the output had the illusion of precision and certainty. If a statistic is composed of a series of poorly defined measures, guesses, misunderstandings, oversimplifications, mismeasurements, or flawed estimates, the resulting conclusion will be flawed." (Daniel J Levitin, "Weaponized Lies", 2017)

"How do you know when a correlation indicates causation? One way is to conduct a controlled experiment. Another is to apply logic. But be careful - it’s easy to get bogged down in semantics." (Daniel J Levitin, "Weaponized Lies", 2017)

"In statistics, the word 'significant' means that the results passed mathematical tests such as t-tests, chi-square tests, regression, and principal components analysis (there are hundreds). Statistical significance tests quantify how easily pure chance can explain the results. With a very large number of observations, even small differences that are trivial in magnitude can be beyond what our models of change and randomness can explain. These tests don’t know what’s noteworthy and what’s not - that’s a human judgment." (Daniel J Levitin, "Weaponized Lies", 2017)

"Infographics are often used by lying weasels to shape public opinion, and they rely on the fact that most people won’t study what they’ve done too carefully." (Daniel J Levitin, "Weaponized Lies", 2017)

"Just because there’s a number on it, it doesn’t mean that the number was arrived at properly. […] There are a host of errors and biases that can enter into the collection process, and these can lead millions of people to draw the wrong conclusions. Although most of us won’t ever participate in the collection process, thinking about it, critically, is easy to learn and within the reach of all of us." (Daniel J Levitin, "Weaponized Lies", 2017)

"Many of us feel intimidated by numbers and so we blindly accept the numbers we’re handed. This can lead to bad decisions and faulty conclusions. We also have a tendency to apply critical thinking only to things we disagree with. In the current information age, pseudo-facts masquerade as facts, misinformation can be indistinguishable from true information, and numbers are often at the heart of any important claim or decision. Bad statistics are everywhere." (Daniel J Levitin, "Weaponized Lies", 2017)

"Measurements must be standardized. There must be clear, replicable, and precise procedures for collecting data so that each person who collects it does it in the same way." (Daniel J Levitin, "Weaponized Lies", 2017)

"Most of us have difficulty figuring probabilities and statistics in our heads and detecting subtle patterns in complex tables of numbers. We prefer vivid pictures, images, and stories. When making decisions, we tend to overweight such images and stories, compared to statistical information. We also tend to misunderstand or misinterpret graphics." (Daniel J Levitin, "Weaponized Lies", 2017)

"One kind of probability - classic probability - is based on the idea of symmetry and equal likelihood […] In the classic case, we know the parameters of the system and thus can calculate the probabilities for the events each system will generate. […] A second kind of probability arises because in daily life we often want to know something about the likelihood of other events occurring […]. In this second case, we need to estimate the parameters of the system because we don’t know what those parameters are. […] A third kind of probability differs from these first two because it’s not obtained from an experiment or a replicable event - rather, it expresses an opinion or degree of belief about how likely a particular event is to occur. This is called subjective probability […]." (Daniel J Levitin, "Weaponized Lies", 2017)

"One way to lie with statistics is to compare things - datasets, populations, types of products - that are different from one another, and pretend that they’re not. As the old idiom says, you can’t compare apples with oranges." (Daniel J Levitin, "Weaponized Lies", 2017)

"Probabilities allow us to quantify future events and are an important aid to rational decision making. Without them, we can become seduced by anecdotes and stories." (Daniel J Levitin, "Weaponized Lies", 2017)

"Samples give us estimates of something, and they will almost always deviate from the true number by some amount, large or small, and that is the margin of error. […] The margin of error does not address underlying flaws in the research, only the degree of error in the sampling procedure. But ignoring those deeper possible flaws for the moment, there is another measurement or statistic that accompanies any rigorously defined sample: the confidence interval." (Daniel J Levitin, "Weaponized Lies", 2017)

"Statistics, because they are numbers, appear to us to be cold, hard facts. It seems that they represent facts given to us by nature and it’s just a matter of finding them. But it’s important to remember that people gather statistics. People choose what to count, how to go about counting, which of the resulting numbers they will share with us, and which words they will use to describe and interpret those numbers. Statistics are not facts. They are interpretations. And your interpretation may be just as good as, or better than, that of the person reporting them to you." (Daniel J Levitin, "Weaponized Lies", 2017)

"The margin of error is how accurate the results are, and the confidence interval is how confident you are that your estimate falls within the margin of error." (Daniel J Levitin, "Weaponized Lies", 2017)

"The most accurate but least interpretable form of data presentation is to make a table, showing every single value. But it is difficult or impossible for most people to detect patterns and trends in such data, and so we rely on graphs and charts. Graphs come in two broad types: Either they represent every data point visually (as in a scatter plot) or they implement a form of data reduction in which we summarize the data, looking, for example, only at means or medians." (Daniel J Levitin, "Weaponized Lies", 2017)

"To be any good, a sample has to be representative. A sample is representative if every person or thing in the group you’re studying has an equally likely chance of being chosen. If not, your sample is biased. […] The job of the statistician is to formulate an inventory of all those things that matter in order to obtain a representative sample. Researchers have to avoid the tendency to capture variables that are easy to identify or collect data on - sometimes the things that matter are not obvious or are difficult to measure." (Daniel J Levitin, "Weaponized Lies", 2017)

"We are a storytelling species, and a social species, easily swayed by the opinions of others. We have three ways to acquire information: We can discover it ourselves, we can absorb it implicitly, or we can be told it explicitly. Much of what we know about the world falls in this last category - somewhere along the line, someone told us a fact or we read about it, and so we know it only second-hand. We rely on people with expertise to tell us." (Daniel J Levitin, "Weaponized Lies", 2017)

"We use the word probability in different ways to mean different things. It’s easy to get swept away thinking that a person means one thing when they mean another, and that confusion can cause us to draw the wrong conclusion." (Daniel J Levitin, "Weaponized Lies", 2017) 

On Structure: Structure in Mathematics III

"Mathematicians create by acts of insight and intuition. Logic then sanctions the conquests of intuition. It is the hygiene that mathematics practices to keep its ideas healthy and strong. Moreover, the whole structure rests fundamentally on uncertain ground, the intuition of humans. Here and there an intuition is scooped out and replaced by a firmly built pillar of thought; however, this pillar is based on some deeper, perhaps less clearly defined, intuition. Though the process of replacing intuitions with precise thoughts does not change the nature of the ground on which mathematics ultimately rests, it does add strength and height to the structure." (Morris Kline, "Mathematics in Western Culture ", 1964)

"The probability concept used in probability theory has exactly the same structure as have the fundamental concepts in any field in which mathematical analysis is applied to describe and represent reality." (Richard von Mises, "Mathematical Theory of Probability and Statistics", 1964)

"The question ‘What is mathematics?’ cannot be answered meaningfully by philosophical generalities, semantic definitions or journalistic circumlocutions. Such characterizations also fail to do justice to music or painting. No one can form an appreciation of these arts without some experience with rhythm, harmony and structure, or with form, color and composition. For the appreciation of mathematics actual contact with its substance is even more necessary." (Richard Courant, "Mathematics in the Modern World", Scientific American Vol. 211 (3), 1964)

"[Game theory is] essentially a structural theory. It uncovers the logical structure of a great variety of conflict situations and describes this structure in mathematical terms. Sometimes the logical structure of a conflict situation admits rational decisions; sometimes it does not." (Anatol Rapoport, "Prisoner's dilemma: A study in conflict and cooperation", 1965)

"The most natural way to give an independence proof is to establish a model with the required properties. This is not the only way to proceed since one can attempt to deal directly and analyze the structure of proofs. However, such an approach to set theoretic questions is unnatural since all our intuition come from our belief in the natural, almost physical model of the mathematical universe." (Paul J Cohen, "Set Theory and the Continuum Hypothesis", 1966)

"The structures with which mathematics deals are more like lace, the leaves of trees, and the play of light and shadow on a human face, than they are like buildings and machines, the least of their representatives. The best proofs in mathematics are short and crisp like epigrams, and the longest have swings and rhythms that are like music. The structures of mathematics and the propositions about them are ways for the imagination to travel and the wings, or legs, or vehicles to take you where you want to go." (Scott Buchanan, "Poetry and Mathematics", 1975)

