On Graphs in Mathematics

"Most of us have some idea of what the word statistics means. We should probably say that it has something to do with tables of figures, diagrams and graphs in economic and scientific publications, with the cost of living [...]  and with a host of other seemingly unrelated matters of concern or unconcern [...] Our answer would be on the right lines. Nor should we be unduly upset if, to start with, we seem a little vague. Statisticians themselves disagree about the definition of the word: over a hundred definitions have been listed." (Walter F  Willcox, "An Improved Method of Measuring Public Health in the United States", Revue de l’lnstitut InternutionuIe de Stutistique  vol. 3 (1), 1935)

"To function in today's society, mathematical literacy - what the British call ‘numeracy' - is as essential as verbal literacy […] Numeracy requires more than just familiarity with numbers. To cope confidently with the demands of today's society, one must be able to grasp the implications of many mathematical concepts - for example, change, logic, and graphs - that permeate daily news and routine decisions - mathematical, scientific, and cultural - provide a common fabric of communication indispensable for modern civilized society. Mathematical literacy is especially crucial because mathematics is the language of science and technology." (National Research Council, "Everybody counts: A report to the nation on the future of mathematics education", 1989)

"Continuous functions can move freely. Graphs of continuous functions can freely branch off at any place, whereas analytic functions coinciding in some neighborhood of a point P cannot branch outside of this neighborhood. Because of this property, continuous functions can mathematically represent wildly changing wind inside a typhoon or a gentle breeze." (Kenji Ueno & Toshikazu Sunada, "A Mathematical Gift, III: The Interplay Between Topology, Functions, Geometry, and Algebra", Mathematical World Vol. 23, 1996)

"Similarly to the graphs of continuous functions, graphs of differentiable (smooth) functions which coincide in a neighborhood of a point P can branch off outside of the neighborhood. Because of this property, differentiable functions can represent smoothly changing natural phenomena." (Kenji Ueno & Toshikazu Sunada, "A Mathematical Gift, III: The Interplay Between Topology, Functions, Geometry, and Algebra", Mathematical World Vol. 23, 1996)

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

"Replacing particles by strings is a naive-sounding step, from which many other things follow. In fact, replacing Feynman graphs by Riemann surfaces has numerous consequences: 1. It eliminates the infinities from the theory. [...] 2. It greatly reduces the number of possible theories. [...] 3. It gives the first hint that string theory will change our notions of spacetime." (Edward Witten, "The Past and Future of String Theory", 2003)

"As geometers study shape, the student of calculus examines change: the mathematics of how an object transforms from one state into another, as when describing the motion of a ball or bullet through space, is rendered pictorial in its graphs’ curves." (Daniel Tammet, "Thinking in Numbers" , 2012)

On Graphs (-1969)

"There is a magic in graphs. The profile of a curve reveals in a flash a whole situation - the life history of an epidemic, a panic, or an era of prosperity. The curve informs the mind, awakens the imagination, convinces." (Henry D Hubbard [in William Brinton's "Graphic Presentation", 1939])

"Graphic methods are very commonly used in business correlation problems. On the whole, carefully handled and skillfully interpreted graphs have certain advantages over mathematical methods of determining correlation in the usual business problems. The elements of judgment and special knowledge of conditions can be more easily introduced in studying correlation graphically. Mathematical correlation is often much too rigid for the data at hand." (John R Riggleman & Ira N Frisbee, "Business Statistics", 1938)

"Graphs are all inclusive. No fact is too slight or too great to plot to a scale suited to the eye. Graphs may record the path of an ion or the orbit of the sun, the rise of a civilization, or the acceleration of a bullet, the climate of a century or the varying pressure of a heart beat, the growth of a business, or the nerve reactions of a child." (Henry D Hubbard [foreword to Willard C Brinton, "Graphic Presentation", 1939)])

"The graphic art depicts magnitudes to the eye. It does more. It compels the seeing of relations. We may portray by simple graphic methods whole masses of intricate routine, the organization of an enterprise, or the plan of a campaign. Graphs serve as storm signals for the manager, statesman, engineer; as potent narratives for the actuary, statist, naturalist; and as forceful engines of research for science, technology and industry. They display results. They disclose new facts and laws. They reveal discoveries as the bud unfolds the flower." (Henry D Hubbard [foreword to Willard C Brinton, "Graphic Presentation", 1939)])

"The graphic language is modern. We are learning its alphabet. That it will develop a lexicon and a literature marvelous for its vividness and the variety of application is inevitable. Graphs are dynamic, dramatic. They may epitomize an epoch, each dot a fact, each slope an event, each curve a history. Wherever there are data to record, inferences to draw, or facts to tell, graphs furnish the unrivalled means whose power we are just beginning to realize and to apply." (Henry D Hubbard [foreword to Willard C Brinton, "Graphic Presentation", 1939)])

"Charts and graphs represent an extremely useful and flexible medium for explaining, interpreting, and analyzing numerical facts largely by means of points, lines, areas, and other geometric forms and symbols. They make possible the presentation of quantitative data in a simple, clear, and effective manner and facilitate comparison of values, trends, and relationships. Moreover, charts and graphs possess certain qualities and values lacking in textual and tabular forms of presentation." (Calvin F Schmid, "Handbook of Graphic Presentation", 1954)

"If one technique of data analysis were to be exalted above all others for its ability to be revealing to the mind in connection with each of many different models, there is little doubt which one would be chosen. The simple graph has brought more information to the data analyst’s mind than any other device. It specializes in providing indications of unexpected phenomena." (John W Tukey, "The Future of Data Analysis", Annals of Mathematical Statistics Vol. 33 (1), 1962)

"The histogram, with its columns of area proportional to number, like the bar graph, is one of the most classical of statistical graphs. Its combination with a fitted bell-shaped curve has been common since the days when the Gaussian curve entered statistics. Yet as a graphical technique it really performs quite poorly. Who is there among us who can look at a histogram-fitted Gaussian combination and tell us, reliably, whether the fit is excellent, neutral, or poor? Who can tell us, when the fit is poor, of what the poorness consists? Yet these are just the sort of questions that a good graphical technique should answer at least approximately." (John W Tukey, "The Future of Processes of Data Analysis", 1965)

"Every graph is at least an indication, by contrast with some common instances of numbers." (John W Tukey, "Data Analysis, Including Statistics", 1968)

"One of the methods making the data intelligible is to represent it by means of graphs and diagrams. The graphic & diagrammatic representation of the data is always appealing to the eye as well as to the mind of the observer." (S P Singh & R P S Verma, "Agricultural Statistics", cca. 1969)

On Graphs (1970-1979)

