On Data: Longitudinal Data

 "Longitudinal data sets are comprised of repeated observations of an outcome and a set of covariates for each of many subjects. One objective of statistical analysis is to describe the marginal expectation of the outcome variable as a function of the covariates while accounting for the correlation among the repeated observations for a given subject." (Scott L Zeger & Kung-Yee Liang, "Longitudinal Data Analysis for Discrete and Continuous Outcomes", Biometrics Vol. 42(1), 1986)

"Longitudinal data sets in which the outcome variable cannot be transformed to be Gaussian are more difficult to analyze for two reasons. First, simple models for the conditional expectation of the outcome do not imply equally simple models for the marginal expectation, as is the case for Gaussian data. Hence, the analyst must choose to model either the marginal or conditional expectation. Second, likelihood analyses often lead to estimators of the regression coefficients which are consistent only when the time dependence is correctly specified." (Scott L Zeger & Kung-Yee Liang, "Longitudinal Data Analysis for Discrete and Continuous Outcomes", Biometrics Vol. 42(1), 1986)

"Longitudinal data comprise repeated observations over time on each of many individuals. Longitudinal data are in contrast to cross-sectional data where only a single response is available for each person. The statistical analysis of longitudinal data presents special opportunities and challenges because the repeated outcomes for one individual tend to be correlated with one another." (Scott L Zeger & Kung‐Yee Liang, "An overview of methods for the analysis of longitudinal data", Statistics in medicine vol. 11, 1992)

"We have two objectives for statistical models of longitudinal data: (1) to adopt the conventional regression tools, which relate the response variables to the explanatory variables; and (2) to account for the within subject correlation." (Scott L Zeger & Kung‐Yee Liang, "An overview of methods for the analysis of longitudinal data", Statistics in medicine vol. 11, 1992)

"Analysis of longitudinal data tends to be simpler because subjects can usually be assumed independent. Valid inferences can be made by borrowing strength across people. That is, the consistency of a pattern across subjects is the basis for substantive conclusions. For this reason, inferences from longitudinal studies can be made more robust to model assumptions than those from time series data, particularly to assumptions about the nature of the correlation." (Peter J Diggle et al, "Analysis of Longitudinal Data", 2002)

"The defining feature of a longitudinal data set is repeated observations on individuals enabling direct study of change. Longitudinal data require special statistical methods because the set of observations on one subject tends to be intercorrelated. This correlation must be taken into account to draw valid scientific inferences." (Peter J Diggle et al, "Analysis of Longitudinal Data", 2002)

Scott L Zeger - Collected Quotes

"Longitudinal data sets are comprised of repeated observations of an outcome and a set of covariates for each of many subjects. One objective of statistical analysis is to describe the marginal expectation of the outcome variable as a function of the covariates while accounting for the correlation among the repeated observations for a given subject." (Scott L Zeger & Kung-Yee Liang, "Longitudinal Data Analysis for Discrete and Continuous Outcomes", Biometrics Vol. 42(1), 1986)

"Longitudinal data sets in which the outcome variable cannot be transformed to be Gaussian are more difficult to analyze for two reasons. First, simple models for the conditional expectation of the outcome do not imply equally simple models for the marginal expectation, as is the case for Gaussian data. Hence, the analyst must choose to model either the marginal or conditional expectation. Second, likelihood analyses often lead to estimators of the regression coefficients which are consistent only when the time dependence is correctly specified." (Scott L Zeger & Kung-Yee Liang, "Longitudinal Data Analysis for Discrete and Continuous Outcomes", Biometrics Vol. 42(1), 1986)

"Statistical models are sometimes misunderstood in epidemiology. Statistical models for data are never true. The question whether a model is true is irrelevant. A more appropriate question is whether we obtain the correct scientific conclusion if we pretend that the process under study behaves according to a particular statistical model." (Scott Zeger, "Statistical reasoning in epidemiology", American Journal of Epidemiology, 1991)

"Statistical models for data are never true. The question whether a model is true is irrelevant. A more appropriate question is whether we obtain the correct scientific conclusion if we pretend that the process under study behaves according to a particular statistical model." (Scott Zeger, "Statistical reasoning in epidemiology", American Journal of Epidemiology, 1991)

"Statistical reasoning is based upon two simple precepts: (1) that natural processes can usefully be described by stochastic models and (2) that by studying apparently haphazard collections of autonomous individuals, one can discover, at a higher level, systematic patterns of potential scientific import." (Scott Zeger, "Statistical reasoning in epidemiology", American Journal of Epidemiology, 1991)

"The rise of statistical reasoning was a key step in the birth of many empirical sciences, especially epidemiology. The ability to focus on the aggregate behavior amidst apparently chaotic variation across autonomous individuals has dramatically increased our understanding of disease processes that affect the health of the public. Simple statistical models based upon the laws of probability provide the language for this population perspective." (Scott Zeger, "Statistical reasoning in epidemiology", American Journal of Epidemiology, 1991)

"Longitudinal data comprise repeated observations over time on each of many individuals. Longitudinal data are in contrast to cross-sectional data where only a single response is available for each person. The statistical analysis of longitudinal data presents special opportunities and challenges because the repeated outcomes for one individual tend to be correlated with one another." (Scott L Zeger & Kung‐Yee Liang, "An overview of methods for the analysis of longitudinal data", Statistics in medicine vol. 11, 1992)

"We have two objectives for statistical models of longitudinal data: (1) to adopt the conventional regression tools, which relate the response variables to the explanatory variables; and (2) to account for the within subject correlation." (Scott L Zeger & Kung‐Yee Liang, "An overview of methods for the analysis of longitudinal data", Statistics in medicine vol. 11, 1992)

On Manifolds: Definitions

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

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

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

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

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

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

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

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

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

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

Occam's Razor = The Law of Parsimony (1500 - 1899)

"We are to admit no more causes of natural things than such as are both true and sufficient to explain their appearances. Therefore, to the same natural effects we must, as far as possible, assign the same causes." (Isaac Newton, "Philosophiæ Naturalis Principia Mathematica" ["Mathematical Principles of Natural Philosophy"], 1687) 

"Entia non sunt multiplicanda praeter necessitatem."
"Entities are not to be multiplied beyond what is necessary." (John Ponce, cca. 17th century)

"Parsimony is enough to make the master of the golden mines as poor as he that has nothing; for a man may be brought to a morsel of bread by parsimony as well as profusion." (Henry Home [Lord Kames] ," Introduction to the Art of Thinking", 1761)

"Mere parsimony is not economy. Expense, and great expense, may be an essential part in true economy." (Edmund Burke, "A Letter to a Noble Lord", 1796)

"It is, after all, a principle of logic not to multiply entities unnecessarily." (Antoine-Laurent Lavoisier, "Réflexions sur le phlogistique", 1862)

"The first obligation of Simplicity is that of using the simplest means to secure the fullest effect." (George H Lewes, "The Principles of Success in Literature", 1865)

"In no case may we interpret an action [of an animal] as the outcome of the exercise of a higher psychical faculty, if it can be interpreted as the outcome of the exercise of one which stands lower in the psychological scale." (Conwy Lloyd Morgan, "An Introduction to Comparative Psychology", 1894) [Morgan's canon, the principle of parsimony in animal research]

"The question is therefore to demonstrate all geometrical truths with the smallest possible number of assumptions." (Augustus de Morgan, "On the Study and Difficulties of Mathematics", 1898)

"Scientists must use the simplest means of arriving at their results and exclude everything not perceived by the senses." (Ernst Mach)

Occam's Razor = The Law of Parsimony (1950 - 1999)

"Nonbeing must in some sense be, otherwise what is it that there is not? This tangled doctrine might be nicknamed Plato's beard; historically it has proved tough, frequently dulling the edge of Occam's razor." (Willard van Orman Quine, "On What There Is" From a Logical Point of View: Nine Logico-Philosophical Essays", 1953)

"[…] the grand aim of all science […] is to cover the greatest possible number of empirical facts by logical deductions from the smallest possible number of hypotheses or axioms.” (Albert Einstein, 1954)

"The principle of parsimony is valid esthetically in that the artist must not go beyond what is needed for his purpose. (Rudolf Arnheim," Art and Visual Perception: A Psychology of the Creative Eye", 1954)

"Our craving for generality has [as one] source […] our preoccupation with the method of science. I mean the method the method of reducing the explanation of natural phenomena to the smallest possible number of primitive natural laws; and, in mathematics, of unifying the treatment of different topics by using a generalization." (Ludwig Wittgenstein, "The Blue and Brown Books", 1958)

"[…] entities must not be reduced to the point of inadequacy and, more generally, that it is in vain to try to do with fewer what requires more." (Karl Menger, "A Counterpart of Occam's Razor in Pure and Applied Mathematics Ontological Uses", Synthese Vol. 12 (4), 1960)

"Let us consider, for a moment, the world as described by the physicist. It consists of a number of fundamental particles which, if shot through their own space, appear as waves, and are thus [...] of the same laminated structure as pearls or onions, and other wave forms called electromagnetic which it is convenient, by Occam’s razor, to consider as travelling through space with a standard velocity. All these appear bound by certain natural laws which indicate the form of their relationship." (G Spencer-Brown, "Laws of Form", 1969)

