On Data Analysis (1970-1979)

"Statistical methods are tools of scientific investigation. Scientific investigation is a controlled learning process in which various aspects of a problem are illuminated as the study proceeds. It can be thought of as a major iteration within which secondary iterations occur. The major iteration is that in which a tentative conjecture suggests an experiment, appropriate analysis of the data so generated leads to a modified conjecture, and this in turn leads to a new experiment, and so on." (George E P Box & George C Tjao, "Bayesian Inference in Statistical Analysis", 1973)

"Almost all efforts at data analysis seek, at some point, to generalize the results and extend the reach of the conclusions beyond a particular set of data. The inferential leap may be from past experiences to future ones, from a sample of a population to the whole population, or from a narrow range of a variable to a wider range. The real difficulty is in deciding when the extrapolation beyond the range of the variables is warranted and when it is merely naive. As usual, it is largely a matter of substantive judgment - or, as it is sometimes more delicately put, a matter of 'a priori nonstatistical considerations'." (Edward R Tufte, "Data Analysis for Politics and Policy", 1974)

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

"The use of statistical methods to analyze data does not make a study any more 'scientific', 'rigorous', or 'objective'. The purpose of quantitative analysis is not to sanctify a set of findings. Unfortunately, some studies, in the words of one critic, 'use statistics as a drunk uses a street lamp, for support rather than illumination'. Quantitative techniques will be more likely to illuminate if the data analyst is guided in methodological choices by a substantive understanding of the problem he or she is trying to learn about. Good procedures in data analysis involve techniques that help to (a) answer the substantive questions at hand, (b) squeeze all the relevant information out of the data, and (c) learn something new about the world." (Edward R Tufte, "Data Analysis for Politics and Policy", 1974)

"Typically, data analysis is messy, and little details clutter it. Not only confounding factors, but also deviant cases, minor problems in measurement, and ambiguous results lead to frustration and discouragement, so that more data are collected than analyzed. Neglecting or hiding the messy details of the data reduces the researcher's chances of discovering something new." (Edward R Tufte, "Data Analysis for Politics and Policy", 1974)

"[...] be wary of analysts that try to quantify the unquantifiable." (Ralph Keeney & Raiffa Howard, "Decisions with Multiple Objectives: Preferences and Value Trade-offs", 1976)

"[...] exploratory data analysis is an attitude, a state of flexibility, a willingness to look for those things that we believe are not there, as well as for those we believe might be there. Except for its emphasis on graphs, its tools are secondary to its purpose." (John W Tukey, [comment] 1979)

On Data Analysis (1990-1999)

"[…] data analysis in the context of basic mathematical concepts and skills. The ability to use and interpret simple graphical and numerical descriptions of data is the foundation of numeracy […] Meaningful data aid in replacing an emphasis on calculation by the exercise of judgement and a stress on interpreting and communicating results." (David S Moore, "Statistics for All: Why, What and How?", 1990)

"Data analysis is rarely as simple in practice as it appears in books. Like other statistical techniques, regression rests on certain assumptions and may produce unrealistic results if those assumptions are false. Furthermore it is not always obvious how to translate a research question into a regression model." (Lawrence C Hamilton, "Regression with Graphics: A second course in applied statistics", 1991)

"Data analysis typically begins with straight-line models because they are simplest, not because we believe reality is inherently linear. Theory or data may suggest otherwise [...]" (Lawrence C Hamilton, "Regression with Graphics: A second course in applied statistics", 1991)

"90 percent of all problems can be solved by using the techniques of data stratification, histograms, and control charts. Among the causes of nonconformance, only one-fifth or less are attributable to the workers." (Kaoru Ishikawa, The Quality Management Journal Vol. 1, 1993)

"Probabilistic inference is the classical paradigm for data analysis in science and technology. It rests on a foundation of randomness; variation in data is ascribed to a random process in which nature generates data according to a probability distribution. This leads to a codification of uncertainly by confidence intervals and hypothesis tests." (William S Cleveland, "Visualizing Data", 1993)

"Visualization is an approach to data analysis that stresses a penetrating look at the structure of data. No other approach conveys as much information. […] Conclusions spring from data when this information is combined with the prior knowledge of the subject under investigation." (William S Cleveland, "Visualizing Data", 1993)

"When the distributions of two or more groups of univariate data are skewed, it is common to have the spread increase monotonically with location. This behavior is monotone spread. Strictly speaking, monotone spread includes the case where the spread decreases monotonically with location, but such a decrease is much less common for raw data. Monotone spread, as with skewness, adds to the difficulty of data analysis. For example, it means that we cannot fit just location estimates to produce homogeneous residuals; we must fit spread estimates as well. Furthermore, the distributions cannot be compared by a number of standard methods of probabilistic inference that are based on an assumption of equal spreads; the standard t-test is one example. Fortunately, remedies for skewness can cure monotone spread as well." (William S Cleveland, "Visualizing Data", 1993)

"A careful and sophisticated analysis of the data is often quite useless if the statistician cannot communicate the essential features of the data to a client for whom statistics is an entirely foreign language." (Christopher J Wild, "Embracing the ‘Wider view’ of Statistics", The American Statistician 48, 1994)

"Science is not impressed with a conglomeration of data. It likes carefully constructed analysis of each problem." (Daniel E Koshland Jr, Science Vol. 263 (5144), [editorial] 1994)

"So we pour in data from the past to fuel the decision-making mechanisms created by our models, be they linear or nonlinear. But therein lies the logician's trap: past data from real life constitute a sequence of events rather than a set of independent observations, which is what the laws of probability demand. [...] It is in those outliers and imperfections that the wildness lurks." (Peter L Bernstein, "Against the Gods: The Remarkable Story of Risk", 1996)

On Data Analysis (2010-2019)

"The discrepancy between our mental models and the real world may be a major problem of our times; especially in view of the difficulty of collecting, analyzing, and making sense of the unbelievable amount of data to which we have access today." (Ugo Bardi, "The Limits to Growth Revisited", 2011)

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

"Data analysis is not generally thought of as being simple or easy, but it can be. The first step is to understand that the purpose of data analysis is to separate any signals that may be contained within the data from the noise in the data. Once you have filtered out the noise, anything left over will be your potential signals. The rest is just details." (Donald J Wheeler," Myths About Data Analysis", International Lean & Six Sigma Conference, 2012)

"The four questions of data analysis are the questions of description, probability, inference, and homogeneity. Any data analyst needs to know how to organize and use these four questions in order to obtain meaningful and correct results. [...] 

THE DESCRIPTION QUESTION: Given a collection of numbers, are there arithmetic values that will summarize the information contained in those numbers in some meaningful way?

THE PROBABILITY QUESTION: Given a known universe, what can we say about samples drawn from this universe? [...] 

THE INFERENCE QUESTION: Given an unknown universe, and given a sample that is known to have been drawn from that unknown universe, and given that we know everything about the sample, what can we say about the unknown universe? [...] 

THE HOMOGENEITY QUESTION: Given a collection of observations, is it reasonable to assume that they came from one universe, or do they show evidence of having come from multiple universes?" (Donald J Wheeler," Myths About Data Analysis", International Lean & Six Sigma Conference, 2012)

"Each systems archetype embodies a particular theory about dynamic behavior that can serve as a starting point for selecting and formulating raw data into a coherent set of interrelationships. Once those relationships are made explicit and precise, the 'theory' of the archetype can then further guide us in our data-gathering process to test the causal relationships through direct observation, data analysis, or group deliberation." (Daniel H Kim, "Systems Archetypes as Dynamic Theories", The Systems Thinker Vol. 24 (1), 2013)

"A complete data analysis will involve the following steps: (i) Finding a good model to fit the signal based on the data. (ii) Finding a good model to fit the noise, based on the residuals from the model. (iii) Adjusting variances, test statistics, confidence intervals, and predictions, based on the model for the noise." (DeWayne R Derryberry, "Basic data analysis for time series with R", 2014)

"The random element in most data analysis is assumed to be white noise - normal errors independent of each other. In a time series, the errors are often linked so that independence cannot be assumed (the last examples). Modeling the nature of this dependence is the key to time series." (DeWayne R Derryberry, "Basic data analysis for time series with R", 2014)

"Statistics is an integral part of the quantitative approach to knowledge. The field of statistics is concerned with the scientific study of collecting, organizing, analyzing, and drawing conclusions from data." (Kandethody M Ramachandran & Chris P Tsokos, "Mathematical Statistics with Applications in R" 2nd Ed., 2015)

