Terry Gannon - Collected Quotes

"In modern mathematics there is a strong tendency towards formulations of concepts that minimise the number and significance of arbitrary choices. This crispness tends to emphasise the naturality of the construction or definition, at the expense sometimes of accessibility. Our mathematics is more conceptual today – more beautiful perhaps – but the cost of less explicitness is the compartmentalism that curses our discipline." (Terry Gannon, "Moonshine Beyond the Monster: The Bridge Connecting Algebra, Modular Forms and Physics", 2006)

"Like moonlight itself, Monstrous Moonshine is an indirect phenomenon. Just as in the theory of moonlight one must introduce the sun, so in the theory of Moonshine one must go well beyond the Monster. Much as a book discussing moonlight may include paragraphs on sunsets or comet tails, so do we discuss fusion rings, Galois actions and knot invariants." (Terry Gannon, "Moonshine Beyond the Monster: The Bridge Connecting Algebra, Modular Forms and Physics", 2006)

"Moonshine concerns the occurrence of modular forms in algebra and physics, and care is taken to avoid analytic complications as much as possible. But spaces here are unavoidably infinite-dimensional, and through this arise subtle but significant points of contact with analysis." (Terry Gannon, "Moonshine Beyond the Monster: The Bridge Connecting Algebra, Modular Forms and Physics", 2006)

"Moonshine forms a way of explaining the mysterious connection between the monster finite group and modular functions from classical number theory. The theory has evolved to describe the relationship between finite groups, modular forms and vertex operator algebras." (Terry Gannon, "Moonshine Beyond the Monster: The Bridge Connecting Algebra, Modular Forms and Physics", 2006)

"Moonshine is interested in the correlation functions of a class of extremely symmetrical and well-behaved quantum field theories called rational conformal field theories - these theories are so special that their correlation functions can be computed exactly and perturbation is not required." (Terry Gannon, "Moonshine Beyond the Monster: The Bridge Connecting Algebra, Modular Forms and Physics", 2006)

"Moonshine is profoundly connected with physics (namely conformal field theory and string theory). String theory proposes that the elementary particles (electrons, photons, quarks, etc.) are vibrational modes on a string of length about 10^−33 cm. These strings can interact only by splitting apart or joining together – as they evolve through time, these (classical) strings will trace out a surface called the world-sheet. Quantum field theory tells us that the quantum quantities of interest (amplitudes) can be perturbatively computed as weighted averages taken over spaces of these world-sheets. Conformally equivalent world-sheets should be identified, so we are led to interpret amplitudes as certain integrals over moduli spaces of surfaces. This approach to string theory leads to a conformally invariant quantum field theory on two-dimensional space-time, called conformal field theory (CFT). The various modular forms and functions arising in Moonshine appear as integrands in some of these genus-1 (‘1-loop’) amplitudes: hence their modularity is manifest." (Terry Gannon, "Moonshine Beyond the Monster: The Bridge Connecting Algebra, Modular Forms and Physics", 2006)

"Physics reduces Moonshine to a duality between two different pictures of quantum field theory: the Hamiltonian one, which concretely gives us from representation theory the graded vector spaces, and another, due to Feynman, which manifestly gives us modularity. In particular, physics tells us that this modularity is a topological effect, and the group SL2(Z) directly arises in its familiar role as the modular group of the torus." (Terry Gannon, "Moonshine Beyond the Monster: The Bridge Connecting Algebra, Modular Forms and Physics", 2006)

"The appeal of Monstrous Moonshine lies in its mysteriousness: it unexpectedly associates various special modular functions with the Monster, even though modular functions and elements of Mare conceptually incommensurable. Now, ‘understanding’ something means to embed it naturally into a broader context. Why is the sky blue? Because of the way light scatters in gases. Why does light scatter in gases the way it does? Because of Maxwell’s equations. In order to understand Monstrous Moonshine, to resolve the mystery, we should search for similar phenomena, and fit them all into the same story." (Terry Gannon, "Moonshine Beyond the Monster: The Bridge Connecting Algebra, Modular Forms and Physics", 2006)

Mathematical Trivia I: Moonshine

"The term 'moonshine' roughly means weird relations between sporadic groups and modular functions (and anything else) similar to this. It was clear to many people that this was just a meaningless coincidence." (Richard E Borcherds, "What is Moonshine?", Proceedings of the International Congress of Mathematicians, 1998)

"Moonshine concerns the occurrence of modular forms in algebra and physics, and care is taken to avoid analytic complications as much as possible. But spaces here are unavoidably infinite-dimensional, and through this arise subtle but significant points of contact with analysis." (Terry Gannon, "Moonshine Beyond the Monster: The Bridge Connecting Algebra, Modular Forms and Physics", 2006)

