On Averages I

“It is difficult to understand why statisticians commonly limit their inquiries to Averages, and do not revel in more comprehensive views. Their souls seem as dull to the charm of variety as that of the native of one of our flat English counties, whose retrospect of Switzerland was that, if its mountains could be thrown into its lakes, two nuisances would be got rid of at once. An Average is but a solitary fact, whereas if a single other fact be added to it, an entire Normal Scheme, which nearly corresponds to the observed one, starts potentially into existence.” (Sir Francis Galton, “Natural Inheritance”, 1889)

“Statistics may rightly be called the science of averages. […] Great numbers and the averages resulting from them, such as we always obtain in measuring social phenomena, have great inertia. […] It is this constancy of great numbers that makes statistical measurement possible. It is to great numbers that statistical measurement chiefly applies.” (Sir Arthur L Bowley, “Elements of Statistics”, 1901)

“[…] the new mathematics is a sort of supplement to language, affording a means of thought about form and quantity and a means of expression, more exact, compact, and ready than ordinary language. The great body of physical science, a great deal of the essential facts of financial science, and endless social and political problems are only accessible and only thinkable to those who have had a sound training in mathematical analysis, and the time may not be very remote when it will be understood that for complete initiation as an efficient citizen of one of the new great complex world wide states that are now developing, it is as necessary to be able to compute, to think in averages and maxima and minima, as it is now to be able to read and to write.” (Herbert G Wells, “Mankind In the Making”, 1906)

“Of itself an arithmetic average is more likely to conceal than to disclose important facts; it is the nature of an abbreviation, and is often an excuse for laziness.” (Arthur L Bowley, “The Nature and Purpose of the Measurement of Social Phenomena”, 1915)

“Averages are like the economic man; they are inventions, not real. When applied to salaries they hide gaunt poverty at the lower end.” (Julia Lathrop, 1919)

“Scientific laws, when we have reason to think them accurate, are different in form from the common-sense rules which have exceptions: they are always, at least in physics, either differential equations, or statistical averages.” (Bertrand A Russell, “The Analysis of Matter”, 1927)

"An average is a single value which is taken to represent a group of values. Such a representative value may be obtained in several ways, for there are several types of averages. […] Probably the most commonly used average is the arithmetic average, or arithmetic mean." (John R Riggleman & Ira N Frisbee, "Business Statistics", 1938)

"Because they are determined mathematically instead of according to their position in the data, the arithmetic and geometric averages are not ascertained by graphic methods." (John R Riggleman & Ira N Frisbee, "Business Statistics", 1938)

“Myth is more individual and expresses life more precisely than does science. Science works with concepts of averages which are far too general to do justice to the subjective variety of an individual life.” (Carl G Jung, “Memories, Dreams, Reflections”, 1963)

“While the individual man is an insoluble puzzle, in the aggregate he becomes a mathematical certainty. You can, for example, never foretell what anyone man will be up to, but you can say with precision what an average number will be up to. Individuals vary, but percentages remain constant. So says the statistician.” (Sir Arthur C Doyle)

On Statistics: Some Historical Definitions (1951 – 1999)

“Statistics is the name for that science and art which deals with uncertain inferences - which uses numbers to find out something about nature and experience.” (Warren Weaver, 1952)

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

"[Statistics] is concerned with things we can count. In so far as things, persons, are unique or ill-defi ned, statistics are meaningless and statisticians silenced; in so far as things are similar and definite - so many workers over 25, so many nuts and bolts made during December - they can be counted and new statistical facts are born.” (Maurice S Bartlett, “Essays on Probability and Statistics”, 1962)

“Statistics is the branch of scientific method which deals with the data obtained by counting or measuring the properties of populations of natural phenomena.” (Sir Maurice G Kendall & Alan Stuart, “The Advanced Theory of Statistics”, 1963)

“Statistics may be defined as the discipline concerned with the treatment of numerical data derived from groups of individuals.” (Peter Armitage, “Statistical Methods in Medical Research”, 1971)

“We provisionally define statistics as the study of how information should be employed to reflect on, and give guidance for action in, a practical situation involving uncertainty.” (Vic Barnett, “Comparative Statistical Inference” 2nd Ed., 1982)

“Statistics is a tool. In experimental science you plan and carry out experiments, and then analyse and interpret the results. To do this you use statistical arguments and calculations. Like any other tool - an oscilloscope, for example, or a spectrometer, or even a humble spanner - you can use it delicately or clumsily, skillfully or ineptly. The more you know about it and understand how it works, the better you will be able to use it and the more useful it will be.” (Roger Barlow, “Statistics: A Guide to the Use of Statistical Methods in the Physical Sciences”, 1989)

“The science of statistics may be described as exploring, analyzing and summarizing data; designing or choosing appropriate ways of collecting data and extracting information from them; and communicating that information. Statistics also involves constructing and testing models for describing chance phenomena. These models can be used as a basis for making inferences and drawing conclusions and, finally, perhaps for making decisions.” (Fergus Daly et al, “Elements of Statistics”, 1995)

“Statistics is a general intellectual method that applies wherever data, variation, and chance appear. It is a fundamental method because data, variation and chance are omnipresent in modern life. It is an independent discipline with its own core ideas rather than, for example, a branch of mathematics. […] Statistics offers general, fundamental, and independent ways of thinking.” (David S Moore, “Statistics among the Liberal Arts”, Journal of the American Statistical Association, 1998)

Further definitions:
1800-1900
1901-1950
2001- …

On Statistics: Some Historical Definitions (1901-1950)

"[…] statistics is the science of the measurement of the social organism, regarded as a whole, in all its manifestations." (Sir Arthur L Bowley, "Elements of Statistics", 1901)

"Statistics may rightly be called the science of averages. […] Great numbers and the averages resulting from them, such as we always obtain in measuring social phenomena, have great inertia. […] It is this constancy of great numbers that makes statistical measurement possible. It is to great numbers that statistical measurement chiefly applies." (Sir Arthur L Bowley, "Elements of Statistics", 1901)

"Statistics may, for instance, be called the science of counting. Counting appears at first sight to be a very simple operation, which any one can perform or which can be done automatically; but, as a matter of fact, when we come to large numbers, e.g., the population of the United Kingdom, counting is by no means easy, or within the power of an individual; limits of time and place alone prevent it being so carried out, and in no way can absolute accuracy be obtained when the numbers surpass certain limits." (Sir Arthur L Bowley, "Elements of Statistics", 1901)

"[...] statistics is the science that, through calculation, leads to an understanding of the characteristics of human societies, and its purpose is the study of masses through the enumeration of the units that compose them." (Armand Julin, "Summary for a Course of Statistics, General and Applied, 1910)

"The science of Statistics is the method of judging collective, natural or social phenomenon from the results obtained from the analysis or enumeration or collection of estimates." (Willford I King, "The Elements of Statistical Method", 1912)

"By Statistics we mean aggregate of facts affected to a marked extent by multiplicity of factors [...] and placed in relation to each other." (Horace Secrist, "An Introduction to Statistical Methods", 1917)

"Statistics may be defined as numerical statements of facts by means of which large aggregates are analyzed, the relations of individual units to their groups are ascertained, comparisons are made between groups, and continuous records are maintained for comparative purposes." (Melvin T Copeland. "Statistical Methods" [in: Harvard Business Studies, Vol. III, Ed. by Melvin T Copeland, 1917])