"Charts and graphs are a method of organizing information for a unique purpose. The purpose may be to inform, to persuade, to obtain a clear understanding of certain facts, or to focus information and attention on a particular problem. The information contained in charts and graphs must, obviously, be relevant to the purpose. For decision-making purposes. information must be focused clearly on the issue or issues requiring attention. The need is not simply for 'information', but for structured information, clearly presented and narrowed to fit a distinctive decision-making context. An advantage of having a 'formula' or 'model' appropriate to a given situation is that the formula indicates what kind of information is needed to obtain a solution or answer to a specific problem." (Cecil H Meyers, "Handbook of Basic Graphs: A modern approach", 1970)

"Data should not be forced into an uncomfortable or improper mold. For example, data that is appropriate for line graphs is not usually appropriate for circle charts and in any case not without some arithmetic transformation. Only graphs that are designed to fit the data can be used profitably." (Cecil H Meyers, "Handbook of Basic Graphs: A modern approach", 1970)

"If two or more data paths ate to appear on the graph. it is essential that these lines be labeled clearly, or at least a reference should be provided for the reader to make the necessary identifications. While clarity seems to be a most obvious goal. graphs with inadequate or confusing labeling do appear in publications, The user should not find identification of data paths troublesome or subject to misunderstanding. The designer normally should place no more than three data paths on the graph to prevent confusion - particularly if the data paths intersect at one or more points on the Cartesian plane." (Cecil H Meyers, "Handbook of Basic Graphs: A modern approach", 1970)

"The numerous design possibilities include several varieties of line graphs that are geared to particular types of problems. The design of a graph should be adapted to the type of data being structured. The data might be percentages, index numbers, frequency distributions, probability distributions, rates of change, numbers of dollars, and so on. Consequently, the designer must be prepared to structure his graph accordingly." (Cecil H Meyers, "Handbook of Basic Graphs: A modern approach", 1970)

"Graphs can have various purposes, such as: (i) to help us perceive and appreciate some broad features of the data, (ii) to let us look behind those broad features and see what else is there. Most kinds of statistical calculation rest on assumptions about the behavior of the data. Those assumptions may be false, and then the calculations may be misleading. We ought always to try to check whether the assumptions are reasonably correct; and if they are wrong we ought to be able to perceive in what ways they are wrong. Graphs are very valuable for these purposes." (Francis J Anscombe, "Graphs in Statistical Analysis", The American Statistician Vol. 27 (1), 1973

"A graph presents a limited number of figures in a bold and forceful manner. To do this it usually must omit a large number of figures available on the subject. The choice of what graphic format to use is largely a matter of deciding what figures have the greatest significance to the intended reader and what figures he can best afford to skip." (Peter H Selby, "Interpreting Graphs and Tables", 1976)

"Just like the spoken or written word, statistics and graphs can lie. They can lie by not telling the full story. They can lead to wrong conclusions by omitting some of the important facts. [...] Always look at statistics with a critical eye, and you will not be the victim of misleading information." (Dyno Lowenstein, "Graphs", 1976)

"Learning to make graphs involves two things:" (l) the techniques of plotting statistics, which might be called the artist's job; and" (2) understanding the statistics. When you know how to work out graphs, all kinds of statistics will probably become more interesting to you." (Dyno Lowenstein, "Graphs", 1976)

"Probably one of the most common misuses (intentional or otherwise) of a graph is the choice of the wrong scale - wrong, that is, from the standpoint of accurate representation of the facts. Even though not deliberate, selection of a scale that magnifies or reduces - even distorts - the appearance of a curve can mislead the viewer." (Peter H Selby, "Interpreting Graphs and Tables", 1976)

"Remember, the primary function of a graph of any kind is to illustrate the relationship between two variables. [...] To draw any graph we must have established some relationship between the two variables. This relationship can be in the form of a formula" (equation is the more mathematical term), as we have just seen, or simply a set of observations, as is common in all types of statistical work. Sometimes we develop set of observations and then try to find an equation that expresses, in mathematical language, the relationship between the two variables." (Peter H Selby, "Interpreting Graphs and Tables", 1976))

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

"There is no more reason to expect one graph to ‘tell all’ than to expect one number to do the same." (John W Tukey, "Exploratory Data Analysis", 1977)

"[...] 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)

"Although advice on how and when to draw graphs is available, we have no theory of statistical graphics […]" (Stephen Fienberg, "The American Statistician", Graphical Methods in Statistics Vol. 13" (4), 1979)

On Graphs (1980-1989)

"We would wish ‘numerate’ to imply the possession of two attributes. The first of these is an ‘at-homeness’ with numbers and an ability to make use of mathematical skills which enable an individual to cope with the practical mathematical demands of his everyday life. The second is ability to have some appreciation and understanding of information which is presented in mathematical terms, for instance in graphs, charts or tables or by reference to percentage increase or decrease." (Cockcroft Committee, "Mathematics Counts: A Report into the Teaching of Mathematics in Schools", 1982)

"An economic justification for computer graphics is that the organization spends an enormous amount of money on data processing, often providing managers with too many reports, too many data, and an overload of information. The report output has to be condensed into a more usable form. The computer graph essentially is the data represented in a structured pictorial form. The role of the graph is to provide meaningful reports. To the extent that it does. it can be justified." (Anker V Andersen, "Graphing Financial Information: How accountants can use graphs to communicate", 1983)

"Graphs are used to meet the need to condense all the available information into a more usable quantity. The selection process of combining and condensing will inevitably produce a less than complete study and will lead the user in certain directions, producing a potential for misleading." (Anker V Andersen, "Graphing Financial Information: How accountants can use graphs to communicate", 1983)

"Graphs can present internal accounting data effectively. Because one of the main functions of the accountant is to communicate accounting information to users. accountants should use graphs, at least to the extent that they clarify the presentation of accounting data. present the data fairly, and enhance management's ability to make a more informed decision. It has been argued that the human brain can absorb and understand images more easily than words and numbers, and, therefore, graphs may be better communicative devices than written reports or tabular statements." (Anker V Andersen, "Graphing Financial Information: How accountants can use graphs to communicate", 1983)

"Understandability implies that the graph will mean something to the audience. If the presentation has little meaning to the audience, it has little value. Understandability is the difference between data and information. Data are facts. Information is facts that mean something and make a difference to whoever receives them. Graphic presentation enhances understanding in a number of ways. Many people find that the visual comparison and contrast of information permit relationships to be grasped more easily. Relationships that had been obscure become clear and provide new insights." (Anker V Andersen, "Graphing Financial Information: How accountants can use graphs to communicate", 1983)

"In the case of graphs, the number of lines which can be included on any one illustration will depend largely on how close the lines are and how often they cross one another. Three or four is likely to be the maximum acceptable number. In some instances, there may be an argument for using several graphs with one line each as opposed to one graph with multiple lines. It has been shown that these two arrangements are equally satisfactory if the user wishes to read off the value of specific points; if, however, he wishes to compare the lines, than the single multi-line graph is superior." (Linda Reynolds & Doig Simmonds, "Presentation of Data in Science" 4th Ed, 1984)