"Moonshine forms a way of explaining the mysterious connection between the monster finite group and modular functions from classical number theory. The theory has evolved to describe the relationship between finite groups, modular forms and vertex operator algebras." (Terry Gannon, "Moonshine Beyond the Monster: The Bridge Connecting Algebra, Modular Forms and Physics", 2006)

"Moonshine is interested in the correlation functions of a class of extremely symmetrical and well-behaved quantum field theories called rational conformal field theories - these theories are so special that their correlation functions can be computed exactly and perturbation is not required." (Terry Gannon, "Moonshine Beyond the Monster: The Bridge Connecting Algebra, Modular Forms and Physics", 2006)

"Moonshine is profoundly connected with physics (namely conformal field theory and string theory). String theory proposes that the elementary particles (electrons, photons, quarks, etc.) are vibrational modes on a string of length about 10^−33 cm. These strings can interact only by splitting apart or joining together – as they evolve through time, these (classical) strings will trace out a surface called the world-sheet. Quantum field theory tells us that the quantum quantities of interest (amplitudes) can be perturbatively computed as weighted averages taken over spaces of these world-sheets. Conformally equivalent world-sheets should be identified, so we are led to interpret amplitudes as certain integrals over moduli spaces of surfaces. This approach to string theory leads to a conformally invariant quantum field theory on two-dimensional space-time, called conformal field theory (CFT). The various modular forms and functions arising in Moonshine appear as integrands in some of these genus-1 (‘1-loop’) amplitudes: hence their modularity is manifest." (Terry Gannon, "Moonshine Beyond the Monster: The Bridge Connecting Algebra, Modular Forms and Physics", 2006)

"Physics reduces Moonshine to a duality between two different pictures of quantum field theory: the Hamiltonian one, which concretely gives us from representation theory the graded vector spaces, and another, due to Feynman, which manifestly gives us modularity. In particular, physics tells us that this modularity is a topological effect, and the group SL2(Z) directly arises in its familiar role as the modular group of the torus." (Terry Gannon, "Moonshine Beyond the Monster: The Bridge Connecting Algebra, Modular Forms and Physics", 2006)

"The appeal of Monstrous Moonshine lies in its mysteriousness: it unexpectedly associates various special modular functions with the Monster, even though modular functions and elements of Mare conceptually incommensurable. Now, ‘understanding’ something means to embed it naturally into a broader context. Why is the sky blue? Because of the way light scatters in gases. Why does light scatter in gases the way it does? Because of Maxwell’s equations. In order to understand Monstrous Moonshine, to resolve the mystery, we should search for similar phenomena, and fit them all into the same story." (Terry Gannon, "Moonshine Beyond the Monster: The Bridge Connecting Algebra, Modular Forms and Physics", 2006)

"The Moonshine mystery itself is still unresolved, despite Borcherd's proof! [...] there are facts about the Monster and Moonshine that we don't understand. [...] The method leading to its discovery, brilliant though it was, gave no clue to the Monster's remarkable properties." (Mark Ronan, "Symmetry and the Monster: One of the greatest quests of mathematics", 2006)

"The term Moonshine [...] has a variety of meanings. It can refer to foolish or naive ideas, but also to the illicit distillation of spirits [...] It gave an impression of dabbling in mysterious matters that might be better left alone, but also had the useful connotation of something shining in reflected light. The true source of light is probably yet to be found, but there were further strange connections to come later [...] The Monster's connections with number theory - the Moonshine connections - had suggested it was a more beautiful and important group of symmetries than first realized. [...] The Moonshine connections between the Monster and number theory have now been placed within a larger theory, but we have yet to grasp the significance of these deep mathematical links with fundamental physics. We have found the Monster, but it remains an enigma. Understanding its full nature is likely to shed light on the very fabric of the universe." (Mark Ronan, "Symmetry and the Monster: One of the greatest quests of mathematics", 2006)

On Mind: Mirrors II

"Conscious apprehension seems to exist […] as happens in a mirror-image when the smooth and bright surface is peaceful." (Plotinus, "Enneads", cca. 270 AD)

"[…] the mind orders nothing by its own motions, but lies merely receptive under the impressions of bodies, reflecting empty images in a mirror in place of reality." (Anicius Manlius Severinus Boethius, "The Consolation of Philosophy", cca. 524)

"In the same way as regards the soul, when that kind of thing in us which mirrors the images of thought and intellect is undisturbed, we see them and know them in a way parallel to sense-perception, along with the prior knowledge that it is intellect and thought that are active. But when this is broken because the harmony of the body is upset, thought and intellect operate without an image, and then intellectual activity takes place without a mind-picture." (Plotinus, "Enneads", cca. 270 AD)