"Statistics may be regarded as (i) the study of populations, (ii) as the study of variation, and (iii) as the study of methods of the reduction of data." (Sir Ronald A Fisher, "Statistical Methods for Research Worker", 1925)

"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)

"[Statistics] is both a science and an art. It is a science in that its methods are basically systematic and have general application; and an art in that their successful application depends to a considerable degree on the skill and special experience of the statistician, and on his knowledge of the field of application, e.g. economics." (Leonard H C Tippett, "Statistics", 1943)

"Statistics is the branch of scientific method which deals with the data obtained by counting or measuring the properties of populations of natural phenomena. In this definition 'natural phenomena' includes all the happenings of the external world, whether human or not " (Sir Maurice G Kendall, "Advanced Theory of Statistics", Vol. 1, 1943)

"To some people, statistics is ‘quartered pies, cute little battleships and tapering rows of sturdy soldiers in diversified uniforms’. To others, it is columns and columns of numerical facts. Many regard it as a branch of economics. The beginning student of the subject considers it to be largely mathematics." (The Editors, "Statistics, The Physical Sciences and Engineering", The American Statistician, Vol. 2, No. 4, 1948) [Link]

Further definitions:
1800-1900
1951-2000
2001- …

On Statistics: Some Modern Definitions (2001 – …)

“Statistics is the branch of mathematics that uses observations and measurements called data to analyze, summarize, make inferences, and draw conclusions based on the data gathered.” (Allan G Bluman, “Probability Demystified”, 2005)

“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)

“Statistics is the art of learning from data. It is concerned with the collection of data, their subsequent description, and their analysis, which often leads to the drawing of conclusions.” (Sheldon M Ross, “Introductory Statistics” 3rd Ed., 2009)

“Statistics is the science of collecting, organizing, analyzing, and interpreting data in order to make decisions.” (Ron Larson & Betsy Farber, “Elementary Statistics: Picturing the World” 5th Ed., 2011)

“Statistics is the discipline of using data samples to support claims about populations.” (Allen B Downey, “Think Stats: Probability and Statistics for Programmers”, 2011)

“Statistics is the scientific discipline that provides methods to help us make sense of data. […] The field of statistics teaches us how to make intelligent judgments and informed decisions in the presence of uncertainty and variation.” (Roxy Peck & Jay L Devore, “Statistics: The Exploration and Analysis of Data” 7th Ed, 2012)

“[…] statistics is a method of pursuing truth. At a minimum, statistics can tell you the likelihood that your hunch is true in this time and place and with these sorts of people. This type of pursuit of truth, especially in the form of an event’s future likelihood, is the essence of psychology, of science, and of human evolution.” (Arthhur Aron et al, "Statistics for Phsychology" 6th Ed., 2012)

“Statistics is the art and science of designing studies and analyzing the data that those studies produce. Its ultimate goal is translating data into knowledge and understanding of the world around us. In short, statistics is the art and science of learning from data.” (Alan Agresti & Christine Franklin, “Statistics: The Art and Science of Learning from Data” 3rd Ed., 2013)

“Statistics is a science that helps us make decisions and draw conclusions in the presence of variability.” (Douglas C Montgomery & George C Runger, “Applied Statistics and Probability for Engineers” 6th Ed., 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)

“Statistics can be defined as a collection of techniques used when planning a data collection, and when subsequently analyzing and presenting data.” (Birger S Madsen, “Statistics for Non-Statisticians”, 2016)

“Statistics is the science of collecting, organizing, and interpreting numerical facts, which we call data. […] Statistics is the science of learning from data.” (Moore McCabe & Alwan Craig, “The Practice of Statistics for Business and Economics” 4th Ed., 2016)

“Statistics is the science of collecting, organizing, summarizing, and analyzing information to draw conclusions or answer questions. In addition, statistics is about providing a measure of confidence in any conclusions.” (Michael Sullivan, “Statistics: Informed Decisions Using Data”, 5th Ed., 2017)

Further definitions:
1800-1900
1901-1950
1951-2000

Statistics – Some Historical Definitions (1800-1900)

“Il y a, dans la statistique, deux choses qui se trouvent continuellement mélangées, und methode et une science. On emploie la statistique comme methode, toutes les fois que l’on compte on que l’on mesure quelque chose, par example, l’éndendue d’un district, le mobre de habitants d’un pays, la quantité ou le prix de certaines denrées, etc. […] Il y a, de plus, une science de la statistique. Elle consiste à savoir réunir les chiffres, les combiner et les calculer, de la manière la plus propre à conduire à des résultats certains. Mais ceci n’est, à propement parler qu’une branche de mathémetiques.“ (Alphonse P de Candolle, “Considérations sur le statistique des délits” 1833)

“There are two aspects of statistics that are continually mixed, the method and the science. Statistics are used as a method, whenever we measure something, for example, the size of a district, the number of inhabitants of a country, the quantity or price of certain commodities, etc. […] There is, moreover, a science of statistics. It consists of knowing how to gather numbers, combine them and calculate them, in the best way to lead to certain results. But this is, strictly speaking, a branch of mathematics." (Alphonse P de Candolle, “Considerations on Crime Statistics”, 1833)

"[statistics is] that department of political science which is concerned in collecting and arranging facts illustrative of the condition and resources of the state. To reason upon such facts and to draw conclusions from them is not within the province of statistics; but is the business of the statesman and of the political economist." (Penny Cyclopedia, 1842)

“Statistics has then for its object that of presenting a faithful representation of a state at a determined epoch.” (Adolphe Quetelet, 1849) 

"Observations and statistics agree in being quantities grouped about a Mean; they differ, in that the Mean of observations is real, of statistics is fictitious. The mean of observations is a cause, as it were the source from which diverging errors emanate. The mean of statistics is a description, a representative quantity put for a whole group, the best representative of the group, that quantity which, if we must in practice put one quantity for many, minimizes the error unavoidably attending such practice. Thus measurements by the reduction of which we ascertain a real time, number, distance are observations. Returns of prices, exports and imports, legitimate and illegitimate marriages or births and so forth, the averages of which constitute the premises of practical reasoning, are statistics. In short, observations are different copies of one original; statistics are different originals affording one ‘generic portrait’. Different measurements of the same man are observations; but measurements of different men, grouped round l’homme moyen, are prima facie at least statistics." (Francis Y Edgeworth, 1885) 

“[Statistics] are the only tools by which an opening can be cut through the formidable thicket of difficulties that bars the path of those who pursue the Science of man.” (Sir Francis Galton, “Natural Inheritance”, 1889)

Further definitions:
1901-1950
1951-2000
2001- …

On Statistics: What is Statistics?

