On Models: On Mathematical Models (1960-1969)

"In fact, the construction of mathematical models for various fragments of the real world, which is the most essential business of the applied mathematician, is nothing but an exercise in axiomatics." (Marshall Stone, cca 1960)

"[...] sciences do not try to explain, they hardly even try to interpret, they mainly make models. By a model is meant a mathematical construct which, with the addition of certain verbal interpretations, describes observed phenomena. The justification of such a mathematical construct is solely and precisely that it is expected to work - that is, correctly to describe phenomena from a reasonably wide area. Furthermore, it must satisfy certain aesthetic criteria - that is, in relation to how much it describes, it must be rather simple.” (John von Neumann, “Method in the physical sciences”, 1961)

"If the system exhibits a structure which can be represented by a mathematical equivalent, called a mathematical model, and if the objective can be also so quantified, then some computational method may be evolved for choosing the best schedule of actions among alternatives. Such use of mathematical models is termed mathematical programming." (George B Dantzig, "Linear Programming and Extensions", 1963)

"Mathematical statistics provides an exceptionally clear example of the relationship between mathematics and the external world. The external world provides the experimentally measured distribution curve; mathematics provides the equation (the mathematical model) that corresponds to the empirical curve. The statistician may be guided by a thought experiment in finding the corresponding equation." (Marshall J Walker, "The Nature of Scientific Thought", 1963)

"Statistics provides a quantitative example of the scientific process usually described qualitatively by saying that scientists observe nature, study the measurements, postulate models to predict new measurements, and validate the model by the success of prediction." (Marshall J Walker, "The Nature of Scientific Thought", 1963)

"After all, without the experiment - either a real one or a mathematical model - there would be no reason for a theory of probability." (Thornton C Fry,"Probability and Its Engineering Uses", 1965)

"Pedantry and sectarianism aside, the aim of theoretical physics is to construct mathematical models such as to enable us, from the use of knowledge gathered in a few observations, to predict by logical processes the outcomes in many other circumstances. Any logically sound theory satisfying this condition is a good theory, whether or not it be derived from 'ultimate' or 'fundamental' truth. It is as ridiculous to deride continuum physics because it is not obtained from nuclear physics as it would be to reproach it with lack of foundation in the Bible." (Clifford Truesdell & Walter Noll, "The Non-Linear Field Theories of Mechanics", 1965)

"A mathematical model is neither an hypothesis nor a theory. Unlike the scientific hypothesis, a model is not verifiable directly by experiment. For all models are both true and false. Almost any plausible proposed relation among aspects of nature is likely to be true in the sense that it occurs (although rarely and slightly). Yet all models leave out a lot and are in that sense false, incomplete, inadequate. The validation of a model is not that it is 'true' but that it generates good testable hypotheses relevant to important problems. A model may be discarded in favor of a more powerful one, but it usually is simply outgrown when the live issues are not any longer those for which it was designed." (Richard Levins, "The Strategy of Model Building in Population Biology", American Scientist 54(4), 1966)

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

"[…] mathematics is not portraying laws inherent in the design of the universe but is merely providing man-made schemes or models which we can use to deduce conclusions about our world only to the extent that the model is a good idealization." (Morris Kline, "Mathematics for the Nonmathematician", 1967)

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

"The mathematical models for many physical systems have manifolds as the basic objects of study, upon which further structure may be defined to obtain whatever system is in question. The concept generalizes and includes the special cases of the cartesian line, plane, space, and the surfaces which are studied in advanced calculus. The theory of these spaces which generalizes to manifolds includes the ideas of differentiable functions, smooth curves, tangent vectors, and vector fields. However, the notions of distance between points and straight lines (or shortest paths) are not part of the idea of a manifold but arise as consequences of additional structure, which may or may not be assumed and in any case is not unique." (Richard L Bishop & Samuel I Goldberg, "Tensor Analysis on Manifolds", 1968)

"The laws of nature 'discovered' by science are merely mathematical or mechanical models that describe how nature behaves, not why, nor what nature 'actually' is. Science strives to find representations that accurately describe nature, not absolute truths. This fact distinguishes science from religion." (George O Abell, "Exploration of the Universe", 1969)

Philip K Dick - Collected Quotes

"[…] anybody with a genuine system of prediction would be using it, not selling it." (Philip K Dick, "Solar Lottery", 1955)

"It’s the highest goal of man - the need to grow and advance [...] to find new things [...] to expand. To spread out, reach areas, experiences, comprehend and live in an evolving fashion. To push aside routine and repetition, to break out of mindless monotony and thrust forward. To keep moving on [...]" (Philip K Dick, "Solar Lottery", 1955)

"Even if all life on our planet is destroyed, there must be other life somewhere which we know nothing of. It is impossible that ours is the only world; there must be world after world unseen by us, in some region or dimension that we simply do not perceive." (Philip K Dick, "The Man in the High Castle", 1962)

"Sometimes one must try anything, he decided. It is no disgrace. On the contrary, it is a sign of wisdom, of recognizing the situation." (Philip K Dick, "The Man in the High Castle", 1962)

"These machines had become old and worn-out, had begun making mistakes; therefore they began to seem almost human." (Philip K Dick & Ray Nelson, "The Ganymede Takeover", 1967)

"A humanoid robot is like any other machine; it can fluctuate between being a benefit and a hazard very rapidly." (Philip K Dick, "Do Androids Dream of Electric Sheep?", 1968)

"No structure, even an artificial one, enjoys the process of entropy. It is the ultimate fate of everything, and everything resists it." (Philip K Dick, "Galactic Pot-Healer", 1969)

"To the paranoid, nothing is a surprise; everything happens exactly as he expected, and sometimes even more so. It all fits into his system. For us, though, there can be no system; maybe all systems - that is, any theoretical, verbal, symbolic, semantic, etc. formulation that attempts to act as an all-encompassing, all-explaining hypothesis of what the universe is about - are manifestations of paranoia. We should be content with the mysterious, the meaningless, the contradictory, the hostile, and most of all the unexplainably warm and giving." (Philip K Dick, "The Android and the Human", [speech] 1972)

"Man and the true God are identical—as the Logos and the true God are - but a lunatic blind creator and his screwed-up world separate man from God. That the blind creator sincerely imagines that he is the true God only reveals the extent of his occlusion." (Philip K Dick, "Valis", 1981)

"Reality is that which when you stop believing in it, it doesn’t go away." (Philip K Dick, "Valis", 1981)

On Models: On Mathematical Models (2010-2019)

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

"System dynamics is an approach to understanding the behaviour of over time. It deals with internal feedback loops and time delays that affect the behaviour of the entire system. It also helps the decision maker untangle the complexity of the connections between various policy variables by providing a new language and set of tools to describe. Then it does this by modeling the cause and effect relationships among these variables." (Raed M Al-Qirem & Saad G Yaseen, "Modelling a Small Firm in Jordan Using System Dynamics", 2010)

"There are actually two sides to the success of mathematics in explaining the world around us (a success that Wigner dubbed ‘the unreasonable effectiveness of mathematics’), one more astonishing than the other. First, there is an aspect one might call ‘active’. When physicists wander through nature’s labyrinth, they light their way by mathematics - the tools they use and develop, the models they construct, and the explanations they conjure are all mathematical in nature. This, on the face of it, is a miracle in itself. […] But there is also a ‘passive’ side to the mysterious effectiveness of mathematics, and it is so surprising that the 'active' aspect pales by comparison. Concepts and relations explored by mathematicians only for pure reasons - with absolutely no application in mind - turn out decades (or sometimes centuries) later to be the unexpected solutions to problems grounded in physical reality!" (Mario Livio, "Is God a Mathematician?", 2011)

"A catastrophe is a universal unfolding of a singular function germ. The singular function germs are called organization centers of the catastrophes. [...] Catastrophe theory is concerned with the mathematical modeling of sudden changes - so called 'catastrophes' - in the behavior of natural systems, which can appear as a consequence of continuous changes of the system parameters. While in common speech the word catastrophe has a negative connotation, in mathematics it is neutral." (Werner Sanns, "Catastrophe Theory" [Mathematics of Complexity and Dynamical Systems, 2012])

"An important aspect of the global theory of dynamical systems is the stability of the orbit structure as a whole. The motivation for the corresponding theory comes from applied mathematics. Mathematical models always contain simplifying assumptions. Dominant features are modeled; supposed small disturbing forces are ignored. Thus, it is natural to ask if the qualitative structure of the set of solutions - the phase portrait - of a model would remain the same if small perturbations were included in the model. The corresponding mathematical theory is called structural stability." (Carmen Chicone, "Stability Theory of Ordinary Differential Equations" [Mathematics of Complexity and Dynamical Systems, 2012])

"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. If you cannot answer the homogeneity question, then you will not know if you have one probability model or many. [...] 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)

"Models do not and need not match reality in all of its aspects and details to be adequate. A mathematical model is usually developed for a specific class of target systems, and its validity is determined relative to its intended applications. A model is considered valid within its intended domain of applicability provided that its predictions in that domain fall within an acceptable range of error, specified prior to the model’s development or identification." (Zoltan Domotor, "Mathematical Models in Philosophy of Science" [Mathematics of Complexity and Dynamical Systems, 2012])