"The noetic act is without parts and has not, so to speak, come out into the open, but remains unobserved within, but the verbal expression unfolds its content and brings it out of the noetic act into the image making power, and so shows the noetic act as if in a mirror, and this is how there is conscious apprehension and persistence and memory of it." (Plotinus, "Enneads", cca. 270 AD)

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

"Let the poet confine his use of individual models to what is necessary to make his subject alive and convincing. As for all the rest, let him rely on the living world as mirrored in his bosom." (Johann Wolfgang von Goethe, 1789)

"The symbol. It is the thing without being the thing, and yet the thing: an image concentrated in the mirror of the mind and yet identical with the object. How inferior is allegory by comparison. Though it may have wit and subtle conceit, it is for the most part rhetorical and conventional. It always improves in proportion to its approach to what we call symbol." (Johann Wolfgang von Goethe, "Addenda on the Paintings of Philostratus", 1820) 

"A human being, what is a human being? Everything and nothing. Through the power of thought it can mirror everything it experiences. Through memory and knowledge it becomes a microcosm, carrying the world within itself. A mirror of things, a mirror of facts. Each human being becomes a little universe within the universe!" (Guy de Maupassant, [in "The Journal of a Madman"] 1851)

"Observation is like a piece of glass, which, as a mirror, must be very smooth, and must be very carefully polished, in order that it may reflect the image pure and undistorted." (Justus von Liebig, "The Study of the Natural Sciences", 1853) 

"In her manifold opportunities Nature has thus helped man to polish the mirror of [man’s] mind, and the process continues. Nature still supplies us with abundance of brain-stretching theoretical puzzles and we eagerly tackle them; there are more worlds to conquer and we do not let the sword sleep in our hand; but how does it stand with feeling? Nature is beautiful, gladdening, awesome, mysterious, wonderful, as ever, but do we feel it as our forefathers did?" (Sir John A Thomson, "The System of Animate Nature", 1920)

"What a lost person needs is a map of the territory, with his own position marked on it so he can see where he is in relation to everything else. Literature is not only a mirror; it is also a map, a geography of the mind. Our literature is one such map, if we can learn to read it as our literature, as the product of who and where we have been. We need such a map desperately, we need to know about here, because here is where we live. For the members of a country or a culture, shared knowledge of their place, their here, is not a luxury but a necessity. Without that knowledge we will not survive." (Margaret Atwood, "Survival: A Thematic Guide to Canadian Literature", 1972)

"Even a tarnished mirror will shine like a jewel if it is polished. A mind which presently is closed by illusions originating from the innate darkness of life is like a tarnished mirror, but once it is polished it will become clear, reflecting the enlightenment of immutable truth." (Nichiren Daishonin,"On Attaining Buddhahood", 1999)

"Thinking involves reasoning about a situation, and to do that we must have some kind of dynamic "model" of the situation in our heads. Any changes we make to this mental model of the world should ideally mirror changes in the real world." (S Ian Robertson, "Problem Solving", 2001)

"'Mental models' are deeply ingrained assumptions, generalizations, or even pictures or images that influence how we understand the world and how we take action. Very often, we are not consciously aware of our mental models or the effects they have on our behavior. […] The discipline of working with mental models starts with turning the mirror inward; learning to unearth our internal pictures of the world, to bring them to the surface and hold them rigorously to scrutiny. It also includes the ability to carry on ‘learningful’ conversations that balance inquiry and advocacy, where people expose their own thinking effectively and make that thinking open to the influence of others.” (Jossey-Bass Publishers, “The Jossey-Bass Reader on Educational Leadership”, 2nd Edi. 2007)

"If intelligence is a capacity that is gradually acquired as a result of development and learning, then a machine that can learn from experience would have, at least in theory, the capacity to carry out intelligent behavior. [...] Humans have created machines that imitate us - that provide mirrors to see ourselves and measure our strength, our intellect, and even our creativity." (Diego Rasskin-Gutman, "Chess Metaphors: Artificial Intelligence and the Human Mind", 2009)

"Language accelerates learning and creation by permitting communication and coordination. A new idea can be spread quickly if someone can explain it and communicate it to others before they have to discover it themselves. But the chief advantage of language is not communication but autogeneration. Language is a trick that allows the mind to question itself; a magic mirror that reveals to the mind what the mind thinks; a handle that turns a mind into a tool." (Kevin Kelly, "What Technology Wants", 2010)

"It is a right, yes a duty, to search in cautious manner for the numbers, sizes, and weights, the norms for everything [God] has created. For He himself has let man take part in the knowledge of these things […] For these secrets are not of the kind whose research should be forbidden; rather they are set before our eyes like a mirror so that by examining them we observe to some extent the goodness and wisdom of the Creator." (Johannes Kepler)