“Statistics is a science which ought to be honourable, the basis of many most important sciences; but it is not to be carried on by steam, this science, any more than others are; a wise hand is requisite for carrying it on. Conclusive facts are inseparable from unconclusive except by a head that already understands and knows.” (Thomas Carlyle, “Critical and Miscellaneous Essays”, 1838)

“To some people, statistics is ‘quartered pies, cute little battleships and tapering rows of sturdy soldiers in diversified uniforms’. To others, it is columns and columns of numerical facts. Many regard it as a branch of economics. The beginning student of the subject considers it to be largely mathematics.” (The Editors, “Statistics, The Physical Sciences and Engineering”, The American Statistician, Vol. 2, No. 4, 1948)

"Statistics is that branch of mathematics which deals with the accumulation and analysis of quantitative data." (David B MacNeil, "Modern Mathematics for the Practical Man", 1963)

“Statistics is the branch of scientific method which deals with the data obtained by counting or measuring the properties of populations of natural phenomena.” (Sir Maurice G Kendall & Alan Stuart, “The Advanced Theory of Statistics”, 1963)

“Statistics may be defined as the discipline concerned with the treatment of numerical data derived from groups of individuals.” (Peter Armitage, “Statistical Methods in Medical Research”, 1971)

“[…] statistics is the science that deals with distributions and proportions in actual (large but finite) classes (also called ‘populations’, ‘aggregates’, ‘ensembles’) of actual things.” (Bas C van Frassen, “The Scientic Image”, 1980)

“We provisionally define statistics as the study of how information should be employed to reflect on, and give guidance for action in, a practical situation involving uncertainty.” (Vic Barnett, “Comparative Statistical Inference” 2nd Ed., 1982)

“[Statistics] is the technology of extracting meaning from data.” (David J Hand, “Statistics: A Very Short Introduction”, 2008)

“[Statistics] is the technology of handling uncertainty.” (David J Hand, “Statistics: A Very Short Introduction”, 2008)

“[…] statistics is the key discipline for predicting the future or for making inferences about the unknown, or for producing convenient summaries of data.” (David J Hand, “Statistics: A Very Short Introduction”, 2008)

“[Statistics: used with a plural verb] facts or data, either numerical or nonnumerical, organized and summarized so as to provide useful and accessible information about a particular subject.” (Neil A Weiss, "Introductory Statistics" 10th Ed., 2017)

“[Statistics: used with a singular verb] the science of organizing and summarizing numerical or nonnumerical information.” (Neil A Weiss, "Introductory Statistics" 10th Ed., 2017)

“Statistics is the science of finding relationships and actionable insights from data.” (Nate Silver)

“Statistics is the science, the art, the philosophy, and the technique of making inferences from the particular to the general.” (John W Tukey)

On Statistics (Trivia)

"Statistics are like the hieroglyphics of ancient Egypt, where the lessons of history, the precepts of wisdom, and the secrets of the future were concealed in mysterious characters." (Moreau de Jonnes, "Elements of Statistics", 1856)

"A knowledge of statistics is like a knowledge of foreign languages or of algebra; it may prove of use at any time under any circumstances." (Sir Arthur L Bowley, "Elements of Statistics", 1901)

"I shall try not to use statistics as a drunken man uses lamp-posts, for support rather than for illumination;" (Andrew Lang [paraphrased in (Francis Yeats-Brown, "Lancer at Large", 1937)])

"Statistical thinking will one day be as necessary for efficient citizenship as the ability to read and write." (Samuel S Wilks 1951) [paraphrasing Herbert G Wells]

"[…] statistical techniques are tools of thought, and not substitutes for thought." (Abraham Kaplan, "The Conduct of Inquiry", 1964)

"The manipulation of statistical formulas is no substitute for knowing what one is doing." (Hubert M Blalock Jr., "Social Statistics" 2nd Ed., 1972)

"It is all too easy to notice the statistical sea that supports our thoughts and actions. If that sea loses its buoyancy, it may take a long time to regain the lost support." (William Kruskal, "Coordination Today: A Disaster or a Disgrace", The American Statistician, Vol. 37, No. 3, 1983)

"’Common sense’ is not common but needs to [be] learnt systematically […]. A ‘simple analysis’ can be harder than it looks […]. All statistical techniques, however sophisticated, should be subordinate to subjective judgment." (Christopher Chatfield, "The Initial Examination of Data", Journal of The Royal Statistical Society, Series A, Vol. 148, 1985)

"Statistics as a science is to quantify uncertainty, not unknown." (Chamont Wang, "Sense and Nonsense of Statistical Inference: Controversy, Misuse, and Subtlety", 1993)

"Statistics is a field of probabilities and sometimes probabilities do not go the way we want." (Jordan Morrow, "Be Data Literate: The data literacy skills everyone needs to succeed", 2021)

"A judicious man uses statistics, not to get knowledge, but to save himself from having ignorance foisted upon him." (Thomas Carlyle)

"Do not put faith in what statistics say until you have carefully considered what they do not say." (William W Watt)

"[Statistics are] the only tools by which an opening can be cut through the formidable thicket of difficulties that bars the path of those who pursue the Science of Man." (Sir Ronald A F Galton)

"The application of efficient statistical procedure has power, but the application of common sense has more." (Jasper Wall)

On Statistics: Unconventional Definitions

“Statistics: The mathematical theory of ignorance.” (Morris Kline)

“Statistics: Fiction in its most uninteresting form.” (Evan Esar, “Esar's Comic Dictionary”, 1943)

“Statistics: The science that can prove everything except the usefulness of statistics.” (Evan Esar, “Esar's Comic Dictionary”, 1943)

“Statistics: The only science that enables different experts using the same figures to draw different conclusions.” (Evan Esar, “Esar's Comic Dictionary”, 1943)

“Statistics: Data of a numerical kind looking for an argument.” (Evan Esar, “Esar's Comic Dictionary”, 1943)

“[Statistics] It is concerned with things we can count.” (Maurice S Bartlett, “Essays on Probability and Statistics”, 1962)

Statistics: The art of dealing with vagueness and with interpersonal difference in decision situations” (Leonard J Savage, “The Foundation of Statistics”, 1954)

“Statistics are the art of stating in precise terms that which one does not know.” (William Kruskal, “Statistics, Moliere, and Henry Adams”, American Scientist Magazine, 1967) [Link]

“Statistics are the refuge of the uninformed.” (Audrey Haber & Richard P Runion, “General Statistics”, 1973)

“Statistics is ’hocuspocus’ with numbers.” (Audrey Haber & Richard P Runion, “General Statistics”, 1973)

“Statistics are the art of lying by means of figures.” (Wilhelm Stekel, “Marriage at the Crossroads”, 1931)

“Statistics: 1. A form of lying that is neither black, white, nor color. 2. An attempt to analyze data-rare and archaic. 3. A disorderly, but not quite random, progress from datum to datum.” (David Durand, “A Dictionary for Statismagicians”, The American Statistician, Vol. 24, No. 3, 1970)

Models vs Facts

“The logical picture of the facts is the thought. […] A picture is a model of reality. In a picture objects have the elements of the picture corresponding to them. The fact that the elements of a picture are related to one another in a determinate way represents that things are related to one another in the same way.” (Ludwig Wittgenstein, “Tractatus Logico-Philosophicus”, 1922)

"It is not impossible that our own Model will die a violent death, ruthlessly smashed by an unprovoked assault of new facts […]. (Clive S Lewis, “The Discarded Image: An Introduction to Medieval and Renaissance Literature”, 1964)

“A model is a useful (and often indispensable) framework on which to organize our knowledge about a phenomenon. […] It must not be overlooked that the quantitative consequences of any model can be no more reliable than the a priori agreement between the assumptions of the model and the known facts about the real phenomenon. When the model is known to diverge significantly from the facts, it is self-deceiving to claim quantitative usefulness for it by appeal to agreement between a prediction of the model and observation.” (John R Philip, 1966)