"In each branch of mathematics it is essential to recognize when two structures are equivalent. For example two sets are equivalent, as far as set theory is concerned, if there exists a bijective function which maps one set onto the other. Two groups are equivalent, known as isomorphic, if there exists a a homomorphism of one to the other which is one-to-one and onto. Two topological spaces are equivalent, known as homeomorphic, if there exists a homeomorphism of one onto the other." (Sydney A Morris, "Topology without Tears", 2011)

On Creativity (Mathematics I)

"Creativity is the heart and soul of mathematics at all levels. The collection of special skills and techniques is only the raw material out of which the subject itself grows. To look at mathematics without the creative side of it, is to look at a black-and-white photograph of a Cezanne; outlines may be there, but everything that matters is missing." (Robert C Buck, "Teaching Machines and Mathematics Programs", American Mathematical Monthly 69, 1962)

"There are, roughly speaking, two kinds of mathematical creativity. One, akin to conquering a mountain peak, consists of solving a problem which has remained unsolved for a long time and has commanded the attention of many mathematicians. The other is exploring new territory." (Mark Kac, "Enigmas Of Chance", 1985)

"Music and higher mathematics share some obvious kinship. The practice of both requires a lengthy apprenticeship, talent, and no small amount of grace. Both seem to spring from some mysterious workings of the mind. Logic and system are essential for both, and yet each can reach a height of creativity beyond the merely mechanical." (Frederick Pratter, "How Music and Math Seek Truth in Beauty", Christian Science Monitor, 1995)

"Mathematics is a fascinating discipline that calls for creativity, imagination, and the mastery of rigorous standards of proof." (John Meier & Derek Smith, "Exploring Mathematics: An Engaging Introduction to Proof", 2017)

"Math is the beautiful, rich, joyful, playful, surprising, frustrating, humbling and creative art that speaks to something transcendental. It is worthy of much exploration and examination because it is intrinsically beautiful, nothing more to say. Why play the violin? Because it is beautiful! Why engage in math? Because it too is beautiful!" (James Tanton, "Thinking Mathematics")

"Mathematics is the summit of human thinking. It has all the creativity and imagination that you can find in all kinds of art, but unlike art-charlatans and all kinds of quacks will not succeed there." (Meir Shalev)

"No discovery has been made in mathematics, or anywhere else for that matter, by an effort of deductive logic; it results from the work of creative imagination which builds what seems to be truth, guided sometimes by analogies, sometimes by an esthetic ideal, but which does not hold at all on solid logical bases. Once a discovery is made, logic intervenes to act as a control; it is logic that ultimately decides whether the discovery is really true or is illusory; its role therefore, though considerable, is only secondary." (Henri Lebesgue)

"The essential feature of mathematical creativity is the exploration, under the pressure of powerful implosive forces, of difficult problems for whose validity and importance the explorer is eventually held accountable by reality." (Alfred Adler)

On Manifolds V (Geometry III)

"Whereas the conception of space and time as a four-dimensional manifold has been very fruitful for mathematical physicists, its effect in the field of epistemology has been only to confuse the issue. Calling time the fourth dimension gives it an air of mystery. One might think that time can now be conceived as a kind of space and try in vain to add visually a fourth dimension to the three dimensions of space. It is essential to guard against such a misunderstanding of mathematical concepts. If we add time to space as a fourth dimension it does not lose any of its peculiar character as time. [...] Musical tones can be ordered according to volume and pitch and are thus brought into a two dimensional manifold. Similarly colors can be determined by the three basic colors red, green and blue… Such an ordering does not change either tones or colors; it is merely a mathematical expression of something that we have known and visualized for a long time. Our schematization of time as a fourth dimension therefore does not imply any changes in the conception of time. [...] the space of visualization is only one of many possible forms that add content to the conceptual frame. We would therefore not call the representation of the tone manifold by a plane the visual representation of the two dimensional tone manifold." (Hans Reichenbach, "The Philosophy of Space and Time", 1928)

"The sequence of numbers which grows beyond any stage already reached by passing to the next number is a manifold of possibilities open towards infinity, it remains forever in the status of creation, but is not a closed realm of things existing in themselves. That we blindly converted one into the other is the true source of our difficulties […]" (Hermann Weyl, "Mathematics and Logic", 1946)

"The first attempts to consider the behavior of so-called 'random neural nets' in a systematic way have led to a series of problems concerned with relations between the 'structure' and the 'function' of such nets. The 'structure' of a random net is not a clearly defined topological manifold such as could be used to describe a circuit with explicitly given connections. In a random neural net, one does not speak of 'this' neuron synapsing on 'that' one, but rather in terms of tendencies and probabilities associated with points or regions in the net." (Anatol Rapoport, "Cycle distributions in random nets", The Bulletin of Mathematical Biophysics 10(3), 1948)

"The main object of study in differential geometry is, at least for the moment, the differential manifolds, structures on the manifolds (Riemannian, complex, or other), and their admissible mappings. On a manifold the coordinates are valid only locally and do not have a geometric meaning themselves." (Shiing-Shen Chern, "Differential geometry, its past and its future", 1970)

"[...] 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)

"It is commonly said that the study of manifolds is, in general, the study of the generalization of the concept of surfaces. To some extent, this is true. However, defining it that way can lead to overshadowing by 'figures' such as geometrical surfaces." (Kenji Ueno & Toshikazu Sunada, "A Mathematical Gift, III: The Interplay Between Topology, Functions, Geometry, and Algebra", Mathematical World Vol. 23, 1996)

"One could also question whether we are looking for a single overarching mathematical structure or a combination of different complementary points of view. Does a fundamental theory of Nature have a global definition, or do we have to work with a series of local definitions, like the charts and maps of a manifold, that describe physics in various 'duality frames'. At present string theory is very much formulated in the last kind of way." (Robbert Dijkgraaf, "Mathematical Structures", 2005)

"Quantum physics, in particular particle and string theory, has proven to be a remarkable fruitful source of inspiration for new topological invariants of knots and manifolds. With hindsight this should perhaps not come as a complete surprise. Roughly one can say that quantum theory takes a geometric object (a manifold, a knot, a map) and associates to it a (complex) number, that represents the probability amplitude for a certain physical process represented by the object." (Robbert Dijkgraaf, "Mathematical Structures", 2005)

"The primary aspects of the theory of complex manifolds are the geometric structure itself, its topological structure, coordinate systems, etc., and holomorphic functions and mappings and their properties. Algebraic geometry over the complex number field uses polynomial and rational functions of complex variables as the primary tools, but the underlying topological structures are similar to those that appear in complex manifold theory, and the nature of singularities in both the analytic and algebraic settings is also structurally very similar." (Raymond O Wells Jr, "Differential and Complex Geometry: Origins, Abstractions and Embeddings", 2017)

"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)

Marcus du Sautoy - Collected Quotes

"Rather mathematicians like to look for patterns, and the primes probably offer the ultimate challenge. When you look at a list of them stretching off to infinity, they look chaotic, like weeds growing through an expanse of grass representing all numbers. For centuries mathematicians have striven to find rhyme and reason amongst this jumble. Is there any music that we can hear in this random noise? Is there a fast way to spot that a particular number is prime? Once you have one prime, how much further must you count before you find the next one on the list? These are the sort of questions that have tantalized generations." (Marcus du Sautoy, "The Music of the Primes", 1998)

"[…] despite their apparent simplicity and fundamental character, prime numbers remain the most mysterious objects studied by mathematicians. In a subject dedicated to finding patterns and order, the primes offer the ultimate challenge." (Marcus du Sautoy, "The Music of the Primes", 2003)

"Despite the unworldly nature of mathematics, mathematicians still have egos that need massaging. Nothing acts as a better drive to the creative process than the thought of the immortality bestowed by having your name attached to a theorem." (Marcus du Sautoy, "The Music of the Primes", 2003)

"The primes have been a constant companion in our exploration of the mathematical world yet they remain the most enigmatic of all numbers. Despite the best efforts of the greatest mathematical minds to explain the modulation and transformation of this mystical music, the primes remain an unanswered riddle." (Marcus du Sautoy, "The Music of the Primes", 2003)

"The concept of proof perhaps marks the true beginning of mathematics as the art of deduction rather than just numerological observation, the point at which mathematical alchemy gave way to mathematical chemistry." (Marcus du Sautoy, "The Music of the Primes", 2004)

"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)