"The dialectical interplay of experiment and theory is a key driving force of modern science. Experimental data do only have meaning in the light of a particular model or at least a theoretical background. Reversely theoretical considerations may be logically consistent as well as intellectually elegant: Without experimental evidence they are a mere exercise of thought no matter how difficult they are. Data analysis is a connector between experiment and theory: Its techniques advise possibilities of model extraction as well as model testing with experimental data." (Achim Zielesny, "From Curve Fitting to Machine Learning" 2^nd Ed., 2016)

"[…] the data itself can lead to new questions too. In exploratory data analysis (EDA), for example, the data analyst discovers new questions based on the data. The process of looking at the data to address some of these questions generates incidental visualizations - odd patterns, outliers, or surprising correlations that are worth looking into further." (Danyel Fisher & Miriah Meyer, "Making Data Visual", 2018)

"Data analysis and data mining are concerned with unsupervised pattern finding and structure determination in data sets. The data sets themselves are explicitly linked as a form of representation to an observational or otherwise empirical domain of interest. 'Structure' has long been understood as symmetry which can take many forms with respect to any transformation, including point, translational, rotational, and many others. Symmetries directly point to invariants, which pinpoint intrinsic properties of the data and of the background empirical domain of interest. As our data models change, so too do our perspectives on analysing data." (Fionn Murtagh, "Data Science Foundations: Geometry and Topology of Complex Hierarchic Systems and Big Data Analytics", 2018)

"Analysis is a two-step process that has an exploratory and an explanatory phase. In order to create a powerful data story, you must effectively transition from data discovery (when you’re finding insights) to data communication (when you’re explaining them to an audience). If you don’t properly traverse these two phases, you may end up with something that resembles a data story but doesn’t have the same effect. Yes, it may have numbers, charts, and annotations, but because it’s poorly formed, it won’t achieve the same results." (Brent Dykes, "Effective Data Storytelling: How to Drive Change with Data, Narrative and Visuals", 2019)

"While visuals are an essential part of data storytelling, data visualizations can serve a variety of purposes from analysis to communication to even art. Most data charts are designed to disseminate information in a visual manner. Only a subset of data compositions is focused on presenting specific insights as opposed to just general information. When most data compositions combine both visualizations and text, it can be difficult to discern whether a particular scenario falls into the realm of data storytelling or not." (Brent Dykes, "Effective Data Storytelling: How to Drive Change with Data, Narrative and Visuals", 2019)

On Data Analysis (2000-2009)

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

"No matter what the data, and no matter how the values are arranged and presented, you must always use some method of analysis to come up with an interpretation of the data. [...] While every data set contains noise, some data sets may contain signals. Therefore, before you can detect a signal within any given data set, you must first filter out the noise." (Donald J Wheeler," Understanding Variation: The Key to Managing Chaos" 2nd Ed., 2000)

"The purpose of analysis is insight. The best analysis is the simplest analysis which provides the needed insight." (Donald J Wheeler, "Understanding Variation: The Key to Managing Chaos" 2nd Ed., 2000)

"Without meaningful data there can be no meaningful analysis. The interpretation of any data set must be based upon the context of those data." (Donald J Wheeler, "Understanding Variation: The Key to Managing Chaos" 2nd Ed., 2000)

"Statistical analysis of data can only be performed within the context of selected assumptions, models, and/or prior distributions. A statistical analysis is actually the extraction of substantive information from data and assumptions. And herein lies the rub, understood well by Disraeli and others skeptical of our work: For given data, an analysis can usually be selected which will result in 'information' more favorable to the owner of the analysis then is objectively warranted." (Stephen B Vardeman & Max D Morris, "Statistics and Ethics: Some Advice for Young Statisticians", The American Statistician vol 57, 2003)

"Exploratory Data Analysis is more than just a collection of data-analysis techniques; it provides a philosophy of how to dissect a data set. It stresses the power of visualisation and aspects such as what to look for, how to look for it and how to interpret the information it contains. Most EDA techniques are graphical in nature, because the main aim of EDA is to explore data in an open-minded way. Using graphics, rather than calculations, keeps open possibilities of spotting interesting patterns or anomalies that would not be apparent with a calculation (where assumptions and decisions about the nature of the data tend to be made in advance)." (Alan Graham, "Developing Thinking in Statistics", 2006)

"It is the aim of all data analysis that a result is given in form of the best estimate of the true value. Only in simple cases is it possible to use the data value itself as result and thus as best estimate." (Manfred Drosg, "Dealing with Uncertainties: A Guide to Error Analysis", 2007)

"Put simply, statistics is a range of procedures for gathering, organizing, analyzing and presenting quantitative data. […] Essentially […], statistics is a scientific approach to analyzing numerical data in order to enable us to maximize our interpretation, understanding and use. This means that statistics helps us turn data into information; that is, data that have been interpreted, understood and are useful to the recipient. Put formally, for your project, statistics is the systematic collection and analysis of numerical data, in order to investigate or discover relationships among phenomena so as to explain, predict and control their occurrence." (Reva B Brown & Mark Saunders, "Dealing with Statistics: What You Need to Know", 2008)

"Data analysis is careful thinking about evidence." (Michael Milton, "Head First Data Analysis", 2009)

"Doing data analysis without explicitly defining your problem or goal is like heading out on a road trip without having decided on a destination." (Michael Milton, "Head First Data Analysis", 2009)

On Data Analysis (1980-1989)

"[...] any hope that we are smart enough to find even transiently optimum solutions to our data analysis problems is doomed to failure, and, indeed, if taken seriously, will mislead us in the allocation of effort, thus wasting both intellectual and computational effort." (John W Tukey, "Choosing Techniques for the Analysis of Data", 1981)

"The fact must be expressed as data, but there is a problem in that the correct data is difficult to catch. So that I always say 'When you see the data, doubt it!' 'When you see the measurement instrument, doubt it!' [...]For example, if the methods such as sampling, measurement, testing and chemical analysis methods were incorrect, data. […] to measure true characteristics and in an unavoidable case, using statistical sensory test and express them as data." (Kaoru Ishikawa, Annual Quality Congress Transactions, 1981)

"Exploratory data analysis, EDA, calls for a relatively free hand in exploring the data, together with dual obligations: (•) to look for all plausible alternatives and oddities - and a few implausible ones, (graphic techniques can be most helpful here) and (•) to remove each appearance that seems large enough to be meaningful - ordinarily by some form of fitting, adjustment, or standardization [...] so that what remains, the residuals, can be examined for further appearances." (John W Tukey, "Introduction to Styles of Data Analysis Techniques", 1982)

"A competent data analysis of an even moderately complex set of data is a thing of trials and retreats, of dead ends and branches." (John W Tukey, Computer Science and Statistics: Proceedings of the 14th Symposium on the Interface, 1983)

"Data in isolation are meaningless, a collection of numbers. Only in context of a theory do they assume significance […]" (George Greenstein, "Frozen Star", 1983)

"Iteration and experimentation are important for all of data analysis, including graphical data display. In many cases when we make a graph it is immediately clear that some aspect is inadequate and we regraph the data. In many other cases we make a graph, and all is well, but we get an idea for studying the data in a different way with a different graph; one successful graph often suggests another." (William S Cleveland, "The Elements of Graphing Data", 1985)

"There are some who argue that a graph is a success only if the important information in the data can be seen within a few seconds. While there is a place for rapidly-understood graphs, it is too limiting to make speed a requirement in science and technology, where the use of graphs ranges from, detailed, in-depth data analysis to quick presentation." (William S Cleveland, "The Elements of Graphing Data", 1985)

"A first analysis of experimental results should, I believe, invariably be conducted using flexible data analytical techniques - looking at graphs and simple statistics - that so far as possible allow the data to 'speak for themselves'. The unexpected phenomena that such a approach often uncovers can be of the greatest importance in shaping and sometimes redirecting the course of an ongoing investigation." (George Box, "Signal to Noise Ratios, Performance Criteria, and Transformations", Technometrics 30, 1988)

"Data analysis is an art practiced by individuals who are skilled at quantitative reasoning and have much experience in looking at numbers and detecting  patterns in data. Usually these individuals have some background in statistics." (David Lubinsky, Daryl Pregibon , "Data analysis as search", Journal of Econometrics Vol. 38 (1–2), 1988)

"Like a detective, a data analyst will experience many dead ends, retrace his steps, and explore many alternatives before settling on a single description of the evidence in front of him." (David Lubinsky & Daryl Pregibon , "Data analysis as search", Journal of Econometrics Vol. 38 (1–2), 1988)

On Data Analysis (1950-1969)

"In every branch of knowledge the progress is proportional to the amount of facts on which to build, and therefore to the facility of obtaining data." (James C Maxwell, [letter to Lewis Campbell] 1851)

"Not even the most subtle and skilled analysis can overcome completely the unreliability of basic data." (Roy D G Allen, "Statistics for Economists", 1951)