“[…] no good model ever accounted for all the facts, since some data was bound to be misleading if not plain wrong. A theory that did fit all the data would have been ‘carpentered’ to do this and would thus be open to suspicion.” (Francis H C Crick, “What Mad Pursuit: A Personal View of Scientific Discovery”, 1988)

“Modeling involves a style of scientific thinking in which the argument is structured by the model, but in which the application is achieved via a narrative prompted by an external fact, an imagined event or question to be answered.” (Uskali Mäki, “Fact and Fiction in Economics: Models, Realism and Social Construction”, 2002)

“We tackle a multifaceted universe one face at a time, tailoring our models and equations to fit the facts at hand. Whatever mechanical conception proves appropriate, that is the one to use. Discovering worlds within worlds, a practical observer will deal with each realm on its own terms. It is the only sensible approach to take.” (Michael Munowitz, “Knowing: The Nature of Physical Law”, 2005)

"There are no surprising facts, only models that are surprised by facts; and if a model is surprised by the facts, it is no credit to that model." (Eliezer S Yudkowsky, "Quantum Explanations", 2008)

“Science does not live with facts alone. In addition to facts, it needs models. Scientific models fulfill two main functions with respect to empirical facts.” (Andreas Bartels [in “Models, Simulations, and the Reduction of Complexity”, Ed. by Ulrich Gähde et al, 2013)

Models vs Theory

“A theory is just a mathematical model to describe the observations.” (Karl Popper)

"A theory is a purely mental image of how something should be." (Adrian Bejan)

“I am of the opinion that the task of the theory consists in constructing a picture of the external world that exists purely internally and must be our guiding star in all thought and experiment.” (Ludwig E Boltzmann)

"[...] we and our models are both part of the universe we are describing. Thus a physical theory is self referencing, like in Gödel’s theorem. One might therefore expect it to be either inconsistent or incomplete. The theories we have so far are both inconsistent and incomplete." (Stephen Hawking, “Godel and the End of the Universe”) [Link]

“With each theory or model, our concepts of reality and of the fundamental constituents of the universe have changed.” (Stephen Hawking & Leonard Mlodinow, “The Grand Design”, 2010)

"A theory has only the alternative of being right or wrong. A model has a third possibility: it may be right, but irrelevant." (Manfred Eigen, 1973)

“Model is used as a theory. It becomes theory when the purpose of building a model is to understand the mechanisms involved in the developmental process. Hence as theory, model does not carve up or change the world, but it explains how change takes place and in what way or manner. This leads to build change in the structures.” (Laxmi Kanta Patnaik, “Model Building in Political Science”, The Indian Journal of Political Science, Vol. 50, No. 2, 1989) [Link]

“A theory is a set of deductively closed propositions that explain and predict empirical phenomena, and a model is a theory that is idealized.” (Jay Odenbaugh, “True Lies: Realism, Robustness, and Models”, Philosophy of Science, Vol. 78, No. 5, 2011) [Link]

“A theory is a good theory if it satisfies two requirements: it must accurately describe a large class of observations on the basis of a model that contains only a few arbitrary elements, and it must make definite predictions about the results of future observations.” (Stephen Hawking, “A Brief History of Time: From Big Bang To Black Holes”, 1988)

On Models II

"The soul never thinks without a picture." (Aristotle)

"We must make a threefold distinction and think of that which becomes, that in which it becomes, and the model which it resembles" (Plato, "Timaeus")

"Rules and models destroy genius and art." (William Hazlitt, "Sketches and Essays", 1839)

"A model is done when nothing else can be taken out." (Freeman J Dyson)

"The purpose of models is not to fit the data but to sharpen the question." (Samuel Karlin, 1983)

"Classical models tell us more than we at first can know." (Karl Popper)

"Models are to be used, but not to be believed." (Henri Theil, "Principles of Econometrics", 1971)

"Do not quench your inspiration and your imagination; do not become the slave of your model." (Vincent van Gogh)

On Models: What is a Model?

“A model is a deliberately simplified representation of a much more complicated situation. […] The idea is to focus on one or two causal or conditioning factors, exclude everything else, and hope to understand how just these aspects of reality work and interact.” (Robert M Solow, “How Did Economics Get That Way and What Way Did It Get?”, Daedalus, Vol. 126, No. 1, 1997) [Link]

“A model is an abstract description of the real world. It is a simple representation of more complex forms, processes and functions of physical phenomena and ideas.” (Moshe F Rubinstein & Iris R Firstenberg, “Patterns of Problem Solving”, 1975)

“A model […] is a story with a specified structure: to explain this catch phrase is to explain what a model is. The structure is given by the logical and mathematical form of a set of postulates, the assumptions of the model. The structure forms an uninterpreted system, in much the way the postulates of a pure geometry are now commonly regarded as doing. The theorems that follow from the postulates tell us things about the structure that may not be apparent from an examination of the postulates alone.” (Allan Gibbard & Hal R. Varian, “Economic Models”, The Journal of Philosophy, Vol. 75, No. 11, 1978) [Link]

“A model is an imitation of reality and a mathematical model is a particular form of representation.” (Ian T Cameron & Katalin Hangos, “Process Modelling and Model Analysis”, 2001)

“A model is an external and explicit representation of part of reality as seen by the people who wish to use that model to understand, to change, to manage, and to control that part of reality in some way or other.” (Michael Pidd, “Just Modeling through: A Rough Guide to Modeling”, Interfaces, Vol. 29, No. 2, 1999) [Link]

“[…] a model is the picture of the real - a short form of the whole. Hence, a model is an abstraction or simplification of a system. It is a technique by which aspects of reality can be 'artificially' represented or 'simulated' and at the same time simplified to facilitate comprehension.” (Laxmi K Patnaik, “Model Building in Political Science”, The Indian Journal of Political Science, Vol. 50, No. 2, 1989) [Link]

“A model is something one tries to construct when one has to describe a complicated situation. A model is therefore an approximate description of reality and invariably involves many simplifying assumptions. […] models are convenient idealisations.” (Ganeschan Venkataraman, “Chandrasekhar and His Limit”, 1992)

“A model isolates one or a few causal connections, mechanisms, or processes, to the exclusion of other contributing or interfering factors - while in the actual world, those other factors make their effects felt in what actually happens. Models may seem true in the abstract, and are false in the concrete. The key issue is about whether there is a bridge between the two, the abstract and the concrete, such that a simple model can be relied on as a source of relevantly truthful information about the complex reality.” (Uskali Mäki, “Fact and Fiction in Economics: Models, Realism and Social Construction”, 2002)

“A model is a representation in that it (or its properties) is chosen to stand for some other entity (or its properties), known as the target system. A model is a tool in that it is used in the service of particular goals or purposes; typically these purposes involve answering some limited range of questions about the target system.” (Wendy S Parker, “Confirmation and Adequacy-for-Purpose in Climate Modelling”, Proceedings of the Aristotelian Society, Supplementary Volumes, Vol. 83, 2009) [Link]

“A model is essentially a calculating engine designed to produce some output for a given input.” (Richard C. Lewontin, “Models, Mathematics and Metaphors”, Synthese, Vol. 15, No. 2, 1963) [Link]