"Simplified description of a real world system in mathematical terms, e. g., by means of differential equations or other suitable mathematical structures." (Benedetto Piccoli, Andrea Tosin, "Vehicular Traffic: A Review of Continuum Mathematical Models" [Mathematics of Complexity and Dynamical Systems, 2012])

"Stated loosely, models are simplified, idealized and approximate representations of the structure, mechanism and behavior of real-world systems. From the standpoint of set-theoretic model theory, a mathematical model of a target system is specified by a nonempty set - called the model’s domain, endowed with some operations and relations, delineated by suitable axioms and intended empirical interpretation." (Zoltan Domotor, "Mathematical Models in Philosophy of Science" [Mathematics of Complexity and Dynamical Systems, 2012])

"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)

"The standard view among most theoretical physicists, engineers and economists is that mathematical models are syntactic (linguistic) items, identified with particular systems of equations or relational statements. From this perspective, the process of solving a designated system of (algebraic, difference, differential, stochastic, etc.) equations of the target system, and interpreting the particular solutions directly in the context of predictions and explanations are primary, while the mathematical structures of associated state and orbit spaces, and quantity algebras – although conceptually important, are secondary." (Zoltan Domotor, "Mathematical Models in Philosophy of Science" [Mathematics of Complexity and Dynamical Systems, 2012])

"Mathematical modeling is a mixed blessing for economics. Mathematical modeling provides real advantages in terms of precision of thought, in seeing how assumptions are linked to conclusions, in generating and communicating insights, in generalizing propositions, and in exporting knowledge from one context to another. In my opinion, these advantages are monumental, far outweighing the costs. But the costs are not zero. Mathematical modeling limits what can be tackled and what is considered legitimate inquiry. You may decide, with experience, that the sorts of models in this book do not help you understand the economic phenomena that you want to understand." (David M Kreps, "Microeconomic Foundations I: Choice and Competitive Markets", 2013) 

"Mathematical symmetry is an idealized model. However, slightly imperfect symmetry requires explanation; it’s not enough just to say ‘it’s asymmetric’." (Ian Stewart, "Symmetry: A Very Short Introduction", 2013)

"To put it simply, we communicate when we display a convincing pattern, and we discover when we observe deviations from our expectations. These may be explicit in terms of a mathematical model or implicit in terms of a conceptual model. How a reader interprets a graphic will depend on their expectations. If they have a lot of background knowledge, they will view the graphic differently than if they rely only on the graphic and its surrounding text." (Andrew Gelman & Antony Unwin, "Infovis and Statistical Graphics: Different Goals, Different Looks", Journal of Computational and Graphical Statistics Vol. 22(1), 2013)

"When confronted with multiple models, I find it revealing to pose the resulting uncertainty as a two-stage lottery. For the purposes of my discussion, there is no reason to distinguish unknown models from unknown parameters of a given model. I will view each parameter configuration as a distinct model. Thus a model, inclusive of its parameter values, assigns probabilities to all events or outcomes within the model’s domain. The probabilities are often expressed by shocks with known distributions and outcomes are functions of these shocks. This assignment of probabilities is what I will call risk. By contrast there may be many such potential models. Consider a two-stage lottery where in stage one we select a model and in stage two we draw an outcome using the model probabilities. Call stage one model ambiguity and stage two risk that is internal to a model." (Lars P Hansen, "Uncertainty Outside and Inside Economic Models", [Nobel lecture] 2013)

“Mathematical modeling is the application of mathematics to describe real-world problems and investigating important questions that arise from it.” (Sandip Banerjee, “Mathematical Modeling: Models, Analysis and Applications”, 2014)

"Models can be: formulations, abstractions, replicas, idealizations, metaphors - and combinations of these. [...] Some mathematical models have been blindly used - their presuppositions as little understood as any legal fine print one ‘agrees to’ but never reads - with faith in their trustworthiness. The very arcane nature of some of the formulations of these models might have contributed to their being given so much credence. If so, we mathematicians have an important mission to perform: to help people who wish to think through the fundamental assumptions underlying models that are couched in mathematical language, making these models intelligible, rather than" (merely) formidable Delphic oracles." (Barry Mazur, "The Authority of the Incomprehensible" , 2014)

"But we also have to know that every model has its limitations. The model of natural numbers and their sums is very successful to determine the number of objects in the union of two different groups of well-distinguished objects. But as a mathematical model, the arithmetic of numbers is not generally true but only validated and confirmed for certain well-controlled situations. […] If a model makes valid predictions in many concrete cases, if it already has been applied and tested successfully in many situations, we have some right to trust in that model. By now, we believe in the model 'natural numbers and their arithmetic' and in its predictions without having to check it every time. We do not expect that the result might be wrong; hence the verification step is not needed any longer for validating the model. If the model had a flaw, it would have been eliminated already in the past." (Alfred S Posamentier & Bernd Thaller, "Numbers: Their tales, types, and treasures", 2015)

"Design is the process of taking something that appears in the mind’s eye, modeling it in one or more of a number of ways, predicting how that thing will behave if it is made, and then making it, sometimes modifying the design as we make it. Design is what engineering is about. Furthermore, modeling is how engineering design is done. This includes mental models, mathematical models, computer models, plans and drawings, written language, and" (sometimes) physical models." (William M Bulleit, "The Engineering Way of Thinking: The Idea", Structure [magazine], 2015)

"[…] the usefulness of mathematics is by no means limited to finite objects or to those that can be represented with a computer. Mathematical concepts depending on the idea of infinity, like real numbers and differential calculus, are useful models for certain aspects of physical reality." (Alfred S Posamentier & Bernd Thaller, "Numbers: Their tales, types, and treasures", 2015)

"There are several reasons why reaction-diffusion systems have been a popular choice among mathematical modelers of spatio-temporal phenomena. First, their clear separation between non-spatial and spatial dynamics makes the modeling and simulation tasks really easy. Second, limiting the spatial movement to only diffusion makes it quite straightforward to expand any existing non-spatial dynamical models into spatially distributed ones. Third, the particular structure of reaction-diffusion equations provides aneasy shortcut in the stability analysis (to be discussed in the next chapter). And finally, despite the simplicity of their mathematical form, reaction-diffusion systems can show strikingly rich, complex spatio-temporal dynamics. Because of these properties, reaction-diffusion systems have been used extensively for modeling self-organization of spatial patterns." (Hiroki Sayama, "Introduction to the Modeling and Analysis of Complex Systems", 2015)

“A mathematical model is a mathematical description (often by means of a function or an equation) of a real-world phenomenon such as the size of a population, the demand for a product, the speed of a falling object, the concentration of a product in a chemical reaction, the life expectancy of a person at birth, or the cost of emission reductions. The purpose of the model is to understand the phenomenon and perhaps to make predictions about future behavior. [...] A mathematical model is never a completely accurate representation of a physical situation - it is an idealization." (James Stewart, “Calculus: Early Transcedentals” 8th Ed., 2016)

"Eventually, mechanical models failed too. They were duly abandoned, and replaced by much more abstract mathematical models. Compared to their predecessors, mathematical models are Spartan affairs. They consist of equations and formulas without the texture, the color, the visual detail - without the rich appeal - of their mechanical relatives. […] But what a mathematical model lacks in charm, it more than makes up for in generality and predictive power." (Hans C von Baeyer, "QBism: The future of quantum physics", 2016)

"The goal of physics is to explain the workings of the nonliving world. At first, philosophers described the properties of real objects: the wandering of planets across the night sky, the formation of ice, or the sound of a lyre. When attention turned to things that couldn’t be seen or measured so easily, physicists invented mechanical models to take the place of real things." (Hans C von Baeyer, "QBism: The future of quantum physics", 2016)

"A model may be defined as a substitute of any object or system. […] A mental image used in thinking is a model, and it is not the real system. A written description of a system is a model that presents one aspect of reality. The simulation model is logically complete and describes the dynamic behaviour of the system. Models can be broadly classified as (a) physical models and (b) abstract models [..] Mental models and mathematical models are examples of abstract models." (Bilash K Bala et al, "System Dynamics: Modelling and Simulation", 2017)

"Data almost always contain uncertainty. This uncertainty may arise from selection of the items to be measured, or it may arise from variability of the measurement process. Drawing general conclusions from data is the basis for increasing knowledge about the world, and is the basis for all rational scientific inquiry. Statistical inference gives us methods and tools for doing this despite the uncertainty in the data. The methods used for analysis depend on the way the data were gathered. It is vitally important that there is a probability model explaining how the uncertainty gets into the data." (William M Bolstad & James M Curran, "Introduction to Bayesian Statistics" 3rd Ed., 2017)

"Different models serve different purposes. Setting up a model involves focusing on features of the phenomenon that are compatible with the methodology being proposed, and neglecting features that are not compatible with it. A mathematical model in applied science explicitly refrains from attempting to be a complete picture of the phenomenon being modeled." (Reuben Hersh, ”Mathematics as an Empirical Phenomenon, Subject to Modeling”, 2017)

"Mathematical modeling is the modern version of both applied mathematics and theoretical physics. In earlier times, one proposed not a model but a theory. By talking today of a model rather than a theory, one acknowledges that the way one studies the phenomenon is not unique; it could also be studied other ways. One's model need not claim to be unique or final. It merits consideration if it provides an insight that isn't better provided by some other model." (Reuben Hersh, ”Mathematics as an Empirical Phenomenon, Subject to Modeling”, 2017)