Mathematical Trivia II: Mirrors I

"It is impossible to disassociate language from science or science from language, because every natural science always involves three things: the sequence of phenomena on which the science is based; the abstract concepts which call these phenomena to mind; and the words in which the concepts are expressed. To call forth a concept a word is needed; to portray a phenomenon a concept is needed. All three mirror one and the same reality." (Antoine-Laurent Lavoisier, "Traite Elementaire de Chimie", 1789)

"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)

"This history constitutes a mirror of past and present conditions in mathematics which can be made to bear on the notational problems now confronting mathematics. The successes and failures of the past will contribute to a more speedy solution of notational problems of the present time." (Florian Cajori, "A History of Mathematical Notations", 1928)

"Given any domain of thought in which the fundamental objective is a knowledge that transcends mere induction or mere empiricism, it seems quite inevitable that its processes should be made to conform closely to the pattern of a system free of ambiguous terms, symbols, operations, deductions; a system whose implications and assumptions are unique and consistent; a system whose logic confounds not the necessary with the sufficient where these are distinct; a system whose materials are abstract elements interpretable as reality or unreality in any forms whatsoever provided only that these forms mirror a thought that is pure. To such a system is universally given the name Mathematics." (Samuel T Sanders, "Mathematics", National Mathematics Magazine, 1937)

"The only possible alternative is simply to keep to immediate experience that consciousness is a singular of which the plural is unknown; that there is only one thing and that what seems to be a plurality is merely a series of different aspects of this one thing, produced by a deception (the Indian MAJA); the same illusion is produced in a gallery of mirrors, and in the same way Gaurisankar and Mt Everest turned out to be the same peak seen from different valleys." (Erwin Schrödinger, "What Is Life?", 1944)

"Mathematical examination problems are usually considered unfair if insoluble or improperly described: whereas the mathematical problems of real life are almost invariably insoluble and badly stated, at least in the first balance. In real life, the mathematician's main task is to formulate problems by building an abstract mathematical model consisting of equations, which will be simple enough to solve without being so crude that they fail to mirror reality. Solving equations is a minor technical matter compared with this fascinating and sophisticated craft of model-building, which calls for both clear, keen common-sense and the highest qualities of artistic and creative imagination." (John Hammersley & Mina Rees, "Mathematics in the Market Place", The American Mathematical Monthly 65, 1958)

"Mathematics is a self-contained microcosm, but it also has the potentiality of mirroring and modeling all the processes of thought and perhaps all of science. It has always had, and continues to an ever increasing degree to have, great usefulness. One could even go so far as to say that mathematics was necessary for man's conquest of nature and for the development of the human race through the shaping of its modes of thinking." (Mark Kac & Stanislaw M Ulam, "Mathematics and Logic", 1968)

 "Every branch of geometry can be defined as the study of properties that are unaltered when a specified figure is given specified symmetry transformations. Euclidian plane geometry, for instance, concerns the study of properties that are 'invariant' when a figure is moved about on the plane, rotated, mirror reflected, or uniformly expanded and contracted. Affine geometry studies properties that are invariant when a figure is 'stretched' in a certain way. Projective geometry studies properties invariant under projection. Topology deals with properties that remain unchanged even when a figure is radically distorted in a manner similar to the deformation of a figure made of rubber." (Martin Gardner, "Aha! Insight", 1978)

"The impossibility of defining absolute motion can be seen as the manifestation of a symmetry known as relativistic invariance. In the same way that parity invariance tells us that we cannot distinguish the mirror-image world from our world, relativistic invariance tells us that it is impossible to decide whether we are at rest or moving steadily." (Anthony Zee, "Fearful Symmetry: The Search for Beauty in Modern Physics", 1986)

"[…] a model is a mathematical representation of the modeler's reality, a way of capturing some aspects of a particular reality within the framework of a mathematical apparatus that provides us with a means for exploring the properties of the reality mirrored in the model." (John L Casti, "Reality Rules: Picturing the world in mathematics", 1992)

The voyage of discovery into our own solar system has taken us from clockwork precision into chaos and complexity. This still unfinished journey has not been easy, characterized as it is by twists, turns, and surprises that mirror the intricacies of the human mind at work on a profound puzzle. Much remains a mystery. We have found chaos, but what it means and what its relevance is to our place in the universe remains shrouded in a seemingly impenetrable cloak of mathematical uncertainty." (Ivars Peterson, "Newton’s Clock", 1993) 

"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)

