On Models: Bad Models

"A bad model is a combination of assertions, some factual, others conjectural, and others plainly false but convenient. [..]  By definition, a bad model does not give power to see-accurately, deeply, or at all-into the actual situation, but only into the assertions embodied in the model. Thus, if the use of a bad model provides insight, it does so not by revealing truth about the world but by revealing its own assumptions and thereby causing its user to go learn something about the world." (James S Hodges, "Six (or So) Things You Can Do with a Bad Model", 1991)

"Just because a model is bad, however, does not mean it is useless. [...] A bad model can be used to construct correct paths from premises to conclusions, but because its relations to reality are questionable, it can only do so in a few ways-at least, ways that permit useful conclusions with respect to reality." (James S Hodges, "Six (or So) Things You Can Do with a Bad Model", 1991)

"Often, though, a policy or systems analyst is stuck with a bad model, that is, one that appeals to the analyst as adequately realistic but which is either: 1) contradicted by some data or is grossly implausible in some aspect it purports to represent, or 2) conjectural, that is, neither supported nor contradicted by data, either because data do not exist or because they are equivocal. [...] A model may have component parts that are not bad, but if, taken as a whole, it meets one of these criteria, it is a bad model." (James S Hodges, "Six (or So) Things You Can Do with a Bad Model", 1991)

"Some readers have argued that the criticism implied by the term 'bad models' is undeserved because they can be used appropriately in some cases. [...] If the logic works, the use is appropriate; if it fails, the use in inappropriate: (Cost effectiveness is a separate issue.) As for the pejorative connotation of the term bad model, perhaps we should admit that many useful models would be embarrassments to scientists, from whom we got the idea of a model, but whose job is to improve the match between models and reality." (James S Hodges, "Six (or So) Things You Can Do with a Bad Model", 1991)

"The definition of a ‘good model’ is when everything inside it is visible, inspectable and testable. It can be communicated effortlessly to others. A ‘bad model’ is a model that does not meet these standards, where parts are hidden, undefined or concealed and it cannot be inspected or tested; these are often labelled black box models." (Hördur V Haraldsson & Harald U Sverdrup,Finding Simplicity in Complexity in Biogeochemical Modelling" [inEnvironmental Modelling: Finding Simplicity in Complexity", Ed. by John Wainwright and Mark Mulligan, 2004])

"It is not always convenient to remember that the right model for a population can fit a sample of data worse than a wrong model - even a wrong model with fewer parameters. We cannot rely on statistical diagnostics to save us, especially with small samples. We must think about what our models mean, regardless of fit, or we will promulgate nonsense." (Leland Wilkinson,The Grammar of Graphics" 2nd Ed., 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)

"An attempt to use the wrong model for a given data set is likely to provide poor results. Therefore, the core principle of discovering outliers is based on assumptions about the structure of the normal patterns in a given data set. Clearly, the choice of the 'normal' model depends highly upon the analyst’s understanding of the natural data patterns in that particular domain." (Charu C Aggarwal,Outlier Analysis", 2013)

"Things which ought to be expected can seem quite extraordinary if you’ve got the wrong model." (David Hand,Significance", 2014)

"Model-building requires much more than just technical knowledge of statistical ideas. It also requires care and judgment, and cannot be reduced to a flowchart, a table of formulas, or a tidy set of numerical summaries that wring every last drop of truth from a data set. There is almost never a single 'right' statistical model for some problem. But there are definitely such things as good models and bad models, and learning to tell the difference is important. Just remember: calling a model good or bad requires knowing both the tool and the task." (James G Scott, "Statistical Modeling: A Gentle Introduction", 2017)

"Overfitting and underfitting are two important factors that could impact the performance of machine-learning models. Overfitting occurs when the model performs well with training data and poorly with test data. Underfitting occurs when the model is so simple that it performs poorly with both training and test data. [...]  When the model does not capture and fit the data, it results in poor performance. We call this underfitting. Underfitting is the result of a poor model that typically does not perform well for any data." (Umesh R Hodeghatta & Umesha Nayak,Business Analytics Using R: A Practical Approach", 2017)

"Mathematicians love math and many non-mathematicians are intimidated by math. This is a lethal combination that can lead to the creation of wildly unrealistic mathematical models. [...] A good mathematical model starts with plausible assumptions and then uses mathematics to derive the implications. A bad model focuses on the math and makes whatever assumptions are needed to facilitate the math." (Gary Smith & Jay Cordes, "The 9 Pitfalls of Data Science", 2019)

"Bad data makes bad models. Bad models instruct people to make ineffective or harmful interventions. Those bad interventions produce more bad data, which is fed into more bad models." (Cory Doctorow,Machine Learning’s Crumbling Foundations", 2021)

On Quantum Mechanics (Unsourced)

"For those who are not shocked when they first come across quantum theory cannot possibly have understood it." (Niels Bohr) 

"I hesitated to think it might be wrong, but I knew that it was rotten. That is to say, one has to find some decent way of expressing whatever truth there is in it." (John S. Bell) 

"[…] if one really understood the central point and its necessity in the construction of the world, one ought to be able to state it in one clear, simple sentence. Until we see the quantum principle with this simplicity we can well believe that we do not know the first thing about the universe, about ourselves, and about our place in the universe." (John A Wheeler)

"The mathematics of quantum mechanics is straightforward, but making the connection between the mathematics and an intuitive picture of the physical world is very hard." (Claude N. Cohen-Tannoudji)

"The world is not as real as we think. […] My personal opinion is that the world is even weirder than what quantum physics tells us." (Anton Zeilinger) 

"Those who are not shocked when they first come across quantum mechanics cannot possibly have understood it." (Niels Bohr)

"Our understanding of quantum mechanics is troubled by the problem of measurement and the problem of nonlocality. [...] It seems to me unlikely that either problem can be solved without a solution to the other, and therefore without a deep adjustment of space-time theory and quantum mechanics to each other." (Abner Shimony)

"String theory is extremely attractive because gravity is forced upon us. All known consistent string theories include gravity, so while gravity is impossible in quantum field theory as we have known it, it is obligatory in string theory." (Edward Witten)