"As mathematicians had gradually got to grips with what symmetry actually meant, they seemed to be gazing upon an endless world filled with a chaotic and infinitely varied range of symmetrical objects." (Marcus du Sautoy, "Symmetry: A Journey into the Patterns of Nature", 2008)

"For both primes and symmetries, zeta functions act as black boxes. They are built from a formula which binds together the numbers you are trying to understand. The hope is that the zeta function will reveal new insights into the numbers of symmetries. It provides a way of getting from part of the mathematical world where chaos seems to reign to a completely different region where one can start to pick out patterns." (Marcus du Sautoy, "Symmetry: A Journey into the Patterns of Nature", 2008)

"For the mathematician, the pattern searcher, understanding symmetry is one of the principal themes in the quest to chart the mathematical world." (Marcus du Sautoy, "Symmetry: A Journey into the Patterns of Nature", 2008)

"Science is about discovery, but it is also about communication. An idea can hardly be said to exist if you do not awaken that same idea in someone else." (Marcus du Sautoy, "Symmetry: A Journey into the Patterns of Nature", 2008)

"Symmetry continues to inform the way we craft words in songs and poetry. From the first cave paintings to modern art, from primitive drumbeats to contemporary music, artists have continually pushed symmetry to the extremes."(Marcus du Sautoy, "Symmetry: A Journey into the Patterns of Nature", 2008) 

"The word ‘symmetry’ conjures to mind objects which are well balanced, with perfect proportions. Such objects capture a sense of beauty and form. The human mind is constantly drawn to anything that embodies some aspect of symmetry. Our brain seems programmed to notice and search for order and structure. Artwork, architecture and music from ancient times to the present day play on the idea of things which mirror each other in interesting ways. Symmetry is about connections between different parts of the same object. It sets up a natural internal dialogue in the shape." (Marcus du Sautoy, "Symmetry: A Journey into the Patterns of Nature", 2008)

"Why, though, is symmetry so pervasive in nature? It is not just a matter of aesthetics. Just as it is for me and mathematics, symmetry in nature is about language. It provides a way for animals and plants to convey a multitude of messages, from genetic superiority to nutritional information. Symmetry is often a sign of meaning, and can therefore be interpreted as a very basic, almost primeval form of communication." (Marcus du Sautoy, "Symmetry: A Journey into the Patterns of Nature", 2008) 

"Just as music is not about reaching the final chord, mathematics is about more than just the result. It is the journey that excites the mathematician. I read and reread proofs in much the same way as I listen to a piece of music: understanding how themes are established, mutated, interwoven and transformed. What people don't realise about mathematics is that it involves a lot of choice: not about what is true or false (I can't make the Riemann hypothesis false if it's true), but from deciding what piece of mathematics is worth ‘listening to’." (Marcus du Sautoy, "Listen by Numbers: Music and Maths", 2011)

"Mathematics has beauty and romance. It's not a boring place to be, the mathematical world. It's an extraordinary place; it's worth spending time there." (Marcus Du Sautoy)

"The reason why we do maths is because it's like poetry. It's about patterns, and that really turned me on. It made me feel that maths was in tune with the other things I liked doing." (Marcus du Sautoy)

On Noise I

"Noise is the most impertinent of all forms of interruption. It is not only an interruption, but also a disruption of thought." (Arthur Schopenhauer, "Parerga and Paralipomena", 1851)

"Mathematics is the predominant science of our time; its conquests grow daily, though without noise; he who does not employ it for himself, will some day find it employed against himself." (Johann F Herbart, Werke, 1890)

"Life pushes its way through this fatalistically determined world like a river flowing upstream. It is a system of utterly improbable order, a message in a world of noise." (Joseph H Rush, "The Dawn of Life", 1957)

"Higher, directed forms of energy (e.g., mechanical, electric, chemical) are dissipated, that is, progressively converted into the lowest form of energy, i.e., undirected heat movement of molecules; chemical systems tend toward equilibria with maximum entropy; machines wear out owing to friction; in communication channels, information can only be lost by conversion of messages into noise but not vice versa, and so forth." (Ludwig von Bertalanffy, "Robots, Men and Minds", 1967)

"To adapt to a changing environment, the system needs a variety of stable states that is large enough to react to all perturbations but not so large as to make its evolution uncontrollably chaotic. The most adequate states are selected according to their fitness, either directly by the environment, or by subsystems that have adapted to the environment at an earlier stage. Formally, the basic mechanism underlying self-organization is the (often noise-driven) variation which explores different regions in the system’s state space until it enters an attractor. This precludes further variation outside the attractor, and thus restricts the freedom of the system’s components to behave independently. This is equivalent to the increase of coherence, or decrease of statistical entropy, that defines self-organization." (Francis Heylighen, "The Science Of Self-Organization And Adaptivity", 1970)

"Probability plays a central role in many fields, from quantum mechanics to information theory, and even older fields use probability now that the presence of 'nois' is officially admitted. The newer aspects of many fields start with the admission of uncertainty." (Richard Hamming, "Methods of Mathematics Applied to Calculus, Probability, and Statistics", 1985)

"An essential element of dynamics systems is a positive feedback that self-enhances the initial deviation from the mean. The avalanche is proverbial. Cities grow since they attract more people, and in the universe, a local accumulation of dust may attract more dust, eventually leading to the birth of a star. Earlier or later, self-enhancing processes evoke an antagonistic reaction. A collapsing stock market stimulates the purchase of shares at a low price, thereby stabilizing the market. The increasing noise, dirt, crime and traffic jams may discourage people from moving into a big city." (Hans Meinhardt, "The Algorithmic Beauty of Sea Shells", 1995)

"Rather mathematicians like to look for patterns, and the primes probably offer the ultimate challenge. When you look at a list of them stretching off to infinity, they look chaotic, like weeds growing through an expanse of grass representing all numbers. For centuries mathematicians have striven to find rhyme and reason amongst this jumble. Is there any music that we can hear in this random noise? Is there a fast way to spot that a particular number is prime? Once you have one prime, how much further must you count before you find the next one on the list? These are the sort of questions that have tantalized generations." (Marcus du Sautoy, "The Music of the Primes", 1998)

"Data are collected as a basis for action. Yet before anyone can use data as a basis for action the data have to be interpreted. The proper interpretation of data will require that the data be presented in context, and that the analysis technique used will filter out the noise."  (Donald J Wheeler, "Understanding Variation: The Key to Managing Chaos" 2nd Ed., 2000)

"Data are generally collected as a basis for action. However, unless potential signals are separated from probable noise, the actions taken may be totally inconsistent with the data. Thus, the proper use of data requires that you have simple and effective methods of analysis which will properly separate potential signals from probable noise." (Donald J Wheeler, "Understanding Variation: The Key to Managing Chaos" 2nd Ed., 2000)

Marvin Minsky - Collected Quotes

"Computer languages of the future will be more concerned with goals and less with procedures specified by the programmer." (Marvin Minsky, "Form and Content in Computer Science", [Turing Award lecture] 1969)

"A memory should induce a state through which we see current reality as an instance of the remembered event - or equivalently, see the past as an instance of the present. […] the system can perform a computation analogous to one from the memorable past, but sensitive to present goals and circumstances." (Marvin Minsky, "K-Linesː A Theory of Memory", 1980) 

"Since we have no systematic way to avoid all the inconsistencies of commonsense logic, each person must find his own way by building a private collection of 'cognitive censors' to suppress the kinds of mistakes he has discovered in the past." (Marvin Minsky, "Jokes and their Relation to the Cognitive Unconscious", 1980) 

"When you, ‘get an idea’, or ‘solve a problem’, or have a ‘memorable experience’, you create what we shall call a K-line. This K-line gets connected to those ‘mental agencies’ that were actively involved in the memorable event. When that K-line is later ‘activated’, it reactivates some of those mental agencies, creating a ‘partial mental state’ resembling the original." (Marvin Minsky, "K-Linesː A Theory of Memory", 1980) 

"If explaining minds seems harder than explaining songs, we should remember that sometimes enlarging problems makes them simpler! The theory of the roots of equations seemed hard for centuries within its little world of real numbers, but it suddenly seemed simple once Gauss exposed the larger world of so-called complex numbers. Similarly, music should make more sense once seen through listeners' minds." (Marvin Minsky, "Music, Mind, and Meaning", 1981)