"The effective communication of information in visual form, whether it be text, tables, graphs, charts or diagrams, requires an understanding of those factors which determine the 'legibility', 'readability' and 'comprehensibility', of the information being presented. By legibility we mean: can the data be clearly seen and easily read? By readability we mean: is the information set out in a logical way so that its structure is clear and it can be easily scanned? By comprehensibility we mean: does the data make sense to the audience for whom it is intended? Is the presentation appropriate for their previous knowledge, their present information needs and their information processing capacities?" (Linda Reynolds & Doig Simmonds, "Presentation of Data in Science" 4th Ed, 1984)

"Clear vision is a vital aspect of graphs. The viewer must be able to visually disentangle the many different items that appear on a graph." (William S Cleveland, "The Elements of Graphing Data", 1985)

"Graphs that communicate data to others often must undergo reduction and reproduction; these processes, if not done with care, can interfere with visual clarity." (William S Cleveland, "The Elements of Graphing Data", 1985)

"Make the data stand out and avoid superfluity are two broad strategies that serve as an overall guide to the specific principles […] The data - the quantitative and qualitative information in the data region - are the reason for the existence of the graph. The data should stand out. […] We should eliminate superfluity in graphs. Unnecessary parts of a graph add to the clutter and increase the difficulty of making the necessary elements - the data - stand out." (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)

"Despite the prevailing use of graphs as metaphors for communicating and reasoning about dependencies, the task of capturing informational dependencies by graphs is not at all trivial." (Judea Pearl, "Probabilistic Reasoning in Intelligent Systems: Network of Plausible Inference", 1988)

"Elementary functions, such as trigonometric functions and rational functions, have their roots in Euclidean geometry. They share the feature that when their graphs are 'magnified' sufficiently, locally they 'look like' straight lines. That is, the tangent line approximation can be used effectively in the vicinity of most points. Moreover, the fractal dimension of the graphs of these functions is always one. These elementary 'Euclidean' functions are useful not only because of their geometrical content, but because they can be expressed by simple formulas. We can use them to pass information easily from one person to another. They provide a common language for our scientific work. Moreover, elementary functions are used extensively in scientific computation, computer-aided design, and data analysis because they can be stored in small files and computed by fast algorithms." (Michael Barnsley, "Fractals Everwhere", 1988)

"To function in today's society, mathematical literacy - what the British call ‘numeracy' - is as essential as verbal literacy […] Numeracy requires more than just familiarity with numbers. To cope confidently with the demands of today's society, one must be able to grasp the implications of many mathematical concepts - for example, change, logic, and graphs - that permeate daily news and routine decisions - mathematical, scientific, and cultural - provide a common fabric of communication indispensable for modern civilized society. Mathematical literacy is especially crucial because mathematics is the language of science and technology." (National Research Council, "Everybody counts: A report to the nation on the future of mathematics education", 1989)

On Graphs (1990-1999)

"Data that are skewed toward large values occur commonly. Any set of positive measurements is a candidate. Nature just works like that. In fact, if data consisting of positive numbers range over several powers of ten, it is almost a guarantee that they will be skewed. Skewness creates many problems. There are visualization problems. A large fraction of the data are squashed into small regions of graphs, and visual assessment of the data degrades. There are characterization problems. Skewed distributions tend to be more complicated than symmetric ones; for example, there is no unique notion of location and the median and mean measure different aspects of the distribution. There are problems in carrying out probabilistic methods. The distribution of skewed data is not well approximated by the normal, so the many probabilistic methods based on an assumption of a normal distribution cannot be applied." (William S Cleveland, "Visualizing Data", 1993)

"A graph is a system of connections expressed by means of commonly accepted symbols. As such, the symbols and symbolic forms used in making graphs are significant. To communicate clearly this symbolism must be acknowledged." (Mary H Briscoe, "Preparing Scientific Illustrations: A guide to better posters, presentations, and publications" 2nd ed., 1995)

"A coordinate is a number or value used to locate a point with respect to a reference point, line, or plane. Generally the reference is zero. […] The major function of coordinates is to provide a method for encoding information on charts, graphs, and maps in such a way that viewers can accurately decode the information after the graph or map has been generated. " (Robert L Harris, "Information Graphics: A Comprehensive Illustrated Reference", 1996)

"Area graphs are generally not used to convey specific values. Instead, they are most frequently used to show trends and relationships, to identify and/or add emphasis to specific information by virtue of the boldness of the shading or color, or to show parts-of-the-whole." (Robert L Harris, "Information Graphics: A Comprehensive Illustrated Reference", 1996)

"Grouped area graphs sometimes cause confusion because the viewer cannot determine whether the areas for the data series extend down to the zero axis. […] Grouped area graphs can handle negative values somewhat better than stacked area graphs but they still have the problem of all or portions of data curves being hidden by the data series towards the front." (Robert L Harris, "Information Graphics: A Comprehensive Illustrated Reference", 1996)

"When analyzing data it is many times advantageous to generate a variety of graphs using the same data. This is true whether there is little or lots of data. Reasons for this are:" (1) Frequently, all aspects of a group of data can not be displayed on a single graph." (2) Multiple graphs generally result in a more in-depth understanding of the information." (3) Different aspects of the same data often become apparent." (4) Some types of graphs cause certain features of the data to stand out better" (5) Some people relate better to one type of graph than another." (Robert L Harris, "Information Graphics: A Comprehensive Illustrated Reference", 1996)

"When approximations are all that are needed, stacked area graphs are usually adequate. When accuracy is desired, this type of graph is generally not used, particularly when the values fluctuate significantly and/or the slopes of the curves are steep." (Robert L Harris, "Information Graphics: A Comprehensive Illustrated Reference", 1996)

On Graphs (2000-2009)

"A bar graph typically presents either averages or frequencies. It is relatively simple to present raw data (in the form of dot plots or box plots). Such plots provide much more information. and they are closer to the original data. If the bar graph categories are linked in some way - for example, doses of treatments - then a line graph will be much more informative. Very complicated bar graphs containing adjacent bars are very difficult to grasp. If the bar graph represents frequencies. and the abscissa values can be ordered, then a line graph will be much more informative and will have substantially reduced chart junk." (Gerald van Belle, "Statistical Rules of Thumb", 2002)

"Stacked bar graphs do not show data structure well. A trend in one of the stacked variables has to be deduced by scanning along the vertical bars. This becomes especially difficult when the categories do not move in the same direction." (Gerald van Belle, "Statistical Rules of Thumb", 2002)