"Quantum mechanics is the weirdest invention of mankind, but also one of the most beautiful. And the beauty of the mathematics underlying the quantum theory implies that we have found something very significant." (Anton Zeilinger)

"What you can show using physics, forces this universe to continue to exist. As long as you're using general relativity and quantum mechanics you are forced to conclude that God exists." (Frank Tipler)

On Quantum Mechanics (2000-)

"Statistical mechanics is the science of predicting the observable properties of a many-body system by studying the statistics of the behaviour of its individual constituents, be they atoms, molecules, photons etc. It provides the link between macroscopic and microscopic states. […] classical thermodynamics. This is a subject dealing with the very large. It describes the world that we all see in our daily lives, knows nothing about atoms and molecules and other very small particles, but instead treats the universe as if it were made up of large-scale continua. […] quantum mechanics. This is the other end of the spectrum from thermodynamics; it deals with the very small. It recognises that the universe is made up of particles: atoms, electrons, protons and so on. One of the key features of quantum mechanics, however, is that particle behaviour is not precisely determined (if it were, it would be possible to compute, at least in principle, all past and future behaviour of particles, such as might be expected in a classical view). Instead, the behaviour is described through the language of probabilities." (A Mike Glazer & Justin S Wark, "Statistical Mechanics: A survival guide", 2001)

"To describe how quantum theory shapes time and space, it is helpful to introduce the idea of imaginary time. Imaginary time sounds like something from science fiction, but it is a well-defined mathematical concept: time measured in what are called imaginary numbers. […] Imaginary numbers can then be represented as corresponding to positions on a vertical line: zero is again in the middle, positive imaginary numbers plotted upward, and negative imaginary numbers plotted downward. Thus imaginary numbers can be thought of as a new kind of number at right angles to ordinary real numbers. Because they are a mathematical construct, they don't need a physical realization […]" (Stephen W Hawking, "The Universe in a Nutshell", 2001)

"Quantum theory introduced uncertainty into physics; not an uncertainty that arises out of mere ignorance but a fundamental uncertainty about the very universe itself. Uncertainty is the price we pay for becoming participators in the universe. Ultimate knowledge may only be possible for ethereal beings who lie outside the universe and observe it from their ivory towers." (F David Peat, "From Certainty to Uncertainty", 2002)

"The quantum world is in a constant process of change and transformation. On the face of it, all possible processes and transformations could take place, but nature’s symmetry principles place limits on arbitrary transformation. Only those processes that do not violate certain very fundamental symmetry principles are allowed in the natural world." (F David Peat, "From Certainty to Uncertainty", 2002)

"String theory was not invented to describe gravity; instead it originated in an attempt to describe the strong interactions, wherein mesons can be thought of as open strings with quarks at their ends. The fact that the theory automatically described closed strings as well, and that closed strings invariably produced gravitons and gravity, and that the resulting quantum theory of gravity was finite and consistent is one of the most appealing aspects of the theory." (David Gross, "Einstein and the Search for Unification", 2005)

"Briefly stated, the orthodox formulation of quantum theory asserts that, in order to connect adequately the mathematically described state of a physical system to human experience, there must be an abrupt intervention in the otherwise smoothly evolving mathematically described state of that system." (Henry P Stapp, "Mindful Universe: Quantum Mechanics and the Participating Observer", 2007)

"The existing general descriptions of quantum theory emphasize puzzles and paradoxes in a way that tend to make non-physicists leery of using in any significant away the profound changes in our understanding of both man and nature wrought by the quantum revolution. Yet in the final analysis quantum mechanics is more understandable than classical mechanics because it is more deeply in line with our common sense ideas about our role in nature than the ‘automaton’ notion promulgated by classical physics." (Henry P Stapp, "Mindful Universe: Quantum Mechanics and the Participating Observer", 2007)

"The objectivist view is that probabilities are real aspects of the universe - propensities of objects to behave in certain ways - rather than being just descriptions of an observer’s degree of belief. For example, the fact that a fair coin comes up heads with probability 0.5 is a propensity of the coin itself. In this view, frequentist measurements are attempts to observe these propensities. Most physicists agree that quantum phenomena are objectively probabilistic, but uncertainty at the macroscopic scale - e.g., in coin tossing - usually arises from ignorance of initial conditions and does not seem consistent with the propensity view." (Stuart J Russell & Peter Norvig, "Artificial Intelligence: A Modern Approach", 2010)

On Quantum Mechanics (1975-1999)

"The 'complete description' that quantum theory claims the wave function to be is a description of physical reality (as in physics). No matter what we are feeling, or thinking about, or looking at, the wave function describes as completely as possible where and when we are doing it. [...] Since the wave function is thought to be a complete description of physical reality and since that which the wave function describes is idea-like as well as matter-like, then physical reality must be both idea-like and matter-like. In other words, the world cannot be as it appears. Incredible as it sounds, this is the conclusion of the orthodox view of quantum mechanics." (Gary Zukav, "The Dancing Wu Li Masters", 1979)

"The conceptual framework of quantum mechanics, supported by massive volumes of experimental data, forces contemporary physicists to express themselves in a manner that sounds, even to the uninitiated, like the language of mystics." (Gary Zukav, "The Dancing Wu Li Masters", 1979)

"The new physics tells us that an observer cannot observe without altering what he sees. Observer and observed are interrelated in a real and fundamental sense. The exact nature of this interrelation is not clear, but there is a growing body of evidence that the distinction between the 'in here' and the 'out there' is illusion." (Gary Zukav, "The Dancing Wu Li Masters", 1979)

"There is another fundamental difference between the old physics and the new physics. The old phvsics assumes that there is an external world which exists apart from us. It further assumes that we can observe measure and speculate about the external world without changing it. According to the old physics the external world is indifferent to us and to our needs. [...] The new physics, quantum mechanics, tells us clearly that it is not possible to observe reality without changing it. If we observe a certain particle collision experiment, not only do we have no way of proving that the result would have been the same if we had not been watching it, all that we know indicates that it would not have been the same, because the result that we got was affected by the fact that we were looking for it." (Gary Zukav, "The Dancing Wu Li Masters", 1979)