"For if as scientists we seek simplicity, then obviously we try the simplest surviving theory first, and retreat from it only when it proves false. Not this course, but any other, requires explanation. If you want to go somewhere quickly, and several alternate routes are equally likely to be open, no one asks why you take the shortest. The simplest theory is to be chosen not because it is the most likely to be true but because it is scientifically the most rewarding among equally likely alternatives. We aim at simplicity and hope for truth." (Nelson Goodman, "Problems and Projects", 1972)

"As glimpsed by physicists, Nature's rules are simple, but also intricate: Different rules are subtly related to each other. The intricate relations between the rules produce interesting effects in many physical situations. [...] Nature's design is not only simple, but minimally so, in the sense that were the design any simpler, the universe would be a much duller place." (Anthony Zee, "Fearful Symmetry: The Search for Beauty in Modern Physics", 1986)

"A mechanistic model has the following advantages: 1. It contributes to our scientific understanding of the phenomenon under study. 2. It usually provides a better basis for extrapolation (at least to conditions worthy of further experimental investigation if not through the entire range of all input variables). 3. It tends to be parsimonious (i. e, frugal) in the use of parameters and to provide better estimates of the response." (George E P Box, "Empirical Model-Building and Response Surfaces", 1987)

"I seek […] to show that - other things being equal - the simplest hypothesis proposed as an explanation of phenomena is more likely to be the true one than is any other available hypothesis, that its predictions are more likely to be true than those of any other available hypothesis, and that it is an ultimate a priori epistemic principle that simplicity is evidence for truth." (Richard Swinburne, "Simplicity as Evidence for Truth", 1997)

"Were it not for Occam's Razor, which always demands simplicity, I'd be tempted to believe that human beings are more influenced by distant causes than immediate ones. This would especially be true of overeducated people, who are capable of thinking past the immediate, of becoming obsessed by the remote. It's the old stuff, the conflicts we've never come to terms with, that sneaks up on us, half forgotten, insisting upon action."(Richard Russo,"Straight Man", 1997)

"It is part of the lore of science that the most parsimonious explanation of observed facts is to be preferred over convoluted and long-winded theories. Ptolemaic epicycles gave way to the Copernican system largely on this premise, and in general, scientific inquiry is governed by the oft-quoted dictum of the medieval cleric William of Occam that 'nunquam ponenda est pluralitas sine necesitate' , which may be paraphrased as 'choose the simplest explanation for the observed facts' ." (Edward Beltrami, "What is Random?: Chaos and Order in Mathematics and Life", 1999)

Occam's Razor = The Law of Parsimony (2000-)

"A smaller model with fewer covariates has two advantages: it might give better predictions than a big model and it is more parsimonious (simpler). Generally, as you add more variables to a regression, the bias of the predictions decreases and the variance increases. Too few covariates yields high bias; this called underfitting. Too many covariates yields high variance; this called overfitting. Good predictions result from achieving a good balance between bias and variance. […] fiding a good model involves trading of fit and complexity." (Larry A Wasserman, "All of Statistics: A concise course in statistical inference", 2004)

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

"The model theory postulates that mental models are parsimonious. They represent what is possible, but not what is impossible, according to assertions. This principle of parsimony minimizes the load on working memory, and so it applies unless something exceptional occurs to overrule it." (Philip N Johnson-Laird, Mental Models, Sentential Reasoning, and Illusory Inferences, [in "Mental Models and the Mind"], 2006)

"Two systems concepts lie at the disposal of the architect to reflect the beauty of harmony: parsimony and variety. The law of parsimony states that given several explanations of a specific phenomenon, the simplest is probably the best. […] On the other hand, the law of requisite variety states that for a system to survive in its environment the variety of choice that the system is able to make must equal or exceed the variety of influences that the environment can impose on the system." (John Boardman & Brian Sauser, "Systems Thinking: Coping with 21st Century Problems", 2008)

"What advantages do diagrams have over verbal descriptions in promoting system understanding? First, by providing a diagram, massive amounts of information can be presented more efficiently. A diagram can strip down informational complexity to its core - in this sense, it can result in a parsimonious, minimalist description of a system. Second, a diagram can help us see patterns in information and data that may appear disordered otherwise. For example, a diagram can help us see mechanisms of cause and effect or can illustrate sequence and flow in a complex system. Third, a diagram can result in a less ambiguous description than a verbal description because it forces one to come up with a more structured description." (Robbie T Nakatsu, "Diagrammatic Reasoning in AI", 2010)

"In my view, the argument from parsimony is really no argument at all - it typically functions only to shut down more interesting discussion. If history is any guide, it's never a good idea to assume that a scientific problem is cornered." (David Eagleman, "Incognito: The Secret Lives of the Brain", 2011)

"Scientists often talk of parsimony (as in 'the simplest explanation is probably correct', also known as Occam’s razor), but we should not get seduced by the apparent elegance of argument from parsimony; this line of reasoning has failed in the past at least as many times as it has succeeded. For example, it is more parsimonious to assume that the sun goes around the Earth, that atoms at the smallest scale operate in accordance with the same rules that objects at larger scales follow, and that we perceive what is really out there. All of these positions were long defended by argument from parsimony, and they were all wrong. In my view, the argument from parsimony is really no argument at all - it typically functions only to shut down more interesting discussion. If history is any guide, it’s never a good idea to assume that a scientific problem is cornered." (David Eagleman, "Incognito: The Secret Lives of the Brain", 2011)

"What can be done with fewer [assumptions] is done in vain with more." (Alan Baker, "Simplicity", The Stanford Encyclopedia of Philosophy, 2012)

Richard Bach - Collected Quotes

"In the path of our happiness shall we find the learning for which we have chosen this lifetime."  (Richard Bach, "Illusions: The Adventures of a Reluctant Messiah", 1977)

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

"There is no such thing as a problem without a gift for you in its hands. You seek problems because you need their gifts." (Richard Bach, "Illusions: The Adventures of a Reluctant Messiah", 1977)

"You teach best what you most need to learn." (Richard Bach, "Illusions: The Adventures of a Reluctant Messiah", 1977)

"The only things that matter are those made of truth and joy, and not of tin and glass." (Richard Bach, :There’s No Such Place as Far Away", 1979)

"There are no mistakes. The events we bring upon ourselves, no matter how unpleasant, are necessary in order to learn what we need to learn; whatever steps we take, they’re necessary to reach the places we’ve chosen to go." (Richard Bach, "The Bridge across Forever", 1984)

"An easy life doesn’t teach us anything. In the end it’s the learning that matters: what we’ve learned and how we’ve grown." (Richard Bach, "One", 1988)

"In every language, from Arabic to Zulu to calligraphy to shorthand to math to music to art to wrought stone, everything from the Unified Field Theory to a curse to a sixpenny nail to an orbiting satellite, anything expressed is a net around some idea." (Richard Bach, "One", 1988)

"No matter how qualified or deserving we are, we will never reach a better life until we can imagine it for ourselves and allow ourselves to have it." (Richard Bach, "One", 1988)

"No one can solve problems for someone whose problem is that they don’t want problems solved." (Richard Bach, "One", 1988)

"The exciting thing about ideas is putting them to work. The moment we try them on our own, launch them away from shore, they switch from what-if to become daring plunges down white rivers, as dangerous and as exhilarating." (Richard Bach, "One", 1988)

"The only way to avoid all frightening choices is to leave society and become a hermit, and that is a frightening choice." (Richard Bach, "One", 1988)

"There is no such thing as a problem without a gift. We seek problems because we need their gifts." (Richard Bach)

Douglas Adams - Collected Quotes

"Far out in the uncharted backwaters of the unfashionable end of the Western spiral arm of the Galaxy lies a small unregarded yellow sun. Orbiting this at a distance of roughly ninety million miles is an utterly insignificant little blue-green planet whose ape-descended life forms are so amazingly primitive that they still think digital watches are a pretty neat idea." (Douglas Adams, "The Hitch-Hiker’s Guide to the Galaxy", [radio series episode] 1978)

"The chances of finding out what really is going on are so absurdly remote that the only thing to do is to say hang the sense of it and just keep yourself occupied."  (Douglas Adams, "The Hitch-Hiker’s Guide to the Galaxy", [radio series episode] 1978)

"There is a theory which states that if ever anyone discovers exactly what the Universe is for and why it is here, it will instantly disappear and be replaced by something even more bizarrely inexplicable. There is another theory which states that this has already happened." (Douglas Adams, "The Hitch-Hiker’s Guide to the Galaxy", [radio series episode]1978)

"You begin to suspect that if there’s any real truth it’s that the entire multidimensional infinity of the Universe is almost certainly being run by a bunch of maniacs." (Douglas Adams, "The Hitch-Hiker’s Guide to the Galaxy", [radio series episode]1978)

"The main reason he had had such a wild and successful life was that he never really understood the significance of anything he did." (Douglas Adams, "The Hitchhiker’s Guide to Galaxy", 1979)