"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)

"The lack of direct control means the outside factors will be affecting the data. There is a danger that the wrong conclusions could be drawn from the experiment due to these uncontrolled outside factors. The important statistical idea of randomization has been developed to deal with this possibility. The unidentified outside factors can be 'averaged out' by randomly assigning each unit to either treatment or control group. This contributes variability to the data. Statistical conclusions always have some uncertainty or error due to variability in the data. We can develop a probability model of the data variability based on the randomization used. Randomization not only reduces this uncertainty due to outside factors, it also allows us to measure the amount of uncertainty that remains using the probability model. Randomization lets us control the outside factors statistically, by averaging out their effects." (William M Bolstad & James M Curran, "Introduction to Bayesian Statistics" 3rd Ed., 2017)

"The scientific method searches for cause-and-effect relationships between an experimental variable and an outcome variable. In other words, how changing the experimental variable results in a change to the outcome variable. Scientific modeling develops mathematical models of these relationships. Both of them need to isolate the experiment from outside factors that could affect the experimental results. All outside factors that can be identified as possibly affecting the results must be controlled." (William M Bolstad & James M Curran, "Introduction to Bayesian Statistics" 3rd Ed., 2017)

"When we use algebraic notation in statistical models, the problem becomes more complicated because we cannot 'observe' a probability and know its exact number. We can only estimate probabilities on the basis of observations." (David S Salsburg, "Errors, Blunders, and Lies: How to Tell the Difference", 2017)

"Some scientists (e.g., econometricians) like to work with mathematical equations; others (e.g., hard-core statisticians) prefer a list of assumptions that ostensibly summarizes the structure of the diagram. Regardless of language, the model should depict, however qualitatively, the process that generates the data - in other words, the cause-effect forces that operate in the environment and shape the data generated." (Judea Pearl & Dana Mackenzie, "The Book of Why: The new science of cause and effect", 2018)

"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)

"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)

On Models: On Mathematical Models (2000-2009)

"The role of graphs in probabilistic and statistical modeling is threefold: (1) to provide convenient means of expressing substantive assumptions; (2) to facilitate economical representation of joint probability functions; and (3) to facilitate efficient inferences from observations." (Judea Pearl,Causality: Models, Reasoning, and Inference", 2000)

"A mathematical model uses mathematical symbols to describe and explain the represented system. Normally used to predict and control, these models provide a high degree of abstraction but also of precision in their application." (Lars Skyttner, "General Systems Theory: Ideas and Applications", 2001)

“A model is an imitation of reality and a mathematical model is a particular form of representation. We should never forget this and get so distracted by the model that we forget the real application which is driving the modelling. In the process of model building we are translating our real world problem into an equivalent mathematical problem which we solve and then attempt to interpret. We do this to gain insight into the original real world situation or to use the model for control, optimization or possibly safety studies." (Ian T Cameron & Katalin Hangos, “Process Modelling and Model Analysis”, 2001)

“Formulation of a mathematical model is the first step in the process of analyzing the behaviour of any real system. However, to produce a useful model, one must first adopt a set of simplifying assumptions which have to be relevant in relation to the physical features of the system to be modelled and to the specific information one is interested in. Thus, the aim of modelling is to produce an idealized description of reality, which is both expressible in a tractable mathematical form and sufficiently close to reality as far as the physical mechanisms of interest are concerned.” (Francois Axisa, “Discrete Systems” Vol. I, 2001)

"[…] interval mathematics and fuzzy logic together can provide a promising alternative to mathematical modeling for many physical systems that are too vague or too complicated to be described by simple and crisp mathematical formulas or equations. When interval mathematics and fuzzy logic are employed, the interval of confidence and the fuzzy membership functions are used as approximation measures, leading to the so-called fuzzy systems modeling." (Guanrong Chen & Trung Tat Pham, "Introduction to Fuzzy Sets, Fuzzy Logic, and Fuzzy Control Systems", 2001)

"Modeling, in a general sense, refers to the establishment of a description of a system (a plant, a process, etc.) in mathematical terms, which characterizes the input-output behavior of the underlying system. To describe a physical system […] we have to use a mathematical formula or equation that can represent the system both qualitatively and quantitatively. Such a formulation is a mathematical representation, called a mathematical model, of the physical system." (Guanrong Chen & Trung Tat Pham, "Introduction to Fuzzy Sets, Fuzzy Logic, and Fuzzy Control Systems", 2001)

"One might think this means that imaginary numbers are just a mathematical game having nothing to do with the real world. From the viewpoint of positivist philosophy, however, one cannot determine what is real. All one can do is find which mathematical models describe the universe we live in. It turns out that a mathematical model involving imaginary time predicts not only effects we have already observed but also effects we have not been able to measure yet nevertheless believe in for other reasons. So what is real and what is imaginary? Is the distinction just in our minds?" (Stephen W Hawking, "The Universe in a Nutshell", 2001)

“What is a mathematical model? One basic answer is that it is the formulation in mathematical terms of the assumptions and their consequences believed to underlie a particular ‘real world’ problem. The aim of mathematical modeling is the practical application of mathematics to help unravel the underlying mechanisms involved in, for example, economic, physical, biological, or other systems and processes.” (John A Adam, “Mathematics in Nature”, 2003)

“Mathematical modeling is as much ‘art’ as ‘science’: it requires the practitioner to (i) identify a so-called ‘real world’ problem (whatever the context may be); (ii) formulate it in mathematical terms (the ‘word problem’ so beloved of undergraduates); (iii) solve the problem thus formulated (if possible; perhaps approximate solutions will suffice, especially if the complete problem is intractable); and (iv) interpret the solution in the context of the original problem.” (John A Adam, “Mathematics in Nature”, 2003)

"Do I claim that everything that is not smooth is fractal? That fractals suffice to solve every problem of science? Not in the least. What I'm asserting very strongly is that, when some real thing is found to be un-smooth, the next mathematical model to try is fractal or multi-fractal. A complicated phenomenon need not be fractal, but finding that a phenomenon is 'not even fractal' is bad news, because so far nobody has invested anywhere near my effort in identifying and creating new techniques valid beyond fractals. Since roughness is everywhere, fractals - although they do not apply to everything - are present everywhere. And very often the same techniques apply in areas that, by every other account except geometric structure, are separate." (Benoît Mandelbrot, "A Theory of Roughness", 2004) 

"Fuzzy models can provide good numerical approximation of functions as well as linguistic information over the behavior of the functions. […] Fuzzy models with embedded linguistic interpretability are useful to extract knowledge from data. This knowledge is represented as a set of IF–THEN rules where the antecedents and the consequences are semantically meaningful." (Jairo Espinosa et al, "Fuzzy Logic, Identification and Predictive Control", 2005)

"Although fiction is not fact, paradoxically we need some fictions, particularly mathematical ideas and highly idealized models, to describe, explain, and predict facts.  This is not because the universe is mathematical, but because our brains invent or use refined and law-abiding fictions, not only for intellectual pleasure but also to construct conceptual models of reality." (Mario Bunge, "Chasing Reality: Strife over Realism", 2006)

"Chaotic system is a deterministic dynamical system exhibiting irregular, seemingly random behavior. Two trajectories of a chaotic system starting close to each other will diverge after some time (such an unstable behavior is often called 'sensitive dependence on initial conditions'). Mathematically, chaotic systems are characterized by local instability and global boundedness of the trajectories. Since local instability of a linear system implies unboundedness (infinite growth) of its solutions, chaotic system should be necessarily nonlinear, i.e., should be described by a nonlinear mathematical model." (Alexander L Fradkov, "Cybernetical Physics: From Control of Chaos to Quantum Control", 2007)

"In order to understand how mathematics is applied to understanding of the real world it is convenient to subdivide it into the following three modes of functioning: model, theory, metaphor. A mathematical model describes a certain range of phenomena qualitatively or quantitatively. […] A (mathematical) metaphor, when it aspires to be a cognitive tool, postulates that some complex range of phenomena might be compared to a mathematical construction." (Yuri I Manin," Mathematics as Metaphor: Selected Essays of Yuri I. Manin", 2007)

"The dichotomy of mathematical vs. statistical modeling says more about the culture of modeling and how different disciplines go about thinking about models than about how we should actually model ecological systems. A mathematician is more likely to produce a deterministic, dynamic process model without thinking very much about noise and uncertainty (e.g. the ordinary differential equations that make up the Lotka-Volterra predator prey model). A statistician, on the other hand, is more likely to produce a stochastic but static model, that treats noise and uncertainty carefully but focuses more on static patterns than on the dynamic processes that produce them (e.g. linear regression)." (Ben Bolker, "Ecological Models and Data in R", 2007)

"A science presents us with representations of the phenomena through artifacts, both abstract, such as theories and mathematical models, and concrete such as graphs, tables, charts, and ‘table-top’ models. These representations do not form a haphazard compilation though any unity, in the range of representations made available, is visible mainly at the more abstract levels." (Bas C van Fraassen, "Scientific Representation: Paradoxes of Perspective", 2008)

"It is impossible to construct a model that provides an entirely accurate picture of network behavior. Statistical models are almost always based on idealized assumptions, such as independent and identically distributed (i.i.d.) interarrival times, and it is often difficult to capture features such as machine breakdowns, disconnected links, scheduled repairs, or uncertainty in processing rates." (Sean Meyn, "Control Techniques for Complex Networks", 2008)