"The phenomena of the subatomic world are so complex that it is by no means certain whether a complete, self-consistent theory will ever be constructed, but one can envisage a series of partly successful models of smaller scope. Each of them would be intended to cover only a part of the observed phenomena and would contain some unexplained aspects, or parameters, but the parameters of one model might be explained by another. Thus more and more phenomena could gradually be covered with ever increasing accuracy by a mosaic of interlocking models whose net number of unexplained parameters keeps decreasing." (Fritjof Capra, "The Turning Point: Science, Society, and the Turning Culture", 1982)

"[…] our present picture of physical reality, particularly in relation to the nature of time, is due for a grand shake up - even greater, perhaps, than that which has already been provided by present-day relativity and quantum mechanics." (Roger Penrose, "The Emperor’s New Mind", 1989)

"There are at least three (overlapping) ways that mathematics may contribute to science. The first, and perhaps the most important, is this: Since the mathematical universe of the mathematician is much larger than that of the physicist, mathematicians are able to go beyond existing frameworks and see geometrical or analytic structures unavailable to tie physicist. Instead of using the particular equations used previously to describe reality, a mathematician has at his disposal an unused world of differential equations, to be studied with no a priori constraints. New scientific phenomena, new discoveries, may thus generated. Understanding of the present knowledge may be deepened via the corresponding deductions. [...] The second way [...] has to do with the consolidation of new physical ideas. One may express this as the proof of consistency of physical theories. [...] mathematical foundations of quantum mechanics with Hilbert space, its operator theory, and corresponding differential equations. [...] The third way [...] is by describing reality in mathematical terms, or by simply constructing a mathematical model." (Steven Smale, "What is chaos?", 1990)

"Quantum mechanics taught that a particle was not a particle but a smudge, a traveling cloud of possibilities […]" (James Gleick, "Genius: The Life and Science of Richard Feynman, Epilogue", 1992)

"While many questions about quantum mechanics are still not fully resolved, there is no point in introducing needless mystification where in fact no problem exists. Yet a great deal of recent writing about quantum mechanics has done just that." (Murray Gell-Mann,"The Quark and the Jaguar", 1994) 

"And of course the space the wave function live in, and (therefore) the space we live in, the space in which any realistic understanding of quantum mechanics is necessarily going to depict the history of the world as playing itself out […] is configuration-space. And whatever impression we have to the contrary (whatever impression we have, say, of living in a three-dimensional space, or in a four dimensional spacetime) is somehow flatly illusory." (David Albert, "Elementary Quantum Metaphysics", 1996)

On Quantum Mechanics (-1974)

"There is thus a possibility that the ancient dream of philosophers to connect all Nature with the properties of whole numbers will some day be realized. To do so physics will have to develop a long way to establish the details of how the correspondence is to be made. One hint for this development seems pretty obvious, namely, the study of whole numbers in modern mathematics is inextricably bound up with the theory of functions of a complex variable, which theory we have already seen has a good chance of forming the basis of the physics of the future. The working out of this idea would lead to a connection between atomic theory and cosmology." (Paul A M Dirac, [Lecture delivered on presentation of the James Scott prize] 1939)

"The great revelation of the quantum theory was that features of discreteness were discovered in the Book of Nature, in a context in which anything other than continuity seemed to be absurd according to the views held until then." (Erwin Schrödinger, "What is Life?", 1944) 

"[In quantum mechanics] we have the paradoxical situation that observable events obey laws of chance, but that the probability for these events itself spreads according to laws which are in all essential features causal laws." (Max Born, "Natural Philosophy of Cause and Chance", 1949)

"Every object that we perceive appears in innumerable aspects. The concept of the object is the invariant of all these aspects. From this point of view, the present universally used system of concepts in which particles and waves appear simultaneously, can be completely justified. The latest research on nuclei and elementary particles has led us, however, to limits beyond which this system of concepts itself does not appear to suffice. The lesson to be learned from what I have told of the origin of quantum mechanics is that probable refinements of mathematical methods will not suffice to produce a satisfactory theory, but that somewhere in our doctrine is hidden a concept, unjustified by experience, which we must eliminate to open up the road." (Max Born, "The Statistical Interpretations of Quantum Mechanics", [Nobel lecture] 1954)

"[...] in quantum mechanics, we are not dealing with an arbitrary renunciation of a more detailed analysis of atomic phenomena, but with a recognition that such an analysis is to principle excluded." (Niels Bohr, "Atomic Theory and Human Knowledge", 1958)

"One might think one could measure a complex dynamical variable by measuring separately its real and pure imaginary parts. But this would involve two measurements or two observations, which would be alright in classical mechanics, but would not do in quantum mechanics, where two observations in general interfere with one another - it is not in general permissible to consider that two observations can be made exactly simultaneously, and if they are made in quick succession the first will usually disturb the state of the system and introduce an indeterminacy that will affect the second." (Ernst C K Stückelberg, "Quantum Theory in Real Hilbert Space", 1960) 

“[…] to the unpreoccupied mind, complex numbers are far from natural or simple and they cannot be suggested by physical observations. Furthermore, the use of complex numbers is in this case not a calculational trick of applied mathematics but comes close to being a necessity in the formulation of quantum mechanics.” (Eugene Wigner, “The Unreasonable Effectiveness of Mathematics in the Natural Sciences”, 1960) 

"It is widely believed that only those who can master the latest quantum mathematics can understand anything of what is happening. That is not so, provided one takes the long view, for no one can see far ahead. Against a historical background, the layman can understand what is involved, for example, in the fascinating challenge of continuity and discontinuity expressed in the antithesis of field and particle." (Lancelot L Whyte, "Essay on Atomism: From Democritus to 1960", 1961)

"It has been generally believed that only the complex numbers could legitimately be used as the ground field in discussing quantum-mechanical operators. Over the complex field, Frobenius' theorem is of course not valid; the only division algebra over the complex field is formed by the complex numbers themselves. However, Frobenius' theorem is relevant precisely because the appropriate ground field for much of quantum mechanics is real rather than complex." (Freeman Dyson, "The Threefold Way. Algebraic Structure of Symmetry Groups and Ensembles in Quantum Mechanics" , Journal of Mathematical Physics Vol. 3, 1962)