"The whole fabric of the space-time continuum is not merely curved, it is in fact totally bent." (Douglas Adams, "The Restaurant at the End of the Universe", 1980)

"One of the interesting things about space [...] is how dull it is." (Douglas Adams, "Life, the Universe, and Everything", 1982)

"Their minds sang with the ecstatic knowledge that either what they were doing was completely and utterly and totally impossible or that physics had a lot of catching up to do." (Douglas Adams, "So Long, and Thanks for All the Fish", 1985)

"The complexities of cause and effect defy analysis." (Douglas Adams, "Dirk Gently's Holistic Detective Agency", 1987)

"The impossible often has a kind of integrity to it which the merely improbable lacks." (Douglas Adams, "The Long Dark Tea-Time of the Soul", 1988)

"Words used carelessly, as if they did not matter in any serious way, often allowed otherwise well-guarded truths to seep through." (Douglas Adams, "The Long Dark Tea-Time of the Soul", 1988)

"Assumptions are the things you don’t know you’re making." (Douglas Adams, "Last Chance to See", 1990)

"Human beings, who are almost unique in having the ability to learn from the experience of others, are also remarkable for their apparent disinclination to do so." (Douglas Adams, "Last Chance to See", 1990)

"It does highlight the irony that everything you go to see is changed by the very action of going to see it, which is the sort of problem which physicists have been wrestling with for most of this century." (Douglas Adams, "Last Chance to See", 1990)

"The great thing about being the only species that makes a distinction between right and wrong is that we can make up the rules for ourselves as we go along." (Douglas Adams, "Last Chance to See", 1990)

"A common mistake that people make when trying to design something completely foolproof was to underestimate the ingenuity of complete fools." (Douglas Adams, "Mostly Harmless", 1992)

"The major difference between a thing that might go wrong and a thing that cannot possibly go wrong is that when a thing that cannot possibly go wrong goes wrong it usually turns out to be impossible to get at or repair." (Douglas Adams, "Mostly Harmless", 1992)

Stanislaw Lem - Collected Quotes

"Insanity, gentlemen, is not a catchall for every human action that involves motives we don’t understand. Insanity has its own structure, its own internal logic." (Stanislaw Lem, "The Investigation", 1959)

"So-called common sense relies on programmed nonperception, concealment, or ridicule of everything that doesn’t fit into the conventional nineteenth century vision of a world that can be explained down to the last detail." (Stanislaw Lem, "The Investigation", 1959)

"What if the world isn’t scattered around us like a jigsaw puzzle - what if it’s like a soup with all kinds of things floating around in it, and from time to time some of them get stuck together by chance to make some kind of whole? What if everything that exists is fragmentary, incomplete, aborted, events with ends but no beginnings, events that only have middles, things that have fronts or rears but not both, with us constantly making categories, seeking out, and reconstructing, until we think we can see total love, total betrayal and defeat, although in reality we are all no more than haphazard fractions. [...] Using religion and philosophy as the cement, we perpetually collect and assemble all the garbage comprised by statistics in order to make sense out of things, to make everything respond in one unified voice like a bell chiming to our glory. But it’s only soup..." (Stanislaw Lem, "The Investigation", 1959)

"Are we not, in the end, a clamorous prelude to the final silence, a marriage bed to engender dust, a universe for microbes, microbes that strive to circumnavigate us? We are as unfathomable, as inscrutable as That which brought us into being, and we choke on our own enigma..." (Stanislaw Lem, "Memoirs Found in a Bathtub", 1961

"How do you expect to communicate with the ocean, when you can’t even understand one another?" (Stanislaw Lem, "Solaris", 1961)

"Man does not create gods, in spite of appearances. The times, the age, impose them on him." (Stanislaw Lem, "Solaris", 1961)

"The more complex a civilization, the more vital to its existence is the maintenance of the flow of information; hence the more vulnerable it becomes to any disturbance in that flow." (Stanislaw Lem, "Memoirs Found in a Bathtub", 1961)

"We take off into the cosmos, ready for anything: for solitude, for hardship, for exhaustion, death. Modesty forbids us to say so, but there are times when we think pretty well of ourselves. And yet, if we examine it more closely, our enthusiasm turns out to be all sham. We don’t want to conquer the cosmos, we simply want to extend the boundaries of Earth to the frontiers of cosmos. [...] We have no need of other worlds. We need mirrors."  (Stanislaw Lem, "Solaris", 1961)

"Where there are no men, there cannot be motives accessible to men." (Stanislaw Lem, "Solaris", 1961)

"Once there lived a certain engineer-cosmogonist who lit stars to dispel the dark." (Stanislaw Lem, ‘"Uranium Earpieces", 1965),

"Science explains the world, but only Art can reconcile us to it." (Stanislaw Lem, "King Globares and the Sages", 1965)

"These dwarfs amass knowledge as others do treasure; for this reason they are called Hoarders of the Absolute. Their wisdom lies in the fact that they collect knowledge but never use it." (Stanislaw Lem, "How Erg the Self-Inducing Slew a Paleface", 1965)

"For years astrophysicists have been racking their brains over the reason for the great difference in the amounts of cosmic dust in various galaxies. The answer, I think, is quite simple: the higher a civilization is, the more dust and refuse it produces. This is a problem more for janitors than for astrophysicists." (Stanislaw Lem, ‘"Let Us Save the Universe (An Open Letter from Ijon Tichy, Space Traveller", 1966)

"By squandering nuclear energy, polluting asteroids and planets, ravaging the Preserve, and leaving litter everywhere we go, we shall ruin outer space and turn it into one big dump. It is high time we came to our senses and enforced the laws. Convinced that every minute of delay is dangerous, I sound the alarm: Let us save the Universe." (Stanislaw Lem, ‘"Let Us Save the Universe (An Open Letter from Ijon Tichy, Space Traveller", 1966)

"Every intelligent creature was curious - and curiosity prompted it to act when something incomprehensible took place."(Stanislaw Lem, "The Hunt", 1968)

"Is there anything more contemptible than Nature? The scientists, the philosophers have always tried to understand Nature, while the thing to do is to destroy it!" (Stanislaw Lem, "The Sanitorium of Dr. Vliperdius", 1971)

On Hypothesis Testing III

 "A little thought reveals a fact widely understood among statisticians: The null hypothesis, taken literally (and that’s the only way you can take it in formal hypothesis testing), is always false in the real world. [...] If it is false, even to a tiny degree, it must be the case that a large enough sample will produce a significant result and lead to its rejection. So if the null hypothesis is always false, what’s the big deal about rejecting it?" (Jacob Cohen, "Things I Have Learned (So Far)", American Psychologist, 1990)

"I believe [...] that hypothesis testing has been greatly overemphasized in psychology and in the other disciplines that use it. It has diverted our attention from crucial issues. Mesmerized by a single all-purpose, mechanized, ‘objective’ ritual in which we convert numbers into other numbers and get a yes-no answer, we have come to neglect close scrutiny of where the numbers come from." (Jacob Cohen, "Things I have learned (so far)", American Psychologist 45, 1990)

"Despite the stranglehold that hypothesis testing has on experimental psychology, I find it difficult to imagine a less insightful means of transitting from data to conclusions." (Geoffrey R Loftus, "On the tyranny of hypothesis testing in the social sciences", Contemporary Psychology 36, 1991)

"How has the virtually barren technique of hypothesis testing come to assume such importance in the process by which we arrive at our conclusions from our data?" (Geoffrey R Loftus, "On the tyranny of hypothesis testing in the social sciences", Contemporary Psychology 36, 1991)

"This remarkable state of affairs [overuse of significance testing] is analogous to engineers’ teaching (and believing) that light consists only of waves while ignoring its particle characteristics - and losing in the process, of course, any motivation to pursue the most interesting puzzles and paradoxes in the field." (Geoffrey R Loftus, "On the tyranny of hypothesis testing in the social sciences", Contemporary Psychology 36, 1991)

"Whereas hypothesis testing emphasizes a very narrow question (‘Do the population means fail to conform to a specific pattern?’), the use of confidence intervals emphasizes a much broader question (‘What are the population means?’). Knowing what the means are, of course, implies knowing whether they fail to conform to a specific pattern, although the reverse is not true. In this sense, use of confidence intervals subsumes the process of hypothesis testing." (Geoffrey R Loftus, "On the tyranny of hypothesis testing in the social sciences", Contemporary Psychology 36, 1991)

"After four decades of severe criticism, the ritual of null hypothesis significance testing - mechanical dichotomous decisions around a sacred .05 criterion - still persist. This article reviews the problems with this practice [...] What’s wrong with [null hypothesis significance testing]? Well, among many other things, it does not tell us what we want to know, and we so much want to know what we want to know that, out of desperation, we nevertheless believe that it does!" (Jacob Cohen, "The earth is round (p<.05)", American Psychologist 49, 1994)

"I argued that hypothesis testing is fundamentally inappropriate for ecological risk assessment, that its use has undesirable consequences for environmental protection, and that preferable alternatives exist for statistical analysis of data in ecological risk assessment. The conclusion of this paper is that ecological risk assessors should estimate risks rather than test hypothesis" (Glenn W Suter, "Abuse of hypothesis testing statistics in ecological risk assessment", Human and Ecological Risk Assessment 2, 1996)