“In order to deal with these phenomena, we abstract from details and attempt to concentrate on the larger picture - a particular set of features of the real world or the structure that underlies the processes that lead to the observed outcomes. Models are such abstractions of reality. Models force us to face the results of the structural and dynamic assumptions that we have made in our abstractions.” (Bruce Hannon and Matthias Ruth, “Dynamic Modeling of Diseases and Pests”, 2009)

On Models: Good Models

"A good model makes the right strategic simplifications. In fact, a really good model is one that generates a lot of understanding from focusing on a very small number of causal arrows." (Robert M Solow, "How Did Economics Get That Way and What Way Did It Get?", Daedalus, Vol. 126, No. 1, 1997) [Link]

"[…] no good model ever accounted for all the facts, since some data was bound to be misleading if not plain wrong. A theory that did fit all the data would have been ‘carpentered’ to do this and would thus be open to suspicion." (Francis H C Crick, "What Mad Pursuit: A Personal View of Scientific Discovery", 1988)

"A model is a good model if it:1. Is elegant2. Contains few arbitrary or adjustable elements3. Agrees with and explains all existing observations4. Makes detailed predictions about future observations that can disprove or falsify the model if they are not borne out." (Stephen Hawking & Leonard Mlodinow, "The Grand Design", 2010)

"One good experiment is worth a thousand models […]; but one good model can make a thousand experiments unnecessary." (David Lloyd & Evgenii I Volkov, "The Ultradian Clock: Timekeeping for Intracelular Dynamics" [in "Complexity, Chaos, and Biological Evolution", Ed. by Erik Mosekilde & Lis Mosekilde, 2013)

"When evaluating a model, at least two broad standards are relevant. One is whether the model is consistent with the data. The other is whether the model is consistent with the ‘real world’." (Kenneth A Bollen, "Structural Equations with Latent Variables", 1989)

"A mathematical model is never a completely accurate representation of a physical situation - it is an idealization. A good model simplifies reality enough to permit mathematical calculations but is accurate enough to provide valuable conclusions. It is important to realize the limitations of the model. In the end, Mother Nature has the final say." (James Stewart, "Calculus: Early Transcedentals" 8th Ed., 2016)

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

"All models are mental projections of our understanding of processes and feedbacks of systems in the real world. The general approach is that models are as good as the system upon which they are based. Models should be designed to answer specific questions and only incorporate the necessary details that are required to provide an answer." (Hördur V Haraldsson & Harald U Sverdrup, "Finding Simplicity in Complexity in Biogeochemical Modelling" [in "Environmental Modelling: Finding Simplicity in Complexity", Ed. by John Wainwright and Mark Mulligan], 2004)

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

On Group Theory: On Symmetry VIII

"Symmetries of a geometric object are traditionally described by its automorphism group, which often is an object of the same geometric class (a topological space, an algebraic variety, etc.). Of course, such symmetries are only a particular type of morphisms, so that Klein’s Erlanger program is, in principle, subsumed by the general categorical approach." (Yuri I Manin, "Topics in Noncommutative Geometry", 1991)

"The recognition of symmetry is intuitive but is often difficult to express in any simple and systematic manner. Group theory is a mathematical device to allow for the analysis of symmetry in a variety of ways." (M Ladd, "Symmetry and Group theory in Chemistry", 1998) 

"One reason for the importance of Riemannian manifolds is that they are generalizations of Euclidean geometry - general enough but not too general. They are still close enough to Euclidean geometry to have a Laplace operator. This is the key to quantum mechanics, heat and waves. The various generalizations of Riemannian manifold [...] do not have a simple natural unambiguous choice of such an operator. [...] Another reason for the prominence of Riemannian manifolds is that the maximal compact subgroup of the general linear group is the orthogonal group. So the least restriction we can make on any geometric structure so that it 'rigidifies' always adds a Riemannian geometry. Moreover, any geometric structure will always permit such a 'rigidification'. [...] Similarly, if we were to pick out a submanifold of the tangent bundle of some manifold, distinguishing tangent vectors, in such a manner that in each tangent space, any two lines could be brought to one another, or any two planes, etc., then the maximal symmetry group we could come up with in a single tangent space which was not the whole general linear group would be the orthogonal group of a Riemannian metric. So Riemannian geometry is the 'least' structure, or most symmetrical one, we can pick, at first order." (Marcel Berger, "A Panoramic View of Riemannian Geometry", 2003)

"Group theory is a branch of mathematics that describes the properties of an abstract model of phenomena that depend on symmetry. Despite its abstract tone, group theory provides practical techniques for making quantitative and verifiable predictions about the behavior of atoms, molecules and solids." (Arthur M Lesk, "Introduction to Symmetry and Group Theory for Chemists", 2004) 

"Group theory is a powerful tool for studying the symmetry of a physical system, especially the symmetry of a quantum system. Since the exact solution of the dynamic equation in the quantum theory is generally difficult to obtain, one has to find other methods to analyze the property of the system. Group theory provides an effective method by analyzing symmetry of the system to obtain some precise information of the system verifiable with observations." (Zhong-Qi Ma, Xiao-Yan Gu, "Problems and Solutions in Group Theory for Physicists", 2004)

"The potential freedom in the choice of a particular mathematical representation of physical objects is loosely called symmetry. In mathematical terms, physical symmetries are intimately related to groups in the sense that symmetry transformations form a group." (Teiko Heinosaari and Mario Ziman, "The Mathematical Language of Quantum Theory: From Uncertainty to Entanglement", 2012)

"Galois and Abel independently discovered the basic idea of symmetry. They were both coming at the problem from the algebra of polynomials, but what they each realized was that underlying the solution of polynomials was a fundamental problem of symmetry. The way that they understood symmetry was in terms of permutation groups. A permutation group is the most fundamental structure of symmetry. […] permutation groups are the master groups of symmetry: every kind of symmetry is encoded in the structure of the permutation group." (Mark C. Chu-Carroll, "Good Math: A Geek’s Guide to the Beauty of Numbers, Logic, and Computation", 2013)

"In a loose analogy, every finite symmetry group can be broken up, in a well-defined manner, into ‘indivisible’ symmetry groups - atoms of symmetry, so to speak. These basic building blocks for finite groups are known as simple groups - not because anything about them is easy, but in the sense of ‘not made up from several parts’. Just as atoms can be combined to build molecules, so these simple groups can be combined to build all finite groups." (Ian Stewart, "Symmetry: A Very Short Introduction", 2013)

"[…] the symmetry group of the infinite logarithmic spiral is an infinite group, with one element for each real number. Two such transformations compose by adding the corresponding angles, so this group is isomorphic to the real numbers under addition." (Ian Stewart, "Symmetry: A Very Short Introduction", 2013)

"[…] the role that symmetry plays is not confined to material objects. Symmetries can also refer to theories and, in particular, to quantum theory. For if the laws of physics are to be invariant under changes of reference frames, the set of all such transformations will form a group. Which transformations and which groups depends on the systems under consideration." (William H Klink & Sujeev Wickramasekara, "Relativity, Symmetry and the Structure of Quantum Theory I: Galilean quantum theory", 2015) 