"The technical analysis of any large collection of data is a task for a highly trained and expensive man who knows the mathematical theory of statistics inside and out. Otherwise the outcome is likely to be a collection of drawings - quartered pies, cute little battleships, and tapering rows of sturdy soldiers in diversified uniforms - interesting enough in the colored Sunday supplement, but hardly the sort of thing from which to draw reliable inferences." (Eric T Bell, "Mathematics: Queen and Servant of Science", 1951)

"If data analysis is to be well done, much of it must be a matter of judgment, and ‘theory’ whether statistical or non-statistical, will have to guide, not command." (John W Tukey, "The Future of Data Analysis", Annals of Mathematical Statistics, Vol. 33 (1), 1962)

"The most important maxim for data analysis to heed, and one which many statisticians seem to have shunned is this: ‘Far better an approximate answer to the right question, which is often vague, than an exact answer to the wrong question, which can always be made precise.’ Data analysis must progress by approximate answers, at best, since its knowledge of what the problem really is will at best be approximate." (John W Tukey, "The Future of Data Analysis", Annals of Mathematical Statistics, Vol. 33, No. 1, 1962)

"The first step in data analysis is often an omnibus step. We dare not expect otherwise, but we equally dare not forget that this step, and that step, and other step, are all omnibus steps and that we owe the users of such techniques a deep and important obligation to develop ways, often varied and competitive, of replacing omnibus procedures by ones that are more sharply focused." (John W Tukey, "The Future of Processes of Data Analysis", 1965)

"The basic general intent of data analysis is simply stated: to seek through a body of data for interesting relationships and information and to exhibit the results in such a way as to make them recognizable to the data analyzer and recordable for posterity. Its creative task is to be productively descriptive, with as much attention as possible to previous knowledge, and thus to contribute to the mysterious process called insight." (John W Tukey & Martin B Wilk, "Data Analysis and Statistics: An Expository Overview", 1966)

"Comparable objectives in data analysis are (l) to achieve more specific description of what is loosely known or suspected; (2) to find unanticipated aspects in the data, and to suggest unthought-of-models for the data's summarization and exposure; (3) to employ the data to assess the (always incomplete) adequacy of a contemplated model; (4) to provide both incentives and guidance for further analysis of the data; and (5) to keep the investigator usefully stimulated while he absorbs the feeling of his data and considers what to do next." (John W Tukey & Martin B Wilk, "Data Analysis and Statistics: An Expository Overview", 1966)

"Data analysis must be iterative to be effective. [...] The iterative and interactive interplay of summarizing by fit and exposing by residuals is vital to effective data analysis. Summarizing and exposing are complementary and pervasive." (John W Tukey & Martin B Wilk, "Data Analysis and Statistics: An Expository Overview", 1966)

"Every student of the art of data analysis repeatedly needs to build upon his previous statistical knowledge and to reform that foundation through fresh insights and emphasis." (John W Tukey, "Data Analysis, Including Statistics", 1968)

"[...] bending the question to fit the analysis is to be shunned at all costs." (John W Tukey, "Analyzing Data: Sanctification or Detective Work?", 1969)

Douglas T Ross - Collected Quotes

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

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

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

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

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

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

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

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

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

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

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

George F Simmons - Collected Quotes

"A special role is played in the theory of metric spaces by the class of open spheres within the class of all open sets. The main feature of their relationship is that the open sets coincide with all unions of open spheres, and it follows from this that the continuity of a mapping can be expressed either in terms of open spheres or in terms of open sets, at our convenience." (George F Simmons, "Introduction to Topology and Modern Analysis", 1963)

"A topological space can be thought of as a set from which has been swept away all structure irrelevant to the continuity of functions defined on it." (George F Simmons, "Introduction to Topology and Modern Analysis", 1963)

"Analysis is primarily concerned with limit processes and continuity, so it is not surprising that mathematicians thinking along these lines soon found themselves studying (and generalizing) two elementary concepts: that of a convergent sequence of real or complex numbers, and that of a continuous function of a real or complex variable." (George F Simmons, "Introduction to Topology and Modern Analysis", 1963)

"Determinants are often advertised to students of elementary mathematics as a computational device of great value and efficiency for solving numerical problems involving systems of linear equations. This is somewhat misleading, for their value in problems of this kind is very limited. On the other hand, they do have definite importance as a theoretical tool. Briefly, they provide a numerical means of distinguishing between singular and non-singular matrices (and operators)." (George F Simmons, "Introduction to Topology and Modern Analysis", 1963)

"Historically speaking, topology has followed two principal lines of development. In homology theory, dimension theory, and the study of manifolds, the basic motivation appears to have come from geometry. In these fields, topological spaces are looked upon as generalized geometric configurations, and the emphasis is placed on the structure of the spaces themselves. In the other direction, the main stimulus has been analysis. Continuous functions are the chief objects of interest here, and topological spaces are regarded primarily as carriers of such functions and as domains over which they can be integrated. These ideas lead naturally into the theory of Banach and Hilbert spaces and Banach algebras, the modern theory of integration, and abstract harmonic analysis on locally compact groups." (George F Simmons, "Introduction to Topology and Modern Analysis", 1963)

"In all candor, we must admit that the intuitive meaning of compactness for topological spaces is somewhat elusive. This concept, however, is so vitally important throughout topology […]" (George F Simmons, "Introduction to Topology and Modern Analysis", 1963)

"In many branches of mathematics - in geometry as well as analysis - it has been found extremely convenient to have available a notion of distance which is applicable to the elements of abstract sets. A metric space (as we define it below) is nothing more than a non-empty set equipped with a concept of distance which is suitable for the treatment of convergent sequences in the set and continuous functions defined on the set." (George F Simmons, "Introduction to Topology and Modern Analysis", 1963) 

"It is extremely helpful to the imagination to have a geometric picture available in terms of which we can visualize sets and operations on sets. […] Diagrammatic thought of this kind is admittedly loose and imprecise; nevertheless, the reader will find it invaluable. No mathematics, however abstract it may appear, is ever carried on without the help of mental images of some kind, and these are often nebulous, personal, and difficult to describe." (George F Simmons, "Introduction to Topology and Modern Analysis", 1963)

"It is sometimes said that mathematics is the study of sets and functions. Naturally, this oversimplifies matters; but it does come as close to the truth as an aphorism can." (George F Simmons, "Introduction to Topology and Modern Analysis", 1963)

"It seems to me that a worthwhile distinction can be drawn between two types of pure mathematics. The first - which unfortunately is somewhat out of style at present - centers attention on particular functions and theorems which are rich in meaning and history, like the gamma function and the prime number theorem, or on juicy individual facts […] The second is concerned primarily with form and structure." (George F Simmons, "Introduction to Topology and Modern Analysis", 1963)

"The essence of analytic geometry lies in the possibility of exploiting this identification by using algebraic tools in geometric arguments and giving geometric interpretations to algebraic calculations." (George F Simmons, "Introduction to Topology and Modern Analysis", 1963)

"The study of sets and functions leads two ways. One path goes down, into the abysses of logic, philosophy, and the foundations of mathematics. The other goes up, onto the highlands of mathematics itself, where these concepts are indispensable in almost all of pure mathematics as it is today." (George F Simmons, "Introduction to Topology and Modern Analysis", 1963)

On Complex Numbers XVIII

"I consider it as one of the most important steps made by Analysis in the last period, that of not being bothered any more by imaginary quantities, and to be able to submit them to calculus, in the same way as the real ones." (Joseph-Louis de Lagrange, [letter to Antonio Lorgna] 1777)

"What should one understand by ∫ ϕx · dx for x = a + bi? Obviously, if we want to start from clear concepts, we have to assume that x passes from the value for which the integral has to be 0 to x = a + bi through infinitely small increments (each of the form x = a + bi), and then to sum all the ϕx · dx. Thereby the meaning is completely determined. However, the passage can take placein infinitely many ways: Just like the realm of all real magnitudes can be conceived as an infinite straight line, so can the realm of all magnitudes, real and imaginary, be made meaningful by an infinite plane, in which every point, determined by abscissa = a and ordinate = b, represents the quantity a+bi. The continuous passage from one value of x to another a+bi then happens along a curve and is therefore possible in infinitely many ways. I claim now that after two different passages the integral ∫ ϕx · dx acquires the same value when ϕx never becomes equal to ∞ in the region enclosed by the two curves representing the two passages."(Carl F Gauss, [letter to Bessel] 1811)

"Without doubt one of the most characteristic features of mathematics in the last century is the systematic and universal use of the complex variable. Most of its great theories received invaluable aid from it, and many owe their very existence to it." (James Pierpont, "History of Mathematics in the Nineteenth Century", Congress of Arts and Sciences Vol. 1, 1905)