"Theoretical physicists live in a classical world, looking out into a quantum-mechanical world. The latter we describe only subjectively, in terms of procedures and results in our classical domain." (John S Bell, "Introduction to the hidden-variable question", 1971)

Steven Smale - Collected Quotes

"A regular curve on a Riemannian manifold is a curve with a continuously turning nontrivial tangent vector. A regular homotopy is a homotopy which at every stage is a regular curve, keeps end points and directions fixed and such that the tangent vector moves continuously with the homotopy. A regular curve is closed if its initial point and tangent  coincides with its end point and tangent." (Steven Smale,"Regular Curves on Riemann Manifolds", 1956)

"The stable manifolds of the critical points of a nice function can be thought of as the cells of a complex while the unstable manifolds are the dual cells. This structure has the advantage over previous structures that both the cells and the duals are differentiably imbedded in M. We believe that nice functions will replace much of the use of С triangulations and combinatorial methods in differential topology." (Steven Smale, "The generalized Poincare conjecture in higher dimensions", Bull. Amer. Math. Soc. 66, 1960)

"Certainly, the problems of combinatorial manifolds and the relationships between combinatorial and differentiable manifolds are legitimate problems in their own right. An example is the questionof existence and uniqueness of differentiable structures on a combinatorial manifold. However, we don't believe such problems are the goal of differential topology itself. This view seems justified by the fact that today one can substantially develop differential topology most simply without any reference to the combinatorial manifolds." (Steven Smale, "A survey of some recent developments in differential topology", 1961)

"It has turned out that the main theorems in differential topology did not depend on developments in combinatorial topology. In fact, the contrary is the case; the main theorems in differential topology inspired corresponding ones in combinatorial topology, or else have no combinatorial counterpart as yet (but there are also combinatorial theorem whose differentiable analogues are false)."  (Steven Smale, "A survey of some recent developments in differential topology", 1961)

"[...] it is clear that differential geometry, analysis and physics prompted the early development of differential topology (it is this that explains our admitted bias toward differential topology, that it lies close to the main stream of  mathematics). On the other hand, the combinatorial approach to manifolds was started because it was believed that these means would afford a useful attack on the differentiable case." (Steven Smale, "A survey of some recent developments in differential topology", 1961)

"Of course, there are a number of important problems left in differential topology that do not reduce in any sense to homotopy theory and topologists can never rest until these are settled. But, on the other hand, it seems that differential topology has reached such a satisfactory stage that, for it to continue its exciting pace, it must look toward the problems of analysis, the sources that led Poincare to its early development." (Steven Smale, "A survey of some recent developments in differential topology", 1961)

"We consider differential topology to be the study of differentiable manifolds and differentiable maps. Then, naturally, manifolds are considered equivalent if they are diffeomorphic, i.e., there exists a differentiable map from one to the other with a differentiable inverse." (Steven Smale, "A survey of some recent developments in differential topology", 1961)

"My definition of global analysis is simply the study of differential equations, both ordinary and partial, on manifolds and vector space bundles. Thus one might consider global analysis as differential equations from a global, or topological point of view." (Steven Smale, "What is global analysis?", American Mathematical Monthly Vol. 76 (1), 1969)

"To abstract the qualitative features of a differential equation on M, the concept of a phase portrait become important. Usually the phase portrait means the picture of the solution curves of the differential equation. [...] Then two differential equations on M have the same phase portrait if they are topologically equivalent. A definition of phase portrait is thus a topological  equivalence class of differential equations on M. A main goal of the qualitative study of ordinary differential equations is to obtain information on the phase portrait of differential equations." (Steven Smale, "What is global analysis?", American Mathematical Monthly Vol. 76 (1), 1969)

"A fairly general procedure for mathematical study of a physical system with explication of the space of states of that system. Now this space of states could reasonably be one of a number of mathematical objects. However, in my mind, a principal candidate for the state space should be a differentiable manifold; and in case the has a finite number of degrees of freedom, then this will be a finite dimensional manifold. Usually associated with physical is the notion of how a state progresses in time. The corresponding object is a dynamical system or a first order ordinary differential equation on the manifold of states." (Stephen Smale, "Personal perspectives on mathematics and mechanics", 1971)

"Related is the idea of structural stability and certain variations. This kind of is a property of a dynamical system itself (not of a or orbit) and asserts that nearby dynamical systems have the same structure. The 'same structure' can be defined in several interesting ways, but the basic idea is that two dynamical systems have 'the same structure' if they have the same gross behavior, or the same qualitative behavior. For example, the original definition of 'same structure' of two dynamical systems was that there was an orbit preserving continuous transformation between them. This yields the definition of structural stability proper. It is a recent theorem that every compact manifold admits structurally stable systems, and almost all gradient dynamical systems are structurally stable. But while there exists a rich set of structurally stable systems, there are also important examples which are not stable, and have good but weaker stability properties." (Stephen Smale, "Personal perspectives on mathematics and mechanics", 1971)

"A criticism commonly made of economic theory is its failure to make predictions of crises in the country or to anticipate correctly unemployment or inflation. One must be cautious in the social sciences about looking toward physics for answers. However, some comparisons with the physical sciences seem profitable in connection with the above criticism. In those sciences, where theory itself is in a far more advanced state, limitations can be seen un a similar way." (Steven Smale, "Dynamics in General Equilibrium Theory", The American Economic Review Vol. 66 (2), 1976) 

"After all of this is said, equilibrium theory will eventually stand or fall, depending on its truth as an important idealization of actual economic systems or as a model with values of justice, of efficient distribution and of utilization of resources." (Steven Smale, "Dynamics in General Equilibrium Theory", The American Economic Review Vol. 66 (2), 1976) 

"In fact even the very best chess players don't analyze very many moves and certainly don't make future commitments. Their experience together with the environment at the moment (the position), some rules of thumb and some other considerations lead to decisions on the playing board." (Steven Smale, "Dynamics in General Equilibrium Theory", The American Economic Review Vol. 66 (2), 1976) 