"The way the mathematics game is played, most variations lie outside the rules, while music can insist on perfect canon or tolerate a casual accompaniment." (Marvin Minsky, "Music, Mind, and Meaning", 1981)

"Theorems often tell us complex truths about the simple things, but only rarely tell us simple truths about the complex ones. To believe otherwise is wishful thinking or ‘mathematics envy’." (Marvin Minsky, "Music, Mind, and Meaning", 1981)

"What is the difference between merely knowing (or remembering, or memorizing) and understanding? […] A thing or idea seems meaningful only when we have several different ways to represent it - different perspectives and different associations […]. Then we can turn it around in our minds, so to speak: however it seems at the moment, we can see it another way and we never come to a full stop. In other words, we can 'think' about it. If there were only one way to represent this thing or idea, we would not call this representation thinking." (Marvin Minsky, "Music, Mind, and Meaning", 1981)

"The hardest problems we have to face do not come from philosophical questions about whether brains are machines or not. There is not the slightest reason to doubt that brains are anything other than machines with enormous numbers of parts that work in perfect accord with physical laws. As far as anyone can tell, our minds are merely complex processes. The serious problems come from our having had so little experience with machines of such complexity that we are not yet prepared to think effectively about them." (Marvin Minsky, 1986) 

"For generations, scientists and philosophers have tried to explain ordinary reasoning in terms of logical principles - with virtually no success. I suspect this enterprise failed because it was looking in the wrong direction: common sense works so well not because it is an approximation of logic; logic is only a small part of our great accumulation of different, useful ways to chain things together." (Marvin Minsky, "The Society of Mind", 1987) 

"Unless we can explain the mind in terms of things that have no thoughts or feelings of their own, we'll only have gone around in a circle." (Marvin Minsky, "The Society of Mind", 1987) 

"Every system that we build will surprise us with new kinds of flaws until those machines become clever enough to conceal their faults from us." (Marvin Minsky, "The Emotion Machine: Commonsense Thinking, Artificial Intelligence, and the Future of the Human Mind", 2006)

"It makes no sense to seek a single best way to represent knowledge - because each particular form of expression also brings its particular limitations. For example, logic-based systems are very precise, but they make it hard to do reasoning with analogies. Similarly, statistical systems are useful for making predictions, but do not serve well to represent the reasons why those predictions are sometimes correct." (Marvin Minsky, "The Emotion Machine: Commonsense Thinking, Artificial Intelligence, and the Future of the Human Mind", 2006)

"The idea that the world exists is like adding an extra term to an equation that doesn’t belong there." (Marvin Minsky)

On Simplicity XII (Complexity vs Simplicity IV)

"Simplicity of structure means organic unity, whether the organism be simple or complex; and hence in all times the emphasis which critics have laid upon Simplicity, though they have not unfrequently confounded it with narrowness of range." (George H Lewes, "The Principles of Success in Literature", 1865)

"The aim of science is to seek the simplest explanations of complex facts. We are apt to fall into the error of thinking that the facts are simple because simplicity is the goal of our quest. The guiding motto in the life of every natural philosopher should be, ‘Seek simplicity and distrust it’." (Alfred N Whitehead, "The Concept of Nature", 1919) 

"In products of the human mind, simplicity marks the end of a process of refining, while complexity marks a primitive stage." (Eric Hoffer, 1954)

"The machinery of the world is far too complex for the simplicity of men." (Jorge L Borges, "Dreamtigers", 1960)

"The ideas need not be complex. Most ideas that are successful are ludicrously simple. Successful ideas generally have the appearance of simplicity because they seem inevitable." (Sol LeWitt, "Paragraphs on Conceptual Art", 1967) 

"Simplicity does not precede complexity, but follows it." (Alan Perlis, "Epigrams on Programming", 1982)

"It is important to emphasize the value of simplicity and elegance, for complexity has a way of compounding difficulties and as we have seen, creating mistakes. My definition of elegance is the achievement of a given functionality with a minimum of mechanism and a maximum of clarity." (Fernando J Corbató, "On Building Systems That Will Fail", 1991)

"When a musical piece is too simple we tend not to like it, finding it trivial. When it is too complex, we tend not to like it, finding it unpredictable - we don't perceive it to be grounded in anything familiar. Music, or any art form […] has to strike the right balance between simplicity and complexity […]" (Daniel Levitin, "This is Your Brain on Music", 2006) 

"Most of the world is of great roughness and infinite complexity. However, the infinite sea of complexity includes two islands of simplicity: one of Euclidean simplicity and a second of relative simplicity in which roughness is present but is the same at all scales." (Benoît Mandelbrot, "The Fractalist", 2012)

"I think there is a profound and enduring beauty in simplicity; in clarity, in efficiency. True simplicity is derived from so much more than just the absence of clutter and ornamentation. It's about bringing order to complexity." (Jonathan Ive, 2013) 

On Spacetime (1850-1874)

"The Mathematics, like language, (of which indeed they may be considered a species,) comprehending under that designation the whole science of number, space, form, time, and motion, as far as it can be expressed in abstract formulas, are evidently not only one of the most useful, but one of the grandest of studies." (Edward Everett, [address] 1857)

"What then is music? [...] It exists between thought and phenomenon, like a twilight medium, it stands between spirit and matter, related to and yet different from both; it is spirit, but spirit governed by time; it is matter, but matter that can manage without space." (Heinrich Heine, "On the French Stage: Intimate letters to August Lewald", 1857)

"All external objects and events which we can contemplate are viewed as having relations of Space, Time, and Number; and are subject to the general conditions which these Ideas impose, as well as to the particular laws which belong to each class of objects and occurrences." (William Whewell, "History of Scientific Ideas" Vol. 1, 1858)

"And so to imagine the action of a man entirely subject to the law of inevitability without any freedom, we must assume the knowledge of an infinite number of space relations, an infinitely long period of time, and an infinite series of causes." (Lev Tolstoy, "War and Peace", 1869)

"Human existence is girt round with mystery: the narrow region of our experience is a small island in the midst of a boundless sea. To add to the mystery, the domain of our earthly existence is not only an island of infinite space, but also in infinite time. The past and the future are alike shrouded from us: we neither know the origin of anything which is, nor its final destination." (John S Mill, "Nature, The Utility of Religion and Theism", 1874)

"In science nothing capable of proof ought to be accepted without proof. Though this demand seems so reasonable yet I cannot regard it as having been met even in […] that part of logic which deals with the theory of numbers. In speaking of arithmetic (algebra, analysis) as a part of logic I mean to imply that I consider the number concept entirely independent of the notions of intuition of space and time, that I consider it an immediate result from the laws of thought." (Richard Dedekind, "Was sind und was sollen die Zahlen?", 1888)

On Abstraction (1990-1999)

"All of engineering involves some creativity to cover the parts not known, and almost all of science includes some practical engineering to translate the abstractions into practice." (Richard W Hamming, "The Art of Probability for Scientists and Engineers", 1991)

"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", 1991)

"Great mathematics seldom comes from idle speculation about abstract spaces and symbols. More often than not it is motivated by definite questions arising in the worlds of nature and humans." (John L Casti, "Reality Rules: Picturing the world in mathematics", 1992)

"The word theory, as used in the natural sciences, doesn’t mean an idea tentatively held for purposes of argument - that we call a hypothesis. Rather, a theory is a set of logically consistent abstract principles that explain a body of concrete facts. It is the logical connections among the principles and the facts that characterize a theory as truth. No one element of a theory [...] can be changed without creating a logical contradiction that invalidates the entire system. Thus, although it may not be possible to substantiate directly a particular principle in the theory, the principle is validated by the consistency of the entire logical structure." (Alan Cromer, "Uncommon Sense: The Heretical Nature of Science", 1993)

"A mental model is not normally based on formal definitions but rather on concrete properties that have been drawn from life experience. Mental models are typically analogs, and they comprise specific contents, but this does not necessarily restrict their power to deal with abstract concepts, as we will see. The important thing about mental models, especially in the context of mathematics, is the relations they represent. […]  The essence of understanding a concept is to have a mental representation or mental model that faithfully reflects the structure of that concept. (Lyn D. English & Graeme S. Halford, "Mathematics Education: Models and Processes", 1995)