Mathematical Trivia I: Monsters

"Logic sometimes makes monsters. For half a century we have seen a mass of bizarre functions which appear to be forced to resemble as little as possible honest functions which serve some purpose. More of continuity, or less of continuity, more derivatives, and so forth. Indeed, from the point of view of logic, these strange functions are the most general; on the other hand those which one meets without searching for them, and which follow simple laws appear as a particular case which does not amount to more than a small corner. In former times when one invented a new function it was for a practical purpose; today one invents them purposely to show up defects in the reasoning of our fathers and one will deduce from them only that. If logic were the sole guide of the teacher, it would be necessary to begin with the most general functions, that is to say with the most bizarre. It is the beginner that would have to be set grappling with this teratologic museum." (Henri Poincaré, 1899)

"The orchard of science is a vast globe-encircling monster, without a map, and known to no one man; indeed, to no group of men fewer than the whole international mass of creative scientists. Within it, each observer clings to his own well-known and well-loved clump of trees. If he looks beyond, it is usually with a guilty sigh." (Isaac Asimov, "View from a Height", 1975)

"What were the needs that led me to single out a few of these monsters, calling them fractals, to add some of their close or distant kin, and then to build a geometric language around them? The original need happens to have been purely utilitarian. That links exist between usefulness and beauty is, of course, well known. What we call the beauty of a flower attracts the insects that will gather and spread its pollen. Thus the beauty of a flower is useful - even indispensable - to the survival of its species. Similarly, it was the attractiveness of the fractal images that first brought them to the attention of many colleagues and then of a wide world." (Benoît B Mandelbrot, "Fractals and an Art for the Sake of Science", Leonardo [Supplemental Issue], 1989)

"The term Moonshine [...] has a variety of meanings. It can refer to foolish or naive ideas, but also to the illicit distillation of spirits [...] It gave an impression of dabbling in mysterious matters that might be better left alone, but also had the useful connotation of something shining in reflected light. The true source of light is probably yet to be found, but there were further strange connections to come later [...] The Monster's connections with number theory - the Moonshine connections - had suggested it was a more beautiful and important group of symmetries than first realized. [...] The Moonshine connections between the Monster and number theory have now been placed within a larger theory, but we have yet to grasp the significance of these deep mathematical links with fundamental physics. We have found the Monster, but it remains an enigma. Understanding its full nature is likely to shed light on the very fabric of the universe." (Mark Ronan, "Symmetry and the Monster: One of the greatest quests of mathematics", 2006)

"The Moonshine mystery itself is still unresolved, despite Borcherd's proof! [...] there are facts about the Monster and Moonshine that we don't understand. [...] The method leading to its discovery, brilliant though it was, gave no clue to the Monster's remarkable properties." (Mark Ronan, "Symmetry and the Monster: One of the greatest quests of mathematics", 2006)

"To the average layperson, mathematics is a mass of abstruse formulae and bizarre technical terms (e.g., perverse sheaves, the monster group, barreled spaces, inaccessible cardinals), usually discussed by academics in white coats in front of a blackboard covered with peculiar symbols. The distinction between mathematics and physics is blurred and that between pure and applied mathematics is unknown. But to the professional, these are three different worlds, different sets of colleagues, with different goals, different standards, and different customs." (David Mumford, ["The Best Writing of Mathematics: 2012"] 2012)

"Infinity is a Loch Ness Monster, capturing the imagination with its awe-inspiring size but elusive nature. Infinity is a dream, a vast fantasy world of endless time and space. Infinity is a dark forest with unexpected creatures, tangled thickets and sudden rays of light breaking through. Infinity is a loop that springs open to reveal an endless spiral." (Eugenia Cheng, "Beyond Infinity: An Expedition to the Outer Limits of Mathematics", 2017)

"The rigid electron is in my view a monster in relation to Maxwell's equations, whose innermost harmony is the principle of relativity." (Hermann Minkowski)

On Mistakes, Blunders and Errors X: Data Science

"Measurement, we have seen, always has an element of error in it. The most exact description or prediction that a scientist can make is still only approximate." (Abraham Kaplan, "The Conduct of Inquiry: Methodology for Behavioral Science", 1964)

"[…] it is not enough to say: 'There's error in the data and therefore the study must be terribly dubious'. A good critic and data analyst must do more: he or she must also show how the error in the measurement or the analysis affects the inferences made on the basis of that data and analysis." (Edward R Tufte, "Data Analysis for Politics and Policy", 1974)

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

"In bagging, generating complementary base-learners is left to chance and to the unstability of the learning method. In boosting, we actively try to generate complementary base-learners by training the next learner boosting on the mistakes of the previous learners." (Ethem Alpaydin, "Introduction to Machine Learning" 2nd Ed, 2010)

"When data is not normal, the reason the formulas are working is usually the central limit theorem. For large sample sizes, the formulas are producing parameter estimates that are approximately normal even when the data is not itself normal. The central limit theorem does make some assumptions and one is that the mean and variance of the population exist. Outliers in the data are evidence that these assumptions may not be true. Persistent outliers in the data, ones that are not errors and cannot be otherwise explained, suggest that the usual procedures based on the central limit theorem are not applicable." (DeWayne R Derryberry, "Basic data analysis for time series with R", 2014)