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

"The real numbers are one of the most audacious idealizations made by the human mind, but they were used happily for centuries before anybody worried about the logic behind them. Paradoxically, people worried a great deal about the next enlargement of the number system, even though it was entirely harmless. That was the introduction of square roots for negative numbers, and it led to the 'imaginary' and 'complex' numbers. A professional mathematican should never leave home without them […]" (Ian Stewart, "Nature's Numbers: The unreal reality of mathematics", 1995)

"Beyond the theory of complex numbers, there is the much greater and grander theory of the functions of a complex variable, as when the complex plane is mapped to the complex plane, complex numbers linking themselves to other complex numbers. It is here that complex differentiation and integration are defined. Every mathematician in his education studies this theory and surrenders to it completely. The experience is like first love." (David Berlinski, "Infinite Ascent: A short history of mathematics", 2005)

"Algebraic geometry uses the geometric intuition which arises from looking at varieties over the complex and real case to deduce important results in arithmetic algebraic geometry where the complex number field is replaced by the field of rational numbers or various finite number fields." (Raymond O Wells Jr, "Differential and Complex Geometry: Origins, Abstractions and Embeddings", 2017)

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

"The very idea of raising a number to an imaginary power may well have seemed to most of the era’s mathematicians like asking the ghost of a late amphibian to jump up on a harpsichord and play a minuet." (David Stipp, "A Most Elegant Equation: Euler's Formula and the Beauty of Mathematics", 2017)

"Today it’s easy to see the beauty of i, thanks, among other things, to its prominence in mathematics’ most beautiful equation. Thus, it may seem strange that it was once regarded as akin to a small waddling gargoyle. Indeed, the simplicity of its definition suggests unpretentious elegance: i is just the square root of −1. But as with many definitions in mathematics, i’s is fraught with provocative implications, and the ones that made it a star in mathematics weren’t apparent until long after it first came on the scene." (David Stipp, "A Most Elegant Equation: Euler's Formula and the Beauty of Mathematics", 2017)

On Critical Points I

"From its beginning critical point theory has been concerned with mutual relations between topology and geometric analysis, including differential geometry. Although it may have seemed to many to have been directed in its initial years toward applications of topology to analysis, one now sees that the road from topology to geometric analysis is a two-way street. Today the methods of critical point theory enter into the foundations of almost all studies of analysis or geometry 'in the large'." (Marston Morse & Stewart S Cairns, "Critical Point Theory in Global Analysis and Differential Topology: An Introduction", 1969)

"The key to making discontinuity emerge from smoothness is the observation that the overall behavior of both static and dynamical systems is governed by what's happening near the critical points. These are the points at which the gradient of the function vanishes. Away from the critical points, the Implicit Function Theorem tells us that the behavior is boring and predictable, linear, in fact. So it's only at the critical points that the system has the possibility of breaking out of this mold to enter a new mode of operation. It's at the critical points that we have the opportunity to effect dramatic shifts in the system's behavior by 'nudging' lightly the system dynamics, one type of nudge leading to a limit cycle, another to a stable equilibrium, and yet a third type resulting in the system's moving into the domain of a 'strange attractor'. It's by these nudges in the equations of motion that the germ of the idea of discontinuity from smoothness blossoms forth into the modern theory of singularities, catastrophes and bifurcations, wherein we see how to make discontinuous outputs emerge from smooth inputs." (John L Casti, "Reality Rules: Picturing the world in mathematics", 1992)

"Catastrophe theory is a local theory, telling us what a function looks like  in a small neighborhood of a critical point; it says nothing about what the function may be doing far away from the singularity. Yet most of the applications of the theory [...]  involve extrapolating these rock-solid, local results to regions that may  well be distant in time and space from the singularity." (John L Casti, "Five Golden Rules", 1995)

"The goal of catastrophe theory is to classify smooth functions with degenerate critical points, just as Morse's Theorem gives us a complete classification for Morse functions. The difficulty, of course, is that there are a lot more ways for critical points to 'go bad' than there are for them to stay 'nice'. Thus, the classification problem is much harder for functions having degenerate critical points, and has not yet been fully carried out for all possible types of degeneracies. Fortunately, though, we can obtain a partial classification for those functions having critical points that are not too bad. And this classification turns out to be sufficient to apply the results to a wide range of phenomena like the predator-prey situation sketched above, in which 'jumps' in the system's biomass can occur when parameters describing the process change only slightly." (John L Casti, "Five Golden Rules", 1995)

"The reason catastrophe theory can tell us about such abrupt changes in a system's behavior is that we usually observe a dynamical system when it's at or near its steady-state, or equilibrium, position. And under various assumptions about the nature of the system's dynamical law of motion, the set of all possible equilibrium states is simply the set of critical points of a smooth function closely related to the system dynamics. When these critical points are nondegenerate, Morse's Theorem applies. But it is exactly when they become degenerate that the system can move sharply from one equilibrium position to another. The Thorn Classification Theorem tells when such shifts will occur and what direction they will take." (John L Casti, "Five Golden Rules", 1995)

"The phenomenon of emergence takes place at critical points of instability that arise from fluctuations in the environment, amplified by feedback loops." (Fritjof Capra, "The Hidden Connections", 2002)

"This spontaneous emergence of order at critical points of instability is one of the most important concepts of the new understanding of life. It is technically known as self-organization and is often referred to simply as ‘emergence’. It has been recognized as the dynamic origin of development, learning and evolution. In other words, creativity-the generation of new forms-is a key property of all living systems. And since emergence is an integral part of the dynamics of open systems, we reach the important conclusion that open systems develop and evolve. Life constantly reaches out into novelty." (Fritjof  Capra, "The Hidden Connections", 2002)

"A commonly accepted principle of systems dynamics is that a quantitative change, beyond a critical point, results in a qualitative change. Accordingly, a difference in degree may become a difference in kind. This doesn't mean that an increased quantity of a given variable will bring a qualitative change in the variable itself. However, when the state of a system depends on a set of variables, a quantitative change in one variable beyond the inflection point will result in a change of phase in the state of the system. This change is a qualitative one, representing a whole new set of relationships among the variables involved." (Jamshid Gharajedaghi, "Systems Thinking: Managing Chaos and Complexity A Platform for Designing Business Architecture" 3rd Ed., 2011)

"This spontaneous emergence of order at critical points of instability, which is often referred to simply as 'emergence', is one of the hallmarks of life. It has been recognized as the dynamic origin of development, learning, and evolution. In other words, creativity-the generation of new forms-is a key property of all living systems." (Fritjof Capra, "The Systems View of Life: A Unifying Vision", 2014)

H Marston Morse - Collected Quotes

"It is possible that analysis in the large may eventually reduce to topology, but not until topology has been greatly broadened. It is equally conceivable that the apparently less general situations which arise with such frequency in problems in analysis in the large may form the canonical cases about which the topology of the future can be built." (Marston Morse, "What is Analysis in the Large?", The American Mathematical Monthly Vol. 49 (6), 1942) 

"A definition is topological if it makes no use of mathematical elements other than those defined in terms of continuous deformations or transformations. Such deformations or transformations take the straightness out of planes and alter lengths and areas." (Marston Morse, "Equilibria in Nature: Stable and Unstable", Proceedings of the American Philosophical Society Vol. 93 (3), 1949)

"Mathematicians are led to new problems not only by way of contact with the world of physical experience but also by introspective study of the methods which they have elected to use. The trend of classical analysis has been to break up the object of study into finer and finer elements without end." (Marston Morse, "Equilibria in Nature: Stable and Unstable", Proceedings of the American Philosophical Society Vol. 93 (3), 1949)

"The concepts of an equilibrium theory are not put forward as a practical method of attaining an economic Utopia. Lack of economic data, knowledge, political and psychological understanding, are too obvious to permit this. Perhaps the greatest value which these considerations have is qualitative in nature. The implications are negative in the sense that the general mathematical theory of equilibria, points to the high a priori probability that any given state of equilibrium is unstable in character." (Marston Morse, "Equilibria in Nature: Stable and Unstable", Proceedings of the American Philosophical Society Vol. 93 (3), 1949)

"There is no conflict between science, philosophy and theology. What conflict there may be is due to a failure of agreement as to the implications of the word 'science'." (Marston Morse, "Science in the Modern World", Mathematics Magazine Vol. 28 (4), 1955)

"This continuity of effort is particularly important in mathematics. It is needed to realize the promise of unity which modern mathematics holds. In no science does it appear truer than- in mathematics that the relatively unexplained universe of known facts can be unified by theories of a general character, built of the bricks of current techniques, if only there could rise enough men of talent with a sense of values that would hold them to their task to the very end." (Marston Morse, "Science in the Modern World", Mathematics Magazine Vol. 28 (4), 1955)