"Music and math together satisfied a sort of abstract 'appetite', a desire that was partly intellectual, partly aesthetic, partly emotional, partly, even, physical." (Edward Rothstein, "Emblems of Mind: The Inner Life of Music and Mathematics", 1995)

"The larger, more detailed and complex the model - the less abstract the abstraction – the smaller the number of people capable of understanding it and the longer it takes for its weaknesses and limitations to be found out." (John Adams, "Risk", 1995)

"The representational nature of maps, however, is often ignored - what we see when looking at a map is not the word, but an abstract representation that we find convenient to use in place of the world. When we build these abstract representations we are not revealing knowledge as much as are creating it." (Alan M MacEachren, "How Maps Work: Representation, Visualization, and Design", 1995)

"Abstract concepts are largely metaphorical." (George Lakoff, "Philosophy in the Flesh: The Embodied Mind and Its Challenge to Western Thought", 1999)

"The abstractions of science are stereotypes, as two-dimensional and as potentially misleading as everyday stereotypes. And yet they are as necessary to the process of understanding as filtering is to the process of perception." (K C Cole, "First You Build a Cloud and Other Reflections on Physics as a Way of Life", 1999)

On Music and Mathematics (2010-2019)

"Just as music is not about reaching the final chord, mathematics is about more than just the result. It is the journey that excites the mathematician. I read and reread proofs in much the same way as I listen to a piece of music: understanding how themes are established, mutated, interwoven and transformed. What people don't realise about mathematics is that it involves a lot of choice: not about what is true or false (I can't make the Riemann hypothesis false if it's true), but from deciding what piece of mathematics is worth ‘listening to’." (Marcus du Sautoy, "Listen by numbers: music and maths", 2011)

"A surprising proportion of mathematicians are accomplished musicians. Is it because music and mathematics share patterns that are beautiful?" (Martin Gardner, The Dover Math and Science Newsletter, 2011)

"Music is a science which should have definite rules; these rules should be drawn from an evident principle; and this principle cannot really be known to us without the aid of mathematics." (Jean-Philippe Rameau, Treatise on Harmony, 2012)

"People think mathematics begins when you write down a theorem followed by a proof. That’s not the beginning, that’s the end. For me the creative place in mathematics comes before you start to put things down on paper, before you try to write a formula. You picture various things, you turn them over in your mind. You’re trying to create, just as a musician is trying to create music, or a poet. There are no rules laid down. You have to do it your own way. But at the end, just as a composer has to put it down on paper, you have to write things down. But the most important stage is understanding. A proof by itself doesn’t give you understanding." (Michael F Atiyah, [interview] 2013)

"Music can capture human emotions to a degree beyond anything that we can convey with equations. (Jim Bell, "The Interstellar Age: Inside the Forty- Year Voyager Mission", 2015)

"[...] the relations between the two [mathematics and music] disciplines were never truly symmetric. Yes, there are many similarities between the two. For example, mathematics and music both depend on an efficient system of notation - a set of written symbols that convey a precise, unambiguous meaning to its practitioners (although in music this is augmented by a large assortment of verbal terms to indicate the more emotional aspects of playing)." (Eli Maor, "Music by the Numbers: From Pythagoras to Schoenberg", 2018)

"The significance of Fourier’s theorem to music cannot be overstated: since every periodic vibration produces a musical sound (provided, of course, that it lies within the audible frequency range), it can be broken down into its harmonic components, and this decomposition is unique; that is, every tone has one, and only one, acoustic spectrum, its harmonic fingerprint. The overtones comprising a musical tone thus play a role somewhat similar to that of the prime numbers in number theory: they are the elementary building blocks from which all sound is made." (Eli Maor, "Music by the Numbers: From Pythagoras to Schoenberg", 2018)

"Ultimately, music is meant to move our souls, to stir our emotions, to arouse us to swing by its rhythms, and this cannot be achieved by mathematical principles alone." (Eli Maor, "Music by the Numbers: From Pythagoras to Schoenberg", 2018)

Note: The quotes have been reordered chronologically.

On Music and Mathematics (2000-2009)

"Like musicians who can read and write complicated scores in a world without sounds, for us mathematics is a source of delight, excitement, and even controversy which are hard to share with non-mathematicians. In our small micro-cosmos we should ever seek the right balance between competition and solidarity, criticism and empathy, exclusion and inclusion." (Gil Kalai, "Combinatorics with a Geometric Flavor: Some Examples", 2000)

"Skills are to mathematics what scales are to music or spelling is to writing. The objective of learning is to write, to play music, or to solve problems -  not just to master skills." (William Briggs, 2005)

"Mathematics can be as effortless as humming a tune, if you know the tune. But our culture does not prepare us for appreciation of mathematics as it does for appreciation of music. Though we start hearing music very early in life, the same cannot be said of mathematics, even though the two subjects are twins. This is a shame; to know music without knowing its mathematics is like hearing a melody without its accompaniment." (Gareth Loy, "Musimathics: The Mathematical Foundations of Music" Vol. 1, 2006)

"We all know what we like in music, painting or poetry, but it is much harder to explain why we like it. The same is true in mathematics, which is, in part, an art form. We can identify a long list of desirable qualities: beauty, elegance, importance, originality, usefulness, depth, breadth, brevity, simplicity, clarity. However, a single work can hardly embody them all; in fact, some are mutually incompatible. Just as different qualities are appropriate in sonatas, quartets or symphonies, so mathematical compositions of varying types require different treatment." (Michael Atiyah, "Mathematics: Art and Science" Bulletin of the AMS 43, 2006)

"Mathematical ideas like number can only be really 'seen' with the 'eyes of the mind' because that is how one 'sees' ideas. Think of a sheet of music which is important and useful but it is nowhere near as interesting, beautiful or powerful as the music it represents. One can appreciate music without reading the sheet of music. Similarly, mathematical notation and symbols on a blackboard are just like the sheet of music; they are important and useful but they are nowhere near as interesting, beautiful or powerful as the actual mathematics (ideas) they represent." (Fiacre O Cairbre, "The Importance of Being Beautiful in Mathematics", IMTA Newsletter 109, 2009)

"Mathematics is the music of reason. To do mathematics is to engage in an act of discovery and conjecture, intuition and inspiration; to be in a state of confusion - not because it makes no sense to you, but because you gave it sense and you still don't understand what your creation is up to; to have a break-through idea; to be frustrated as an artist; to be awed and overwhelmed by an almost painful beauty; to be alive, damn it." (Paul Lockhart, "A Mathematician's Lament", 2009)

"There is nothing as dreamy and poetic, nothing as radical, subversive, and psychedelic, as mathematics. It is every bit as mind blowing as cosmology or physics (mathematicians conceived of black holes long before astronomers actually found any), and allows more freedom of expression than poetry, art, or music (which depends heavily on properties of the physical universe). Mathematics is the purest of the arts, as well as the most misunderstood." (Paul Lockhart, "A Mathematician's Lament", 2009)

"In our travels, we have come across many equations - math for understanding the universe, for making music, for mapping stars, and also for tipping, which is important." (Libba Bray, "Going Bovine", 2009)

Note: The quotes have been reordered chronologically.

On Music and Mathematics (1950-1974)

"There exists a passion for comprehension, just as there exists a passion for music. That passion is rather common in children but gets lost in most people later on. Without this passion, there would be neither mathematics nor natural science." (Albert Einstein, Scientific American Vol. 182 (4), 1950)

"Mathematics, like music and poetry, is a creation of the mind; [...] the primary task of the mathematician, like that of any other artist, is to extend man's mental horizon by representation and interpretation." (Graham Sutton, "Mathematics in Action", 1954)

"Is it possible to breach this wall, to present mathematics in such a way that the spectator may enjoy it? Cannot the enjoyment of mathematics be extended beyond the small circle of those who are ‘mathematically gifted’? Indeed, only a few are mathematically gifted in the sense that they are endowed with the talent to discover new mathematical facts. But by the same token, only very few are musically gifted in that they are able to compose music. Nevertheless, there are many who can understand and perhaps reproduce music, or who at least enjoy it. We believe that the number of people who can understand simple mathematical ideas is not relatively smaller than the number of those who are commonly called musical, and that their interest will be stimulated if only we can eliminate the aversion toward mathematics that so many have acquired from childhood experiences." (Hans Rademacher & Otto Toeplitz, "The Enjoyment of Mathematics", 1957)