"Choose scales wisely, as they have a profound influence on the interpretation of graphs. Not all scales require that zero be included, but bar graphs and other graphs where area is judged do require it." (Naomi B Robbins, "Creating More effective Graphs", 2005)

"Graphs are for the forest and tables are for the trees. Graphs give you the big picture and show you the trends; tables give you the details." (Naomi B Robbins, "Creating More effective Graphs", 2005)

"Graphs are pictorial representations of numerical quantities. It therefore seems reasonable to expect that the visual impression we get when looking at a graph is proportional to the numbers that the graph represents. Unfortunately, this is not always the case." (Naomi B Robbins, "Creating More effective Graphs", 2005)

"One graph is more effective than another if its quantitative information can be decoded more quickly or more easily by most observers. […] This definition of effectiveness assumes that the reason we draw graphs is to communicate information - but there are actually many other reasons to draw graphs." (Naomi B Robbins, "Creating More effective Graphs", 2005)

"Cleverly drawn pictures can sometimes disguise or render invisible what is there. At other times, they can make you see things that are not really there. It is helpful to be aware of how these illusions are achieved, as some of the illusionist’s 'tricks of the trade' can also be found in distortions used in graphs and diagrams." (Alan Graham, "Developing Thinking in Statistics", 2006)

"Sparklines work at intense resolutions, at the level of good typography and cartography. [...] Just as sparklines are like words, so then distributions of sparklines on a page are like sentences and paragraphs. The graphical idea here is make it word-like and typographic - an idea that leads to reasonable answers for most questions about sparkline arrangements." (Edward R Tufte, "Beautiful Evidence", 2006)

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

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

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

"Plotting data is a useful first stage to any analysis and will show extreme observations together with any discernible patterns. In addition the relative sizes of categories are easier to see in a diagram" (bar chart or pie chart) than in a table. Graphs are useful as they can be assimilated quickly, and are particularly helpful when presenting information to an audience. Tables can be useful for displaying information about many variables at once, while graphs can be useful for showing multiple observations on groups or individuals. Although there are no hard and fast rules about when to use a graph and when to use a table, in the context of a report or a paper it is often best to use tables so that the reader can scrutinise the numbers directly." (Jenny Freeman et al, "How to Display Data", 2008)

On Graphs (2010-2019)

"Dealing with a circular visualization and trying to compare its radial portions is always problematic. When designing with data, the story should always be told as clearly as possible. To do so, it is often best to avoid round charts and graphs." (Brian Suda, "A Practical Guide to Designing with Data", 2010)

"Diagrams are information graphics that are made up primarily of geometric shapes, such as rectangles, circles, diamonds, or triangles, that are typically" (but not always) interconnected by lines or arrows. One of the major purposes of a diagram is to show how things, people, ideas, activities, etc. interrelate and interconnect. Unlike quantitative charts and graphs, diagrams are used to show interrelationships in a qualitative way." (Robbie T Nakatsu, "Diagrammatic Reasoning in AI", 2010)

"Good graphic design is not a panacea for bad copy, poor layout or misleading statistics. If any one of these facets are feebly executed it reflects poorly on the work overall, and this includes bad graphs and charts." (Brian Suda, "A Practical Guide to Designing with Data", 2010)

"So what is the difference between a chart or graph and a visualization? […] a chart or graph is a clean and simple atomic piece; bar charts contain a short story about the data being presented. A visualization, on the other hand, seems to contain much more ʻchart junkʼ, with many sometimes complex graphics or several layers of charts and graphs. A visualization seems to be the super-set for all sorts of data-driven design." (Brian Suda, "A Practical Guide to Designing with Data", 2010)

"Sparklines are compact line graphs that do not have a quantitative scale. They are meant to provide a quick sense of a metric's movement or trend, usually over time. They are more expressive than arrows, which only indicate change from the prior period and do not qualify the degree of change. Sparklines are significantly more compact than normal line graphs but are precise." (Wayne W Eckerson, "Performance Dashboards: Measuring, Monitoring, and Managing Your Business", 2010)

"The biggest difference between line graphs and sparklines is that a sparkline is compact with no grid lines. It isnʼt meant to give precise values; rather, it should be considered just like any other word in the sentence. Its general shape acts as another term and lends additional meaning in its context. The driving forces behind these compact sparklines are speed and convenience." (Brian Suda, "A Practical Guide to Designing with Data", 2010)

"Sparklines are compact line graphs that do not have a quantitative scale. They are meant to provide a quick sense of a metric’s movement or trend, usually over time. They are more expressive than arrows, which only indicate change from the prior period and do not qualify the degree of change. Sparklines are significantly more compact than normal line graphs but are precise." (Wayne W Eckerson, "Performance Dashboards: Measuring, Monitoring, and Managing Your Business" 2nd Ed., 2011)

"Diagrams furnish only approximate information. They do not add anything to the meaning of the data and, therefore, are not of much use to a statistician or research worker for further mathematical treatment or statistical analysis. On the other hand, graphs are more obvious, precise and accurate than the diagrams and are quite helpful to the statistician for the study of slopes, rates of change and estimation," (interpolation and extrapolation), wherever possible." (S C Gupta & Indra Gupta, "Business Statistics", 2013)

"Line graphs that show more than one line can be useful for making comparisons, but sometimes it is important to discuss each individual line. By using sparklines evaluators can call attention to and discuss individual cases. Sparklines can be embedded within a sentence to illustrate a trend and help stakeholders better understand the data. Evaluators can use this simple visualization when creating reports." (Christopher Lysy, "Developments in Quantitative Data Display and Their Implications for Evaluation", 2013)

"Visualization is what happens when you make the jump from raw data to bar graphs, line charts, and dot plots. […] In its most basic form, visualization is simply mapping data to geometry and color. It works because your brain is wired to find patterns, and you can switch back and forth between the visual and the numbers it represents. This is the important bit. You must make sure that the essence of the data isn’t lost in that back and forth between visual and the value it represents because if you can’t map back to the data, the visualization is just a bunch of shapes." (Nathan Yau, "Data Points: Visualization That Means Something", 2013)

"Graphs can help us interpret data and draw inferences. They can help us see tendencies, patterns, trends, and relationships. A picture can be worth not only a thousand words, but a thousand numbers. However, a graph is essentially descriptive - a picture meant to tell a story. As with any story, bumblers may mangle the punch line and the dishonest may lie." (Gary Smith, "Standard Deviations", 2014)

"Graphs should not be mere decoration, to amuse the easily bored. A useful graph displays data accurately and coherently, and helps us understand the data. Chartjunk, in contrast, distracts, confuses, and annoys. Chartjunk may be well-intentioned, but it is misguided. It may also be a deliberate attempt to mystify." (Gary Smith, "Standard Deviations", 2014)