"I contend that the general acceptance of statistical hypothesis testing is one of the most unfortunate aspects of 20th century applied science. Tests for the identity of population distributions, for equality of treatment means, for presence of interactions, for the nullity of a correlation coefficient, and so on, have been responsible for much bad science, much lazy science, and much silly science. A good scientist can manage with, and will not be misled by, parameter estimates and their associated standard errors or confidence limits." (Marks Nester, "A Myopic View and History of Hypothesis Testing", 1996)

"Statistical hypothesis testing is commonly used inappropriately to analyze data, determine causality, and make decisions about significance in ecological risk assessment,[...] It discourages good toxicity testing and field studies, it provides less protection to ecosystems or their components that are difficult to sample or replicate, and it provides less protection when more treatments or responses are used. It provides a poor basis for decision-making because it does not generate a conclusion of no effect, it does not indicate the nature or magnitude of effects, it does address effects at untested exposure levels, and it confounds effects and uncertainty[...]. Risk assessors should focus on analyzing the relationship between exposure and effects[...]."  (Glenn W Suter, "Abuse of hypothesis testing statistics in ecological risk assessment", Human and Ecological Risk Assessment 2, 1996)

On Hypothesis Testing II

"Small wonder that students have trouble [with statistical hypothesis testing]. They may be trying to think." (W Edwards Deming, "On probability as a basis for action", American Statistician 29, 1975)

"Tests appear to many users to be a simple way to discharge the obligation to provide some statistical treatment of the data." (H V Roberts, "For what use are tests of hypotheses and tests of significance",  Communications in Statistics [Series A], 1976)

"In practice, of course, tests of significance are not taken seriously." (Louis Guttman, "The illogic of statistical inference for cumulative science", Applied Stochastic Models and Data Analysis, 1985)

"Most readers of The American Statistician will recognize the limited value of hypothesis testing in the science of statistics. I am not sure that they all realize the extent to which it has become the primary tool in the religion of Statistics." (David Salsburg, The Religion of Statistics as Practiced in Medical Journals, "The American Statistician" 39, 1985)

"Since a point hypothesis is not to be expected in practice to be exactly true, but only approximate, a proper test of significance should almost always show significance for large enough samples. So the whole game of testing point hypotheses, power analysis notwithstanding, is but a mathematical game without empirical importance." (Louis Guttman, "The illogic of statistical inference for cumulative science", Applied Stochastic Models and Data Analysis, 1985

"We shall marshal arguments against [significance] testing, leading to the conclusion that it be abandoned by all substantive science and not just by educational research and other social sciences which have begun to raise voices against the virtual tyranny of this branch of inference in the academic world." (Louis Guttman, "The illogic of statistical inference for cumulative science", Applied Stochastic Models and Data Analysis, 1985)

"Analysis of variance [...] stems from a hypothesis-testing formulation that is difficult to take seriously and would be of limited value for making final conclusions." (Herman Chernoff, Comment,  The American Statistician 40(1), 1986)

"We are better off abandoning the use of hypothesis tests entirely and concentrating on developing continuous measures of toxicity which can be used for estimation." (David Salsburg, "Statistics for Toxicologists", 1986)

"Beware of the problem of testing too many hypotheses; the more you torture the data, the more likely they are to confess, but confessions obtained under duress may not be admissible in the court of scientific opinion." (Stephen M Stigler, "Neutral Models in Biology", 1987)

On Hypothesis Testing I

"Statistics is the fundamental and most important part of inductive logic. It is both an art and a science, and it deals with the collection, the tabulation, the analysis and interpretation of quantitative and qualitative measurements. It is concerned with the classifying and determining of actual attributes as well as the making of estimates and the testing of various hypotheses by which probable, or expected, values are obtained. It is one of the means of carrying on scientific research in order to ascertain the laws of behavior of things - be they animate or inanimate. Statistics is the technique of the Scientific Method." (Bruce D Greenschields & Frank M Weida, "Statistics with Applications to Highway Traffic Analyses", 1952)

"The peculiarity of [...] statistical hypotheses is that they are not conclusively refutable by any experience." (Richard B Braithwaite, "Scientific Explanation: A Study of the Function of Theory, Probability and Law in Science", 1953)

"Tests of the null hypothesis that there is no difference between certain treatments are often made in the analysis of agricultural or industrial experiments in which alternative methods or processes are compared. Such tests are [...] totally irrelevant. What are needed are estimates of magnitudes of effects, with standard errors." (Francis J Anscombe, "Discussion on Dr. David’s and Dr. Johnson’s Paper", Journal of the Royal Statistical Society B 18, 1956)

"[...] the tests of null hypotheses of zero differences, of no relationships, are frequently weak, perhaps trivial statements of the researcher’s aims [...] in many cases, instead of the tests of significance it would be more to the point to measure the magnitudes of the relationships, attaching proper statements of their sampling variation. The magnitudes of relationships cannot be measured in terms of levels of significance." (Leslie Kish, "Some statistical problems in research design", American Sociological Review 24, 1959)

"In view of our long-term strategy of improving our theories, our statistical tactics can be greatly improved by shifting emphasis away from over-all hypothesis testing in the direction of statistical estimation. This always holds true when we are concerned with the actual size of one or more differences rather than simply in the existence of differences." (David A Grant, "Testing the null hypothesis and the strategy and tactics of investigating theoretical models", Psychological Review 69, 1962)

"[...] we need to get on with the business of generating [...] hypotheses and proceed to do investigations and make inferences which bear on them, instead of [...] testing the statistical null hypothesis in any number of contexts in which we have every reason to suppose that it is false in the first place." (David Bakan, "The test of significance in psychological research", Psychological Bulletin 66, 1966)

"All testing, all confirmation and disconfirmation of a hypothesis takes place already within a system. And this system is not a more or less arbitrary and doubtful point of departure for all our arguments; no it belongs to the essence of what we call an argument. The system is not so much the point of departure, as the element in which our arguments have their life." (Ludwig Wittgenstein, "On Certainty", 1969)

"Science consists simply of the formulation and testing of hypotheses based on observational evidence; experiments are important where applicable, but their function is merely to simplify observation by imposing controlled conditions." (Henry L Batten, "Evolution of the Earth", 1971)

"[...] the statistical power of many psychological studies is ridiculously low. This is a self-defeating practice: it makes for frustrated scientists and inefficient research. The investigator who tests a valid hypothesis but fails to obtain significant results cannot help but regard nature as untrustworthy or even hostile." (Amos Tversky & Daniel Kahneman, "Belief in the law of small numbers", Psychological Bulletin 76(2), 1971) 

"Decision-making problems (hypothesis testing) involve situations where it is desired to make a choice among various alternative decisions (hypotheses). Such problems can be viewed as generalized state estimation problems where the definition of state has simply been expanded." (Fred C Scweppe, "Uncertain dynamic systems", 1973)

"Hypothesis testing can introduce the need for multiple models for the multiple hypotheses and,' if appropriate, a priori probabilities. The one modeling aspect of hypothesis testing that has no estimation counterpart is the problem of specifying the hypotheses to be considered. Often this is a critical step which influences both performance arid the difficulty of implementation." (Fred C Scweppe, "Uncertain dynamic systems", 1973)

"Pattern recognition can be viewed as a special case of hypothesis testing. In pattern recognition, an observation z is to be used to decide what pattern caused it. Each possible pattern can be viewed as one hypothesis. The main problem in pattern recognition is the development of models for the z corresponding to each pattern (hypothesis)." (Fred C Scweppe, "Uncertain dynamic systems", 1973)

"The term hypothesis testing arises because the choice as to which process is observed is based on hypothesized models. Thus hypothesis testing could also be called model testing. Hypothesis testing is sometimes called decision theory. The detection theory of communication theory is a special case." (Fred C Scweppe, "Uncertain dynamic systems", 1973)

Douglas T Ross - Collected Quotes

"Automatic design has the computer do too much and the human do too little, whereas automatic programming has the human do too much and the computer do too little. Both techniques are important, but are not representative for what we wish to mean by computer-aided design." (Douglas T Ross, "Computer-Aided Design: A Statement of Objectives", 1960)

"Computer-aided design is not automatic design, although it must include many automatic design features. By automatic design we mean design procedures which are capable of being completely specified in a form which a computer can execute without human intervention." (Douglas T Ross, "Computer-Aided Design: A Statement of Objectives", 1960)

"It is very difficult to define what is meant by computer-aided design since the complete definition is, in fact, the sum and substance of the total project effort which has only begun. It is much easier to describe, what is not computer-aided design as we mean it." (Douglas T Ross, "Computer-Aided Design: A Statement of Objectives", 1960)

"The objective of the Computer-Aided Design Project is to evolve a machine systems which will permit the human designer and the computer to work together on creative design problems."  (Douglas T Ross, "Computer-Aided Design: A Statement of Objectives", 1960)