"What is special about 'general equilibrium theory' as opposed to economic theory in general? For me the importance of general equilibrium theory lies in its traditions which are deeper than any other part of economic theory. These traditions, which of course derive from actual economic history, explain why equilibrium theory has played such a central role and possesses such depth in content. Many of the procedures and  mathematical methods used in other parts of economics grew out of developments in general equilibrium theory. Also  equilibrium theory plays an important role in communication within the economics profession." (Steven Smale, "Dynamics in General Equilibrium Theory", The American Economic Review Vol. 66 (2), 1976) 

"A diffeomorphism is a differentiable function between manifolds with a differentiable inverse. In particular Φ is one-to-one and onto; thus for example an allocation has a unique supporting price system, and this association is smooth over all optimal allocations." ((Steven Smale, "Global Analysis and Economics VI: Geometrical analysis of Pareto Optima and price equilibria under classical hypotheses", 1974)

"Dynamical systems have two kinds of classical attractors which persist under small perturbations of the differential equations. These are the stable equilibria and the stable nontrivial periodic solutions or oscillators. An important development of recent times is a new kind of attractor which is robust in the sense that its properties persist under perturbations of the differential equation (it is structurally stable). These new attractors are sometimes called strange attractors." (Steven Smale, "On the Problem of Reviving the Ergodic Hypothesis of Boltzmann and Birkhoff", [in "The Mathematics of Time"] 1980)

"Let me start by posing what I tike to call 'the fundamental problem of equilibrium theory': how is economic equilibrium attained? A dual question more commonly raised is: why is economic equilibrium stable? Behind these questions lie the problem of modeling economic processes and introducing dynamics into equilibrium theory. A successful attack here would give greater validity to equilibrium theory. It may be however that a resolution of this fundamental problem will require a recasting of the foundations of equilibrium theory." (Steven Smale, "Some Dynamics Questions in Mathematical Economics", 1980)

"Chaos is a characteristic of dynamics, that is, of time  evolution of a set of states of nature. Let me take time to be measured in discrete units. A state of nature will be idealized as a point in the two-dimensional plane." (Steven Smale, "What is chaos?", 1990)

"Here is the result of the horseshoe analysis that I found on that Copacabana beach. Consider all the points which, under the horseshoe mapping, stay in the square, i.e., don't drift out of our field of vision. These 'visual motions' correspond precisely to the set of all coin-flipping experiments! This discovery demonstrates the occurrence of unpredictability in fully non-linear motion and gives a mechanism of how determinism produces uncertainty."(Steven Smale, "What is chaos?", 1990)

"Instead of a state of nature evolving according to a mathematical fomula, the evolution is given geometrically. The full advantage of the geometrical point of view is beginning to appear. The more traditional way of dealing with dynamics was with the use of algebraic expressions. But a description given by formulae would be cumbersome. That form of description wouldn't have led me to insights or to perceptive analysis. My background as a topologist, trained to bend objects like squares, helped to make it possible to see the horseshoe." (Steven Smale, "What is chaos?", 1990)

"It is astounding how important scientific ideas can get lost, even when they are aired by leading mathematicians of the preceding decades." (Steven Smale, "What is chaos?", 1990)

"Let us tum to the question, 'What is chaos?' Certainly the typical answer, 'sensitive dependence on initial conditions', is reasonable. But to be relevant to physics, more is required: the property should be robust or preserved under perturbations of the system. A response &lat involves a mechanism and meets the criteria is: 'Chaos is the presence of a (transversal) homoclinic point'." (Steven Smale, "What is chaos?", 1990)

"Summarizing, we have indicated why homoclinic points imply chaos. The converse is less formalizable, but I believe it is basically true. However,  there is a stronger notion of chaos, which we might call full chaos, where the set of initial conditions with sensitive dependence has positive or even full measure. In this case the corresponding homoclinic point could well be associated with a chaotic attractor, such as the Lorenz attractor. " (Steven Smale, "What is chaos?", 1990)

"The horseshoe is a natural consequence of a geometrical way of looking at the equations of Cartwright-Littlewood and Levinson. It helps understand the mechanism of chaos, and explain the widespread unpredictability in dynamics." (Steven Smale, "What is chaos?", 1990)

"The laws of chance, with good reason, have traditionally been expressed in terms of flipping a coin. Guessing whether heads or tails is the outcome of a coin toss is the paradigm of pure chance. On the other hand it is a deterministic process that governs the whole motion of a real coin, and hence the result, heads or tails, depends only on very subtle factors of the initiation of the toss. This is 'sensitive dependence on initial conditions'." (Steven Smale, "What is chaos?", 1990)

"There are at least three (overlapping) ways that mathematics may contribute to science. The first, and perhaps the most important, is this: Since the mathematical universe of the mathematician is much larger than that of the physicist, mathematicians are able to go beyond existing frameworks and see geometrical or analytic structures unavailable to tie physicist. Instead of using the particular equations used previously to describe reality, a mathematician has at his disposal an unused world of differential equations, to be studied with no a priori constraints. New scientific phenomena, new discoveries, may thus generated. Understanding of the present knowledge may be deepened via the corresponding deductions. [...] The second way [...] has to do with the consolidation of new physical ideas. One may express this as the proof of consistency of physical theories. [...] mathematical foundations of quantum mechanics with Hilbert space, its operator theory, and corresponding differential equations. [...] The third way [...] is by describing reality in mathematical terms, or by simply constructing a mathematical model." (Steven Smale, "What is chaos?", 1990)

"This construction, the horseshoe, has some consequences. First, it yields the fact that homoclinic points do exist and gives a direct construction of them. Second, one obtains such a useful analysis of a general transversal homoclinic point that many properties follow, including sensitive dependence on initial conditions - 'a large number', anyway. Third, one can prove robustness of the horseshoe in a strong global sense structural stability)." (Steven Smale, "What is chaos?", 1990)