"[...] tables interact with our verbal system, graphs interact with our visual system, which is faster at processing information." (Cole N Knaflic, "Storytelling with Data: A Data Visualization Guide for Business Professionals", 2015)

"The most accurate but least interpretable form of data presentation is to make a table, showing every single value. But it is difficult or impossible for most people to detect patterns and trends in such data, and so we rely on graphs and charts. Graphs come in two broad types: Either they represent every data point visually" (as in a scatter plot) or they implement a form of data reduction in which we summarize the data, looking, for example, only at means or medians." (Daniel J Levitin, "Weaponized Lies", 2017)

"When visuals are applied to data, they can enlighten the audience to insights that they wouldn’t see without charts or graphs. Many interesting patterns and outliers in the data would remain hidden in the rows and columns of data tables without the help of data visualizations. They connect with our visual nature as human beings and impart knowledge that couldn’t be obtained as easily using other approaches that involve just words or numbers." (Brent Dykes, "Effective Data Storytelling: How to Drive Change with Data, Narrative and Visuals", 2019)

On Continuity I (Calculus)

"Since [...] nature is a principle of motion and mutation [...] it is necessary that we should not be ignorant of what motion is [...] But motion appears to belong to things continuous; and the infinite first presents itself to the view in that which is continuous. [..] frequently [...] those who define the continuous, employ the nature or the infinite, as if that which is divisible to infinity is continuous." (Aristotle, "Physics", cca. 350 BC)

"Things [...] are some of them continuous [...] which are properly and peculiarly called 'magnitudes'; others are discontinuous, in a side-by-side arrangement, and, as it were, in heaps, which are called 'multitudes,' a flock, for instance, a people, a heap, a chorus, and the like. Wisdom, then, must be considered to be the knowledge of these two forms. Since, however, all multitude and magnitude are by their own nature of necessity infinite - for multitude starts from a definite root and never ceases increasing; and magnitude, when division beginning with a limited whole is carried on, cannot bring the dividing process to an end [...] and since sciences are always sciences of limited things, and never of infinites, it is accordingly evident that a science dealing with magnitude [...] or with multitude [...] could never be formulated. […] A science, however, would arise to deal with something separated from each of them, with quantity, set off from multitude, and size, set off from magnitude." (Nicomachus, cca. 100 AD) 

"Since the nature of differentials […] consists in their being infinitely small and infinitely changeable up to zero, in being only quantitates evanescentes, evanescentia divisibilia, they will be always smaller than any given quantity whatsoever. In fact, some difference which one can assign between two magnitudes which only differ by a differential, the continuous and imperceptible variability of that infinitely small differential, even at the very point of becoming zero, always allows one to find a quantity less than the proposed difference." (Johann Bernoulli, cca. 1692–1702)

"There are two famous labyrinths where our reason very often goes astray. One concerns the great question of the free and the necessary, above all in the production and the origin of Evil. The other consists in the discussion of continuity, and of the indivisibles which appear to be the elements thereof, and where the consideration of the infinite must enter in." (Gottfried W Leibniz, "Theodicy: Essays on the Goodness of God and Freedom of Man and the Origin of Evil", 1710)

"Every quantity which, keeping the same expression, increases or diminishes continually (non per saltum), is called a variable, and that which, with the same expression, keeps the same value, is called fixed or constant." (Pierre Varignon, "Eclaircissemens sur l'Analyse des Infinimens Petits", 1725)

"In fact, a similar principle of hardness cannot exist; it is a chimera which offends that general law which nature constantly observes in all its operations; I speak of that immutable and perpetual order, established since the creation of the Universe, that can be called the LAW OF CONTINUITY, by virtue of which everything that takes place, takes place by infinitely small degrees. It seems that common sense dictates that no change can take place at a jump; natura non operatur per saltion; nothing can pass from one extreme to the other without passing through all the degrees in between." (Johann Bernoulli, "Discours sur les Loix de la Communication du Mouvement", 1727)

"The Law of Continuity, as we here deal with it, consists in the idea that [...] any quantity, in passing from one magnitude to another, must pass through all intermediate magnitudes of the same class. The same notion is also commonly expressed by saying that the passage is made by intermediate stages or steps; [...] the idea should be interpreted as follows: single states correspond to single instants of time, but increments or decrements only to small areas of continuous time." (Roger J Boscovich, "Philosophiae Naturalis Theoria Redacta Ad Unicam Legera Virium in Natura Existentium", 1758)

"Any change involves at least two conditions, one preceding and one following, which are distinct from one another in such a way that the difference between the former and the latter can be established. Now the law of continuity prohibits the thing which is being changed to transcend abruptly from the former to the latter. It must pass through an intermediate condition which is as little distinct from the previous as from the subsequent one. And because the difference between this intermediate condition and the previous condition can be established still, there must be an intermediate condition between these two as well, and this must continue in the same way, until the difference between the previous condition and the one immediately succeeding it vanishes. As long as the set of these intermediate conditions can be established, every difference between one and the next can be established as well: hence their set must become larger than any given set if these differences shall vanish, and thus we imagine infinitely many conditions where one differs from the next to an infinitely small degree." (Abraham G Kästner, "Anfangsgründe der Analysis des Unendlichen" [Beginnings of the Analysis of the Infinite"], 1766)

"It is held because of this law in particular, that no change may occur suddenly, but rather that every change always passes by infinitely small stages, of which the trajectory of a point in a curved line provides a first example." (Abraham G Kästner, "Anfangsgründe der Analysis des Unendlichen" ["Beginnings of the Analysis of the Infinite"], 1766)

"Whoever wishes to extend this law [of continuity] to the real must justify his inferences by a law other than that, the suspicion remaining that he took images for things." (Abraham G Kästner, "Anfangsgründe der Analysis des Unendlichen" ["Beginnings of the Analysis of the Infinite"], 1766)

"[continuity] could be only appearance, and in this case Euler’s entire argument against the atoms would disappear; for one would be justified to apply the law of continuity only where experience shows that it agrees with the phenomena. [...]. The law of continuity thus belongs to the clothes of things which we must need rely on wherever reality seems impenetrably cloaked with it, but which we do not consider to be reality itself, and which we may still less cloak with things which do not serve us to see them." (Johann S T Gehler, "Physikalisches Wörterbuch" Bd. 4, 1798)

"Logic sometimes makes monsters. For half a century we have seen a mass of bizarre functions which appear to be forced to resemble as little as possible honest functions which serve some purpose. More of continuity, or less of continuity, more derivatives, and so forth. Indeed, from the point of view of logic, these strange functions are the most general; on the other hand those which one meets without searching for them, and which follow simple laws appear as a particular case which does not amount to more than a small corner. In former times when one invented a new function it was for a practical purpose; today one invents them purposely to show up defects in the reasoning of our fathers and one will deduce from them only that. If logic were the sole guide of the teacher, it would be necessary to begin with the most general functions, that is to say with the most bizarre. It is the beginner that would have to be set grappling with this teratologic museum." (Henri Poincaré, 1899)