"There are two kinds of mistakes that an inappropriate inductive bias can lead to: underfitting and overfitting. Underfitting occurs when the prediction model selected by the algorithm is too simplistic to represent the underlying relationship in the dataset between the descriptive features and the target feature. Overfitting, by contrast, occurs when the prediction model selected by the algorithm is so complex that the model fits to the dataset too closely and becomes sensitive to noise in the data." (John D Kelleher et al, "Fundamentals of Machine Learning for Predictive Data Analytics: Algorithms, Worked Examples, and Case Studies", 2015)

"[...] data often has some errors, outliers and other strange values, but these do not necessarily need to be individually identified and excluded. It also points to the benefits of using summary measures that are not unduly affected by odd observations [...] are known as robust measures, and include the median and the inter-quartile range." (David Spiegelhalter, "The Art of Statistics: Learning from Data", 2019)

"Statistical models have two main components. First, a mathematical formula that expresses a deterministic, predictable component, for example the fitted straight line that enables us to make a prediction [...]. But the deterministic part of a model is not going to be a perfect representation of the observed world [...] and the difference between what the model predicts, and what actually happens, is the second component of a model and is known as the residual error - although it is important to remember that in statistical modelling, ‘error’ does not refer to a mistake, but the inevitable inability of a model to exactly represent what we observe." (David Spiegelhalter, "The Art of Statistics: Learning from Data", 2019)

"There are many ways for error to creep into facts and figures that seem entirely straightforward. Quantities can be miscounted. Small samples can fail to accurately reflect the properties of the whole population. Procedures used to infer quantities from other information can be faulty. And then, of course, numbers can be total bullshit, fabricated out of whole cloth in an effort to confer credibility on an otherwise flimsy argument. We need to keep all of these things in mind when we look at quantitative claims. They say the data never lie - but we need to remember that the data often mislead." (Carl T Bergstrom & Jevin D West, "Calling Bullshit: The Art of Skepticism in a Data-Driven World", 2020)

On Mistakes, Blunders and Errors IX: Artificial Intelligence

"Because the subject matter of cybernetics is the propositional or informational aspect of the events and objects in the natural world, this science is forced to procedures rather different from those of the other sciences. The differentiation, for example, between map and territory, which the semanticists insist that scientists shall respect in their writings must, in cybernetics, be watched for in the very phenomena about which the scientist writes. Expectably, communicating organisms and badly programmed computers will mistake map for territory; and the language of the scientist must be able to cope with such anomalies." (Gregory Bateson, "Steps to an Ecology of Mind", 1972)

"To expect truth to come from thinking signifies that we mistake the need to think with the urge to know." (Hannah Arendt, "The Life of the Mind", 1977)

"There is a tendency to mistake data for wisdom, just as there has always been a tendency to confuse logic with values, intelligence with insight. Unobstructed access to facts can produce unlimited good only if it is matched by the desire and ability to find out what they mean and where they lead." (Norman Cousins, "Human Options : An Autobiographical Notebook", 1981)

"The other buzzword that epitomizes a bias toward substitution is 'big data'. Today’s companies have an insatiable appetite for data, mistakenly believing that more data always creates more value. But big data is usually dumb data. Computers can find patterns that elude humans, but they don’t know how to compare patterns from different sources or how to interpret complex behaviors. Actionable insights can only come from a human analyst (or the kind of generalized artificial intelligence that exists only in science fiction)." (Peter Thiel & Blake Masters, "Zero to One: Notes on Startups, or How to Build the Future", 2014)

"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)

"There are two kinds of mistakes that an inappropriate inductive bias can lead to: underfitting and overfitting. Underfitting occurs when the prediction model selected by the algorithm is too simplistic to represent the underlying relationship in the dataset between the descriptive features and the target feature. Overfitting, by contrast, occurs when the prediction model selected by the algorithm is so complex that the model fits to the dataset too closely and becomes sensitive to noise in the data."(John D Kelleher et al, "Fundamentals of Machine Learning for Predictive Data Analytics: Algorithms, Worked Examples, and Case Studies", 2015)

"The chief weakness of the machine is that it will not learn by its mistakes. The only way to improve its play is by improving the program. Some thought has been given to designing a program that would develop its own improvements in strategy with increasing experience in play. Although it appears to be theoretically possible, the methods thought of so far do not seem to be very practical. One possibility is to devise a program that would change the terms and coefficients involved in the evaluation function on the basis of the results of games the machine had already played. Small variations might be introduced in these terms, and the values would be selected to give the greatest percentage of wins." (Claude E Shannon)