"There is beauty in discovery. There is mathematics in music, a kinship of science and poetry in the description of nature, and exquisite form in a molecule. Attempts to place different disciplines in different camps are revealed as artificial in the face of the unity of knowledge. All illiterate men are sustained by the philosopher, the historian, the political analyst, the economist, the scientist, the poet, the artisan, and the musician." (Glenn T Seaborg, 1958)

"Languages, literature, art, music, history: are all self-evident helps - and even mathematics and sciences, by training memory and demanding the analytical approach, are helpful by indirection." (Irene Dunne, "If You Want Success", [Screenland Interview] 1961)

"Science begins with the world we have to live in, accepting its data and trying to explain its laws. From there, it moves toward the imagination: it becomes a mental construct, a model of a possible way of interpreting experience. The further it goes in this direction, the more it tends to speak the language of mathematics, which is really one of the languages of the imagination, along with literature and music." (Northrop Frye, "The Educated Imagination", 1963)

"The question ‘What is mathematics?’ cannot be answered meaningfully by philosophical generalities, semantic definitions or journalistic circumlocutions. Such characterizations also fail to do justice to music or painting. No one can form an appreciation of these arts without some experience with rhythm, harmony and structure, or with form, color and composition. For the appreciation of mathematics actual contact with its substance is even more necessary." (Richard Courant, "Mathematics in the Modern World", Scientific American Vol. 211 (3), 1964)

"It seems to me now that mathematics is capable of an artistic excellence as great as that of any music, perhaps greater; not because the pleasure it gives (although very pure) is comparable, either in intensity or in the number of people who feel it, to that of music, but because it gives in absolute perfection that combination, characteristic of great art, of godlike freedom, with the sense of inevitable destiny; because, in fact, it constructs an ideal world where everything is perfect and yet true." (Bertrand Russell, "Autobiography", 1967)

"The fact is that there are few more ‘popular’ subjects than mathematics. Most people have some appreciation of mathematics, just as most people can enjoy a pleasant tune; and there are probably more people really interested in mathematics than in music. Appearances may suggest the contrary, but there are easy explanations. Music can be used to stimulate mass emotion, while mathematics cannot; and musical incapacity is recognized (no doubt rightly) as mildly discreditable [...]" (Godfrey H Hardy, "A Mathematician’s Apology", 1967)

"The syntax and the grammar of the language of music are not capricious; they are dictated by the texture and organization of the deep levels of the mind, so with mathematics." (H E Huntley, "The Divine Proportion", 1970)

Note: The quotes have been reordered chronologically.

On Music and Mathematics (1900-1949)

"Music is an order of mystic, sensuous mathematics. A sounding mirror, an aural mode of motion, it addresses itself on the formal side to the intellect, in its content of expression it appeals to the emotions." (James Huneker, "Chopin: The Man and His Music", 1900)

"Mathematics, rightly viewed, possesses not only truth, but supreme beauty - a beauty cold and austere, like that of sculpture, without appeal to any part of our weaker nature, without the gorgeous trappings of painting or music, yet sublimely pure, and capable of a stern perfection such as only the greatest art can show." (Bertrand Russell, 'The Study of Mathematics", 1902)

"We study art because we receive pleasure from the great works of the masters, and probably we appreciate them the more because we have dabbled a little in pigments or in clay. We do not expect to be composers, or poets, or sculptors, but we wish to appreciate music and letters and the fine arts, and to derive pleasure from them and be uplifted by them. […] So it is with geometry. We study it because we derive pleasure from contact with a great and ancient body of learning that has occupied the attention of master minds during the thousands of years in which it has been perfected, and we are uplifted by it." (David E Smith, "The Teaching of Geometry", 1911)

"Every intelligent musician should be familiar with the physical laws which underline his art." (Clarence G Hamilton, "Sound and Its Relation to Music", 1912)

"Translating mathematics into ordinary language is like translating music. It cannot be done. One could describe in detail a sheet of music and tell the shape of each note and where it is placed on the staff, but that would not convey any idea of how it would sound when played. So, too, I suppose that even the most complicated equation could be described in common words, but it would be so verbose and involved that nobody could get the sense of it." (Edwin E Slosson, "Chats on Science", 1924)

"[…] mathematics, accessible in its full depth only to the very few, holds a quite peculiar position amongst the creation of the mind. It is a science of the most rigorous kind, like logic but more comprehensive and very much fuller; it is a true art, along with sculpture and music, as needing the guidance of inspiration and as developing under great conventions of form […]" (Oswald Spengler, "The Decline of the West" Vol. 1, 1926)

"The best proofs in mathematics are short and crisp like epigrams, and the longest have swings and rhythms that are like music." (Scott Buchanan, "Poetry and Mathematics", 1929)

"What had already been done for music by the end of the eighteenth century has at last been begun for the pictorial arts. Mathematics and physics furnished the means in the form of rules to be followed and to be broken. In the beginning it is wholesome to be concerned with the functions and to disregard the finished form. Studies in algebra, in geometry, in mechanics characterize teaching directed towards the essential and the functional, in contrast to apparent. One learns to look behind the façade, to grasp the root of things. One learns to recognize the undercurrents, the antecedents of the visible. One learns to dig down, to uncover, to find the cause, to analyze." (Paul Klee, "Bauhaus prospectus", 1929)

"If all the arts aspire to the condition of music, all the sciences aspire to the condition of mathematics." (George Santayana, "Some Turns of Thought in Modern Philosophy: Five Essays", 1933)

"[…] mathematics is like music, freely exploring the possibilities of form. And yet, notoriously, mathematics holds true of things; hugs and permeates them far more closely than does confused and inconstant human perception; so that the dream of many exasperated critics of human error has been to assimilate all science to mathematics, so as to make knowledge safe by making it, as Locke wished, direct perception of the relations between ideas […]" (George Santayana, "The Realm of Truth: Book Third of Realms of Being", 1937)

"Mathematizing may well be a creative activity of man, like language or music, of primary originality, whose historical decisions defy complete objective rationalizations." (Hermann Weyl, "Obituary for David Hilbert", Royal Society Biographies Vol. 4, 1944)

Note: The quotes have been reordered chronologically.

On Simplicity VIII (Simplicity & Beauty)

"Number theory is revealed in its entire simplicity and natural beauty when the field of arithmetic is extended to the imaginary numbers" (Carl F Gauss, "Disquisitiones arithmeticae" ["Arithmetical Researches"], 1801)

"The researcher worker, in his efforts to express the fundamental laws of Nature in mathematical form, should strive mainly for mathematical beauty. He should still take simplicity into consideration in a subordinate way to beauty. […] It often happens that the requirements of simplicity and beauty are the same, but where they clash the latter must take precedence." (Paul A M Dirac, "The Relation Between Mathematics and Physics", Proceedings of the Royal Society , Volume LIX, 1939)

"The line that describes the beautiful is elliptical. It has simplicity and constant change. It cannot be described by a compass, and it changes direction at every one of its points." (Rudolf Arnheim, "Entropy and Art: An Essay on Disorder and Order", 1974)

"The equations of physics have in them incredible simplicity, elegance and beauty. That in itself is sufficient to prove to me that there must be a God who is responsible for these laws and responsible for the universe" (Paul C W Davies, 1984)

"It is not merely the truth of science that makes it beautiful, but its simplicity." (Walker Percy, "Signposts in a Strange Land", 1991)

"Elegance and simplicity should remain important criteria in judging mathematics, but the applicability and consequences of a result are also important, and sometimes these criteria conflict. I believe that some fundamental theorems do not admit simple elegant treatments, and the proofs of such theorems may of necessity be long and complicated. Our standards of rigor and beauty must be sufficiently broad and realistic to allow us to accept and appreciate such results and their proofs. As mathematicians we will inevitably use such theorems when it is necessary in the practice our trade; our philosophy and aesthetics should reflect this reality." (Michael Aschbacher, "Highly complex proofs and implications", 2005)

"We all know what we like in music, painting or poetry, but it is much harder to explain why we like it. The same is true in mathematics, which is, in part, an art form. We can identify a long list of desirable qualities: beauty, elegance, importance, originality, usefulness, depth, breadth, brevity, simplicity, clarity. However, a single work can hardly embody them all; in fact, some are mutually incompatible. Just as different qualities are appropriate in sonatas, quartets or symphonies, so mathematical compositions of varying types require different treatment." (Michael F Atiyah, "Mathematics: Art and Science" Bulletin of the AMS 43, 2006)