"The Calculus required continuity, and continuity was supposed to require the infinitely little; but nobody could discover what the infinitely little might be." (Bertrand Russell, "Mysticism and Logic and Other Essays", cca. 1910)

"The course of the values of a continuous function is determined at all points of an interval, if only it is determined for all rational points of this interval." (Felix Klein, "Elementary Mathematics from a Higher Standpoint" Vol III: "Precision Mathematics and Approximation Mathematics", 1928)

"A common and very powerful constraint is that of continuity. It is a constraint because whereas the function that changes arbitrarily can undergo any change, the continuous function can change, at each step, only to a neighbouring value." (W Ross Ashby, "An Introduction to Cybernetics", 1956)

"As a simple trick, the discrete can often be carried over into the continuous, in a way suitable for practical purposes, by making a graph of the discrete, with the values shown as separate points. It is then easy to see the form that the changes will take if the points were to become infinitely numerous and close together." (W Ross Ashby, "An Introduction to Cybernetics", 1956)

"The mathematical theory of continuity is based, not on intuition, but on the logically developed theories of number and sets of points." (Carl B Boyer, "The History of the Calculus and Its Conceptual Development", 1959)

"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)

"Continuous functions can move freely. Graphs of continuous functions can freely branch off at any place, whereas analytic functions coinciding in some neighborhood of a point P cannot branch outside of this neighborhood. Because of this property, continuous functions can mathematically represent wildly changing wind inside a typhoon or a gentle breeze." (Kenji Ueno & Toshikazu Sunada, "A Mathematical Gift, III: The Interplay Between Topology, Functions, Geometry, and Algebra", Mathematical World Vol. 23, 1996)

"If we know when a sequence approaches a point or, as we say, converges to a point, we can define a continuous mapping from one metric space to another by using the property that a converging sequence is mapped to the corresponding converging sequence." (Kenji Ueno & Toshikazu Sunada, "A Mathematical Gift, III: The Interplay Between Topology, Functions, Geometry, and Algebra", Mathematical World Vol. 23, 1996)

"Intuitively speaking, a visual representation associated with the concept of continuity is the property that a near object is sent to a corresponding near object, that is, a convergent sequence is sent to a corresponding convergent sequence." (Kenji Ueno & Toshikazu Sunada, "A Mathematical Gift, III: The Interplay Between Topology, Functions, Geometry, and Algebra", Mathematical World Vol. 23, 1996)

"If you assume continuity, you can open the well-stocked mathematical toolkit of continuous functions and differential equations, the saws and hammers of engineering and physics for the past two centuries (and the foreseeable future)." (Benoît Mandelbrot, "The (Mis)Behaviour of Markets: A Fractal View of Risk, Ruin and Reward", 2004)

"Continuity is the rigorous formulation of the intuitive concept of a function that varies with no abrupt breaks or jumps. A function is a relationship in which every value of an independent variable - say x - is associated with a value of a dependent variable - say y. Continuity of a function is sometimes expressed by saying that if the x-values are close together, then the y-values of the function will also be close. But if the question 'How close?' is asked, difficulties arise." (Erik Gregersen [Ed.], "Math Eplained: The Britannica Guide to Analysis and Calculus", 2011)

On Networks XVII (Semantic Networks I)

"In comparison with Predicate Calculus encoding s of factual knowledge, semantic nets seem more natural and understandable. This is due to the one-to-one correspondence between nodes and the concepts they denote, to the clustering about a particular node of propositions about a particular thing, and to the visual immediacy of 'interrelationships' between concepts, i.e., their connections via sequences of propositional links." (Lenhart K Schubert, "Extending the Expressive Power of Semantic Networks", Artificial Intelligence 7, 1976)

"[…] semantic nets [are defined] as graphical analogues of data structures representing "facts" in a computer system for understanding natural language." (Lenhart K Schubert," "Extending the Expressive Power of Semantic Networks", Artificial Intelligence 7, 1976)

"The advantage of semantic networks over standard logic is that some selected set of the possible inferences can be made in a specialized and efficient way. If these correspond to the inferences that people make naturally, then the system will be able to do a more natural sort of reasoning than can be easily achieved using formal logical deduction." (Avron Barr, Natural Language Understanding, AI Magazine Vol. 1 (1), 1980)

"We define a semantic network as 'the collection of all the relationships that concepts have to other concepts, to percepts, to procedures, and to motor mechanisms' of the knowledge." (John F Sowa, "Conceptual Structures", 1984)

"[…] semantic nets fail to be distinctive in the way they (1) represent propositions, (2) cluster information for access, (3) handle property inheritance, and (4) handle general inference; in other words, they lack distinctive representational properties (i.e., 1) and distinctive computational properties (i.e., 2-4). Certain propagation mechanisms, notably 'spreading activation', 'intersection search', or 'inference propagation' have sometimes been regarded as earmarks of semantic nets, but since most extant semantic nets lack such mechanisms, they cannot be considered criterial in current usage." (Lenhart K Schubert, "Semantic Nets are in the Eye of the Beholder", 1990)

"[…] the representational and computational strategies employed in semantic net systems are abstractly equivalent to those employed in virtually all state-of-the-art systems incorporating a substantial propositional knowledge base, whether they are described as logic-based, frame-based, rule-based, or some-thing else." (Lenhart K Schubert, "Semantic Nets are in the Eye of the Beholder", 1990)

"A semantic network or net represents knowledge as a net-like graph. An idea, event, situation or object almost always has a composite structure; this is represented in a semantic network by a corresponding structure of nodes (drawn as circles or boxes) representing conceptual units, and directed links (drawn as arrows between the nodes) representing the relations between the units. […] An abstract (graph-theoretic) network can be diagrammed, defined mathematically, programmed in a computer, or hard-wired electronically. It becomes semantic when you assign a meaning to each node and link. Unlike specialized networks and diagrams, semantic networks aim to represent any kind of knowledge which can be described in natural language. A semantic network system includes not only the explicitly stored net structure but also methods for automatically deriving from that a much larger structure or body of implied knowledge." (Fritz Lehman, "Semantic Networks",  Computers & Mathematics with Applications Vol. 23 (2-5), 1992)

"The essential idea of semantic networks is that the graph-theoretic structure of relations and. abstractions can be used for inference as well as understanding. […] A semantic network is a discrete structure as is any linguistic description. Representation of the continuous 'outside world' with such a structure is necessarily incomplete, and requires decisions as to which information is kept and which is lost." (Fritz Lehman, "Semantic Networks",  Computers & Mathematics with Applications Vol. 23 (2-5), 1992)