"At least in my own case, understanding mathematics doesn't come from reading or even listening. It comes from rethinking what I see or hear. I must redo the mathematics in the context of my particular background. And that background consists of many threads, some strong, some weak, some algebraic, some visual. My background is stronger in geometric analysis, but following a sequence of formulae gives me trouble. I tend to be slower than most mathematicians to understand an argument. The mathematical literature is useful in that it provides clues, and one can often use these clues to put together a cogent picture. When I have reorganized the mathematics in my own terms, then I feel an understanding, not before." (Stephen Smale, "Finding a Horseshoe on the Beaches of Rio", 2000)

"It is more a philosophy than mathematics, and even as a philosophy it doesn't explain the real world [...] as mathematics, it brings together two of the most basic ideas in modern math: the study of dynamic systems and the study of the singularities of maps. Together, they cover a very wide area - but catastrophe theory brings them together in an arbitrary and constrained way." (Stephen Smale)

On Mapping XI

"A logarithm is a mapping that allows you to multiply by adding." (George Lakoff & Rafael E Nuñez, "Where Mathematics Come From: How the Embodied Mind Brings Mathematics into Being", 2000)

"As an abstract mathematical function, log maps every positive real number onto a corresponding real number, and maps every product of positive real numbers onto a sum of real numbers. Of course, there can be no table for such a mapping, because it would be infinitely long. But abstractly, such a mapping can be characterized as outlined here. These constraints completely and uniquely determine every possible value of the mapping. But the constraints do not in provide an algorithm for computing such mappings for al1 the real numbers. Approximations to values for real numbers can be made to any degree of accuracy required by doing arithmetic operations on rational numbers." (George Lakoff & Rafael E Nuñez, "Where Mathematics Come From: How the Embodied Mind Brings Mathematics into Being", 2000)

"[…] most earlier attempts to construct a theory of complexity have overlooked the deep link between it and networks. In most systems, complexity starts where networks turn nontrivial. No matter how puzzled we are by the behavior of an electron or an atom, we rarely call it complex, as quantum mechanics offers us the tools to describe them with remarkable accuracy. The demystification of crystals-highly regular networks of atoms and molecules-is one of the major success stories of twentieth-century physics, resulting in the development of the transistor and the discovery of superconductivity. Yet, we continue to struggle with systems for which the interaction map between the components is less ordered and rigid, hoping to give self-organization a chance." (Albert-László Barabási, "Linked: How Everything Is Connected to Everything Else and What It Means for Business, Science, and Everyday Life", 2002)

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

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

"A continuous function preserves closeness of points. A discontinuous function maps arbitrarily close points to points that are not close. The precise definition of continuity involves the relation of distance between pairs of points. […] continuity, a property of functions that allows stretching, shrinking, and folding, but preserves the closeness relation among points." (Robert Messer & Philip Straffin, "Topology Now!", 2006)

"A function is also sometimes referred to as a map or a mapping. This terminology is common in mathematics, but less so in physics or other scientific fields. The idea of a mapping is useful if one wants to think of a function as acting on an entire set of input values." (David P Feldman,"Chaos and Fractals: An Elementary Introduction", 2012)

"To mathematicians who study them, moduli schemes are just as real as the regular objects in the world. […] The key idea is that an ordinary object can be studied using the set of functions on the object. […] Secondly, you can do algebra with these functions - that is, you can add or multiply two such functions and get a third function. This step makes the set of these functions into a ring. […] Then the big leap comes: If you start with any ring - that is, any set of entities that can be added and multiplied subject to the usual rules, you simply and brashly declare that this creates a new kind of geometric object. The points of the object can be given by maps from the ring to the real numbers, as in the example of the pot. But they may also be given by maps to other fields. A field is a special sort of ring in which division is possible." (David Mumford, ["The Best Writing of Mathematics: 2012"] 2012)

"Quantum theory can be thought of as the science of constructing wavefunctions and extracting predictions of measurable outcomes from them. […] The wavefunction is a little bit like a map - the best possible kind of map. It encodes all that can be said about a quantum system." (Hans C von Baeyer, "QBism: The future of quantum physics", 2016)

"Making a map is the physical production including conceptualization and design. Mapping is the mental interpretation of the world and although it must precede the map, it does not necessarily result in a map artifact. Mapping defined in mathematics is the correspondence between each element of a given set with each element of another. Similarly in linguistics emphasis is on the correspondence between associated elements of different types. For designers all drawings are maps - they represent relationships between objects, places and ideas." (Winifred E Newman, "Data Visualization for Design Thinking: Applied Mapping", 2017)

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

"A neural-network algorithm is simply a statistical procedure for classifying inputs (such as numbers, words, pixels, or sound waves) so that these data can mapped into outputs. The process of training a neural-network model is advertised as machine learning, suggesting that neural networks function like the human mind, but neural networks estimate coefficients like other data-mining algorithms, by finding the values for which the model’s predictions are closest to the observed values, with no consideration of what is being modeled or whether the coefficients are sensible." (Gary Smith & Jay Cordes, "The 9 Pitfalls of Data Science", 2019)

On Maps (1950-1959)

"A fundamental value in the scientific outlook is concern with the best available map of reality. The scientist will always seek a description of events which enables him to predict most by assuming least. He thus already prefers a particular form of behavior. If moralities are systems of preferences, here is at least one point at which science cannot be said to be completely without preferences. Science prefers good maps." (Anatol Rapoport, "Science and the goals of man: a study in semantic orientation", 1950)

"No map contains all the information about the territory it represents. The road map we get at the gasoline station may show all the roads in the state, but it will not as a rule show latitude and longitude. A physical map goes into details about the topography of a country but is indifferent to political boundaries. Furthermore, the scale of the map makes a big difference. The smaller the scale the less features will be shown." (Anatol Rapoport, "Science and the goals of man: a study in semantic orientation", 1950)

"Many proofs in mathematics have been actually found by extremely roundabout processes. A man starts to prove this theorem and he finds that he wanders all over the map. He starts off and prove a good many results which don’t seem to be leading anywhere and then eventually ends up by the back door on the solution of the given problem." (Claude E Shannon, "Creative Thinking", 1952)

"Good design looks right. It is simple (clear and uncomplicated). Good design is also elegant, and does not look contrived. A map should be aesthetically pleasing, thought provoking, and communicative."  (Arthur H Robinson, "Elements of Cartography", 1953)