"Mechanical drawings and blueprints are not mere pictures, but a complete and rich language. In blueprint language, scientific, mathematical, and geometric formulations, notations, mensurations, and naming do not merely describe an object or process, they actually model it. Because of broad differences in subject, purpose, roles, and the needs of the people who use them, many forms of blueprint have evolved, but all rigorously present well structured information in understandable form." (Douglas T Ross, "Structured analysis (SA): A language for communicating ideas", IEEE Transactions on Software Engineering Vol. 3 No. 1, 1977)

"Structured analysis (SA) combines blueprint-like graphic language with the nouns and verbs of any other language to provide a hierarchic, top-down, gradual exposition of detail in the form of an SA model. The things and happenings of a subject are expressed in a data decomposition and an activity decomposition, both of which employ the same graphic building block, the SA box, to represent a part of a whole. SA arrows, representing input, output, control, and mechanism, express the relation of each part to the whole." (Douglas T Ross, "Structured analysis (SA): A language for communicating ideas", IEEE Transactions on Software Engineering Vol. 3 No. 1, 1977)

"The natural law of good communications takes the following, quite different, form in SA: Everything worth saying about anything worth saying something about must be expressed in six or fewer pieces." (Douglas T Ross, "Structured analysis (SA): A language for communicating ideas", IEEE Transactions on Software Engineering Vol. 3 No. 1, 1977)

"There are certain basic, known principles about how people's minds go about the business of understanding, and communicating understanding by means of language, which have been known and used for many centuries. No matter how these principles are addressed, they always end up with hierarchic decomposition as being the heart of good storytelling." (Douglas T Ross, "Structured analysis (SA): A language for communicating ideas", IEEE Transactions on Software Engineering Vol. 3 No. 1, 1977)

"We never have any understanding of any subject matter except in terms of our own mental constructs of ‘things’ and ‘happenings’ of that subject matter." (Douglas T Ross, "Structured analysis (SA): A language for communicating ideas", IEEE Transactions on Software Engineering Vol. 3 No. 1, 1977)

"A general theme for what I'm trying to convey and what actually drove me and my very industrious and creative project members over all these years, is… that there is much more to it than pictures. It has to be a picture language. There has to be meaning there, and the meaning is useful. You're trying to solve problems. So it really comes down to man machine problem solving. Better means of communication and expression is what always has driven our work." (Douglas T Ross, "Retrospectives: The Early Years in Computer Graphics at at MIT", Lincoln Lab and Harvard, 1989)

"There is a rigorous science, just waiting to be recognized and developed, which encompasses the whole of 'the software problem,' as defined, including the hardware, software, languages, devices, logic, data, knowledge, users, users, and effectiveness, etc. for end-users, providers, enablers, commissioners, and sponsors, alike." (Douglas T Ross,1989)

George B Dyson - Collected Quotes

"An Internet search engine is a finite-state, deterministic machine, except at those junctures where people, individually and collectively, make a nondeterministic choice as to which results are selected as meaningful and given a click. These clicks are then immediately incorporated into the state of the deterministic machine, which grows ever so incrementally more knowledgeable with every click. This is what Turing defined as an oracle machine."  (George B Dyson, "Turing's Cathedral: The Origins of the Digital Universe", 2012)

"If life, by some chance, happens to have originated, and survived, elsewhere in the universe, it will have had time to explore an unfathomable diversity of forms. Those best able to survive the passage of time, adapt to changing environments, and migrate across interstellar distances will become the most widespread. A life form that assumes digital representation, for all or part of its life cycle, will be able to travel at the speed of light." (George B Dyson, "Turing's Cathedral: The Origins of the Digital Universe", 2012)

"In our universe, we measure time with clocks, and computers have a 'clock speed', but the clocks that govern the digital universe are very different from the clocks that govern ours. In the digital universe, clocks exist to synchronize the translation between bits that are stored in memory (as structures in space) and bits that are communicated by code (as sequences in time). They are clocks more in the sense of regulating escapement than in the sense of measuring time."(George B Dyson, "Turing's Cathedral: The Origins of the Digital Universe", 2012)

"It is characteristic of objects of low complexity that it is easier to talk about the object than produce it and easier to predict its properties than to build it." (George B Dyson, "Turing's Cathedral: The Origins of the Digital Universe", 2012)

"Sixty-some years ago, biochemical organisms began to assemble digital computers. Now digital computers are beginning to assemble biochemical organisms. Viewed from a distance, this looks like part of a life cycle. But which part? Are biochemical organisms the larval phase of digital computers? Or are digital computers the larval phase of biochemical organisms?" (George B Dyson, "Turing's Cathedral: The Origins of the Digital Universe", 2012)

"Life evolved, so far, by making use of the viral cloud as a source of backup copies and a way to rapidly exchange genetic code. Life may be better adapted to the digital universe than we think." (George B Dyson, "Turing's Cathedral: The Origins of the Digital Universe", 2012)

"Mathematical reasoning may be regarded rather schematically as the exercise of a combination of two faculties, which we may call intuition and ingenuity [...] (George B Dyson, "Turing's Cathedral: The Origins of the Digital Universe", 2012)

"Monte Carlo is able to discover practical solutions to otherwise intractable problems because the most efficient search of an unmapped territory takes the form of a random walk. Today’s search engines, long descended from their ENIAC-era ancestors, still bear the imprint of their Monte Carlo origins: random search paths being accounted for, statistically, to accumulate increasingly accurate results. The genius of Monte Carlo - and its search-engine descendants - lies in the ability to extract meaningful solutions, in the face of overwhelming information, by recognizing that meaning resides less in the data at the end points and more in the intervening paths." (George B Dyson, "Turing's Cathedral: The Origins of the Digital Universe", 2012)

"Music allows us to assemble temporal sequences into mental scaffolding that transcends the thinness of time in which we live." (George B Dyson, "Turing's Cathedral: The Origins of the Digital Universe", 2012)

"One of the facets of extreme originality is not to regard as obvious the things that lesser minds call obvious," (George B Dyson, "Turing's Cathedral: The Origins of the Digital Universe", 2012)

"Over long distances, it is expensive to transport structures, and inexpensive to transmit sequences. Turing machines, which by definition are structures that can be encoded as sequences, are already propagating themselves, locally, at the speed of light. The notion that one particular computer resides in one particular location at one time is obsolete. (George Dyson, "Turing's Cathedral: The Origins of the Digital Universe", 2012) 

"Random search can be more efficient than nonrandom search - something that Good and Turing had discovered at Bletchley Park. A random network, whether of neurons, computers, words, or ideas, contains solutions, waiting to be discovered, to problems that need not be explicitly defined." (George B Dyson, "Turing's Cathedral: The Origins of the Digital Universe", 2012)

"The brain is a statistical, probabilistic system, with logic and mathematics running as higher-level processes. The computer is a logical, mathematical system, upon which higher-level statistical, probabilistic systems, such as human language and intelligence, could possibly be built." (George B Dyson, "Turing's Cathedral: The Origins of the Digital Universe", 2012)

"The good news is that, as Leibniz suggested, we appear to live in the best of all possible worlds, where the computable functions make life predictable enough to be survivable, while the noncomputable functions make life (and mathematical truth) unpredictable enough to remain interesting, no matter how far computers continue to advance."  (George B Dyson, "Turing's Cathedral: The Origins of the Digital Universe", 2012)

"The fundamental, indivisible unit of information is the bit. The fundamental, indivisible unit of digital computation is the transformation of a bit between its two possible forms of existence: as structure (memory) or as sequence (code). This is what a Turing Machine does when reading a mark (or the absence of a mark) on a square of tape, changing its state of mind accordingly, and making (or erasing) a mark somewhere else." (George B Dyson, "Turing's Cathedral: The Origins of the Digital Universe", 2012)

"The genius of Monte Carlo - and its search-engine descendants - lies in the ability to extract meaningful solutions, in the face of overwhelming information, by recognizing that meaning resides less in the data at the end points and more in the intervening paths." (George B Dyson, "Turing's Cathedral: The Origins of the Digital Universe", 2012)

"The paradox of artificial intelligence is that any system simple enough to be understandable is not complicated enough to behave intelligently, and any system complicated enough to behave intelligently is not simple enough to understand. The path to artificial intelligence, suggested Turing, is to construct a machine with the curiosity of a child, and let intelligence evolve." (George B Dyson, "Turing's Cathedral: The Origins of the Digital Universe", 2012)

"The question of whether something is feasible in a type belongs to a higher logical type. It is characteristic of objects of low complexity that it is easier to talk about the object than produce it and easier to predict its properties than to build it. But in the complicated parts of formal logic it is always one order of magnitude harder to tell what an object can do than to produce the object." (George B Dyson, "Turing's Cathedral: The Origins of the Digital Universe", 2012)

"There are two kinds of creation myths: those where life arises out of the mud, and those where life falls from the sky. In this creation myth, computers arose from the mud, and code fell from the sky." (George B Dyson, "Turing's Cathedral: The Origins of the Digital Universe", 2012)

"Where does meaning come in? If everything is assigned a number, does this diminish the meaning in the world? What Gödel (and Turing) proved is that formal systems will, sooner or later, produce meaningful statements whose truth can be proved only outside the system itself. This limitation does not confine us to a world with any less meaning. It proves, on the contrary, that we live in a world where higher meaning exists." (George B Dyson, "Turing's Cathedral: The Origins of the Digital Universe", 2012)