"Mathematics are the result of mysterious powers which no one understands, and which the unconscious recognition of beauty must play an important part. Out of an infinity of designs a mathematician chooses one pattern for beauty's sake and pulls it down to earth." (Marston Morse, 1959)

"From its beginning critical point theory has been concerned with mutual relations between topology and geometric analysis, including differential geometry. Although it may have seemed to many to have been directed in its initial years toward applications of topology to analysis, one now sees that the road from topology to geometric analysis is a two-way street. Today the methods of critical point theory enter into the foundations of almost all studies of analysis or geometry 'in the large'." (Marston Morse & Stewart S Cairns, "Critical Point Theory in Global Analysis and Differential Topology: An Introduction", 1969)

"Mathematicians are finding that the study of global analysis or differential topology requires a knowledge not only of the separate techniques of analysis, differential geometry, topology, and algebra, but also a deeper understanding of how these fields can join forces." (Marston Morse & Stewart S Cairns, "Critical Point Theory in Global Analysis and Differential Topology: An Introduction", 1969)

"But mathematics is the sister, as well as the servant, of the arts and is touched with the same madness and genius." (Marston Morse)

"Discovery in mathematics is not a matter of logic. It is rather the result of mysterious powers which no one understands, and in which unconscious recognition of beauty must play an important part. Out of an infinity of designs, a mathematician chooses one pattern for beauty's sake and pulls it down to earth." (Marston Morse)

"Mathematicians of today are perhaps too exuberant in their desire to build new logical foundations for everything. Forever the foundation and never the cathedral." (Marston Morse)

"Most convincing to me of the spiritual relations between mathematics and music, is my own very personal experience. Composing in an amateurish way, I get exactly the same elevation from a prelude that has come to me at the piano, as I do from a new idea that has come to me in mathematics." (Marston Morse)

"The creative scientist lives in a 'wildness of logic,' where reason is the handmaiden and not the master." (Marston Morse)

Augustin-Louis Cauchy - Collected Quotes

"As for methods I have sought to give them all the rigour that one requires in geometry, so as never to have recourse to the reasons drawn from the generality of algebra." (Augustin-Louis Cauchy, "Cours d'analyse", 1821)

"Thus, let us be persuaded that there are other truths than those of algebra, realities other than sensible objects. Let us cultivate ardently the mathematical sciences, without trying to extend them beyond their domain; and let us not imagine that one can tackle history with formulae, or use theorems of algebra or of differential calculus as an assent to morals.16 (Augustin-Louis Cauchy, "Cours d’analyse de l’École Royale Polytechnique", 1821)

"When variable quantities are so tied to each other that, given the values of some of them, we can deduce the values of all the others, we usually conceive these various quantities expressed in terms of one of them, which then bears the name independent variable; and the other quantities expressed in terms of the independent variable are what we call functions of that variable." (Augustin-Louis Cauchy, "Cours d’analyse de l’École Royale Polytechnique", 1821)

"When variable quantities are so tied to each other that, given the values of some of them, we can deduce the values of all the others, we usually conceive these various quantities expressed in terms of several of them, which then bear the name independent variables; and the remaining quantities expressed in terms of the independent variables, are what we call functions of these same variables." (Augustin-Louis Cauchy, "Cours d’analyse de l’École Royale Polytechnique", 1821)

"[…] a function of the variable x will be continuous between two limits a and b of this variable if between two limits the function has always a value which is unique and finite, in such a way that an infinitely small increment of this variable always produces an infinnitely small increment of the function itself." (Augustin-Louis Cauchy, "Mémoire sur les fonctions continues" ["Memoir on continuous functions"], 1844)

"We completely repudiate the symbol √-1, abandoning it without regret because we do not know what this alleged symbolism signifies nor what meaning to give to it." (Augustin-Louis Cauchy, 1847)

"[...] very often the laws derived by physicists from a large number of observations are not rigorous, but approximate." (Augustin-Louis Cauchy, "Sept leçons de physique" ["Seven lessons of Physics"], Bureau du Journal Les Mondes, 1868)

Calculus II: Integral Calculus

"I see with much pleasure that you are working on a large work on the integral Calculus [...] The reconciliation of the methods which you are planning to make, serves to clarify them mutually, and what they have in common contains very often their true metaphysics; this is why that metaphysics is almost the last thing that one discovers. The spirit arrives at the results as if by instinct; it is only on reflecting upon the route that it and others have followed that it succeeds in generalising the methods and in discovering its metaphysics." (Pierre-Simon Laplace [letter to Sylvestre F Lacroix] 1792)

"The effects of heat are subject to constant laws which cannot be discovered without the aid of mathematical analysis. The object of the theory is to demonstrate these laws; it reduces all physical researches on the propagation of heat, to problems of the integral calculus, whose elements are given by experiment. No subject has more extensive relations with the progress of industry and the natural sciences; for the action of heat is always present, it influences the processes of the arts, and occurs in all the phenomena of the universe." (Jean-Baptiste-Joseph Fourier, "The Analytical Theory of Heat", 1822)

"If one looks at the different problems of the integral calculus which arise naturally when he wishes to go deep into the different parts of physics, it is impossible not to be struck by the analogies existing. Whether it be electrostatics or electrodynamics, the propagation of heat, optics, elasticity, or hydrodynamics, we are led always to differential equations of the same family." (Henri Poincaré, "Sur les Equations aux Dérivées Partielles de la Physique Mathématique", American Journal of Mathematics Vol. 12, 1890)

"Everyone who understands the subject will agree that even the basis on which the scientific explanation of nature rests is intelligible only to those who have learned at least the elements of the differential and integral calculus, as well as analytical geometry." (Felix Klein, Jahresbericht der Deutsche Mathematiker Vereinigung Vol. 1, 1902)

"The chief difficulty of modern theoretical physics resides not in the fact that it expresses itself almost exclusively in mathematical symbols, but in the psychological difficulty of supposing that complete nonsense can be seriously promulgated and transmitted by persons who have sufficient intelligence of some kind to perform operations in differential and integral calculus […]" (Celia Green, "The Decline and Fall of Science", 1976)

"The acceptance of complex numbers into the realm of algebra had an impact on analysis as well. The great success of the differential and integral calculus raised the possibility of extending it to functions of complex variables. Formally, we can extend Euler's definition of a function to complex variables without changing a single word; we merely allow the constants and variables to assume complex values. But from a geometric point of view, such a function cannot be plotted as a graph in a two-dimensional coordinate system because each of the variables now requires for its representation a two-dimensional coordinate system, that is, a plane. To interpret such a function geometrically, we must think of it as a mapping, or transformation, from one plane to another." (Eli Maor, "e: The Story of a Number", 1994)

"By studying analytic functions using power series, the algebra of the Middle Ages was connected to infinite operations (various algebraic operations with infinite series). The relation of algebra with infinite operations was later merged with the newly developed differential and integral calculus. These developments gave impetus to early stages of the development of analysis. In a way, we can say that analyticity is the notion that first crossed the boundary from finite to infinite by passing from polynomials to infinite series. However, algebraic properties of polynomial functions still are strongly present in analytic functions." (Kenji Ueno & Toshikazu Sunada, "A Mathematical Gift, III: The Interplay Between Topology, Functions, Geometry, and Algebra", Mathematical World Vol. 23, 1996)

"Thus, calculus proceeds in two phases: cutting and rebuilding. In mathematical terms, the cutting process always involves infinitely fine subtraction, which is used to quantify the differences between the parts. Accordingly, this half of the subject is called differential calculus. The reassembly process always involves infinite addition, which integrates the parts back into the original whole. This half of the subject is called integral calculus." (Steven H Strogatz, "Infinite Powers: The Story of Calculus - The Most Important Discovery in Mathematics", 2019)

"This method of subjecting the infinite to algebraic manipulations is called differential and integral calculus. It is the art of numbering and measuring with precision things the existence of which we cannot even conceive. Indeed, would you not think that you are being laughed at, when told that there are lines infinitely great which form infinitely small angles? Or that a line which is straight so long as it is finite would, by changing its direction infinitely little, become an infinite curve? Or that there are infinite squares, infinite cubes, and infinities of infinities, one greater than another, and that, as compared with the ultimate infinitude, those which precede it are as nought. All these things at first appear as excess of frenzy; yet, they bespeak the great scope and subtlety of the human spirit, for they have led to the discovery of truths hitherto undreamt of." (Voltaire)

On Structure: Structure in Mathematics I

"The first step, whenever a practical problem is set before a mathematician, is to form the mathematical hypothesis. It is neither needful nor practical that we should take account of the details of the structure as it will exist. We have to reason about a skeleton diagram in which bearings are reduced to points, pieces to lines, etc. and [in] which it is supposed that certain relations between motions are absolutely constrained, irrespective of forces." (Charles S Peirce, "Report on Live Loads", cca. 1895)