"Therefore, mathematical ecology does not deal directly with natural objects. Instead, it deals with the mathematical objects and operations we offer as analogs of nature and natural processes. These mathematical models do not contain all information about nature that we may know, but only what we think are the most pertinent for the problem at hand. In mathematical modeling, we have abstracted nature into simpler form so that we have some chance of understanding it. Mathematical ecology helps us understand the logic of our thinking about nature to help us avoid making plausible arguments that may not be true or only true under certain restrictions. It helps us avoid wishful thinking about how we would like nature to be in favor of rigorous thinking about how nature might actually work. (John Pastor, "Mathematical Ecology of Populations and Ecosystems", 2008)

"Much of the recorded knowledge of physics and engineering is written in the form of mathematical models. These mathematical models form the foundations of our understanding of the universe we live in. Furthermore, nearly all of the existing technology, in one way or another, rests on these models. To the extent that we are surrounded by evidence of the technology working and being reliable, human confidence in the validity of the underlying mathematical models is all but unshakable." (Jerzy A Filar, "Mathematical Models", 2009)

"To understand, how noise is related to scale-freeness, we have to do some mathematics again. Noise is usually characterized by a mathematical trick. The seemingly random fluctuation of the signal is regarded as a sum of sinusoidal waves. The components of the million waves giving the final noise structure are characterized by their frequency. To describe noise, we plot the contribution (called spectral density) of the various waves we use to model the noise as a function of their frequency. This transformation is called a Fourier transformation [...]" (Péter Csermely, "Weak Links: The Universal Key to the Stabilityof Networks and Complex Systems", 2009)

On Literature: On Space (From Fiction to Science-Fiction)

"Hasheesh helped a great deal, and once sent him to a part of space where form does not exist, but where glowing gases study the secrets of existence. And a violet-coloured gas told him that this part of space was outside what he had called infinity. The gas had not heard of planets and organisms before, but identified Kuranes merely as one from the infinity where matter, energy, and gravitation exist." (Howard P Lovecraft, "Celephais", 1922)

"Man has natural three-dimensional limits, and he also has four-dimensional ones, considering time as an extension. When he reaches those limits, he ceases to grow and mature, and forms rigidly within the mold of those limiting walls. It is stasis, which is retrogression unless all else stands still as well. A man who reaches his limits is tending toward subhumanity. Only when he becomes superhuman in time and space can immortality become practical." (Henry Kuttner & C L Moore, "Time Enough", 1946)

"There are and have been worlds and cultures without end, each nursing the proud illusion that it is unique in space and time. There have been men without number suffering from the same megalomania; men who imagined themselves unique, irreplaceable, irreproducible. There will be more [...] more plus infinity." (Alfred Bester, "The Demolished Man", 1953)

"The 'romance' of space - drivel written in the old days. When you’re not blasting, you float in a cramped hotbox, crawl through dirty mazes of greasy pipe and cable to tighten a lug, scratch your arms and bark your shins, get sick and choked up because no gravity helps your gullet get the food down." (Walter M Miller Jr., "Death of a Spaceman", 1954)

"He jaunted up the geodesic lines of space-time to an Elsewhere and an Elsewhen. He arrived in chaos. He hung in a precarious para-Now for a moment and then tumbled back into chaos." (Alfred Bester, "The Stars My Destination", 1956)

"There is a fifth dimension beyond that which is known to Man. It is a dimension as vast as space and as timeless as infinity. It is the middle ground between light and shadow, between science and superstition, and it lies between the pit of man’s fears and the summit of his knowledge. This is the dimension of imagination. It is an area which we call ... The Twilight Zone." (Rod Serling, "The Twilight Zone" [TV series] 1959)

"It was a place without a single feature of the space-time matrix that he knew. It was a place where nothing yet had happened - an utter emptiness. There was neither light nor dark: there was nothing here but emptiness. There had never been anything in this place, nor was anything ever intended to occupy this place [...]" (Clifford D Simak, "Time is the Simplest Thing", 1961)

"We’re free out here, really free for the first time. We’re floating, literally. Gravity can’t bow our backs or break our arches or tame our ideas. You know, it’s only out here that stupid people like us can really think. The weightlessness gets our thoughts and we can sort them. Ideas grow out here like nowhere else - it’s the right environment for them. Anyone can get into space, if he wants to hard enough. The ticket is a dream." (Fritz Leiber," The Beat Cluster", 1961)

"[...] for the 5th dimension [...] you can travel through space without having to go the long way around.. .In other words a straight line is not the shortest distance between two points." (Madeleine L'Engle, "A Wrinkle in Time", 1962)

"The mathematicians and physics men have their mythology; they work alongside the truth, never touching it; their equations are false But the things work. Or, when gross error appears, they invent new ones; they drop the theory of waves In universal ether and imagine curved space." (Robinson Jeffers, "The Beginning and the End and Other Poems, The Great Wound", 1963) 

"Beyond a critical point within a finite space, freedom diminishes as numbers increase. This is as true of humans as it is of gas molecules in a sealed flask. The human question is not how many can possibly survive within the system, but what kind of existence is possible for those who do survive." (Frank Herbert, "Dune", 1965)

"[...] the universe was scrawled [...] along all its dimensions [...] space didn't exist and perhaps had never existed." (Italo Calvino, "A Sign in Space", 1965)

"His vessel found itself between two vortices of gravitation called Bakhrida and Scintilla; Bakhrida speeds up time, Scintilla on the other hand slows it down, and between them lies a zone of stagnation, in which the present, becalmed, flows neither backward nor forward. There Heptodius froze alive, and remains to this day, along with the countless frigates and galleons of other astromariners, pirates, and spaceswashers, not aging in the least, suspended in the silence and excruciating boredom that is Eternity." (Stanislaw Lem, "How Erg the Self-Inducing Slew a Paleface", 1965)

"We cannot predict the new forces, powers, and discoveries that will be disclosed to us when we reach the other planets and set up new laboratories in space. They are as much beyond our vision today as fire or electricity would be beyond the imagination of a fish." (Arthur C Clarke, "Space and the Spirit of Man", 1965)

"Someday, the real masters of space would be machines, not men - and he was neither. Already conscious of his destiny, he took a somber pride in his unique loneliness - the first immortal midway between two orders of creation.

He would, after all, be an ambassador; between the old and the new - between the creatures of carbon and the creatures of metal who must one day supersede them.

Both would have need of him in the troubled centuries that lay ahead." (Arthur C Clarke, "A Meeting with Medusa", 1971)

"It is tempting to wonder if our present universe, large as it is and complex though it seems, might not be merely the result of a very slight random increase in order over a very small portion of an unbelievably colossal universe which is virtually entirely in heat-death. Perhaps we are merely sliding down a gentle ripple that has been set up, accidently and very temporarily, in a quiet pond, and it is only the limitation of our own infinitesimal range of viewpoint in space and time that makes it seem to ourselves that we are hurtling down a cosmic waterfall of increasing entropy, a waterfall of colossal size and duration." (Isaac Asimov, 1976)

"If mankind were to continue in other than the present barbarism, a new path must be found, a new civilization based on some other method than technology. Space is an illusion, and time as well. There is no such factor as either time or space. We have been blinded by our own cleverness, blinded by false perceptions of those qualities that we term eternity and infinity. There is another factor that explains it all, and once this universal factor is recognized, everything grows simple. There is no longer any mystery, no longer any wonder, no longer any doubt; for the simplicity of it all lies before us [...]" (Clifford D Simak,"A Heritage of Stars", 1977)

"The catastrophe story, whoever may tell it, represents a constructive and positive act by the imagination rather than a negative one, an attempt to confront the terrifying void of a patently meaningless universe by challenging it at its own game. [. . .] Each one of these fantasies represents an arraignment of the finite, an attempt to dismantle the formal structure of time and space which the universe wraps around us at the moment we first achieve consciousness." (James G Ballard, "Cataclysms and Dooms" 1977)

"An infinity of universes swim in superspace, all passing through their own cycles of birth and death; some are novel, others repetitious; some produce macrolife, others do not; still others are lifeless. In time, macrolife will attempt to reach out from its cycles to other space-time bubbles, perhaps even to past cycles, which leave their echoes in superspace, and might be reached. In all these ambitions, only the ultimate pattern of development is unknown, drawing macrolife toward some future transformation still beyond its view. There are times when the oldest macrolife senses that vaster intelligences are peering in at it from some great beyond [...]" (George Zebrowski, "Macrolife: A Mobile Utopia", 1979)

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

"The dimension of the imagination is much more complex than those of time and space, which are very junior dimensions indeed." (Terry Pratchett, "The Colour of Magic", 1983)

"History too has an inertia. In the four dimensions of spacetime, particles (or events) have directionality; mathematicians, trying to show this, draw what they call ‘world lines' on graphs. In human affairs, individual world lines form a thick tangle, curling out of the darkness of prehistory and stretching through time: a cable the size of Earth itself, spiraling round the sun on a long curved course. That cable of tangled world lines is history. Seeing where it has been, it is clear where it is going 0 it is a matter of simple extrapolation." (Kim S Robinson, "Red Mars", 1992)