On Mistakes, Blunders and Errors VIII: Physics

"In time they [physicists] hoped to devise a model which would reproduce all the phenomena of physics, and so make it possible to predict them all. […] To-day we not only have no perfect model, but we know that it is of no use to search for one - it could have no intelligible meaning for us. For we have found out that nature does not function in a way that can be made comprehensible to the human mind through models or pictures. […] Although we can never devise a pictorial representation which shall be both true to nature and intelligible to our minds, we may still be able to make partial aspects of the truth comprehensible through pictorial representations or parables. As the whole truth does not admit of intelligible representation, every such pictorial representation or parable must fail somewhere. The physicist of the last generation was continually making pictorial representations and parables, and also making the mistake of treating the half-truths of pictorial representations and parables as literal truths." (James H Jeans, "Physics and Philosophy" 3rd Ed., 1943)

"In physics it is usual to give alternative theoretical treatments of the same phenomenon. We construct different models for different purposes, with different equations to describe them. Which is the right model, which the 'true' set of equations? The question is a mistake. One model brings out some aspects of the phenomenon; a different model brings out others. Some equations give a rougher estimate for a quantity of interest, but are easier to solve. No single model serves all purposes best." (Nancy Cartwright, "How the Laws of Physics Lie", 1983)

"Our mistake is not that we take our theories too seriously, but that we do not take them seriously enough. It is always hard to realize that these numbers and equations we play with at our desks have something to do with the real world." (Steven Weinberg, "The First Three Minutes", 1977)

"In physics it is usual to give alternative theoretical treatments of the same phenomenon. We construct different models for different purposes, with different equations to describe them. Which is the right model, which the 'true' set of equations? The question is a mistake. One model brings out some aspects of the phenomenon; a different model brings out others. Some equations give a rougher estimate for a quantity of interest, but are easier to solve. No single model serves all purposes best." (Nancy Cartwright, "How the Laws of Physics Lie", 1983)

"It is a testimony to the power of education that classical mechanics could operate for so long under a mistaken conception. Teaching and research concentrated on integrable systems, each feeding the other, until in the end we had no longer the tools nor the interest for studying nonintegrable systems." (Ivar Ekeland, "The Best of All Possible Worlds", 2006)

“This is often the way it is in physics - our mistake is not that we take our theories too seriously, but that we do not take them seriously enough. It is always hard to realize that these numbers and equations we play with at our desks have something to do with the real world." (Heinrich Hertz)

"There is a fundamental error in separating the parts from the whole, the mistake of atomizing what should not be atomized. Unity and complementarity constitute reality." (Werner K Heisenberg)

On Mistakes, Blunders and Errors VII: NLP

"Knowledge being to be had only of visible and certain truth, error is not a fault of our knowledge, but a mistake of our judgment, giving assent to that which is not true." (John Locke, "An Essay Concerning Human Understanding", 1689)

"Most mistakes in philosophy and logic occur because the human mind is apt to take the symbol for the reality." (Albert Einstein, "Cosmic Religion: With Other Opinions and Aphorisms", 1931)

"The most pervasive paradox of the human condition which we see is that the processes which allow us to survive, grow, change, and experience joy are the same processes which allow us to maintain an impoverished model of the world - our ability to manipulate symbols, that is, to create models. So the processes which allow us to accomplish the most extraordinary and unique human activities are the same processes which block our further growth if we commit the error of mistaking the model of the world for reality." (Richard Bandler & John Grinder, "The Structure of Magic", 1975)

"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)

"All our language is composed of brief little dreams; and the wonderful thing is that we sometimes make of them strangely accurate and marvelously reasonable thoughts. […] What should we be without the help of that which does not exist? Very little. And our unoccupied minds would languish if fables, mistaken notions, abstractions, beliefs, and monsters, hypotheses, and the so-called problems of metaphysics did not people with beings and objectless images our natural depths and darkness. Myths are the souls of our actions and our loves. We cannot act without moving towards a phantom. We can love only what we create." (Paul Valéry, "The Outlook for Intelligence", 1962)

"To expect truth to come from thinking signifies that we mistake the need to think with the urge to know." (Hannah Arendt, "The Life of the Mind", 1977)

"To assume you know someone well enough that you can and do predict their behavior and mental perspective is a gross and often tragic mistake, for it eliminates that person's freedom to create his or her own opinion and drastically affects the emerging picture of the relationship." (Meredith L Young-Sowers, "Agartha: a journey to the stars", 1984) 

"Don't mistake a solution method for a problem definition - especially if it’s your own solution method." (Donald C Gause & Gerald M Weinberg, "Are Your Lights On?", 2011)