"The theory of groups is considered the language par excellence to study symmetry in science; it provides the mathematical formalism needed to tackle symmetry in a precise way. The aim of this chapter, therefore, is to lay the foundations of abstract group theory." (Pieter Thyssen & Arnout Ceulemans, "Shattered Symmetry: Group Theory from the Eightfold Way to the Periodic Table", 2017) 

"The universe is an enormous direct product of representations of symmetry groups." (Steven Weinberg)

5 Books 10 Quotes II: Nature and the Quest for Models

Clifford A Pickover, "The Loom of God: Mathematical Tapestries at the Edge of Time", 1997
	"I do not know if God is a mathematician, but mathematics is the loom upon which God weaves the fabric of the universe. [...] The fact that reality can be described or approximated by simple mathematical expressions suggests to me that nature has mathematics at its core."
“In many ways, the mathematical quest to understand infinity parallels mystical attempts to understand God. Both religions and mathematics attempt to express the relationships between humans, the universe, and infinity. Both have arcane symbols and rituals, and impenetrable language. Both exercise the deep recesses of our mind and stimulate our imagination. Mathematicians, like priests, seek ‘ideal’, immutable, nonmaterial truths and then often try to apply theses truth in the real world.”
Fritjof Capra, "The Tao of Physics: An Exploration of the Parallels between Modern Physics and Eastern Mysticism", 1976
	“If physics leads us today to a world view which is essentially mystical, it returns, in a way, to its beginning, 2,500 years ago. […] Eastern thought and, more generally, mystical thought provide a consistent and relevant philosophical background to the theories of contemporary science; a conception of the world in which scientific discoveries can be in perfect harmony with spiritual aims and religious beliefs. The two basic themes of this conception are the unity and interrelation of all phenomena and the intrinsically dynamic nature of the universe. The further we penetrate into the submicroscopic world, the more we shall realize how the modern physicist, like the Eastern mystic, has come to see the world as a system of inseparable, interacting and ever-moving components with the observer being an integral part of this system.”
“Whenever the Eastern mystics express their knowledge in words - be it with the help of myths, symbols, poetic images or paradoxical statements-they are well aware of the limitations imposed by language and 'linear' thinking. Modern physics has come to take exactly the same attitude with regard to its verbal models and theories. They, too, are only approximate and necessarily inaccurate. They are the counterparts of the Eastern myths, symbols and poetic images, and it is at this level that I shall draw the parallels. The same idea about matter is conveyed, for example, to the Hindu by the cosmic dance of the god Shiva as to the physicist by certain aspects of quantum field theory. Both the dancing god and the physical theory are creations of the mind: models to describe their authors' intuition of reality.”
Mario Livio, "Is God a Mathematician?", 2011
	“The reality is that without mathematics, modern-day cosmologists could not have progressed even one step in attempting to understand the laws of nature. Mathematics provides the solid scaffolding that holds together any theory of the universe. […] Mathematics appears to be almost too effective in describing and explaining not only the cosmos at large, but even some of the most chaotic of human enterprises.”
“There are actually two sides to the success of mathematics in explaining the world around us (a success that Wigner dubbed ‘the unreasonable effectiveness of mathematics’), one more astonishing than the other. First, there is an aspect one might call ‘active’. When physicists wander through nature’s labyrinth, they light their way by mathematics—the tools they use and develop, the models they construct, and the explanations they conjure are all mathematical in nature. This, on the face of it, is a miracle in itself. […] But there is also a ‘passive’ side to the mysterious effectiveness of mathematics, and it is so surprising that the “active” aspect pales by comparison. Concepts and relations explored by mathematicians only for pure reasons—with absolutely no application in mind—turn out decades (or sometimes centuries) later to be the unexpected solutions to problems grounded in physical reality!”
Ian Stewart & Martin Golubitsky, “Fearful Symmetry: Is God a Geometer?”, 1992
	“Scientists use mathematics to build mental universes. They write down mathematical descriptions - models - that capture essential fragments of how they think the world behaves. Then they analyse their consequences. This is called 'theory'. They test their theories against observations: this is called 'experiment'. Depending on the result, they may modify the mathematical model and repeat the cycle until theory and experiment agree. Not that it's really that simple; but that's the general gist of it, the essence of the scientific method.”
“Nature behaves in ways that look mathematical, but nature is not the same as mathematics. Every mathematical model makes simplifying assumptions; its conclusions are only as valid as those assumptions.”
Peter Coles, “Hawking and the Mind of God”, 2000
	“[…] the search for a Theory of Everything also raises interesting philosophical questions. Some physicists, [Stephen] Hawking among them, would regard the construction of a Theory of Everything as being, in some sense, reading the mind of God. Or at least unravelling the inner secrets of physical reality. Others simply argue that a physical theory is just a description of reality, rather like a map.”
“To look at the development of physics since Newton is to observe a struggle to define the limits of science. Part of this process has been the intrusion of scientific methods and ideas into domains that have traditionally been the province of metaphysics or religion. In this conflict, Hawking’s phrase ‘to know the Mind of God’ is just one example of a border infringement. But by playing the God card, Hawking has cleverly fanned the flames of his own publicity, appealing directly to the popular allure of the scientist-as-priest.” Previous Post <<||>> Next Post

On Symmetry VII (Nature)

"Nature builds up by her refined and invisible architecture, with a delicacy eluding our conception, yet with a symmetry and beauty which we are never weary of admiring." (Sir John F W Herschel, "The Cabinet of Natural Philosophy", 1831)

"[…] the lifeless symmetry of architecture, however beautiful the design and proportion, no man would be so mad as to put in competition with the animated charms of nature." (Fanny Burney, "Evelina", 1909)

"The essential vision of reality presents us not with fugitive appearances but with felt patterns of order which have coherence and meaning for the eye and for the mind. Symmetry, balance and rhythmic sequences express characteristics of natural phenomena: the connectedness of nature - the order, the logic, the living process. Here art and science meet on common ground." (Gyorgy Kepes, "The New Landscape: In Art and Science", 1956)

"[…] nature, at the fundamental level, does not just prefer symmetry in a physical theory; nature demands it." (Jennifer T Thompson, "Beyond Einstein: The Cosmic Quest for the Theory of the Universe", 1987)

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

"[…] in all things that live there are certain irregularities and deficiencies which are not only signs of life, but sources of beauty. No human face is exactly the same in its lines on each side, no leaf perfect in its lobes, no branch in its symmetry. All admit irregularity as they imply change; […]" (John Ruskin, "The Stones of Venice: The Sea Stories", 2013)

"We find, therefore, under this orderly arrangement, a wonderful symmetry in the universe, and a definite relation of harmony in the motion and magnitude of the orbs, of a kind that is not possible to obtain in any other way." (Johannes Kepler)

"Nature builds up her refined and invisible architecture, with a delicacy eluding our conception, yet with a symmetry and beauty which we are never weary of admiring." (John Herschel)

"The most general law in nature is equity-the principle of balance and symmetry which guides the growth of forms along the lines of the greatest structural efficiency." (Herbert Read)

"The secret of nature is symmetry. When searching for new and more fundamental laws of nature, we should search for new symmetries." (David Gross)

"The universe is built on a plan the profound symmetry of which is somehow present in the inner structure of our intellect." (Paul Valéry)

On Symmetry VI (Symmetry and Perception)