"Once we overcome our fear of being tiny, we find ourselves on the threshold of a vast and awesome Universe that utterly dwarfs - in time, in space, and in potential - the tidy anthropocentric proscenium of our ancestors." (Carl Sagan, "Pale Blue Dot: A Vision of the Human Future in Space", 1994)

On Literature: On Mazes (From Fiction to Science-Fiction)

"A mighty maze! but not without a plan [...]" (Alexander Pope, "An Essay on Man", 1733-34)

"Language gradually varies, and with it fade away the writings of authors who have flourished their allotted time; otherwise, the creative powers of genius would overstock the world, and the mind would be completely bewildered in the endless mazes of literature." (Washington Irving, "The Sketch Book of Geoffrey Crayon", 1819–1820)

"The 'romance' of space - drivel written in the old days. When you’re not blasting, you float in a cramped hotbox, crawl through dirty mazes of greasy pipe and cable to tighten a lug, scratch your arms and bark your shins, get sick and choked up because no gravity helps your gullet get the food down." (Walter M Miller Jr., "Death of a Spaceman", 1954)

"A promise is a direction taken, a self-limitation of choice. [...] if no direction is taken, if one goes nowhere, no change will occur. One’s freedom to choose and to change will be unused, exactly as if one were in jail, a jail of one’s own building, a maze in which no one way is better than any other." (Ursula K. Le Guin, "The Dispossessed: An Ambiguous Utopia", 1974)

"The best maze is the mind.' (Ursula K Le Guin, "Mazes", 1975)

"Any path that narrows future possibilities may become a lethal trap. Humans are not threading their way through a maze; they scan a vast horizon filled with unique opportunities. The narrowing viewpoint of the maze should appeal only to creatures with their noses buried in the sand." (Frank Herbert, "Children of Dune", 1976)

"There is in Fantastica a certain place from which one can go anywhere and which can be reached from anywhere. We call it the Temple of a Thousand Doors. No one has ever seen it from outside. The inside is a maze of doors. Anyone wishing to know it must dare to enter it." (Michael Ende, "The Neverending Story", 1979)

"The truth of life is that every year we get farther away from the essence that is born within us. We get shouldered with burdens, some of them good, some of them not so good. Things happen to us. Loved ones die. People get in wrecks and get crippled. People lose their way, for one reason or another. It's not hard to do, in this world of crazy mazes. Life itself does its best to take that memory of magic away from us." (Robert McCammon, "Boy's Life", 1991)

"The maze itself is the sum of a man's life: choices he makes, dreams he hangs on to. And there at the center, there's a legendary man who had been killed over and over again countless times, but always clawed his way back to life. The man returned for the last time and vanquished all his oppressors in a tireless fury. He built a house. Around that house he built a maze so complicated, only he could navigate through it. I reckon he'd seen enough of fighting." (The Adversary" Westworld (TV series), 2016)

"We depend on nature not only for our physical survival. We also need nature to show us the way home, the way out of the prison of our own minds. We got lost in doing, thinking, remembering, anticipating–lost in a maze of complexity and a world of problems." (Eckhart Tolle, "Stillness Speaks", 2003)

On Literature: On Forces (From Fiction to Science-Fiction)

"Every probability - and most of our common, working beliefs are probabilities - is provided with buffers at both ends, which break the force of opposite opinions clashing against it […]" (Oliver W Holmes, "The Autocrat of the Breakfast-Table", 1891) 

"The forceps of our minds are clumsy forceps, and crush the truth a little in taking hold of it." (Herbert G Wells, "Scepticism of the Instrument: A Modern Utopia", 1905)

"No honest historian can take part with - or against - the forces he has to study. To him even the extinction of the human race should be merely a fact to be grouped with other vital statistics." (Henry Adams, "The Education of Henry Adams", 1907)

"The humans have a curious force they call ambition. It drives them, and, through them, it drives us. This force which keeps them active, we lack. Perhaps, in time, we machines will acquire it." (John Wyndham, "The Lost Machine", 1932)

"It was in this world that we found in its most striking form a social disease which is perhaps the commonest of all world-diseases—namely, the splitting of the  population into two mutually unintelligible castes through the influence of economic forces." (Olaf Stapledon, "Star Maker", 1937)

"Scientific study and reflection had taught us that the known universe of three dimensions embraces the merest fraction of the whole cosmos of substance and energy. In this case an overwhelming preponderance of evidence from numerous authentic sources pointed to the tenacious existence of certain forces of great power and, so far as the human point of view is concerned, exceptional malignancy." (Howard P Lovecraft, "The Shunned House", 1937)

"It's not the machine itself that does the trick. The machine merely acts as an intermediary between the sensitive and the spiritual force. It is an extension of the sensitive. It magnifies the capability of the sensitive and acts as a link of some sort. It enables the sensitive to perform his function." (Clifford D Simak, "Way Station", 1963)

"There was a comfort in the thought, a strange sort of personal comfort in being able to believe that some intelligence might have solved the riddle of that mysterious equation of the universe. And how, perhaps, that mysterious equation might tie in with the spiritual force that was idealistic brother to time and space and all those other elemental factors that held the universe together." (Clifford D Simak, "Way Station", 1963)

"We cannot predict the new forces, powers, and discoveries that will be disclosed to us when we reach the other planets and set up new laboratories in space. They are as much beyond our vision today as fire or electricity would be beyond the imagination of a fish." (Arthur C Clarke, "Space and the Spirit of Man", 1965)

"The universe is full of matter and force. Yet in all that force, amongst all the bulks and gravities, the rains of cosmic light, the bombardment of energy - how little spirit, how small the decimal points of intelligence." (Ray Bradbury et al, "Mars and the Mind of Man", 1973)

"The force of gravity-though it is the first force with which we are acquainted, and though it is always with us, and though it is the one with a strength we most thoroughly appreciate-is by far the weakest known force in nature. It is first and rearmost." (Isaac Asimov, 1976)

"Belief is a force. It’s a weak force, by comparison with gravity; when it comes to moving mountains, gravity wins every time. But it still exists." (Terry Pratchett, "Pyramids", 1989)

"Of all the forces in the universe, the hardest to overcome is the force of habit. Gravity is easy-peasy by comparison." (Terry Pratchett, "Johnny and the Dead", 1993)

"Grief can have a quality of profound healing because we are forced to a depth of feeling that is usually below the threshold of awareness." (Stephen Levine, "Who Dies?: An Investigation of Conscious Living and Conscious Dying", 2012)

"Physics admits of a lovely unification, not just at the level of fundamental forces, but when considering its extent and implications. Classifications like "optics" or "thermodynamics" are just straitjackets, preventing physicists from seeing countless intersections." (Ted Chiang, "Arrival: Film tie-in", 2016)

On Measures (-1849)

"Numbers prime to one another are those which are measured by a unit alone as a common measure." (Euclid, "The Elements", Book VII)

"Measure, time and number are nothing but modes of thought or rather of imagination." (Baruch Spinoza, [Letter to Ludvicus Meyer] 1663)

"To measure motion, space is as necessary to be considered as time. [... They] are made use of to denote the position of finite: real beings, in respect one to another, in those infinite uniform oceans of duration and space." (John Locke, "An Essay Concerning Human Understanding", 1689)

"Most of our philosophical instruments are measures of effects. The progress made in natural philosophy increases every day by the number of these measures; by these it still continues to be improved." (George Adams, "Lectures on Natural and Experimental Philosophy" Vol. 1, 1794)

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

"The measure of the probability of an event is the ratio of the number of cases favourable to that event, to the total number of cases favourable or contrary, and all equally possible, or all of which have the same chance." (Siméon-Denis Poisson, "Recherches sur la Probabilités des Jugemens" ["An Investigation of the Laws of Thought"], 1837)

"Yet time and space are but inverse measures of the force of the soul. The spirit sports with time." (Ralph W Emerson, "Essays", 1841)

On Measures (1850-1899)

"Measure consists in the superposition of the magnitudes to be compared; it therefore requires a means of using one magnitude as the standard for another. In the absence of this, two magnitudes can only be compared when one is a part of the other; in which case we can only determine the more or less and not the how much." (Bernhard Riemann, "On the hypotheses which lie at the foundation of geometry", 1854)

"The purely formal sciences, logic and mathematics, deal with such relations which are independent of the definite content, or the substance of the objects, or at least can be. In particular, mathematics involves those relations of objects to each other that involve the concept of size, measure, number." (Hermann Hankel, "Theorie der Complexen Zahlensysteme", 1867)

"The union of the mathematician with the poet, fervor with measure, passion with correctness, this surely is the ideal." (William James, "Clifford's Lectures and Essays", 1879)

"Nothing that we can measure is inconceivably large or inconceivably small in physical science." (William T Kelvin, 1883)

"She [Nature] works with reference to no measure of time, no limit of space, and with an abundance of material not expressed by exhaustless." (John Burroughs, "Birds and Poets With Other Papers", 1884)

"I call a sign which stands for something merely because it resembles it, an icon. Icons are so completely substituted for their objects as hardly to be distinguished from them. Such are the diagrams of geometry. A diagram, indeed, so far as it has a general signification, is not a pure icon; but in the middle part of our reasonings we forget that abstractness in great measure, and the diagram is for us the very thing. So in contemplating a painting, there is a moment when we lose the consciousness that it is not the thing, the distinction of the real and the copy disappears, and it is for the moment a pure dream, - not any particular existence, and yet not general. At that moment we are contemplating an icon." (Charles S Peirce, "On The Algebra of Logic : A Contribution to the Philosophy of Notation" in The American Journal of Mathematics 7, 1885)