"The design process involves a series of operations. In map design, it is convenient to break this sequence into three stages. In the first stage, you draw heavily on imagination and creativity. You think of various graphic possibilities, consider alternative ways." (Arthur H Robinson, "Elements of Cartography", 1953)

"If words are not things, or maps are not the actual territory, then, obviously, the only possible link between the objective world and the linguistic world is found in structure, and structure alone. The only usefulness of a map or a language depends on the similarity of structure between the empirical world and the map-languages. If the structure is not similar, then the traveller or speaker is led astray, which, in serious human life-problems, must become always eminently harmful. If the structures are similar, then the empirical world becomes 'rational' to a potentially rational being, which means no more than that verbal, or map-predicted characteristics, which follow up the linguistic or mapstructure, are applicable to the empirical world." (Alfred Korzybski, "Science and Sanity: An Introduction to Non-Aristotelian Systems and General Semantics", 1958)

On Maps (1960-1969)

"Category theory is an embodiment of Klein’s dictum that it is the maps that count in mathematics. If the dictum is true, then it is the functors between categories that are important, not the categories. And such is the case. Indeed, the notion of category is best excused as that which is necessary in order to have the notion of functor. But the progression does not stop here. There are maps between functors, and they are called natural transformations." (Peter Freyd, "The theories of functors and models", 1965)

"Design problems - generating or discovering alternatives - are complex largely because they involve two spaces, an action space and a state space, that generally have completely different structures. To find a design requires mapping the former of these on the latter. For many, if not most, design problems in the real world systematic algorithms are not known that guarantee solutions with reasonable amounts of computing effort. Design uses a wide range of heuristic devices - like means-end analysis, satisficing, and the other procedures that have been outlined - that have been found by experience to enhance the efficiency of search. Much remains to be learned about the nature and effectiveness of these devices." (Herbert A Simon, "The Logic of Heuristic Decision Making", [in "The Logic of Decision and Action"], 1966)

"A diagram thus enables us to discover the internal categorization which characterizes the information being processed in a much shorter time than does a map. […] A diagram permits the rapid and precise internal processing of information having three components, but it does not permit introducing the information into a universal system of visual memorization and geographic comparison. It is a closed graphic system, limited solely to the information being processed. […] In a diagram, one begins by attributing a meaning to the planar dimensions, then one plots the correspondences." (Jacques Bertin, "Semiology of graphics", 1967)

"Adaptive system - whether on the biological, psychological, or sociocultural level - must manifest (1) some degree of 'plasticity' and 'irritability' vis-a-vis its environment such that it carries on a constant interchange with acting on and reacting to it; (2) some source or mechanism for variety, to act as a potential pool of adaptive variability to meet the problem of mapping new or more detailed variety and constraints in a changeable environment; (3) a set of selective criteria or mechanisms against which the 'variety pool' may be sifted into those variations in the organization or system that more closely map the environment and those that do not; and (4) an arrangement for preserving and/or propagating these 'successful' mappings." (Walter F Buckley," Sociology and modern systems theory", 1967)

"To analyse graphic representation precisely, it is helpful to distinguish it from musical, verbal and mathematical notations, all of which are perceived in a linear or temporal sequence. The graphic image also differs from figurative representation essentially polysemic, and from the animated image, governed by the laws of cinematographic time. Within the boundaries of graphics fall the fields of networks, diagrams and maps. The domain of graphic imagery ranges from the depiction of atomic structures to the representation of galaxies and extends into the spheres of topography and cartography."  (Jacques Bertin, "Semiology of graphics", 1967)

"A manifold can be given by specifying the coordinate ranges of an atlas, the images in those coordinate ranges of the overlapping parts of the coordinate domains, and the coordinate transformations for each of those overlapping domains. When a manifold is specified in this way, a rather tricky condition on the specifications is needed to give the Hausdorff property, but otherwise the topology can be defined completely by simply requiring the coordinate maps to be homeomorphisms." (Richard L Bishop & Samuel I Goldberg, "Tensor Analysis on Manifolds", 1968)

"Scientific research was much like prospecting: you went out and you hunted, armed with your maps and your instruments, but in the end your preparations did not matter, or even your intuition. You needed your luck, and whatever benefits accrued to the diligent, through sheer, grinding hard work." (Michael Crichton, "The Andromeda Strain", 1969)

On Maps (1970-1979)

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

"Everything we think we know about the world is a model. Every word and every language is a model. All maps and statistics, books and databases, equations and computer programs are models. So are the ways I picture the world in my head - my mental models. None of these is or ever will be the real world. […] Our models usually have a strong congruence with the world. That is why we are such a successful species in the biosphere. Especially complex and sophisticated are the mental models we develop from direct, intimate experience of nature, people, and organizations immediately around us." (Donella Meadows, "Limits to Growth", 1972)

"To do science is to search for repeated patterns, not simply to accumulate facts, and to do the science of geographical ecology is to search for patterns of plants and animal life that can be put on a map." (Robert H. MacArthur, "Geographical Ecology", 1972)

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

"A person is changed by the contingencies of reinforcement under which he behaves; he does not store the contingencies. In particular, he does not store copies of the stimuli which have played a part in the contingencies. There are no 'iconic representations' in his mind; there are no 'data structures stored in his memory'; he has no 'cognitive map' of the world in which he has lived. He has simply been changed in such a way that stimuli now control particular kinds of perceptual behavior." (Burrhus F Skinner, "About behaviorism", 1974)

"The ‘culture’ of a group or class, is the peculiar and distinctive ‘way of life’ of the group or class, the meanings, values and ideas embodied in institutions, in social relations, in systems of beliefs, in mores and customs, in the uses of objects and material life. Culture is the distinctive shapes in which this material and social organization of life expresses itself. A culture includes the ‘maps of meaning’ which make things intelligible to its members. These ‘maps of meaning’ are not simply carried around in the head: they are objectivated in the patterns of social organization and relationship through which the individual becomes a ‘social individual’. Culture is the way the social relations of a group are structured and shaped: but it is also the way those shapes are experienced, understood and interpreted." (John Clark et al "Subcultures, Cultures and Class", 1975)