"For thought raised on specialization the most potent objection to the possibility of a universal organizational science is precisely its universality. Is it ever possible that the same laws be applicable to the combination of astronomic worlds and those of biological cells, of living people and the waves of the ether, of scientific ideas and quanta of energy? [...] Mathematics provide a resolute and irrefutable answer: yes, it is undoubtedly possible, for such is indeed the case. Two and two homogenous separate elements amount to four such elements, be they astronomic systems or mental images, electrons or workers; numerical structures are indifferent to any element, there is no place here for specificity." (Alexander Bogdanov, "Tektology: The Universal Organizational Science" Vol. I, 1913)

"Once a statement is cast into mathematical form it may be manipulated in accordance with [mathematical] rules and every configuration of the symbols will represent facts in harmony with and dependent on those contained in the original statement. Now this comes very close to what we conceive the action of the brain structures to be in performing intellectual acts with the symbols of ordinary language. In a sense, therefore, the mathematician has been able to perfect a device through which a part of the labor of logical thought is carried on outside the central nervous system with only that supervision which is requisite to manipulate the symbols in accordance with the rules." (Horatio B Williams, "Mathematics and the Biological Sciences", Bulletin of the American Mathematical Society Vol. 38, 1927)

"Physics is the attempt at the conceptual construction of a model of the real world and its lawful structure." (Albert Einstein, [letter to Moritz Schlick] 1931)

"Today's scientists have substituted mathematics for experiments, and they wander off through equation after equation, and eventually build a structure which has no relation to reality." (Nikola Tesla, "Radio Power Will Revolutionize the World", Modern Mechanics and Inventions, 1934)

"Men of science belong to two different types - the logical and the intuitive. Science owes its progress to both forms of minds. Mathematics, although a purely logical structure, nevertheless makes use of intuition. " (Alexis Carrel, "Man the Unknown", 1935)

"Statistics is a scientific discipline concerned with collection, analysis, and interpretation of data obtained from observation or experiment. The subject has a coherent structure based on the theory of Probability and includes many different procedures which contribute to research and development throughout the whole of Science and Technology." (Egon Pearson, 1936)

Mathematics through Students' Eyes I

"Often I have considered the fact that most of the difficulties which block the progress of students trying to learn analysis stem from this: that although they understand little of ordinary algebra, still they attempt this more subtle art. From this it follows not only that they remain on the fringes, but in addition they entertain strange ideas about the concept of the infinite, which they must try to use." (Leonhard Euler, "Introduction to Analysis of the Infinite", 1748)

"The first thing to be attended to in reading any algebraical treatise, is the gaining a perfect understanding of the different processes there exhibited, and of their connection with one another. This cannot be attained by a mere reading of the book, however great the attention which may be given. It is impossible, in a mathematical work, to fill up every process in the manner in which it must be filled up in the mind of the student before he can be said to have completely mastered it. Many results must be given of which the details are suppressed, such are the additions, multiplications, extractions of the square root, etc., with which the investigations abound. These must not be taken on trust by the student, but must be worked by his own pen, which must never be out of his hand, while engaged in any algebraical process." (Augustus de Morgan, "On the Study and Difficulties of Mathematics", 1830)

"This science, Geometry, is one of indispensable use and constant reference, for every student of the laws of nature; for the relations of space and number are the alphabet in which those laws are written. But besides the interest and importance of this kind which geometry possesses, it has a great and peculiar value for all who wish to understand the foundations of human knowledge, and the methods by which it is acquired. For the student of geometry acquires, with a degree of insight and clearness which the unmathematical reader can but feebly imagine, a conviction that there are necessary truths, many of them of a very complex and striking character; and that a few of the most simple and self-evident truths which it is possible for the mind of man to apprehend, may, by systematic deduction, lead to the most remote and unexpected results." (William Whewell, "The Philosophy of the Inductive Sciences", 1858)

"Besides accustoming the student to demand complete proof, and to know when he has not obtained it, mathematical studies are of immense benefit to his education by habituating him to precision. It is one of the peculiar excellencies of mathematical discipline, that the mathematician is never satisfied with à peu près. He requires the exact truth." (John S Mill, "An Examination of Sir William Hamilton's Philosophy", 1865)

"Therefore, the great business of the scientific teacher is, to imprint the fundamental, irrefragable facts of his science, not only by words upon the mind, but by sensible impressions upon the eye, and ear, and touch of the student, in so complete a manner, that every term used, or law enunciated, should afterwards call up vivid images of the particular structural, or other, facts which furnished the demonstration of the law, or the illustration of the term." (Thomas H Huxley, "Lay Sermons, Addresses and Reviews", 1870)

"Thought is symbolical of Sensation as Algebra is of Arithmetic, and because it is symbolical, is very unlike what it symbolises. For one thing, sensations are always positive; in this resembling arithmetical quantities. A negative sensation is no more possible than a negative number. But ideas, like algebraic quantities, may be either positive or negative. However paradoxical the square of a negative quantity, the square root of an unknown quantity, nay, even in imaginary quantity, the student of Algebra finds these paradoxes to be valid operations. And the student of Philosophy finds analogous paradoxes in operations impossible in the sphere of Sense. Thus although it is impossible to feel non-existence, it is possible to think it; although it is impossible to frame an image of Infinity, we can, and do, form the idea, and reason on it with precision." (George H Lewes "Problems of Life and Mind", 1873)

"The most striking characteristic of the written language of algebra and of the higher forms of the calculus is the sharpness of definition, by which we are enabled to reason upon the symbols by the mere laws of verbal logic, discharging our minds entirely of the meaning of the symbols, until we have reached a stage of the process where we desire to interpret our results. The ability to attend to the symbols, and to perform the verbal, visible changes in the position of them permitted by the logical rules of the science, without allowing the mind to be perplexed with the meaning of the symbols until the result is reached which you wish to interpret, is a fundamental part of what is called analytical power. Many students find themselves perplexed by a perpetual attempt to interpret not only the result, but each step of the process. They thus lose much of the benefit of the labor-saving machinery of the calculus and are, indeed, frequently incapacitated for using it." (Thomas Hill, "Uses of Mathesis", Bibliotheca Sacra Vol. 32 (127), 1875)

On Facts (-1799)

"While those whom devotion to abstract discussions has rendered unobservant of the facts are too ready to dogmatize on the basis of a few observations." (Aristotle, "De Caelo" ["On the Heavens"], cca. 350 BC)

"Everything we hear is an opinion, not a fact. Everything we see is a perspective, not the truth." (Marcus Aurelius, "Meditations", cca. 2nd century)

"The Syllogism consists of propositions, propositions consist of words, words are symbols of notions. Therefore if the notions themselves (which is the root of the matter) are confused and over-hastily abstracted from the facts, there can be no firmness in the superstructure. Our only hope therefore lies in a true induction." (Francis Bacon, "The New Organon", 1620)

"There are two kinds of truths: those of reasoning and those of fact. The truths of reasoning are necessary and their opposite is impossible; the truths of fact are contingent and their opposites are possible." (Gottfried W Leibniz, "Monadology", 1714)

"We have three principal means: observation of nature, reflection, and experiment. Observation gathers the facts reflection combines them, experiment verifies the result of the combination. It is essential that the observation of nature be assiduous, that reflection be profound, and that experimentation be exact. Rarely does one see these abilities in combination. And so, creative geniuses are not common." (Denis Diderot, "On the Interpretation of Nature", 1753)

"Facts, observations, experiments - these are the materials of a great edifice, but in assembling them we must combine them into classes, distinguish which belongs to which order and to which part of the whole each pertains." (Antoine L Lavoisier, "Mémoires de l’Académie Royale des Sciences", 1777)

"The impossibility of separating the nomenclature of a science from the science itself is owing to this, that every branch of physical science must consist of three things: the series of facts which are the objects of the science, the ideas which represent these facts, and the words by which these ideas are expressed. Like three impressions of the same seal, the word ought to produce the idea, and the idea to be a picture of the fact." (Antoine L Lavoisier, "Elements of Chemistry in a New Systematic Order", 1790)

"We must trust to nothing but facts: These are presented to us by Nature, and cannot deceive. We ought, in every instance, to submit our reasoning to the test of experiment, and never to search for truth but by the natural road of experiment and observation." (Antoin-Laurent de Lavoisiere, "Elements of Chemistry", 1790)