On Measure (1900-1949)

"Let us notice first of all, that every generalization implies in some measure the belief in the unity and simplicity of nature." (Jules H Poincaré, "Science and Hypothesis", 1905)

"So completely is nature mathematical that some of the more exact natural sciences, in particular astronomy and physics, are in their theoretic phases largely mathematical in character, while other sciences which have hitherto been compelled by the complexity of their phenomena and the inexactitude of their data to remain descriptive and empirical, are developing towards the mathematical ideal, proceeding upon the fundamental assumption that mathematical relations exist between the forces and the phenomena, and that nothing short, of the discovery and formulations of these relations would constitute definitive knowledge of the subject. Progress is measured by the closeness of the approximation to this ideal formulation." (Jacob W A Young, "The Teaching of Mathematics", 1907)

"Just as data gathered by an incompetent observer are worthless - or by a biased observer, unless the bias can be measured and eliminated from the result - so also conclusions obtained from even the best data by one unacquainted with the principles of statistics must be of doubtful value." (William F White, "A Scrap-Book of Elementary Mathematics: Notes, Recreations, Essays", 1908)

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

"Our environment may and should mean something towards us which is not to be measured with the tools of the physicist or described by the metrical symbols of the mathematician." (Arthur S Eddington, "Science and the Unseen World", 1929)

"Search for measurable elements among your phenomena, and then search for relations between these measures of physical quantities." (Alfred N Whitehead, "Science and the Modern World", 1929)

"The discoveries in physical science, the triumphs in invention, attest the value of the process of trial and error. In large measure, these advances have been due to experimentation." (Louis Brandeis, "Judicial opinions", 1932)

"[…] reality is a system, completely ordered and fully intelligible, with which thought in its advance is more and more identifying itself. We may look at the growth of knowledge […] as an attempt by our mind to return to union with things as they are in their ordered wholeness. […] and if we take this view, our notion of truth is marked out for us. Truth is the approximation of thought to reality […] Its measure is the distance thought has travelled […] toward that intelligible system […] The degree of truth of a particular proposition is to be judged in the first instance by its coherence with experience as a whole, ultimately by its coherence with that further whole, all comprehensive and fully articulated, in which thought can come to rest." (Brand Blanshard, "The Nature of Thought" Vol. II, 1939)

"Hence the awkward expression ‘negative entropy’ can be replaced by a better one: entropy, taken with the negative sign, is itself a measure of order. Thus the device by which an organism maintains itself stationary at a fairly high level of orderliness ( = fairly low level of entropy) really consists in continually sucking orderliness from its environment." (Erwin Schrödinger, "What is Life?", 1944)

On Measures (1950-1974)

"Just as entropy is a measure of disorganization, the information carried by a set of messages is a measure of organization. In fact, it is possible to interpret the information carried by a message as essentially the negative of its entropy, and the negative logarithm of its probability. That is, the more probable the message, the less information it gives. Clichés, for example, are less illuminating than great poems." (Norbert Wiener, "The Human Use of Human Beings", 1950)

"The belief in science has replaced in large measure, the belief in God. Even where religion was regarded as compatible with science, it was modified by the mentality of the believer in scientific truth." (Hans Reichenbach, "The Rise of Scientific Philosophy", 1951)

"Beauty had been born, not, as we so often conceive it nowadays, as an ideal of humanity, but as measure, as the reduction of the chaos of appearances to the precision of linear symbols. Symmetry, balance, harmonic division, mated and mensurated intervals - such were its abstract characteristics." (Herbert Read, "Icon and Idea: The Function of Art in the Development of Human Consciousness", 1955)

"Religion and science are the two conjugated faces of phases of one and the same act of complete knowledge - the only one which can embrace the past and future of evolution so as to contemplate, measure and fulfill them." (Pierre T de Chardin, "The Phenomenon of Man", 1955)

"Since we are assured that the all-wise Creator has observed the most exact proportions of number, weight and measure in the make of all things, the most likely way therefore to get any insight into the nature of those parts of the Creation which come within our observation must in all reason be to number, weigh and measure." (Stephen Hales, "Vegetable Staticks", 1961)

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

"If our model is to be at all realistic, it will also need to be rather complex, It will in fact be too complex for easy handling by the traditional analytic measures, even after suitable simplifications." (Charles P Bonini, "Simulation of Information and Decision System in the Firm" , 1963)

"It is a commonplace of modern technology that there is a high measure of certainty that problems have solutions before there is knowledge of how they are to be solved." (John K Galbraith, "The New Industrial State", 1967)

"The aim of science is not so much to search for truth, or even truths, as to classify our knowledge and to establish relations between observable phenomena in order to be able to predict the future in a certain measure and to explain the sequence of phenomena in relation to ourselves." (Pierre L du Noüy, "Between Knowing and Believing", 1967)

"The 'flow of information' through human communication channels is enormous. So far no theory exists, to our knowledge, which attributes any sort of unambiguous measure to this 'flow'." (Anatol Rapoport, "Modern Systems Research for the Behavioral Scientist", 1969)

On Measures (1975-1999)

 "The amount of information conveyed by the message increases as the amount of uncertainty as to what message actually will be produced becomes greater. A message which is one out of ten possible messages conveys a smaller amount of information than a message which is one out of a million possible messages. The entropy of communication theory is a measure of this uncertainty and the uncertainty, or entropy, is taken as the measure of the amount of information conveyed by a message from a source. The more we know about what message the source will produce, the less uncertainty, the less the entropy, and the less the information." (John R Pierce, "An Introduction to Information Theory: Symbols, Signals and Noise", 1979)

"The term closed loop-learning process refers to the idea that one learns by determining what s desired and comparing what is actually taking place as measured at the process and feedback for comparison. The difference between what is desired and what is taking place provides an error indication which is used to develop a signal to the process being controlled." (Harold Chestnut, 1984)

"Just like a computer, we must remember things in the order in which entropy increases. This makes the second law of thermodynamics almost trivial. Disorder increases with time because we measure time in the direction in which disorder increases."  (Stephen Hawking, "A Brief History of Time", 1988)

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

"It has long been appreciated by science that large numbers behave differently than small numbers. Mobs breed a requisite measure of complexity for emergent entities. The total number of possible interactions between two or more members accumulates exponentially as the number of members increases. At a high level of connectivity, and a high number of members, the dynamics of mobs takes hold. " (Kevin Kelly, "Out of Control: The New Biology of Machines, Social Systems and the Economic World", 1995)

"Quantify. If whatever it is you’re explaining has some measure, some numerical quantity attached to it, you’ll be much better able to discriminate among competing hypotheses. What is vague and qualitative is open to many explanations." (Carl Sagan, "The Demon-Haunted World: Science as a Candle in the Dark", 1995)

"Yet everything has a beginning, everything comes to an end, and if the universe actually began in some dense explosion, thus creating time and space, so time and space are themselves destined to disappear, the measure vanishing with the measured, until with another ripple running through the primordial quantum field, something new arises from nothingness once again." (David Berlinski, "A Tour of the Calculus", 1995)

"Probabilities aren't just numbers, and they aren't just frequencies-on-average. They are also rather like a substance that flows, dividing according to the likelihood of various outcomes, subdividing when several trials are performed in succession, and adding together when several outcomes are combined to give an event. This is a metaphor, but an accurate and powerful one. It is, in a sense, the metaphor that mathematicians formalise when they offer a definition of probability. In this sense, probability behaves like volume, mass, or area. The technical term is 'measure'. The technical definition of probability is 'a measure such that various nice things happen'. Probability is a quantity that flows through the conceptual maze of possible events, and it behaves just like water flowing through pipes." (Ian Stewart, "The Magical Maze: Seeing the World Through Mathematical Eyes", 1997)

On Measures (2000-2009)

"One measure of the depth of a physical theory is the extent to which it poses serious challenges to aspects of our worldview that had previously seemed immutable." (Brian Greene, "The Elegant Universe: Superstrings, Hidden Dimensions, and the Quest, for the Ultimate Theory", 2000)

"[…] interval mathematics and fuzzy logic together can provide a promising alternative to mathematical modeling for many physical systems that are too vague or too complicated to be described by simple and crisp mathematical formulas or equations. When interval mathematics and fuzzy logic are employed, the interval of confidence and the fuzzy membership functions are used as approximation measures, leading to the so-called fuzzy systems modeling." (Guanrong Chen & Trung Tat Pham, "Introduction to Fuzzy Sets, Fuzzy Logic, and Fuzzy Control Systems", 2001)

"One might think this means that imaginary numbers are just a mathematical game having nothing to do with the real world. From the viewpoint of positivist philosophy, however, one cannot determine what is real. All one can do is find which mathematical models describe the universe we live in. It turns out that a mathematical model involving imaginary time predicts not only effects we have already observed but also effects we have not been able to measure yet nevertheless believe in for other reasons. So what is real and what is imaginary? Is the distinction just in our minds?" (Stephen W Hawking, "The Universe in a Nutshell", 2001)

"The total disorder in the universe, as measured by the quantity that physicists call entropy, increases steadily steadily as we go from past to future. On the other hand, the total order in the universe, as measured by the complexity and permanence of organized structures, also increases steadily as we go from past to future." (Freeman J Dyson, [Page-Barbour lecture], 2004)