"Nature uses digital computing for generation-to-generation information storage, combinatorics, and error correction but relies on analog computing for real-time intelligence and control." (George B Dyson, Analogia: The Emergence of Technology Beyond Programmable Control", 2020)

A Picture's Worth

"The drawing shows me at a glance what would be spread over ten pages in a book." (Ivan Turgenev, 1862) [2]

"Sometimes, half a dozen figures will reveal, as with a lighting-flash, the importance of a subject which ten thousand labored words with the same purpose in view, had left at last but dim and uncertain." (Mark Twain, "Life on the Mississippi", 1883) 

"One good picture is worth many pages of written description." (William Sproston Caine, 1891) [2]

"One look is worth a thousand words" (Kathleen Caffyn, 1903) 

"Use a picture. It's worth a thousand words." (Arthur Brisbane, The Post-Standard, 1911)

"One Look Is Worth A Thousand Words" ([advertisement] 1913)

"A picture is worth ten thousand words. If you can’t see the truth in these pictures you are among the vast majority that must learn only by experience." (Arthur Brisbane, 1915)

"One picture is worth ten thousand words." (Frederick R Barnard, Printer’s Ink, 1921)

"One Picture Worth Ten Thousand Words" ([Chinese proverb] 1927)

"In many instances, a picture is indeed worth a thousand words. To make this true in more diverse circumstances, much more creative effort is needed to pictorialize the output from data analysis. Naive pictures are often extremely helpful, but more sophisticated pictures can be both simple and even more informative." (John W Tukey & Martin B Wilk, "Data Analysis and Statistics: An Expository Overview", 1966)

"One word is worth a thousand pictures. If it's the right word." (Edward Abbey, "Beyond the Wall: Essays from the Outside", 1984)

"A picture may be worth a thousand words, a formula is worth a thousand pictures." (Edsger Dijkstra, [conference at ETH Zurich] 1994)

"A magnificent picture is never worth a thousand perfect words." (John Dunning, "The Bookman's Wake", 1995)

"A picture tells a thousand words. But you get a thousand pictures from someone's voice." (Paul Fleischman, "Seek", 2001)

"If a picture is worth a thousand words, a metaphor is worth a thousand pictures." (Daniel H Pink, "A Whole New Mind: Why Right-Brainers Will Rule the Future", 2005)

"A picture may be worth a thousand words, but not all pictures are readable, interpretable, meaningful, or relevant." (Kristen Sosulski, "Data Visualization Made Simple: Insights into Becoming Visual", 2018)

"A good metaphor is worth a thousand pictures." (Anon) 

"As the Chinese say, 1001 words is worth more than a picture." (John McCarthy [source]) 

References:
[1] Wikipedia (2024) A picture is worth a thousand words [link]
[2] Quote Investigator (2022) A Picture Is Worth Ten Thousand Words [link] 
[3] SQL-Troubles (2024) Charts vs. Thousand Words [link]

David G Green - Collected Quotes

"Although it might be intuitively apparent that a system is complex, defining complexity has proved difficult to pin down with numerous definitions on record. As yet there is no agreed theory of complexity. Much of the mathematics is intractable and computer simulation plays a major part." (Terry R J Bossomaier & David G Green, 2000) 

"Although many natural phenomena may result from the interaction of complex entities, the details of the components may be unimportant. In the discussion of neural networks, the individual neuron turns out to be a highly sophisticated biological system. But the collective properties of neurons may be captured by spin-glass models, in which the neuron is simplified to a binary quantity […] " (Terry R J Bossomaier & David G Green, 2000) 

"Interaction: the other major source of complexity is the interaction of many autonomous, adaptive agents. Again, there are many questions to ask about the agents, the nature of the interaction and the circumstances in which complex surface phenomena result. (Terry R J Bossomaier & David G. Green, 2000)

"Iteration: fractals and chaos result from repetition of simple operations. These generating rules produce complex phenomena. There are many interesting questions to ask about how to describe the processes, how to measure the resulting complexity, whether we can work backwards from the reult to the rules and so on. (Terry R J Bossomaier & David G. Green, 2000)

"Of course what we would all like to see is a general theory of complex systems or complexity. Despite several promising candidates the selection process is still under way. Maybe there is no universal theory, but there are certainly common paradigms and methods which have proved to be useful across a wide area." (Terry R J Bossomaier & David G Green, 2000) 

" The evident power of simple heuristics […] teaches us the important lesson that global behavior patterns, and social organization, can emerge out of local interactions. Organisms do not necessarily need to have an over-riding plan nor do they require awareness of the large-scale. Grand patterns and processes can emerge as the nett effect of small-scale, local behavior.

"If entropy must increase, then how is it possible (say) for all the variety of the living world to persist? The usual answer to the above question is that living systems are open systems, not closed, so the law does not apply locally. However this answer is somewhat unsatisfying. In effect all systems are open systems, since everything interacts with its surroundings to some degree." (David G Green, 2000) 

"The really crucial question in multi-object systems is whether local interactions do grow into large-scale patterns." (David G Green, 2000) 

"The self-similarity on different scales arises because growth often involves iteration of simple, discrete processes (e.g. branching). These repetitive processes can often be summarized as sets of simple rules." (David G Green, 2000) 

Paul Klee - Collected Quotes

"Our initial perplexity before nature is explained by our seeing at first the small outer branches and not penetrating to the main branches or the trunk. But once this is realized, one will perceive a repetition of the whole law even in the outermost leaf and turn it to good use." (Paul Klee, [diary entry] 1904)

"When looking at any significant work of art, remember that a more significant one probably has had to be sacrificed." (Paul Klee, [diary entry] 1904)

"The beautiful, which is perhaps inseparable from art, is not after all tied to the subject, but to the pictorial representation. In this way and in no other does art overcome the ugly without avoiding it." (Paul Klee, [diary entry] 1905)

"Things are not quite so simple with 'pure' art as it is dogmatically claimed. In the final analysis, a drawing simply is no longer a drawing, no matter how self-sufficient its execution may be. It is a symbol, and the more profoundly the imaginary lines of projection meet higher dimensions, the better. In this sense I shall never be a pure artist as the dogma defines him. We higher creatures are also mechanically produced children of God, and yet intellect and soul operate within us in completely different dimensions." (Paul Klee, [diary entry] 1905)

"Nature can afford to be prodigal in everything, the artist must be frugal down to the last detail."  Paul Klee, [diary entry] 1909)

"First of all, the art of living; then as my ideal profession, poetry and philosophy, and as my real profession, plastic arts; in the last resort, for lack of income, illustrations." (Paul Klee, cca. 1910

"Graphic work as the expressive movement of the hand holding the recording pencil.... is so fundamentally different from dealing with tone and color that one can use this technique quite well in the dark, even in the blackest night. On the other hand, tone (movement from light to dark) presupposes some light, and color presupposes a great deal of light." (Paul Klee, 1912)

"We document, explain, justify, construct, organize: these are good things, but we do not succeed in coming to the whole [...]. But we may as well calm down: construction is not absolute. Our virtue is this: by cultivating the exact we have laid the foundations for a science of art, including the unknown X." (Paul Klee, "Statement of 1917"

"A tendency toward the abstract is inherent in linear expression: graphic imagery being confined to outlines has a fairy-like quality and at the same time can achieve great precision." (Paul Klee, "Creative Credo", 1920)

"Things appear to assume a broader and more diversified meaning, often seemingly contradicting the rational experience of yesterday. There is a striving to emphasize the essential character of the accidental." (Paul Klee, "Creative Credo", 1920)

"For the artist communication with nature remains the most essential condition. The artist is human; himself nature; part of nature within natural space." (Paul Klee, 1923)

"It is possible that a picture will move far away from Nature and yet find its way back to reality. The faculty of memory, experience at a distance produces pictorial associations." (Paul Klee, cca. 1925)

"Thought is the medially between earth and world. The broader the magnitude of his reach, the more painful man's tragic limitation. Thought is the medially between earth and world. The broader the magnitude of his reach, the more painful man's tragic limitation. To get where motion is interminate." ( Paul Klee, "Pedagogical Sketch Book, 1925)

"The longer a line, the more of the time element it contains. Distance is time whereas a surface is apprehended more in terms of the moment." (Paul Klee, "Exact Experiments in the Realm of Art", 1927)

"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)

"Art should be like a holiday: something to give a man the opportunity to see things differently and to change his point of view." (Paul Klee)

"It is interesting to observe how real the object remains, in spite of all abstractions." (Paul Klee)

Richard B Braithwaite - Collected Quotes

"It has been a fortunate fact in the modern history of physical science that the scientist constructing a new theoretical system has nearly always found that the mathematics [...] required [...] had already been worked out by pure mathematicians for their own amusement [...] The moral for statesmen would seem to be that, for proper scientific 'planning' , pure mathematics should be endowed fifty years ahead of scientists." (Richard B Braithwaite, "Scientific Explanation: A Study of the Function of Theory, Probability and Law in Science", 1953)