"The great organizing principle of thought is abstraction. By assigning particular things to abstract categories we are able to dispense with irrelevant detail and yet instantly draw copious conclusions about a thing due to its membership in various categories. Semantic networks specify the structure of interrelated abstract categories and use this structure to draw conclusions." (Fritz Lehman, "Semantic Networks",  Computers & Mathematics with Applications Vol. 23 (2-5), 1992)

On Symbols (2000-2009)

"Precision is greatly enhanced by the human capacity to symbolize. Symbols can be devised to stand for mathematical ideas, entities, operations, and relations. Symbols also permit precise and repeatable calculation." (George Lakoff & Rafael E Nuñez, "Where Mathematics Come From: How the Embodied Mind Brings Mathematics into Being, 2000)

"The motion of the mind is conveyed along a cloud of meaning. There is this paradox that we get to meaning only when we strip the meaning from symbols." (David Berlinski, "The Advent of the Algorithm: The Idea that Rules the World", 2000)

"A symbol is a mental representation regarding the internal reality referring to its object by a convention and produced by the conscious interpretation of a sign. In contrast to signals, symbols may be used every time if the receiver has the corresponding representation. Symbols also relate to feelings and thus give access not only to information but also to the communicator’s motivational and emotional state. The use of symbols makes it possible for the organism using it to evoke in the receiver the same response it evokes in himself. To communicate with symbols is to use a language." (Lars Skyttner, "General Systems Theory: Ideas and Applications", 2001)

"In the definition of meaning, it is assumed that both the source and receiver have previously coded (and stored) signals of the same or similar referents, such that the messages may have meaning and relate to behaviour. That is, the used symbols must have the same signification for both sender and receiver. If not, the receiver will create a different mental picture than intended by the transmitter. Meaning is generated by individuals in a process of social interaction with a more or less common environment. It is a relation subsisting within a field of experience and appears as an emergent property of a symbolic representation when used in culturally accepted interaction. The relation between the symbolic representation and its meaning is random. Of this, however, the mathematical theory has nothing to say. If human links in the chain of communication are missing, of course no questions of meaning will arise." (Lars Skyttner, "General Systems Theory: Ideas and Applications", 2001)

"A person thinking in the nonverbal mode is actually thinking with the meaning of the language in the form of mental pictures of the concepts and ideas it contains. Nonverbal thought doesn't require literacy. An illiterate person can communicate without knowing what the symbols look like. [...] Literacy, then, is established as the person learns how the symbols look and becomes able to recognize them as representing certain things or concepts." (Ronald D Davis & Eldon M Braun, "The Gift of Learning", 2003)

"Science does not speak of the world in the language of words alone, and in many cases it simply cannot do so. The natural language of science is a synergistic integration of words, diagrams, pictures, graphs, maps, equations, tables, charts, and other forms of visual and mathematical expression. […] [Science thus consists of] the languages of visual representation, the languages of mathematical symbolism, and the languages of experimental operations." (Jay Lemke, "Teaching all the languages of science: Words, symbols, images and actions", 2003)

"I often told the fanatics of realism that there is no such thing as realism in art: it only exists in the mind of the observer. Art is a symbol, a thing conjuring up reality in our mental image. That is why I don't see any contradiction between abstract and figurative art either." (Antoni Tàpies, "Tàpies, Werke auf Papier 1943 – 2003", 2004)

"A symbol is an object, act, or event that conveys meaning to others. Symbols can be considered a rich, non-verbal language that vibrantly conveys the organization’s important values concerning how people relate to one another and interact with the environment" (Richard L Daft & Dorothy Marcic, "Understanding Management" 5th Ed., 2006)

"But because of the way in which depictions represent, there is a correspondence between parts and spatial relations of the representation and those of the object; this structural mapping, which confers a type of resemblance, underlies the way images convey specific content. In this respect images are like pictures. Unlike words and symbols, depictions are not arbitrarily paired with what they represent." (Stephen Kosslyn et al," The Case for Mental Imagery", 2006)

"Imagination has the creative task of making symbols, joining things together in such a way that they throw new light on each other and on everything around them. The imagination is a discovering faculty, a faculty for seeing relationships, for seeing meanings that are special and even quite new." (Thomas Merton, "Angelic Mistakes: The Art of Thomas Merton", 2006)

"[...] the scientific models of concrete things are symbolic rather than iconic: they are systems of propositions, not pictures. Besides, such models are seldom if ever completely accurate, if only because they involve more or less brutal simplifications, such as pretending that a metallic surface is smooth, a crystal has no impurities, a biopopulation has a single predator, or a market is in equilibrium.  These are all fictions. However, they are stylizations rather than wild fantasies. Hence, introducing and using them to account for real existents does not commit us to fictionism, just as defending the role of experience need not make us empiricists, nor is admitting the role of intuition enough to qualify as intuitionist." (Mario Bunge, "Chasing Reality: Strife over Realism", 2006)

"But notice, a subatomic particle is itself a holon [hole/parts]. And so is a cell. And so is a symbol, and an image, and a concept. What all of those entities are, before they are anything else, is a holon. So the world is not composed of atoms or symbols or cells or concepts. It is composed of holons." (Ken Wilber, "A Brief History of Everything", 2007)

"Language use is a curious behavior, but once the transition to language is made, literature is a likely consequence, since it is linked to the dynamic of the linguistic symbol through the functioning of the imagination." (Russell Berman, "Fiction Sets You Free: Literature, Liberty and Western Culture", 2007)

"Images and pictures […] have played a key role in shaping our scientific picture of the world. […] Carefully constructed families of pictures can act as a calculus all their own. Like any successful systems of symbols, with an appropriate grammar they enlarge the number of things that we can do without consciously thinking." (John D Barrow, "Cosmic Imagery: Key Images in the History of Science", 2008)

"How are we to explain the contrast between the matter-of-fact way in which √-1 and other imaginary numbers are accepted today and the great difficulty they posed for learned mathematicians when they first appeared on the scene? One possibility is that mathematical intuitions have evolved over the centuries and people are generally more willing to see mathematics as a matter of manipulating symbols according to rules and are less insistent on interpreting all symbols as representative of one or another aspect of physical reality. Another, less self-congratulatory possibility is that most of us are content to follow the computational rules we are taught and do not give a lot of thought to rationales." (Raymond S Nickerson, "Mathematical Reasoning: Patterns, Problems, Conjectures, and Proofs", 2009)

"Mathematicians are sometimes described as living in an ideal world of beauty and harmony. Instead, our world is torn apart by inconsistencies, plagued by non sequiturs and, worst of all, made desolate and empty by missing links between words, and between symbols and their referents; we spend our lives patching and repairing it. Only when the last crack disappears are we rewarded by brief moments of harmony and joy." (Alexandre V Borovik, "Mathematics under the Microscope: Notes on Cognitive Aspects of Mathematical Practice", 2009)