"All models (whether mental or those turned into computer maps/models) are developed using a particular lens of what we value - what we think is important to understand, or what performance we wish to develop or improve. Although organizations can build forum models focusing on the performance measure du jour, they would be well advised to use a systemic or integral framework for what to include." (Peggy Holman et al, "The Change Handbook", 2007)

"The total disorder in the universe, as measured by the quantity that physicists call entropy, increases steadily as we go from past to future. On the other hand, the total order in the universe, as measured by the complexity and permanence of organized structures, also increases steadily as we go from past to future." (Freeman J Dyson, "A Many-Colored Glass: Reflections on the Place of Life in the Universe", 2007)

"This new model of development would be based clearly on the goal of sustainable human well-being. It would use measures of progress that clearly acknowledge this goal. It would acknowledge the importance of ecological sustainability, social fairness, and real economic efficiency. Ecological sustainability implies recognizing that natural and social capital are not infinitely substitutable for built and human capital, and that real biophysical limits exist to the expansion of the market economy." (Robert Costanza, "Toward a New Sustainable Economy", 2008)

On Literature: On Infinitesimals (From Fiction to Science-Fiction)

"Science has gone down into the mines and coal-pits, and before the safety-lamp the Gnomes and Genii of those dark regions have disappeared [...] Sirens, mermaids, shining cities glittering at the bottom of quiet seas and in deep lakes, exist no longer; but in their place, Science, their destroyer, shows us whole coasts of coral reef constructed by the labours of minute creatures; points to our own chalk cliffs and limestone rocks as made of the dust of myriads of generations of infinitesimal beings that have passed away; reduces the very element of water into its constituent airs, and re-creates it at her pleasure." (Charles Dickens, Poetry of Science, [Book review of Robert Hunt]1848)

"A modern branch of mathematics, having achieved the art of dealing with the infinitely small, can now yield solutions in other more complex problems of motion, which used to appear insoluble. This modern branch of mathematics, unknown to the ancients, when dealing with problems of motion, admits the conception of the infinitely small, and so conforms to the chief condition of motion (absolute continuity) and thereby corrects the inevitable error which the human mind cannot avoid when dealing with separate elements of motion instead of examining continuous motion. In seeking the laws of historical movement just the same thing happens. The movement of humanity, arising as it does from innumerable human wills, is continuous. To understand the laws of this continuous movement is the aim of history." (Lev N Tolstoy, "War and Peace", 1867)

"Arriving at infinitesimals, mathematics, the most exact of sciences, abandons the process of analysis and enters on the new process of the integration of unknown, infinitely small, quantities." (Lev N Tolstoy, "War and Peace", 1867)

"Only by reducing this element of free will to the infinitesimal, that is, by regarding it as an infinitely small quantity, can we convince ourselves of the absolute inaccessibility of the causes, and then instead of seeking causes, history will take the discovery of laws as its problem." (Lev N Tolstoy, "War and Peace", 1867)

"Only by taking infinitesimally small units for observation (the differential of history, that is, the individual tendencies of men) and attaining to the art of integrating them (that is, finding the sum of these infinitesimals) can we hope to arrive at the laws of history."  (Lev N Tolstoy, "War and Peace", 1867)

"To-day alone was real. Never was man brought into contact with reality save through the evanescent emotions and sensations of that single moment, that infinitesimal fraction of a second, which was passing now - and it was in the insignificance of this moment, precisely, that religious persons must believe. So ran the teachings of all dead and lingering faiths alike. Here was, perhaps, only another instance of mankind's abhorrence of actualities; and man's quaint dislike of facing reality was here disguised as a high moral principle. That was why all art, which strove to make the sensations of a moment soul-satisfying, was dimly felt to be irreligious. For art performed what religion only promised." (James B Cabell, "The Cream of the Jest", 1917)

"Measured objectively, what a man can wrest from Truth by passionate striving is utterly infinitesimal. But the striving frees us from the bonds of the self and makes us comrades of those who are the best and the greatest." (Albert Einstein, cca. 1920s)

"It is venturesome to think that a coordination of words (philosophies are nothing more than that) can resemble the universe very much. It is also venturesome to think that of all these illustrious coordinations, one of them - at least in an infinitesimal way - does not resemble the universe a bit more than the others." (Jorge L Borges, "Discussion", 1932)

"To be so closely caught up in the teeth of things that they kill you, no matter how infinitesimally kill you, is, truly, to be a poet: and to be a poet in fact it is additionally necessary that you should possess the tongues and instruments with which to record this series of infinitesimal deaths." (George Barker,"Therefore All Poems Are Elegies", 1940)

"An ugliness unfurled in the moonlight and soft shadow and suffused the whole world. If I were an amoeba, he thought, with an infinitesimal body, I could defeat ugliness. A man isn’t tiny or giant enough to defeat anything." (Yukio Mishima, "The Sailor Who Fell from Grace with the Sea", 1963)

"The creative element in the mind of man [...] emerges in as mysterious a fashion as those elementary particles which leap into momentary existence in great cyclotrons, only to vanish again like infinitesimal ghosts." (Loren Eiseley, "The Night Country", 1971)

"It is tempting to wonder if our present universe, large as it is and complex though it seems, might not be merely the result of a very slight random increase in order over a very small portion of an unbelievably colossal universe which is virtually entirely in heat-death. Perhaps we are merely sliding down a gentle ripple that has been set up, accidently and very temporarily, in a quiet pond, and it is only the limitation of our own infinitesimal range of viewpoint in space and time that makes it seem to ourselves that we are hurtling down a cosmic waterfall of increasing entropy, a waterfall of colossal size and duration." (Isaac Asimov, 1976)

"I wonder if being sane means disregarding the chaos that is life, pretending only an infinitesimal segment of it is reality." (Rabih Alameddine, "Koolaids: The Art of War", 1998)

On Literature: On Infinite (From Fiction to Science-Fiction)

"We know that there is an infinite, and we know not its nature. As we know it to be false that numbers are finite, it is therefore true that there is a numerical infinity. But we know not of what kind; it is untrue that it is even, untrue that it is odd; for the addition of a unit does not change its nature; yet it is a number, and every number is odd or even (this certainly holds of every finite number). Thus, we may quite well know that there is a God without knowing what He is." (Blaise Pascal, "Pensées", 1670)

"Who then understands the reciprocal flux and reflux of the infinitely great and the infinitely small, the echoing of causes in the abysses of being, and the avalanches of creation?" (Victor Hugo, "Saint Denis", 1862) 

"Phenomena may well be suspected of anything, are capable of anything. Hypothesis proclaims the infinite; that is what gives hypothesis its greatness. Beneath the surface fact it seeks the real fact. It asks creation for her thoughts, and then for her second thoughts. The great scientific discoverers are those who hold nature suspect." (Victor Hugo, "The Toilers of the Sea", 1866)

"A modern branch of mathematics, having achieved the art of dealing with the infinitely small, can now yield solutions in other more complex problems of motion, which used to appear insoluble. This modern branch of mathematics, unknown to the ancients, when dealing with problems of motion, admits the conception of the infinitely small, and so conforms to the chief condition of motion (absolute continuity) and thereby corrects the inevitable error which the human mind cannot avoid when dealing with separate elements of motion instead of examining continuous motion. In seeking the laws of historical movement just the same thing happens. The movement of humanity, arising as it does from innumerable human wills, is continuous. To understand the laws of this continuous movement is the aim of history." (Lev N Tolstoy, "War and Peace", 1867)

"Arriving at infinitesimals, mathematics, the most exact of sciences, abandons the process of analysis and enters on the new process of the integration of unknown, infinitely small, quantities." (Lev N Tolstoy, "War and Peace", 1867)

"One microscopic glittering point; then another; and another, and still another; they are scarcely perceptible, yet they are enormous. This light is a focus; this focus, a star; this star, a sun; this sun, a universe; this universe, nothing. Every number is zero in the presence of the infinite." (Victor Hugo, "The Toilers of the Sea", 1874)

"Science says the first word on everything, and the last word on nothing." (Victor Hugo, "Things of the Infinite: Intellectual Autobiography", 1907)

"Hasheesh helped a great deal, and once sent him to a part of space where form does not exist, but where glowing gases study the secrets of existence. And a violet-coloured gas told him that this part of space was outside what he had called infinity. The gas had not heard of planets and organisms before, but identified Kuranes merely as one from the infinity where matter, energy, and gravitation exist." (Howard P Lovecraft, "Celephais", 1922)

"Revolution is everywhere, in everything. It is infinite. There is no final revolution, no final number. The social revolution is only one of an infinite number of numbers; the law of revolution is not a social law, but an immeasurably  greater one. It is a cosmic, universal law - like the laws of the conservation of energy and of the dissipation of energy (entropy)." (Yevgeny Zamiatin, "On Literature, Revolution, Entropy, and Other Matters", 1923)

"The function of man’s highest faculty, his reason, consists precisely of the continuous limitation of infinity, the breaking up of infinity into convenient, easily digestible portions - differentials. This is precisely what lends my field, mathematics, its divine beauty." (Yevgeny Zamiatin, "We", 1924)