"[…] the speculative propositions of mathematics do not relate to facts; […] all that we are convinced of by any demonstration in the science, is of a necessary connection subsisting between certain suppositions and certain conclusions. When we find these suppositions actually take place in a particular instance, the demonstration forces us to apply the conclusion. Thus, if I could form a triangle, the three sides of which were accurately mathematical lines, I might affirm of this individual figure, that its three angles are equal to two right angles; but as the imperfection of my senses puts it out of my power to be, in any case, certain of the exact correspondence of the diagram which I delineate, with the definitions given in the elements of geometry, I never can apply with confidence to a particular figure, a mathematical theorem." (Dugald Stewart, "Elements of the Philosophy of the Human Mind", 1792)

"It has never yet been supposed, that all the facts of nature, and all the means of acquiring precision in the computation and analysis of those facts, and all the connections of objects with each other, and all the possible combinations of ideas, can be exhausted by the human mind." (Nicolas de Condorcet, "Outlines Of An Historical View Of The Progress Of The Human Mind", 1795)

Catastrophe Theory I

"[...] if the behavior points for the entire control surface are plotted and then connected, they form a smooth surface: the behavior surface. The surface has an overall slope from high values where rage predominates to low values in the region where fear is the prevailing state of mind, but the slope is not its most distinctive feature. Catastrophe theory reveals that in the middle of the surface there must be a smooth double fold, creating a pleat without creases, which grows narrower from the front of the surface to the back and eventually disappears in a singular point where the three sheets of the pleat come together. It is the pleat that gives the model its most interesting characteristics. All the points on the behavior surface represent the most probable behavior [...], with the exception of those on the middle sheet, which represent least probable behavior. Through catastrophe theory we can deduce the shape of the entire surface from the fact that the behavior is bimodal for some control points." (E Cristopher Zeeman, "Catastrophe Theory", Scientific American, 1976)

"Catastrophe Theory is - quite likely - the first coherent attempt (since Aristotelian logic) to give a theory on analogy. When narrow-minded scientists object to Catastrophe Theory that it gives no more than analogies, or metaphors, they do not realise that they are stating the proper aim of Catastrophe Theory, which is to classify all possible types of analogous situations." (René F Thom," La Théorie des catastrophes: État présent et perspective", 1977)

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

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

"But there is another kind of change, too, change that is less suited to mathematical analysis: the abrupt bursting of a bubble, the discontinuous transition from ice at its melting point to water at its freezing point, the qualitative shift in our minds when we 'get' a pun or a play on words. Catastrophe theory is a mathematical language created to describe and classify this second type of change. It challenges scientists to change the way they think about processes and events in many fields." (Alexander Woodcock & Monte Davis, "Catastrophe Theory", 1978)

"Catastrophe theory is a controversial new way of thinking about change - change in a course of events, change in an object's shape, change in a system's behavior, change in ideas themselves. Its name suggests disaster, and indeed the theory can be applied to literal catastrophes such as the collapse of a bridge or the downfall of an empire. But it also deals with changes as quiet as the dancing of sunlight on the bottom of a pool of water and as subtle as the transition from waking to sleep." (Alexander Woodcock & Monte Davis, "Catastrophe Theory", 1978)

"Catastrophes are often stimulated by the failure to feel the emergence of a domain, and so what cannot be felt in the imagination is experienced as embodied sensation in the catastrophe. (William I Thompson, "Gaia, a Way of Knowing: Political Implications of the New Biology", 1987)

"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])

"Catastrophe theory can be thought of as a link between classical analysis, dynamical systems, differential topology (including singularity theory), modern bifurcation theory and the theory of complex systems. [...] The name ‘catastrophe theory’ is used for a combination of singularity theory and its applications. [...] From the didactical point of view, there are two main positions for courses in catastrophe theory at university level: Trying to teach the theory as a perfect axiomatic system consisting of exact definitions, theorems and proofs or trying to teach mathematics as it can be developed from historical or from natural problems." (Werner Sanns, "Catastrophe Theory" [Mathematics of Complexity and Dynamical Systems, 2012])

"Classification is only one of the mathematical aspects of catastrophe theory. Another is stability. The stable states of natural systems are the ones that we can observe over a longer period of time. But the stable states of a system, which can be described by potential functions and their singularities, can become unstable if the potentials are changed by perturbations. So stability problems in nature lead to mathematical questions concerning the stability of the potential functions." (Werner Sanns, "Catastrophe Theory" [Mathematics of Complexity and Dynamical Systems, 2012])

Hiroki Sayama - Collected Quotes

"A giant component is a connected component whose size is on the same order of magnitude as the size of the whole network. Network percolation is the appearance of such a giant component in a random graph, which occurs when the average node degree is above 1." (Hiroki Sayama, "Introduction to the Modeling and Analysis of Complex Systems", 2015)

"A good model is simple, valid, and robust. Simplicity of a model is really the key essence of what modeling is all about. The main reason why we want to build a model is that we want to have a shorter, simpler description of reality. [...] Validity of a model is how closely the model’s prediction agrees with the observed reality. This is of utmost importance from a practical viewpoint. If your model’s prediction doesn’t reasonably match the observation, the model is not representing reality and is probably useless. [...] finally, robustness of a model is how insensitive the model’s prediction is to minor variations of model assumptions and/or parameter settings. This is important because there are always errors when we create assumptions about, or measure parameter values from, the real world. If the prediction made by your model is sensitive to their minor variations, then the conclusion derived from it is probably not reliable." (Hiroki Sayama, "Introduction to the Modeling and Analysis of Complex Systems", 2015)

"A model is a simplified representation of a system. It can be conceptual, verbal, diagrammatic, physical, or formal (mathematical)." (Hiroki Sayama, "Introduction to the Modeling and Analysis of Complex Systems", 2015)

"A network (or graph) consists of a set of nodes (or vertices, actors) and a set of edges (or links, ties) that connect those nodes. [...] The size of a network is characterized by the numbers of nodes and edges in it." (Hiroki Sayama, "Introduction to the Modeling and Analysis of Complex Systems", 2015)

"Bifurcation is a qualitative, topological change of a system’s phase space that occurs when some parameters are slightly varied across their critical thresholds. Bifurcations play important roles in many real-world systems as a switching mechanism. […] There are two categories of bifurcations. One is called a local bifurcation, which can be characterized by a change in the stability of equilibrium points. It is called local because it can be detected and analyzed only by using localized information around the equilibrium point. The other category is called a global bifurcation, which occurs when non-local features of the phase space, such as limit cycles (to be discussed later), collide with equilibrium points in a phase space. This type of bifurcation can’t be characterized just by using localized information around the equilibrium point."  (Hiroki Sayama, "Introduction to the Modeling and Analysis of Complex Systems", 2015)

"Chaos can be understood as a dynamical process in which microscopic information hidden in the details of a system’s state is dug out and expanded to a macroscopically visible scale (stretching), while the macroscopic information visible in the current system’s state is continuously discarded (folding)." (Hiroki Sayama, "Introduction to the Modeling and Analysis of Complex Systems", 2015)

"Chaos is a long-term behavior of a nonlinear dynamical system that never falls in any static or periodic trajectories. [It] looks like a random fluctuation, but still occurs in completely deterministic, simple dynamical systems. [It] exhibits sensitivity to initial conditions. [It] occurs when the period of the trajectory of the system’s state diverges to infinity. [It] occurs when no periodic trajectories are stable." (Hiroki Sayama, "Introduction to the Modeling and Analysis of Complex Systems", 2015)

"Complex systems are networks made of a number of components that interact with each other, typically in a nonlinear fashion. Complex systems may arise and evolve through self-organization, such that they are neither completely regular nor completely random, permitting the development of emergent behavior at macroscopic scales." (Hiroki Sayama, "Introduction to the Modeling and Analysis of Complex Systems", 2015)

"Dynamics of a linear system are decomposable into multiple independent one-dimensional exponential dynamics, each of which takes place along the direction given by an eigenvector. A general trajectory from an arbitrary initial condition can be obtained by a simple linear superposition of those independent dynamics." (Hiroki Sayama, "Introduction to the Modeling and Analysis of Complex Systems", 2015)

"Emergence is a nontrivial relationship between the properties of a system at microscopic and macroscopic scales. Macroscopic properties are called emergent when it is hard to explain them simply from microscopic properties." (Hiroki Sayama, "Introduction to the Modeling and Analysis of Complex Systems", 2015)

"Equilibrium points are important for both theoretical and practical reasons. Theoretically, they are key points in the system’s phase space, which serve as meaningful references when we understand the structure of the phase space. And practically, there are many situations where we want to sustain the system at a certain state that is desirable for us. In such cases, it is quite important to know whether the desired state is an equilibrium point, and if it is, whether it is stable or unstable." (Hiroki Sayama, "Introduction to the Modeling and Analysis of Complex Systems", 2015)