"Trust is fundamental to leading others into the dark, since trust enables fear to be 'actionable' as courage rather than actionable as anger. Since the bedrock of trust is faith that all will be OK within uncertainty, leaders’ fundamental role is to ultimately lead themselves. Research has found that successful leaders share three behavioral traits: they lead by example, admit their mistakes, and see positive qualities in others. All three are linked to spaces of play. Leading by example creates a space that is trusted - and without trust, there is no play. Admitting mistakes is to celebrate uncertainty. Seeing qualities in others is to encourage diversity." (Beau Lotto, "Deviate: The Science of Seeing Differently", 2017)

On Mistakes, Blunders and Errors VI: Mind

"Mathematical analysis is as extensive as nature itself; it defines all perceptible relations, measures times, spaces, forces, temperatures; this difficult science is formed slowly, but it preserves every principle which it has once acquired; it grows and strengthens itself incessantly in the midst of the many variations and errors of the human mind. It's chief attribute is clearness; it has no marks to express confused notations. It brings together phenomena the most diverse, and discovers the hidden analogies which unite them." (J B Joseph Fourier, "The Analytical Theory of Heat", 1822)

"It has often been said that, to make discoveries, one must be ignorant. This opinion, mistaken in itself, nevertheless conceals a truth. It means that it is better to know nothing than to keep in mind fixed ideas based on theories whose confirmation we constantly seek, neglecting meanwhile everything that fails to agree with them." (Claude Bernard, "An Introduction to the Study of Experimental Medicine", 1865)

"All experience attests the strength of the tendency to mistake mental abstractions, even negative ones, for substantive realities; and the Permanent Possibilities of sensation which experience guarantees arc so extremely unlike in many of their properties to actual sensations, that since we are capable of imagining something which transcends sensations, there is a great natural probability that we should suppose these to be it." (Hippolyte Taine, "On intelligence", 1871)

"Perfect readiness to reject a theory inconsistent with fact is a primary requisite of the philosophic mind. But it, would be a mistake to suppose that this candour has anything akin to fickleness; on the contrary, readiness to reject a false theory may be combined with a peculiar pertinacity and courage in maintaining an hypothesis as long as its falsity is not actually apparent. (William S Jevons, "The Principles of Science", 1874)

"Most mistakes in philosophy and logic occur because the human mind is apt to take the symbol for the reality." (Albert Einstein, "Cosmic Religion: With Other Opinions and Aphorisms", 1931)

"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)

"There exists, if I am not mistaken, an entire world which is the totality of mathematical truths, to which we have access only with our mind, just as a world of physical reality exists, the one like the other independent of ourselves, both of divine creation." (Charles Hermite, The Mathematical Intelligencer, Vol. 5, No. 4, 1983)

"In specific cases, we think by applying mental rules, which are similar to rules in computer programs. In most of the cases, however, we reason by constructing, inspecting, and manipulating mental models. These models and the processes that manipulate them are the basis of our competence to reason. In general, it is believed that humans have the competence to perform such inferences error-free. Errors do occur, however, because reasoning performance is limited by capacities of the cognitive system, misunderstanding of the premises, ambiguity of problems, and motivational factors. Moreover, background knowledge can significantly influence our reasoning performance. This influence can either be facilitation or an impedance of the reasoning process." (Carsten Held et al, "Mental Models and the Mind", 2006)

"A border is a completely imaginary line on a paper or cybernetic map that has no genuine counterpart in the real world. Do not mistake it for a property line. It is possible, in some instances, for a border to be congruent with a property line, but they are not the same thing at all. One represents the geographical limit of a military and political claim to authority over a given territory. The other is part of the description of something - in this case, land - lawfully owned by an individual or a voluntary and contractual association of individuals." (L Neil Smith, "Only Nixon", 2010)

"The human mind delights in finding pattern - so much so that we often mistake coincidence or forced analogy for profound meaning. No other habit of thought lies so deeply within the soul of a small creature trying to make sense of a complex world not constructed for it." (Stephen J Gould, "The Flamingo's Smile: Reflections in Natural History", 2010)

"Mental models represent possibilities, and the theory of mental models postulates three systems of mental processes underlying inference: (0) the construction of an intensional representation of a premise’s meaning – a process guided by a parser; (1) the building of an initial mental model from the intension, and the drawing of a conclusion based on heuristics and the model; and (2) on some occasions, the search for alternative models, such as a counterexample in which the conclusion is false. System 0 is linguistic, and it may be autonomous. System 1 is rapid and prone to systematic errors, because it makes no use of a working memory for intermediate results. System 2 has access to working memory, and so it can carry out recursive processes, such as the construction of alternative models." (Sangeet Khemlania & P.N. Johnson-Laird, "The processes of inference", Argument and Computation, 2012)