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

"[…] there is an irreducible difference between the world and our experience of it. We as human beings do not operate directly on the world. Each of us creates a representation of the world in which we live - that is, we create a map or model which we use to generate our behavior. Our representation of the world determines to a large degree what our experience of the world will be, how we will perceive the world, what choices we will see available to us as we live in the world." (Richard Bandler & John Grinder, "The Structure of Magic", 1975)

"A cognitive map is a specific way of representing a person's assertions about some limited domain, such as a policy problem. It is designed to capture the structure of the person's causal assertions and to generate the consequences that follow front this structure. […]  a person might use his cognitive map to derive explanations of the past, make predictions for the future, and choose policies in the present." (Robert M Axelrod, "Structure of Decision: The cognitive maps of political elites", 1976)

"As we experience space, and construct representations of it, we know that it will be continuous, everything is somewhere, and no matter what other characteristics objects do not share, they always share relative location, that is, spatiality; hence the desirability of equating knowledge with space, an intellectual space. This assures an organization and basis for predictability, which are shared by absolutely everyone. This proposition appears to be so fundamental that apparently it is simply adopted a priori." (Arthur H Robinson & Barbara B Petchenik, "The Nature of Maps: Essays toward Understanding Maps and Mapping", 1976)

"Mapping is based on systems of assumptions, on logic, on human needs, and on human cognitive characteristics, very little of which has been recognized or discussed in cartography." (Arthur H Robinson & Barbara B Petchenik, "The Nature of Maps: Essays toward Understanding Maps and Mapping", 1976)

"The concepts a person uses are represented as points, and the causal links between these concepts are represented as arrows between these points. This gives a pictorial representation of the causal assertions of a person as a graph of points and arrows. This kind of representation of assertions as a graph will be called a cognitive map. The policy alternatives, all of the various causes and effects, the goals, and the ultimate utility of the decision maker can all be thought of as concept variables, and represented as points in the cognitive map. The real power of this approach ap pears when a cognitive map is pictured in graph form; it is then relatively easy to see how each of the concepts and causal relation ships relate to each other, and to see the overall structure of the whole set of portrayed assertions." (Robert Axelrod, "The Cognitive Mapping Approach to Decision Making" [in "Structure of Decision: The Cognitive Maps of Political Elites"], 1976)

"What you may call a graph, someone else may call a chart, for both terms are used for the same thing. Actually, however. the word 'chart' was originally used only for navigation maps and diagrams. Most people agree that it is best to leave the term 'chart' to the navigators." (Dyno Lowenstein, "Graphs", 1976)

"A map seems the type of conceptual object, yet the interesting thing is the grotesquely token foot it keeps in the world of the physical, having the unreality without the far-fetched appropriateness of the edibles in Communion, being a picture to the degree that the sacrament is a meal. For a feeling of thorough transcendence such unobvious relations between the model and the representation seem essential, and the flimsy connection between acres of soil and their image on the map makes reading one an erudite act." (Robert Harbison, "Eccentric Spaces", 1977)

"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)

"The types of graphics used in operating a business fall into three main categories: diagrams, maps, and charts. Diagrams, such as organization diagrams, flow diagrams, and networks, are usually intended to graphically portray how an activity should be, or is being, accomplished, and who is responsible for that accomplishment. Maps such as route maps, location maps, and density maps, illustrate where an activity is, or should be, taking place, and what exists there. [...] Charts such as line charts, column charts, and surface charts, are normally constructed to show the businessman how much and when. Charts have the ability to graphically display the past, present, and anticipated future of an activity. They can be plotted so as to indicate the current direction that is being followed in relationship to what should be followed. They can indicate problems and potential problems, hopefully in time for constructive corrective action to be taken." (Robert D Carlsen & Donald L Vest, "Encyclopedia of Business Charts", 1977)

"Because of its foundation in topology, catastrophe theory is qualitative, not quantitative. Just as geometry treated the properties of a triangle without regard to its size, so topology deals with properties that have no magnitude, for example, the property of a given point being inside or outside a closed curve or surface. This property is what topologists call 'invariant' -it does not change even when the curve is distorted. A topologist may work with seven-dimensional space, but he does not and cannot measure (in the ordinary sense) along any of those dimensions. The ability to classify and manipulate all types of form is achieved only by giving up concepts such as size, distance, and rate. So while catastrophe theory is well suited to describe and even to predict the shape of processes, its descriptions and predictions are not quantitative like those of theories built upon calculus. Instead, they are rather like maps without a scale: they tell us that there are mountains to the left, a river to the right, and a cliff somewhere ahead, but not how far away each is, or how large." (Alexander Woodcock & Monte Davis, "Catastrophe Theory", 1978)

"'Catastrophe theory' denotes both a purely mathematical discipline describing certain singularities of smooth maps, as well as the concerted effort to apply these theorems to a wide variety of problems in fields ranging from linguistics and psychology to embryology, evolution, physics, and engineering." (Héctor J Sussmann & Raphael S Zahler, "Catastrophe Theory as Applied to the Social and Biological Sciences: A Critique" Synthese Vol. 37 (2), 1978)

"[...] it seems (to many) that we cannot account for perception unless we suppose it provides us with an internal image (or model or map) of the external world, and yet what good would that image do us unless we have an inner eye to perceive it, and how are we to explain its capacity for perception? It also seems (to many) that understanding a heard sentence must be somehow translating it into some internal message, but how will this message be understood: by translating it into something else? The problem is an old one, and let’s call it Hume’s Problem, for while he did not state it explicitly, he appreciated its force and strove mightily to escape its clutches. (Daniel Dennett, "Brainstorms: Philosophical essays on mind and psychology", 1978)

"Mathematical equations and literary phrases are useful but they are no substitute for the spatial eloquence of the map." (Arthur H Robinson, "Uniqueness of the Map", American Cartographer Vol. 5 (1), 1978)

"The cognitive map is not a picture or image which 'looks like' what it represents; rather, it is an information structure from which map-like images can be reconstructed and from which behaviour dependent upon place information can be generated." (John O'Keefe & Lynn Nadel, "The Hippocampus as a Cognitive Map", 1978)