“Symmetry is a characteristic of the human mind.” (Alexander Pushkin, 1825)

“Symmetry is what we see at a glance; based on the fact that there is no reason for any difference, and based also on the face of man; whence it happens that symmetry is only wanted in breadth, not in height or depth.” (Blaise Pascal, “Pensées”, 1670)

"It is the harmony of the diverse parts, their symmetry, their happy balance; in a word it is all that introduces order, all that gives unity, that permits us to see clearly and to comprehend at once both the ensemble and the details." (Henri Poincaré, “The Future of Mathematics”, Monist, Vol. 20)

“[…] admiration for elegant symmetry never dies […]” (Robert Kaplan & Ellen Kaplan, „The Art of the Infinite: The Pleasures of Mathematics”, 2003)

“The word ‘symmetry’ conjures to mind objects which are well balanced, with perfect proportions. Such objects capture a sense of beauty and form. The human mind is constantly drawn to anything that embodies some aspect of symmetry. Our brain seems programmed to notice and search for order and structure. Artwork, architecture and music from ancient times to the present day play on the idea of things which mirror each other in interesting ways. Symmetry is about connections between different parts of the same object. It sets up a natural internal dialogue in the shape.” (Marcus du Sautoy, “Symmetry: A Journey into the Patterns of Nature”, 2008)

“Symmetry is the means by which shape is converted into memory.” (Michael Leyton, “Symmetry, Causality, Mind”, 1992)

“The desire for symmetry, for balance, for rhythm in form as well as in sound, is one of the most inveterate of human instincts.” (Edith Wharton)

On Symmetry V (Symmetry vs. Asymmetry I)

“Symmetry may have its appeal but it is inherently stale. Some kind of imbalance is behind every transformation.” (Marcelo Gleiser, “A Tear at the Edge of Creation: A Radical New Vision for Life in an Imperfect Universe”, 2010)

“Rut seldom is asymmetry merely the absence of symmetry. Even in asymmetric designs one feels symmetry as the norm from which one deviates under the influence of forces of non-formal character.” (Hermann Weyl, “Symmetry”, 1952)

“[…] in all things that live there are certain irregularities and deficiencies which are not only signs of life, but sources of beauty. No human face is exactly the same in its lines on each side, no leaf perfect in its lobes, no branch in its symmetry. All admit irregularity as they imply change; […]” (John Ruskin, “The Stones of Venice: The Sea Stories”, 2013)

“Nature is never perfectly symmetric. Nature's circles always have tiny dents and bumps. There are always tiny fluctuations, such as the thermal vibration of molecules. These tiny imperfections load Nature's dice in favour of one or other of the set of possible effects that the mathematics of perfect symmetry considers to be equally possible.” (Ian Stewart & Martin Golubitsky, “Fearful Symmetry: Is God a Geometer?”, 1992)

“In every symmetrical system every deformation that tends to destroy the symmetry is complemented by an equal and opposite deformation that tends to restore it. […] One condition, therefore, though not an absolutely sufficient one, that a maximum or minimum of work corresponds to the form of equilibrium, is thus applied by symmetry.” (Ernst Mach, “The Science of Mechanics: A Critical and Historical Account of Its Development”, 1893)

“Symmetry is a fundamental organizing principle of shape. It helps in classifying and understanding patterns in mathematics, nature, art, and, of course, poetry. And often the counterpoint to symmetry — the breaking or interruption of symmetry - is just as important in creative endeavors.” (Marcia Birken & Anne C. Coon, “Discovering Patterns in Mathematics and Poetry”, 2008)

“An asymmetry in the present is understood as having originated from a past symmetry.” (Michael Leyton, “Symmetry, Causality, Mind”, 1992)

“Approximate symmetry is a softening of the hard dichotomy between symmetry and asymmetry. The extent of deviation from exact symmetry that can still be considered approximate symmetry will depend on the context and the application and could very well be a matter of personal taste.” (Joe Rosen, “Symmetry Rules: How Science and Nature Are Founded on Symmetry”, 2008)

“[…] asymmetry can be defined only relative to symmetry, and vice versa. Asymmetric elements in paintings or buildings are most effective when superimposed against a background of symmetry.” (Alan Lightman, “The Accidental Universe: The World You Thought You Knew”, 2014)

“Chaos demonstrates that deterministic causes can have random effects […] There's a similar surprise regarding symmetry: symmetric causes can have asymmetric effects. […] This paradox, that symmetry can get lost between cause and effect, is called symmetry-breaking. […] From the smallest scales to the largest, many of nature's patterns are a result of broken symmetry; […]” (Ian Stewart & Martin Golubitsky, “Fearful Symmetry: Is God a Geometer?”, 1992)

On Symmetry IV (Mathematics and Symmetry I)

"The mathematical sciences particularly exhibit order, symmetry, and limitation; and these are the greatest forms of the beautiful." (Aristotle, "Metaphysics", cca. 350 BC)

"Those who assert that the mathematical sciences make no affirmation about what is fair or good make a false assertion; for they do speak of these and frame demonstrations of them in the most eminent sense of the word. […] Of what is fair, however, the most important species are order and symmetry, and that which is definite, which the mathematical sciences make manifest in a most eminent degree." (Aristotle, "Metaphysics", cca. 350)

"Mathematics has beauties of its own - a symmetry and proportion in its results, a lack of superfluity, an exact adaptation of means to ends, which is exceedingly remarkable and to be found elsewhere only in the works of the greatest beauty." (Jacob W A Young, "The Teaching of Mathematics", 1907)

"Nature seems to take advantage of the simple mathematical representations of the symmetry laws. When one pauses to consider the elegance and the beautiful perfection of the mathematical reasoning involved and contrast it with the complex and far-reaching physical consequences, a deep sense of respect for the power of the symmetry laws never fails to develop." (Chen Ning Yang, "The Law of Parity Conservation and Other Symmetry Laws of Physics", [Nobel lecture] 1957)

"The study of symmetry was born out of art and mathematics; art as the comprehension of the beauty of nature and mathematics as the comprehension of the world's harmony. " (N F Ovchinnikov, "Principles of Preservation", 1966)

"Symmetries abound in nature, in technology, and - especially - in the simplified mathematical models we study so assiduously. Symmetries complicate things and simplify them. They complicate them by introducing exceptional types of behavior, increasing the number of variables involved, and making vanish things that usually do not vanish. They simplify them by introducing exceptional types of behavior, increasing the number of variables involved, and making vanish things that usually do not vanish. They violate all the hypotheses of our favorite theorems, yet lead to natural generalizations of those theorems. It is now standard to study the 'generic' behavior of dynamical systems. Symmetry is not generic. The answer is to work within the world of symmetric systems and to examine a suitably restricted idea of genericity." (Ian Stewart, "Bifurcation with symmetry", 1988)

"Nature behaves in ways that look mathematical, but nature is not the same as mathematics. Every mathematical model makes simplifying assumptions; its conclusions are only as valid as those assumptions. The assumption of perfect symmetry is excellent as a technique for deducing the conditions under which symmetry-breaking is going to occur, the general form of the result, and the range of possible behaviour. To deduce exactly which effect is selected from this range in a practical situation, we have to know which imperfections are present" (Ian Stewart & Martin Golubitsky, "Fearful Symmetry: Is God a Geometer?", 1992)