"[...] no batch of observations, however large, either definitively rejects or definitively fails to reject the hypothesis H0." (Richard B Braithwaite, "Scientific Explanation: A Study of the Function of Theory, Probability and Law in Science", 1953)

"The peaks of science may appear to be floating in the clouds, but their foundations are in the hard facts of experience." (Richard B Braithwaite, "Scientific Explanation: A Study of the Function of Theory, Probability and Law in Science", 1953)

"The peculiarity of [...] statistical hypotheses is that they are not conclusively refutable by any experience." (Richard B Braithwaite, "Scientific Explanation: A Study of the Function of Theory, Probability and Law in Science", 1953)

"The ultimate justification for any scientific belief will depend upon the main purpose for which we think scientifically - that of predicting and thereby controlling the future." (Richard B Braithwaite, "Scientific Explanation: A Study of the Function of Theory, Probability and Law in Science", 1953)

"The world is not made up of empirical facts with the addition of the laws of nature: what we call the laws of nature are conceptual devices by which we organize our empirical knowledge and predict the future." (Richard B Braithwaite, "Scientific Explanation: A Study of the Function of Theory, Probability and Law in Science", 1953)

Howard Wainer - Collected Quotes

"Although arguments can be made that high data density does not imply that a graphic will be good, nor one with low density bad, it does reflect on the efficiency of the transmission of information. Obviously, if we hold clarity and accuracy constant, more information is better than less. One of the great assets of graphical techniques is that they can convey large amounts of information in a small space." (Howard Wainer, "How to Display Data Badly", The American Statistician Vol. 38(2), 1984) 

"The essence of a graphic display is that a set of numbers having both magnitudes and an order are represented by an appropriate visual metaphor - the magnitude and order of the metaphorical representation match the numbers. We can display data badly by ignoring or distorting this concept." (Howard Wainer, "How to Display Data Badly", The American Statistician Vol. 38(2), 1984)

"The standard error of most statistics is proportional to 1 over the square root of the sample size. God did this, and there is nothing we can do to change it." (Howard Wainer, "Improving Tabular Displays, With NAEP Tables as Examples and Inspirations", Journal of Educational and Behavioral Statistics Vol 22 (1), 1997)

"[…] a graph is nothing but a visual metaphor. To be truthful, it must correspond closely to the phenomena it depicts: longer bars or bigger pie slices must correspond to more, a rising line must correspond to an increasing amount. If a graphical depiction of data does not faithfully follow this principle, it is almost sure to be misleading. But the metaphoric attachment of a graphic goes farther than this. The character of the depiction ism a necessary and sufficient condition for the character of the data. When the data change, so too must their depiction; but when the depiction changes very little, we assume that the data, likewise, are relatively unchanging. If this convention is not followed, we are usually misled." (Howard Wainer, "Graphic Discovery: A trout in the milk and other visuals" 2nd, 2008)

"A graphic display has many purposes, but it achieves its highest value when it forces us to see what we were not expecting." (Howard Wainer, "Graphic Discovery: A trout in the milk and other visuals" 2nd, 2008)

"Nothing that had been produced before was even close. Even today, after more than two centuries of graphical experience, Playfair’s graphs remain exemplary standards for clearcommunication of quantitative phenomena. […] Graphical forms were available before Playfair, but they were rarely used to plot empirical information." (Howard Wainer, "Graphic Discovery: A trout in the milk and other visuals" 2nd, 2008)

"Oftentimes a statistical graphic provides the evidence for a plausible story, and the evidence, though perhaps only circumstantial, can be quite convincing. […] But such graphical arguments are not always valid. Knowledge of the underlying phenomena and additional facts may be required." (Howard Wainer, "Graphic Discovery: A trout in the milk and other visuals" 2nd, 2008)

"Placing a fact within a context increases its value greatly. […] . An efficacious way to add context to statistical facts is by embedding them in a graphic. Sometimes the most helpful context is geographical, and shaded maps come to mind as examples. Sometimes the most helpful context is temporal, and time-based line graphs are the obvious choice. But how much time? The ending date (today) is usually clear, but where do you start? The starting point determines the scale. […] The starting point and hence the scale are determined by the questions that we expect the graph to answer." (Howard Wainer, "Graphic Discovery: A trout in the milk and other visuals" 2nd, 2008)

"Simpson’s Paradox can occur whenever data are aggregated. If data are collapsed across a subclassification (such as grades, race, or age), the overall difference observed may not represent what is going on. Standardization can help correct this, but nothing short of random assignment of individuals to groups will prevent the possibility of yet another subclassificatiion, as yet unidentified, changing things around again. But I believe that knowing of the possibility helps us, so that we can contain the enthusiasm of our impulsive first inferences." (Howard Wainer, "Graphic Discovery: A trout in the milk and other visuals" 2nd, 2008)

"The appearance, and hence the perception, of any statistical graphic is massively influenced by the choice of scale. If the scale of the vertical axis is too narrow relative to the scale of the horizontal axis, random meanders look meaningful." (Howard Wainer, "Graphic Discovery: A trout in the milk and other visuals" 2nd, 2008)

"The difficult task of properly setting the scale of a graph remains difficult but not mysterious. There is agreement among experts spanning two hundred years. The default option should be to choose a scale that fills the plot with data. We can deviate from this under circumstances when it is better not to fill the plot with data, but those circumstances are usually clear. It is important to remember that the sin of using too small a scale is venial; the viewer can correct it. The sin of using too large a scale cannot be corrected without access to the original data; it can be mortal." (Howard Wainer, "Graphic Discovery: A trout in the milk and other visuals" 2nd, 2008)

"Usually the effectiveness of a good display increases with the complexity of the data. When there are only a few points, almost anything will do; even a pie chart with only three or four categories is usually comprehensible." (Howard Wainer, "Graphic Discovery: A trout in the milk and other visuals" 2nd, 2008)

"Thus when we look at, or prepare, a time-based statistical graphic, it is important to ask what is the right time scale, the right context, for the questions of greatest interest. The answer to this question is sometimes complex, but the very act of asking it provides us with some protection against surprises." (Howard Wainer, "Graphic Discovery: A trout in the milk and other visuals" 2nd, 2008)

"The only thing we know for sure about a missing data point is that it is not there, and there is nothing that the magic of statistics can do change that. The best that can be managed is to estimate the extent to which missing data have influenced the inferences we wish to draw." (Howard Wainer, "14 Conversations About Three Things", Journal of Educational and Behavioral Statistics Vol. 35(1, 2010)

"For an analyst to willfully avoid learning about the science is akin to malfeasance. Of course, it is likely that a deep understanding both of the science and of data analytic methods does not reside in the same person. When it does not, data analysis should be done jointly. It is my understanding that data mining is not often done as a team. This is unfortunate, for then it is too easy to miss what might have been found." (Howard Wainer, Comment, Journal of Computational and Graphical Statistics Vol. 20(1), 2011)

"Too often there is a disconnect between the people who run a study and those who do the data analysis. This is as predictable as it is unfortunate. If data are gathered with particular hypotheses in mind, too often they (the data) are passed on to someone who is tasked with testing those hypotheses and who has only marginal knowledge of the subject matter. Graphical displays, if prepared at all, are just summaries or tests of the assumptions underlying the tests being done. Broader displays, that have the potential of showing us things that we had not expected, are either not done at all, or their message is not able to be fully appreciated by the data analyst." (Howard Wainer, Comment, Journal of Computational and Graphical Statistics Vol. 20(1), 2011)

Clay Helberg - Collected Quotes

"Another key element in making informative graphs is to avoid confounding design variation with data variation. This means that changes in the scale of the graphic should always correspond to changes in the data being represented." (Clay Helberg, "Pitfalls of Data Analysis (or How to Avoid Lies and Damned Lies)", 1995) 

"Another trouble spot with graphs is multidimensional variation. This occurs where two-dimensional figures are used to represent one-dimensional values. What often happens is that the size of the graphic is scaled both horizontally and vertically according to the value being graphed. However, this results in the area of the graphic varying with the square of the underlying data, causing the eye to read an exaggerated effect in the graph." (Clay Helberg, "Pitfalls of Data Analysis (or How to Avoid Lies and Damned Lies)", 1995) 

"It may be helpful to consider some aspects of statistical thought which might lead many people to be distrustful of it. First of all, statistics requires the ability to consider things from a probabilistic perspective, employing quantitative technical concepts such as 'confidence', 'reliability', 'significance'. This is in contrast to the way non-mathematicians often cast problems: logical, concrete, often dichotomous conceptualizations are the norm: right or wrong, large or small, this or that." (Clay Helberg, "Pitfalls of Data Analysis (or How to Avoid Lies and Damned Lies)", 1995) 

"[...] many non-mathematicians hold quantitative data in a sort of awe. They have been lead to believe that numbers are, or at least should be, unquestionably correct." (Clay Helberg, "Pitfalls of Data Analysis (or How to Avoid Lies and Damned Lies)", 1995) 