"Mean-field approximation is a technique that ignores spatial relationships among components. It works quite well for systems whose parts are fully connected or randomly interacting with each other. It doesn’t work if the interactions are local or non-homogeneous, and/or if the system has a non-uniform pattern of states. In such cases, you could still use mean-field approximation as a preliminary, “zeroth-order” approximation, but you should not derive a final conclusion from it." (Hiroki Sayama, "Introduction to the Modeling and Analysis of Complex Systems", 2015)

"[...] nonlinearity means that the outputs of a system are not given by a linear combination of the inputs. In the context of system behavior, the inputs and outputs can be the current and next states of the system, and if their relationship is not linear, the system is called a nonlinear system." (Hiroki Sayama, "Introduction to the Modeling and Analysis of Complex Systems", 2015)

"One of the unique features of typical CA [ cellular automata] models is that time, space, and states of cells are all discrete. Because of such discreteness, the number of all possible state-transition functions is finite, i.e., there are only a finite number of “universes” possible in a given CA setting. Moreover, if the space is finite, all possible configurations of the entire system are also enumerable. This means that, for reasonably small CA settings, one can conduct an exhaustive search of the entire rule space or phase space to study the properties of all the 'parallel universes'." (Hiroki Sayama, "Introduction to the Modeling and Analysis of Complex Systems", 2015)

"Self-organization is a dynamical process by which a system spontaneously forms nontrivial macroscopic structures and/or behaviors over time." (Hiroki Sayama, "Introduction to the Modeling and Analysis of Complex Systems", 2015)

"The challenge in developing a model becomes particularly tough when it comes to the modeling of complex systems, because their unique properties (networks, nonlinearity, emergence, self-organization, etc.) are not what we are familiar with. We usually think about things on a single scale in a step-by-step, linear chain of reasoning, in which causes and effects are clearly distinguished and discussed sequentially. But this approach is not suitable for understanding complex systems where a massive amount of components are interacting with each other interdependently to generate patterns over a broad range of scales. Therefore, the behavior of complex systems often appears to contradict our everyday experiences." (Hiroki Sayama, "Introduction to the Modeling and Analysis of Complex Systems", 2015)

"The sensitivity of chaotic systems to initial conditions is particularly well known under the moniker of the 'butterfly effect', which is a metaphorical illustration of the chaotic nature of the weather system in which 'a flap of a butterfly’s wings in Brazil could set off a tornado in Texas'. The meaning of this expression is that, in a chaotic system, a small perturbation could eventually cause very large-scale difference in the long run." (Hiroki Sayama, "Introduction to the Modeling and Analysis of Complex Systems", 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)

"Trajectories of a deterministic dynamical system will never branch off in its phase space (though they could merge), because if they did, that would mean that multiple future states were possible, which would violate the deterministic nature of the system. No branching means that, once you specify an initial state of the system, the trajectory that follows is uniquely determined too. You can visually inspect where the trajectories are going in the phase space visualization. They may diverge to infinity, converge to a certain point, or remain dynamically changing yet stay in a confined region in the phase space from which no outgoing trajectories are running out. Such a converging point or a region is called an attractor." (Hiroki Sayama, "Introduction to the Modeling and Analysis of Complex Systems", 2015)

"Variable rescaling is a technique to eliminate parameters from your model without losing generality. The basic idea is this: Variables that appear in your model represent quantities that are measured in some kind of units, but those units can be arbitrarily chosen without changing the dynamics of the system being modeled. This must be true for all scientific quantities that have physical dimensions - switching from inches to centimeters shouldn’t cause any change in how physics works! This means that you have the freedom to choose any convenient unit for each variable, some of which may simplify your model equations." (Hiroki Sayama, "Introduction to the Modeling and Analysis of Complex Systems", 2015)

On Complex Numbers XIX (Euler's Formula II)

"The equation e^πi+1 = 0 is true only by virtue of a large number of profound connections across many fields. It is true because of what it means! And it means what it means because of all those metaphors and blends in the conceptual system of a mathematician who understands what it means. To show why such an equation is true for conceptual reasons is to give what we have called an idea analysis of the equation." (George Lakoff & Rafael E Nuñez, "Where Mathematics Come From: How the Embodied Mind Brings Mathematics into Being", 2000)

"The equation e^πi =-1 says that the function w= e^z, when applied to the complex number πi as input, yields the real number -1 as the output, the value of w. In the complex plane, πi is the point [0,π) - π on the i-axis. The function w=e^z maps that point, which is in the z-plane, onto the point (-1, 0) - that is, -1 on the x-axis-in the w-plane. […] But its meaning is not given by the values computed for the function w=e^z. Its meaning is conceptual, not numerical. The importance of  e^πi =-1 lies in what it tells us about how various branches of mathematics are related to one another - how algebra is related to geometry, geometry to trigonometry, calculus to trigonometry, and how the arithmetic of complex numbers relates to all of them." (George Lakoff & Rafael E Nuñez, "Where Mathematics Come From: How the Embodied Mind Brings Mathematics into Being", 2000)

"The significance of e^πi+1 = 0 is thus a conceptual significance. What is important is not just the numerical values of e, π, i, 1, and 0 but their conceptual meaning. After all, e, π, i, 1, and 0 are not just numbers like any other numbers. Unlike, say, 192,563,947.9853294867, these numbers have conceptual meanings in a system of common, important nonmathematical concepts, like change, acceleration, recurrence, and self-regulation.

They are not mere numbers; they are the arithmetizations of concepts. When they are placed in a formula, the formula incorporates the ideas the function expresses as well as the set of pairs of complex numbers it mathematically determines by virtue of those ideas." (George Lakoff & Rafael E Nuñez, "Where Mathematics Come From: How the Embodied Mind Brings Mathematics into Being", 2000)

"We will now turn to e^πi+1 = 0. Our approach will be there as it was here. e^πi+1 = 0 uses the conceptual structure of all the cases we have discussed so far - trigonometry, the exponentials, and the complex numbers. Moreover, it puts together all that conceptual structure. In other words, all those metaphors and blends are simultaneously activated and jointly give rise to inferences that they would not give rise to separately. Our job is to see how e^πi+1 = 0 is a precise consequence that arises when the conceptual structure of these three domains is combined to form a single conceptual blend." (George Lakoff & Rafael E Nuñez, "Where Mathematics Come From: How the Embodied Mind Brings Mathematics into Being", 2000)

"[…] the equation’s five seemingly unrelated numbers (e, i, π, 1, and 0) fit neatly together in the formula like contiguous puzzle pieces. One might think that a cosmic carpenter had jig-sawed them one day and mischievously left them conjoined on Euler’s desk as a tantalizing hint of the unfathomable connectedness of things.[…] when the three enigmatic numbers are combined in this form, e^iπ, they react together to carve out a wormhole that spirals through the infinite depths of number space to emerge smack dab in the heartland of integers." (David Stipp, "A Most Elegant Equation: Euler's Formula and the Beauty of Mathematics", 2017)

"Thus, while feelings may be the essence of subjectivity, they are by no means part of our weaker nature - the valences they automatically generate are integral to our thought processes and without them we’d simply be lost. In particular, we’d have no sense of beauty at all, much less be able to feel (there’s that word again) that we’re in the presence of beauty when contemplating a work such as Euler’s formula. After all, e^iπ + 1 = 0 can give people limbic-triggered goosebumps when they first peer with understanding into its depths." (David Stipp, "A Most Elegant Equation: Euler's Formula and the Beauty of Mathematics", 2017)

"Today, Euler’s formula is a tool as basic to electrical engineers and physicists as the spatula is to short-order cooks. It’s arguable that the formula’s ability to simplify the design and analysis of circuits contributed to the accelerating pace of electrical innovation during the twentieth century." (David Stipp, "A Most Elegant Equation: Euler's Formula and the Beauty of Mathematics", 2017)

"[…] when the three enigmatic numbers are combined in this form, e^iπ, they react together to carve out a wormhole that spirals through the infinite depths of number space to emerge smack dab in the heartland of integers." (David Stipp, "A Most Elegant Equation: Euler's Formula and the Beauty of Mathematics", 2017)

"Euler’s formula - although deceptively simple - is actually staggeringly conceptually difficult to apprehend in its full glory, which is why so many mathematicians and scientists have failed to see its extraordinary scope, range, and ontology, so powerful and extensive as to render it the master equation of existence, from which the whole of mathematics and science can be derived, including general relativity, quantum mechanics, thermodynamics, electromagnetism and the strong and weak nuclear forces! It’s not called the God Equation for nothing. It is much more mysterious than any theistic God ever proposed." (Thomas Stark, "God Is Mathematics: The Proofs of the Eternal Existence of Mathematics", 2018)