"Humans seem to have an inbuilt need to impose mathematical order, symmetry and cause-and-effect relationships on a natural world that often may not work in that way at all." (Michael Hanlon, "10 Questions Science Can’t Answer (Yet): A Guide to the Scientific Wilderness", 2007)

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

"Symmetry is a vast subject, significant in art and nature. Mathematics lies at its root, and it would be hard to find a better one on which to demonstrate the working of the mathematical intellect." (Hermann Weyl)

On Symmetry III (Mathematicians I)

“To a mathematician, an object possesses symmetry if it retains its form after some transformation. A circle, for example, looks the same after any rotation; so a mathematician says that a circle is symmetric, even though a circle is not really a pattern in the conventional sense - something made up from separate, identical bits. Indeed the mathematician generalizes, saying that any object that retains its form when rotated - such as a cylinder, a cone, or a pot thrown on a potter's wheel - has circular symmetry.” (Ian Stewart & Martin Golubitsky, “Fearful Symmetry: Is God a Geometer?”, 1992)

"Guided only by their feeling for symmetry, simplicity, and generality, and an indefinable sense of the fitness of things, creative mathematicians now, as in the past, are inspired by the art of mathematics rather than by any prospect of ultimate usefulness." (Eric Temple Bell)

"What makes a great mathematician? A feel for form, a strong sense of what is important. Möbius had both in abundance. He knew that topology was important. He knew that symmetry is a fundamental and powerful mathematical principle. The judgment of posterity is clear: Möbius was right.” (Ian Stewart)

“Mathematicians attach great importance to the elegance of their methods and their results. This is not pure dilettantism. What is it indeed that gives us the feeling of elegance in a solution, in a demonstration? It is the harmony of the diverse parts, their symmetry, their happy balance; in a word it is all that introduces order, all that gives unity, that permits us to see clearly and to comprehend at once both the ensemble and the details.” (Henri Poincaré, “The Future of Mathematics”, Monist Vol. 20, 1910)

“The mathematician has, above all things, an eye for symmetry […]) (James C Maxwell, 1871)
“For the mathematician, the pattern searcher, understanding symmetry is one of the principal themes in the quest to chart the mathematical world.” (Marcus du Sautoy, “Symmetry: A Journey into the Patterns of Nature”, 2008)

“Perhaps the simplest way to explain symmetry is to follow the operational approach used by mathematicians: a symmetry is a motion. That is, suppose you have an object and pick it up, move it around, and set it down. If it is impossible to distinguish between the object in its original and final positions, we say that it has a symmetry.” (Michael Field & Martin Golubitsky, “Symmetry in Chaos: A Search for Pattern in Mathematics, Art, and Nature” 2nd Ed, 2009)

On Symmetry II (Beauty and Symmetry I)

“Beauty is rather a light that plays over the symmetry of things than that symmetry itself.” (Plotinus)

“Proportion, or symmetry, is the basis of beauty; propriety, of grace.” (Henry Fuseli)

"Symmetry, as wide or as narrow as you may define its meaning, is one idea by which man through the ages has tried to comprehend and create order, beauty and perfection." (Hermann Weyl, “Symmetry”, 1952)

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

“The fact is that the beautiful, humanly speaking, is merely form considered in its simplest aspect, in its most perfect symmetry, in its most entire harmony with our make-up.” (Victor Hugo, “Cromwell”, 1909)

“Whereas symmetry can create beauty, its breaking does not necessarily destroy beauty; instead, it may even create another kind of beauty.” (Guozhen Wu, “Nonlinearity and Chaos in Molecular Vibrations”, 2005)

“In the nonmathematical sense, symmetry is associated with regularity in form, pleasing proportions, periodicity, or a harmonious arrangement; thus it is frequently associated with a sense of beauty. In the geometric sense, symmetry may be more precisely analyzed. We may have, for example, an axis of symmetry, a center of symmetry, or a plane of symmetry, which define respectively the line, point, or plane about which a figure or body is symmetrical. The presence of these symmetry elements, usually in combinations, is responsible for giving form to many compositions; the reproduction of a motif by application of symmetry operations can produce a pattern that is pleasing to the senses.” (Hans H Jaffé & ‎Milton Orchin, “Symmetry in Chemistry”, 2002)

“Beauty is our weapon against nature; by it we make objects, giving them limit, symmetry, proportion. Beauty halts and freezes the melting flux of nature.” (Camille Paglia)

“Symmetry creates harmony and beauty in an object.” (Aleksandr P Dubrov, “Symmetry of Biorhythms and Reactivity”, 1989)

“To a considerable degree science consists in originating the maximum amount of information with the minimum expenditure of energy. Beauty is the cleanness of line in such formulations along with symmetry, surprise, and congruence with other prevailing beliefs.” (Edward O Wilson, “Biophilia”, 1984)

On Symmetry I

“Symmetry is what we see at a glance; based on the fact that there is no reason for any difference, and based also on the face of man; whence it happens that symmetry is only wanted in breadth, not in height or depth.” (Blaise Pascal, “Pensées”, 1670)

“Symmetry is evidently a kind of unity in variety, where a whole is determined by the rhythmic repetition of similar.” (George Santayana, “The Sense of Beauty”, 1896)

“By the word symmetry […] one thinks of an external relationship between pleasing parts of a whole; mostly the word is used to refer to parts arranged regularly against one another around a centre. We have […] observed [these parts] one after the other, not always like following like, but rather a raising up from below, a strength out of weakness, a beauty out of ordinariness.” (Goethe)

“In everyday language, the words 'pattern' and 'symmetry' are used almost interchangeably, to indicate a property possessed by a regular arrangement of more-or-less identical units […]” (Ian Stewart & Martin Golubitsky, “Fearful Symmetry: Is God a Geometer?”, 1992)

“In the one sense symmetric means something like well-proportioned, well-balanced, and symmetry denotes that sort of concordance of several parts by which they integrate into a whole. Beauty is bound up with symmetry.” (Hermann Weyl, “Symmetry”, 1952)

“A thing is symmetrical if there is something you can do to it so that after you have finished doing it, it looks the same as before.” (Hermann Weyl, “Symmetry”, 1952)

“[…] a symmetry isn't a thing; it's a transformation. Not any old transformation, though: a symmetry of an object is a transformation that leaves it apparently unchanged.” (Ian Stewart & Martin Golubitsky, “Fearful Symmetry: Is God a Geometer?”, 1992)

“A symmetry is a set of transformations applied to a structure, such that the transformations preserve the properties of the structure.” (Philip Dorrell, “What is Music?: Solving a Scientific Mystery”, 2004)

“Symmetry is the representation of two or more equivalent or balanced elements with respect to a common origin, position, or axis.” (David Smith, “The Symmetry Solution: A Modern View of Biblical Prophecy”, 2009)

“Symmetry is basically a geometrical concept. Mathematically it can be defined as the invariance of geometrical patterns under certain operations. But when abstracted, the concept applies to all sorts of situations. It is one of the ways by which the human mind recognizes order in nature. In this sense symmetry need not be perfect to be meaningful. Even an approximate symmetry attracts one's attention, and makes one wonder if there is some deep reason behind it.” (Eguchi Tohru & ‎K Nishijima , “Broken Symmetry: Selected Papers Of Y Nambu”, 1995)