"Most statistical models assume error free measurement, at least of independent (predictor) variables. However, as we all know, measurements are seldom if ever perfect. Particularly when dealing with noisy data such as questionnaire responses or processes which are difficult to measure precisely, we need to pay close attention to the effects of measurement errors. Two characteristics of measurement which are particularly important in psychological measurement are reliability and validity." (Clay Helberg, "Pitfalls of Data Analysis (or How to Avoid Lies and Damned Lies)", 1995) 

"Remember that a p-value merely indicates the probability of a particular set of data being generated by the null model - it has little to say about the size of a deviation from that model (especially in the tails of the distribution, where large changes in effect size cause only small changes in p-values)." (Clay Helberg, "Pitfalls of Data Analysis (or How to Avoid Lies and Damned Lies)", 1995)

"There are a number of ways that statistical techniques can be misapplied to problems in the real world. Three of the most common hazards are designing experiments with insufficient power, ignoring measurement error, and performing multiple comparisons." (Clay Helberg, "Pitfalls of Data Analysis (or How to Avoid Lies and Damned Lies)", 1995)

"We can consider three broad classes of statistical pitfalls. The first involves sources of bias. These are conditions or circumstances which affect the external validity of statistical results. The second category is errors in methodology, which can lead to inaccurate or invalid results. The third class of problems concerns interpretation of results, or how statistical results are applied (or misapplied) to real world issues." (Clay Helberg, "Pitfalls of Data Analysis (or How to Avoid Lies and Damned Lies)", 1995) 

References:
[1] Clay Helberg, "Pitfalls of Data Analysis (or How to Avoid Lies and Damned Lies)", 1995 [link]

On Probability (2000 - )

"In the laws of probability theory, likelihood distributions are fixed properties of a hypothesis. In the art of rationality, to explain is to anticipate. To anticipate is to explain." (Eliezer S. Yudkowsky, "A Technical Explanation of Technical Explanation", 2005)

"I have always thought that statistical design and sampling from populations should be the first courses taught, but all elementary courses I know of start with statistical methods or probability. To me, this is putting the cart before the horse!" (Walter Federer, "A Conversation with Walter T Federer", Statistical Science Vol 20, 2005)

"For some scientific data the true value cannot be given by a constant or some straightforward mathematical function but by a probability distribution or an expectation value. Such data are called probabilistic. Even so, their true value does not change with time or place, making them distinctly different from  most statistical data of everyday life." (Manfred Drosg, "Dealing with Uncertainties: A Guide to Error Analysis", 2007)

"In fact, H [entropy] measures the amount of uncertainty that exists in the phenomenon. If there were only one event, its probability would be equal to 1, and H would be equal to 0 - that is, there is no uncertainty about what will happen in a phenomenon with a single event because we always know what is going to occur. The more events that a phenomenon possesses, the more uncertainty there is about the state of the phenomenon. In other words, the more entropy, the more information." (Diego Rasskin-Gutman, "Chess Metaphors: Artificial Intelligence and the Human Mind", 2009)

"The four questions of data analysis are the questions of description, probability, inference, and homogeneity. [...] Descriptive statistics are built on the assumption that we can use a single value to characterize a single property for a single universe. […] Probability theory is focused on what happens to samples drawn from a known universe. If the data happen to come from different sources, then there are multiple universes with different probability models.  [...] Statistical inference assumes that you have a sample that is known to have come from one universe." (Donald J Wheeler," Myths About Data Analysis", International Lean & Six Sigma Conference, 2012)

"When statisticians, trained in math and probability theory, try to assess likely outcomes, they demand a plethora of data points. Even then, they recognize that unless it’s a very simple and controlled action such as flipping a coin, unforeseen variables can exert significant influence." (Zachary Karabell, "The Leading Indicators: A short history of the numbers that rule our world", 2014)

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

On Probability (1975 - 1999)

"Of course, we know the laws of trial and error, of large numbers and probabilities. We know that these laws are part of the mathematical and mechanical fabric of the universe, and that they are also at play in biological processes. But, in the name of the experimental method and out of our poor knowledge, are we really entitled to claim that everything happens by chance, to the exclusion of all other possibilities?" (Albert Claude, "The Coming of Age of the Cell", Science, 1975)

"We often use the ideas of chance, likelihood, or probability in everyday language. For example, 'It is unlikely to rain today', 'The black horse will probably win the next race', or 'A playing card selected at random from a pack is unlikely to be the ace of spades' . Each of these remarks, if accepted at face value, is likely to reflect the speaker's expectation based on experience gained in the same position, or similar positions, on many  previous occasions. In order to be quantitative about probability, we focus on this aspect of repeatable situations." (Peter Lancaster, "Mathematics: Models of the Real World", 1976)

"The theory of probability is the only mathematical tool available to help map the unknown and the uncontrollable. It is fortunate that this tool, while tricky, is extraordinarily powerful and convenient." (Benoit Mandelbrot, "The Fractal Geometry of Nature", 1977)

"In decision theory, mathematical analysis shows that once the sampling distribution, loss function, and sample are specified, the only remaining basis for a choice among different admissible decisions lies in the prior probabilities. Therefore, the logical foundations of decision theory cannot be put in fully satisfactory form until the old problem of arbitrariness (sometimes called 'subjectiveness') in assigning prior probabilities is resolved." (Edwin T Jaynes, "Prior Probabilities", 1978)

"Another reason for the applied statistician to care about Bayesian inference is that consumers of statistical answers, at least interval estimates, commonly interpret them as probability statements about the possible values of parameters. Consequently, the answers statisticians provide to consumers should be capable of being interpreted as approximate Bayesian statements." (Donald B Rubin, "Bayesianly justifiable and relevant frequency calculations for the applied statistician", Annals of Statistics 12(4), 1984)

"In the path-integral formulation, the essence of quantum physics may be summarized with two fundamental rules: (1). The classical action determines the probability amplitude for a specific chain of events to occur, and (2) the probability that either one or the other chain of events occurs is determined by the probability amplitudes corresponding to the two chains of events. Finding these rules represents a stunning achievement by the founders of quantum physics." (Anthony Zee, "Fearful Symmetry: The Search for Beauty in Modern Physics", 1986)

"In the design of experiments, one has to use some informal prior knowledge. How does one construct blocks in a block design problem for instance? It is stupid to think that use is not made of a prior. But knowing that this prior is utterly casual, it seems ludicrous to go through a lot of integration, etc., to obtain ‘exact’ posterior probabilities resulting from this prior. So, I believe the situation with respect to Bayesian inference and with respect to inference, in general, has not made progress. Well, Bayesian statistics has led to a great deal of theoretical research. But I don’t see any real utilizations in applications, you know. Now no one, as far as I know, has examined the question of whether the inferences that are obtained are, in fact, realized in the predictions that they are used to make." (Oscar Kempthorne, "A conversation with Oscar Kempthorne", Statistical Science vol. 10, 1995)

"Events may appear to us to be random, but this could be attributed to human ignorance about the details of the processes involved." (Brain S Everitt, "Chance Rules", 1999)

On Convergence IV

"Once more, an invariably-recurring lesson of geological history, at whatever point its study is taken up: the lesson of the almost infinite slowness of the modification of living forms. The lines of the pedigrees of living things break off almost before they begin to converge." (Thomas H Huxley, On the Formation of Coal, 1870)

"Analytic functions are those that can be represented by a power series, convergent within a certain region bounded by the so-called circle of convergence. Outside of this region the analytic function is not regarded as given a priori ; its continuation into wider regions remains a matter of special investigation and may give very different results, according to the particular case considered." (Felix Klein, "Sophus Lie", [lecture] 1893)

"Particular landforms or surface morphologies may be generated, in some cases, by several different processes, sets of environmental controls, or developmental histories. This convergence to similar forms despite variations in processes and controls is called equifinality." (Jonathan Phillips, "Simplexity and the Reinvention of Equifinality", Geographical Analysis Vol. 29 (1), 1997)

"The underlying reason for convergence seems to be that all organisms are under constant scrutiny of natural selection and are also subject to the constraints of the physical and chemical factors that severely limit the action of all inhabitants of the biosphere. Put simply, convergence shows that in a real world not all things are possible." (Simon C Morris, "The Crucible of Creation", 1998)

"Equifinality is the principle which states that morphology alone cannot be used to reconstruct the mode of origin of a landform on the grounds that identical landforms can be produced by a number of alternative processes, process assemblages or process histories. Different processes may lead to an apparent similarity in the forms produced. For example, sea-level change, tectonic uplift, climatic change, change in source of sediment or water or change in storage may all lead to river incision and a convergence of form." (Olav Slaymaker, "Equifinality", 2004)

"Convergence is, in my opinion, not only deeply fascinating but, curiously, it is as often overlooked. More importantly, it hints at the existence of a deeper structure to biology. It helps us to delineate a metaphorical map across which evolution must navigate. In this sense the Darwinian mechanisms and the organic substrate we call life are really a search engine to discover particular solutions, including intelligence and - risky thought - perhaps deeper realities?" (Simon C Morris,  "Aliens like us?", Astronomy and Geophysics Vol. 46 (4), 2005)

"Sometimes the most important fit statistic you can get is ‘convergence not met’ - it can tell you something is wrong with your model." (Oliver Schabenberger, "Applied Statistics in Agriculture Conference", 2006)