"In mathematics, beauty is a very important ingredient. Beauty exists in mathematics as in architecture and other things. It is a difficult thing to define but it is something you recognise when you see it. It certainly has to have elegance, simplicity, structure and form. All sorts of things make up real beauty. There are many different kinds of beauty and the same is true of mathematical theorems. Beauty is an important criterion in mathematics because basically there is a lot of choice in what you can do in mathematics and science. It determines what you regard as important and what is not." (Michael Atiyah, 2009)

"The beauty in the laws of physics is the fantastic simplicity that they have." (John A Wheeler)

"The man of science will acts as if this world were an absolute whole controlled by laws independent of his own thoughts or act; but whenever he discovers a law of striking simplicity or one of sweeping universality or one which points to a perfect harmony in the cosmos, he will be wise to wonder what role his mind has played in the discovery, and whether the beautiful image he sees in the pool of eternity reveals the nature of this eternity, or is but a reflection of his own mind." (Tobias Dantzig)

Jean Dieudonné - Collected Quotes

"We have not begun to understand the relationship between combinatorics and conceptual mathematics." (Jean Dieudonné, "A Panorama of Pure Mathematics: As Seen by N. Bourbaki", 1982)

"A mathematician, then, will be defined in what follows as someone who has published the proof of at least one non-trivial theorem." (Jean Dieudonné, "Mathematics and Mathematicians", 1992)

"Engineers, always looking for optimal values for the measures of magnitudes which interest them, think of mathematicians as custodians of a fund of formulae, to be supplied to them on demand." (Jean Dieudonné, "Mathematics - The Music of Reason", 1992)

"In the flowering of a mathematical talent social environment has an important part to play." (Jean Dieudonné, "Mathematics - The Music of Reason", 1992)

"The life of a mathematician is dominated by an insatiable curiosity, a desire bordering on passion to solve the problems he is studying." (Jean Dieudonné, "Mathematics - The Music of Reason", 1992)

"[...] there is no criterion for appreciation which does not vary from one epoch to another and from one mathematician to another. [...] These divergences in taste recall the quarrels aroused by works of art, and it is a fact that mathematicians often discuss among themselves whether a theorem is more or less ‚beautiful‘. This never fails to surprise practitioners of other sciences: for them the sole criterion is the 'truth' of a theory or formula." (Jean Dieudonné, "Mathematics - The Music of Reason", 1992)

"On foundations we believe in the reality of mathematics, but of course, when philosophers attack us with their paradoxes, we rush to hide behind formalism and say 'mathematics is just a combination of meaningless symbols’ […]. Finally we are left in peace to go back to our mathematics and do it as we have always done, with the feeling each mathematician has that he is working with something real. The sensation is probably an illusion, but it is very convenient." (Jean Dieudonné)

Discovery in Mathematics (2000-2019)

"Mathematics is about truth: discovering the truth, knowing the truth, and communicating the truth to others. It would be a great mistake to discuss mathematics without talking about its relation to the truth, for truth is the essence of mathematics. In its search for the purity of truth, mathematics has developed its own language and methodologies - its own way of paring down reality to an inner essence and capturing that essence in subtle patterns of thought. Mathematics is a way of using the mind with the goal of knowing the truth, that is, of obtaining certainty." (William Byers, "How Mathematicians Think", 2007)

"There are two aspects of proof to be borne in mind. One is that it is our lingua franca. It is the mathematical mode of discourse. It is our tried-and true methodology for recording discoveries in a bullet-proof fashion that will stand the test of time. The second, and for the working mathematician the most important, aspect of proof is that the proof of a new theorem explains why the result is true. In the end what we seek is new understanding, and ’proof’ provides us with that golden nugget." (Steven G Krantz, "The Proof is in the Pudding", 2007)

"[…] a proof is a device of communication. The creator or discoverer of this new mathematical result wants others to believe it and accept it." (Steven G Krantz, "The Proof is in the Pudding", 2007)

"I enjoy mathematics so much because it has a strange kind of unearthly beauty. There is a strong feeling of pleasure, hard to describe, in thinking through an elegant proof, and even greater pleasure in discovering a proof not previously known." (Martin Gardner, 2008)

"Mathematics is the music of reason. To do mathematics is to engage in an act of discovery and conjecture, intuition and inspiration; to be in a state of confusion - not because it makes no sense to you, but because you gave it sense and you still don't understand what your creation is up to; to have a break-through idea; to be frustrated as an artist; to be awed and overwhelmed by an almost painful beauty; to be alive, damn it." (Paul Lockhart, "A Mathematician's Lament", 2009)

"What is the basis of this interest in beauty? Is it the same in both mathematics and science? Is it rational, in either case, to expect or demand that the products of the discipline satisfy such a criterion? Is there an underlying assumption that the proper business of mathematics and science is to discover what can be discovered about reality and that truth - mathematical and physical - when seen as clearly as possible, must be beautiful? If the demand for beauty stems from some such assumption, is the assumption itself an article of blind faith? If such an assumption is not its basis, what is?" (Raymond S Nickerson, "Mathematical Reasoning: Patterns, Problems, Conjectures, and Proofs", 2010)

"[…] intuition is a very important factor in the psychology of mathematics, in the sense that mathematicians spend a great deal of time exploring guesses and checking out hunches in their efforts to discover and prove new theorems." (Raymond S Nickerson, "Mathematical Reasoning: Patterns, Problems, Conjectures, and Proofs", 2010)

Discovery in Mathematics (1950-1974)

"We are driven to conclude that science, like mathematics, is a system of axioms, assumptions, and deductions; it may start from being, but later leaves it to itself, and ends in the formation of a hypothetical reality that has nothing to do with existence; or it is the discovery of an ideal being which is, of course, present in what we call actuality, and renders it an existence for us only by being present in it." (Poolla T Raju, "Idealistic Thought of India", 1953)

"The result of the mathematician's creative work is demonstrative reasoning, a proof; but the proof is discovered by plausible reasoning, by guessing. If the learning of mathematics reflects to any degree the invention of mathematics, it must have a place for guessing, for plausible inference." (George Pólya, "Induction and Analogy in Mathematics", 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)

"The progress of science is the discovery at each step of a new order which gives unity to what had seemed unlike." (Jacob Bronowski, "Science and Human Values", 1956)

"Is it possible to breach this wall, to present mathematics in such a way that the spectator may enjoy it? Cannot the enjoyment of mathematics be extended beyond the small circle of those who are ‘mathematically gifted’? Indeed, only a few are mathematically gifted in the sense that they are endowed with the talent to discover new mathematical facts. But by the same token, only very few are musically gifted in that they are able to compose music. Nevertheless, there are many who can understand and perhaps reproduce music, or who at least enjoy it. We believe that the number of people who can understand simple mathematical ideas is not relatively smaller than the number of those who are commonly called musical, and that their interest will be stimulated if only we can eliminate the aversion toward mathematics that so many have acquired from childhood experiences." (Hans Rademacher & Otto Toeplitz, "The Enjoyment of Mathematics", 1957)

"The heart of all major discoveries in the physical sciences is the discovery of novel methods of representation and so of fresh techniques by which inferences can be drawn - and drawn in ways which fit the phenomena under investigation." (Stephen Toulmin, "The Philosophy of Science", 1957)

“Many people think of mathematics itself as a static art - a body of eternal truth that was discovered by a few ancient, shadowy figures, and upon which engineers and scientists can draw as needed.” (Paul Halmos, “Innovation in Mathematics”, Scientific American Vol. 199 (3) , 1958) 

"There is beauty in discovery. There is mathematics in music, a kinship of science and poetry in the description of nature, and exquisite form in a molecule. Attempts to place different disciplines in different camps are revealed as artificial in the face of the unity of knowledge. All illiterate men are sustained by the philosopher, the historian, the political analyst, the economist, the scientist, the poet, the artisan, and the musician." (Glenn T Seaborg, 1958)

"The enormous usefulness of mathematics in natural sciences is something bordering on the mysterious, and there is no rational explanation for it. It is not at all natural that ‘laws of nature’ exist, much less that man is able to discover them. The miracle of the appropriateness of the language of mathematics for the formulation of the laws of physics is a wonderful gift which we neither understand nor deserve." (Eugene P Wigner, "The Unreasonable Effectiveness of Mathematics in the Natural Sciences," 1960)