"Your ancestor did not believe in a uniform, absolute time. He believed in an. infinite series of times, in a growing, dizzying net of divergent, convergent and parallel times. This network of times which approached one another, forked, broke off, or were unaware of one another for centuries, embraces all possibilities of time." (Jorge Luis Borges, "The Garden of Forking Paths", 1941)

"And time itself? Time was a never-ending medium that stretched into the future and the past - except there was no future and no past, but an infinite number of brackets, extending either way, each bracket enclosing its single phase of the Universe." (Clifford D Simak,"Ring Around the Sun", 1954)

"There is a fifth dimension beyond that which is known to Man. It is a dimension as vast as space and as timeless as infinity. It is the middle ground between light and shadow, between science and superstition, and it lies between the pit of man’s fears and the summit of his knowledge. This is the dimension of imagination. It is an area which we call ... The Twilight Zone." (Rod Serling, "The Twilight Zone" [TV series] 1959)

"When they [radio astronomers] grew weary at their electronic listening posts, when their eyes grew dim with looking at unrevealing dials and studying uneventful graphs, they could step outside their concrete cells and renew their dull spirits in communion with the giant mechanism they commanded, the silent, sensing instrument in which the smallest packets of energy, the smallest waves of matter, were detected in their headlong, eternal flight across the universe. It was the stethoscope with which they took the pulse of the all and noted the birth and death of stars, the probe with which, here on an insignificant planet of an undistinguished star on the edge of its galaxy, they explored the infinite." (James Gunn, "The Listeners", 1968)

"Well it's a matter of continuity. Most people's lives have ups and downs that are gradual, a sinuous curve with first derivatives at every point. They're the ones who never get struck by lightning. No real idea of cataclysm at all. But the ones who do get hit experience a singular point. a discontinuity in the curve of life - do you know what the time rate of change is at a cusp? Infinity, that's what! A-and right across the point, it's minus infinity! How's that for sudden change, eh?" (Thomas Pynehon, "Gravity's Rainbow", 1973)

"If mankind were to continue in other than the present barbarism, a new path must be found, a new civilization based on some other method than technology. Space is an illusion, and time as well. There is no such factor as either time or space. We have been blinded by our own cleverness, blinded by false perceptions of those qualities that we term eternity and infinity. There is another factor that explains it all, and once this universal factor is recognized, everything grows simple. There is no longer any mystery, no longer any wonder, no longer any doubt; for the simplicity of it all lies before us [...]" (Clifford D Simak,"A Heritage of Stars", 1977)

"It is impossible to import things into an infinite area, there being no outside to import things in from." (Douglas N Adams, "The Original Hitchhiker Radio Script, Fit the Fifth" , 1978)

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

"An infinity of universes swim in superspace, all passing through their own cycles of birth and death; some are novel, others repetitious; some produce macrolife, others do not; still others are lifeless. In time, macrolife will attempt to reach out from its cycles to other space-time bubbles, perhaps even to past cycles, which leave their echoes in superspace, and might be reached. In all these ambitions, only the ultimate pattern of development is unknown, drawing macrolife toward some future transformation still beyond its view. There are times when the oldest macrolife senses that vaster intelligences are peering in at it from some great beyond [...]" (George Zebrowski, "Macrolife: A Mobile Utopia", 1979)

"In a perfectly rational universe, infinities turn back on themselves [...]" (George Zebrowski, "Is Science Rational?", OMNI Magazine, 1994)

"Science, when it runs up against infinities, seeks to eliminate them, because a proliferation of entities is the enemy of explanation." (George Zebrowski, "Time Is Nothing But A Clock" , OMNI Magazine Vol. 17 (1), 1994)

On Structures: On Mazes (1950-1999)

"In mathematics […] we find two tendencies present. On the one hand, the tendency towards abstraction seeks to crystallise the logical relations inherent in the maze of materials [….] being studied, and to correlate the material in a systematic and orderly manner. On the other hand, the tendency towards intuitive understanding fosters a more immediate grasp of the objects one studies, a live rapport with them, so to speak, which stresses the concrete meaning of their relations." (David Hilbert, "Geometry and the imagination", 1952)

"Our language can be seen as an ancient city: a maze of little streets and squares, of old and new houses, and of houses with additions from various periods; and this surrounded by a multitude of new boroughs with straight regular streets and uniform houses." (Ludwig Wittgenstein, "Philosophical Investigations", 1953)

"In presenting a mathematical argument the great thing is to give the educated reader the chance to catch on at once to the momentary point and take details for granted: two trivialities omitted can add up to an impasse). The unpractised writer, even after the dawn of a conscience, gives him no such chance; before he can spot the point he has to tease his way through a maze of symbols of which not the tiniest suffix can be skipped." (John E Littlewood, "A mathematicians's miscellany", 1953)

"Man develops his way of anticipating events by construing, by scratching out his channels of thought. Thus he builds his own maze. His runways are the constructs he forms, each a two-way street, each essentially a pair of alternatives between which he can choose." (George A Kelly, "Man's construction of his alternatives", Assessment of human motives, 1958)

"With the help of physical theories we try to find our way through the maze of observed facts, to order and understand the world of our sense impressions." (Leopold Infeld, "The Evolution of Physics, Physics and Reality", 1961) 

"It is not that we propose a theory and Nature may shout NO; rather, we propose a maze of theories, and Nature may shout INCONSISTENT." (Imre Lakatos, "Falsification and the Methodology of Scientific Research Programmes", [in I. Lakatos and A. Musgrave (eds.), "Criticism and the Growth of Knowledge: Proceedings of the International Colloquium in the Philosophy of Science"] 1965)

"Depth First Search is especially appropriate for threading mazes, because it is possible to use it without having a map of the maze. It involves only local rules at nodes, plus a record of nodes and edges already used, so you can explore the graph and traverse it as you go. The name indicates the basic idea: give top priority to pushing deeper into the maze. The number of steps required is at most twice the number of passages in the maze."  (Ian Stewart, "The Magical Maze: Seeing the world through mathematical eyes", 1997)

"One of the best definitions of mathematics is 'the science of patterns'. Mathematics is how we detect, analyse, and classify regular patterns - be they numerical, geometric, or of some other kind. But what is a pattern? A pattern is a landmark in the magical maze. It's one of those things that you recognise when you see it, but it's not so easy to pin down the concept of a pattern once and for all with a neat, tidy, compact characterisation. In fact, the entire development of mathematics can be seen as a slow and erratic broadening of what we accept under the term 'pattern'." (Ian Stewart, "The Magical Maze: Seeing the World Through Mathematical Eyes", 1997)

"One very effective approach is to represent all the possible actions as a maze, and try to find a route through it. It is a logical maze rather than a real one, touched with that magic genius of mathematical transformations in which a problem that seems unassailable in one form becomes trivial in another, logically equivalent one. The idea is to represent the problem in a visual manner, using a diagram called a graph. A graph consists of a number of nodes (dots) linked by edges (lines), possibly with arrows on them. Each 'state' of the puzzle - position of the items of produce relative to the river - is represented by a node. Each 'legal' move between states is represented as an edge joining the corresponding nodes. If necessary, arrows can be added to the edges to show which is the starting state and which is the end state. The solution of the puzzle then reduces to tracing a path through its graph, starting from the initial state of the problem and finishing at the desired final state. The graph is a kind of conceptual map of the puzzle - a maze of possible states whose passages are the edges of the graph and whose junctions are its nodes." (Ian Stewart, "The Magical Maze: Seeing the World Through Mathematical Eyes", 1997)

"The analogy with threading a maze runs deeper than games and puzzles. It illuminates the whole of mathematics. Indeed, one way to think about mathematics is as an exercise in threading an elaborate, infinitely large maze. A logical maze. A maze of ideas, whose pathways represent 'lines of thought' from one idea to another. A maze which, despite its apparent complexity, has a definite 'geography', to which mathematicians are unusually attuned." (Ian Stewart, "The Magical Maze: Seeing the World Through Mathematical Eyes", 1997)

"We wrote down all the states and legal moves (here it turned out to be helpful to have a systematic notation, but that's not essential). Then we formed a graph whose nodes correspond to states and whose edges correspond to legal moves. The solution of the puzzle is then a path through the graph that joins the start to the finish. Such a path is usually obvious to the eye, provided the puzzle is sufficiently simple for the entire graph to be drawn. Puzzles of this type are really mazes, for a maze is just a graph drawn in a slightly different fashion. Metaphorically, they are logical mazes - you have to find the right sequence of moves to solve them. The graph turns the logical maze into a genuine maze, turning the metaphor into reality. The fact that solving the real maze also solves the logical maze is one of the magical features of the maze that is mathematics." (Ian Stewart, "The Magical Maze: Seeing the World Through Mathematical Eyes", 1997)

"When a book is being written, it is a maze of possibilities, most of which are never realised. Reading the resulting book, once all decisions have been taken, is like tracing one particular path through that maze. The writer's job is to choose that path, define it clearly, and make it as smooth as possible for those who follow. Mathematics is much the same. Mathematical ideas form a network. The interconnections between ideas are logical deductions. If we assume this, then that must follow - a logical path from this to that. When mathematicians try to understand a problem, they have to thread a maze of logic. The body of knowledge that we call mathematics is a catalogue of interesting excursions through the logical maze."(Ian Stewart, "The Magical Maze: Seeing the world through mathematical eyes", 1997)