"No mathematical idea has ever been published in the way it was discovered. Techniques have been developed and are used, if a problem has been solved, to turn the solution procedure upside down, or if it is a larger complex of statements and theories, to turn definitions into propositions, and propositions into definitions, the hot invention into icy beauty. This then if it has affected teaching matter, is the didactical inversion, which as it happens may be anti-didactical." (Hans Freudenthal, "The Concept and the Role of the Model in Mathematics and Natural and Social Sciences", 1961)

"It is impossible to overstate the importance of problems in mathematics. It is by means of problems that mathematics develops and actually lifts itself by its own bootstraps. […] Every new discovery in mathematics, results from an attempt to solve some problem."   (Howard W Eves, "A Survey of Geometry", 1963)

"Mathematics is a creation of the mind. To begin with, there is a collection of things, which exist only in the mind, assumed to be distinguishable from one another; and there is a collection of statements about these things, which are taken for granted. Starting with the assumed statements concerning these invented or imagined things, the mathematician discovers other statements, called theorems, and proves them as necessary consequences. This, in brief, is the pattern of mathematics. The mathematician is an artist whose medium is the mind and whose creations are ideas." (Hubert S Wall, "Creative Mathematics", 1963)

"The introduction and gradual acceptance of concepts that have no immediate counterparts in the real world certainly forced the recognition that mathematics is a human, somewhat arbitrary creation, rather than an idealization of the realities in nature, derived solely from nature. But accompanying this recognition and indeed propelling its acceptance was a more profound discovery - mathematics is not a body of truths about nature." (Morris Kline, "Mathematical Thought from Ancient to Modern Times" Vol. III, 1972)

"Discovery is a double relation of analysis and synthesis together. As an analysis, it probes for what is there; but then, as a synthesis, it puts the parts together in a form by which the creative mind transcends the bare limits, the bare skeleton, that nature provides."(Jacob Bronowski, "The Ascent of Man", 1973)

On Beauty: Beauty and Mathematics (2000-2019)

"Where we find certainty and truth in mathematics we also find beauty. Great mathematics is characterized by its aesthetics. Mathematicians delight in the elegance, economy of means, and logical inevitability of proof. It is as if the great mathematical truths can be no other way. This light of logic is also reflected back to us in the underlying structures of the physical world through the mathematics of theoretical physics." (F David Peat, "From Certainty to Uncertainty", 2002)

“Pure mathematics was characterized by an obsession with proof, rigor, beauty, and elegance, and sought its foundations in the disembodied worlds of logic or intuition. Far from being coextensive with physics, pure mathematics could be ‘applied’ only after it had been made foundationally secure by the purists.” (Andrew Warwick, “Masters of Theory: Cambridge and the rise of mathematical physics”, 2003)

“Elegance and simplicity should remain important criteria in judging mathematics, but the applicability and consequences of a result are also important, and sometimes these criteria conflict. I believe that some fundamental theorems do not admit simple elegant treatments, and the proofs of such theorems may of necessity be long and complicated. Our standards of rigor and beauty must be sufficiently broad and realistic to allow us to accept and appreciate such results and their proofs. As mathematicians we will inevitably use such theorems when it is necessary in the practice our trade; our philosophy and aesthetics should reflect this reality.” (Michael Aschbacher, “Highly complex proofs and implications”, 2005)

"Still, in the end, we find ourselves drawn to the beauty of the patterns themselves, and the amazing fact that we humans are smart enough to prove even a feeble fraction of all possible theorems about them. Often, greater than the contemplation of this beauty for the active mathematician is the excitement of the chase. Trying to discover first what patterns actually do or do not occur, then finding the correct statement of a conjecture, and finally proving it - these things are exhilarating when accomplished successfully. Like all risk-takers, mathematicians labor months or years for these moments of success." (Avner Ash & Robert Gross, "Fearless Symmetry: Exposing the hidden patterns of numbers", 2006)

"Arguments for the Riemann hypothesis often include its widespread ramifications and appeals to mathematical beauty; however, we also have a large corpus of hard facts. With the advent of powerful computational tools over the last century, mathematicians have increasingly turned to computational evidence to support conjectures, and the Riemann hypothesis is no exception." (Peter Borwein et al, "The Riemann Hypothesis: A Resource for the Afficionado and Virtuoso Alike", 2007)

"I think that the beauty of mathematics lies in uncovering the hidden simplicity and complexity that coexist in the rigid logical framework that the subject imposes." (David Ruelle, "The Mathematician's Brain", 2007)

"[...] it is while doing mathematical research that one truly comes to see the beauty of mathematics. It faces you in those moments when the underlying simplicity of a question appears and its meaningless complications can be forgotten. In those moments a piece of a colossal logical structure is illumi￾nated, and some of the meaning hidden in the nature of things is finally revealed." (David Ruelle, "The Mathematician's Brain", 2007)

“The immediate evidence from the natural world may seem to be chaotic and without any inner regularity, but mathematics reveals that under the surface the world of nature has an unexpected simplicity - an extraordinary beauty and order.” (William Byers, “How Mathematicians Think”, 2007)

“I enjoy mathematics so much because it has a strange kind of unearthly beauty. There is a strong feeling of pleasure, hard to describe, in thinking through an elegant proof, and even greater pleasure in discovering a proof not previously known.” (Martin Gardner, 2008)

“In mathematics, beauty is a very important ingredient. Beauty exists in mathematics as in architecture and other things. It is a difficult thing to define but it is something you recognise when you see it. It certainly has to have elegance, simplicity, structure and form. All sorts of things make up real beauty. There are many different kinds of beauty and the same is true of mathematical theorems. Beauty is an important criterion in mathematics because basically there is a lot of choice in what you can do in mathematics and science. It determines what you regard as important and what is not.” (Michael Atiyah, 2009)

“Mathematics is the music of reason. To do mathematics is to engage in an act of discovery and conjecture, intuition and inspiration; to be in a state of confusion - not because it makes no sense to you, but because you gave it sense and you still don't understand what your creation is up to; to have a break-through idea; to be frustrated as an artist; to be awed and overwhelmed by an almost painful beauty; to be alive, damn it.” (Paul Lockhart, A Mathematician's Lament, 2009)

“Mathematicians seek a certain kind of beauty. Perhaps mathematical beauty is a constant - as far as the contents of mathematics are concerned - and yet the forms this beauty takes are certainly cultural. And while the history of mathematics surely is many stranded, one of its most important strands is formed by such cultural forms of mathematical beauty.” (Reviel Netz, “Ludic Proof: Greek Mathematics and the Alexandrian Aesthetic”, 2009)

“Mathematicians are sometimes described as living in an ideal world of beauty and harmony. Instead, our world is torn apart by inconsistencies, plagued by non sequiturs and, worst of all, made desolate and empty by missing links between words, and between symbols and their referents; we spend our lives patching and repairing it. Only when the last crack disappears are we rewarded by brief moments of harmony and joy.” (Alexandre V Borovik, “Mathematics under the Microscope: Notes on Cognitive Aspects of Mathematical Practice”, 2009)

“What is the basis of this interest in beauty? Is it the same in both mathematics and science? Is it rational, in either case, to expect or demand that the products of the discipline satisfy such a criterion? Is there an underlying assumption that the proper business of mathematics and science is to discover what can be discovered about reality and that truth - mathematical and physical - when seen as clearly as possible, must be beautiful? If the demand for beauty stems from some such assumption, is the assumption itself an article of blind faith? If such an assumption is not its basis, what is?” (Raymond S Nickerson, “Mathematical Reasoning:  Patterns, Problems, Conjectures, and Proofs”, 2010)

"You do not study mathematics because it helps you build a bridge. You study mathematics because it is the poetry of the universe. Its beauty transcends mere things." (Jonathan D Farley, 2011)

"Mathematics leads to simple and beautiful conclusions by ignoring and idealizing various factors in order to abstract. However, reality is both more complex and discrete, and in many cases mathematical conclusions are only approximately valid. There is also the risk that the assumptions used for valuation models differ from the market (model risk), as well as issues with the reliability of the parameters used in the model (parameter risk)." (Kenichi Watanabe, "The Role of Mathematics in Finance: Applied Mathematics and Risk", 2013)