"Mathematical ideas like number can only be really 'seen' with the 'eyes of the mind' because that is how one 'sees' ideas. Think of a sheet of music which is important and useful but it is nowhere near as interesting, beautiful or powerful as the music it represents. One can appreciate music without reading the sheet of music. Similarly, mathematical notation and symbols on a blackboard are just like the sheet of music; they are important and useful but they are nowhere near as interesting, beautiful or powerful as the actual mathematics (ideas) they represent." (Fiacre O Cairbre, "The Importance of Being Beautiful in Mathematics", IMTA Newsletter 109, 2009)

On Graph Theory II

"Graph theory is often under attack, and so are its practitioners. We are accused of being shallow, knowing and using no real mathematics, and tackling problems of little interest, whose solutions are easy if not trivial. Although these criticisms are usually made by people unsympathetic to everything combinatorial, there is a grain of truth in these accusations - perhaps even more than a grain. In graph theory we do write too many papers, sometimes we do tackle problems that are too easy, and we have a tendency to become wrapped up in our circle of ideas and problems, unconcerned about the rest of mathematics. However, I am convinced that these are mostly teething problems." (Béla Bollobás, "The Future of Graph Theory", [in "Quo Vadis, Graph Theory?"] 1993)

"The four-color map theorem is an assertion about graph theory, which is the study of discrete points and the lines that connect them; each point is called a vertex and each line is called an edge." (Alexander Humez et al, "Zero to Lazy Eight: The romance of numbers", 1993)

"Perhaps the greatest strength of graph theory is the abundance of natural and beautiful problems waiting to be solved. […] Paradoxically, much of what is wrong with graph theory is due to this richness of problems. It is all too easy to find new problems based on no theory whatsoever, and to solve the first few cases by straightforward methods. Unfortunately, in some instances the problems are unlikely to lead anywhere […]" (Béla Bollobás, "The Future of Graph Theory", [in "Quo Vadis, Graph Theory?"] 1993)

"Graphs are one of the unifying themes of computer science - an abstract representation that describes the organization of transportation systems, human interactions, and telecommunication networks. That so many different structures can be modeled using a single formalism is a source of great power to the educated programmer." (Steven S Skiena, "The Algorithm Design Manual", 1997)

"A graph in mathematics is a set of nodes and a set of edges between pairs of those nodes; the edges are ordered or nonordered pairs, or a relation, that defines the pairs of nodes for which the relation being examined is valid. […] The edges can either be undirected or directed; directed edges depict a relation that requires the nodes to be ordered while an undirected edge defines a relation in which no ordering of the edges is implied." (Dennis M Buede, "The Engineering Design of Systems: Models and methods", 2009)

"A graph enables us to visualize a relation over a set, which makes the characteristics of relations such as transitivity and symmetry easier to understand. […] Notions such as paths and cycles are key to understanding the more complex and powerful concepts of graph theory. There are many degrees of connectedness that apply to a graph; understanding these types of connectedness enables the engineer to understand the basic properties that can be defined for the graph representing some aspect of his or her system. The concepts of adjacency and reachability are the first steps to understanding the ability of an allocated architecture of a system to execute properly." (Dennis M Buede, "The Engineering Design of Systems: Models and methods", 2009)

"First, what are the 'graphs' studied in graph theory? They are not graphs of functions as studied in calculus and analytic geometry. They are (usually finite) structures consisting of vertices and edges. As in geometry, we can think of vertices as points (but they are denoted by thick dots in diagrams) and of edges as arcs connecting pairs of distinct vertices. The positions of the vertices and the shapes of the edges are irrelevant: the graph is completely specified by saying which vertices are connected by edges. A common convention is that at most one edge connects a given pair of vertices, so a graph is essentially just a pair of sets: a set of objects." (John Stillwell, "Mathematics and Its History", 2010)

"Infinite reasoning is likewise essential for graph theory. The field had its origins in topology, and it is still relevant there, but it has expanded extraordinarily far in other directions. Graph theory today is exploring the boundaries of finite provability first exposed by Gödel’s incompleteness theorem." (John Stillwell, "Mathematics and Its History", 2010)

"The most naive branch of combinatorics is graph theory, a subject that is visual and easily grasped, yet rich in connections with other parts of mathematics." (John Stillwell, "Mathematics and Its History", 2010)

"Graphs are among the most important abstract data structures in computer science, and the algorithms that operate on them are critical to modern life. Graphs have been shown to be powerful tools for modeling complex problems because of their simplicity and generality." (Jeremy Kepner & John Gilbert [Eds],"Graph Algorithms in the Language of Linear Algebra", 2011)

On Graph Theory I

"I am not content with algebra, in that it yields neither the shortest proofs nor the most beautiful constructions of geometry. Consequently, in view of this, I consider that we need yet another kind of analysis, geometric or linear, which deals directly with position, as algebra deals with magnitude." (Gottfried Leibniz, [letter to Christiaan Huygens] 1670)

"A problem was posed to me about an island in the city of Königsberg, surrounded by a river spanned by seven bridges, and I was asked whether someone could traverse the separate bridges in a connected walk in such a way that each bridge is crossed only once. I was informed that hitherto no-one had demonstrated the possibility of doing this, or shown that it is impossible. This question is so banal, but seemed to me worthy of attention in that not geometry, nor algebra, nor even the art of counting was sufficient to solve it. In view of this, it occurred to me to wonder whether it belonged to the geometry of position, which Leibniz had once so much longed for. And so, after some deliberation, I obtained a simple, yet completely established, rule with whose help one can immediately decide for all examples of this kind, with any number of bridges in any arrangement, whether or not such a round trip is possible […]" (Leonard Euler, [letter to Giovanni Marinoni] 1736)

"Graph theory is the study of sets of points that are joined by lines." (Martin Gardner, "Aha! Insight", 1978)

"At the other end of the spectrum is, for example, graph theory, where the basic object, a graph, can be immediately comprehended. One will not get anywhere in graph theory by sitting in an armchair and trying to understand graphs better. Neither is it particularly necessary to read much of the literature before tackling a problem: it is of course helpful to be aware of some of the most important techniques, but the interesting problems tend to be open precisely because the established techniques cannot easily be applied." (Timothy Gowers, "The two cultures of mathematics", 2000)

"Euler's proof that in Königsberg there is no path crossing all seven bridges only once was based on a simple observation. Nodes with an odd number of links must be either the starting or the end point of the journey. A continuous path that goes through all the bridges can have only one starting and one end point. Thus, such a path cannot exist on a graph that has more than two nodes with an odd number of links. As the Königsberg graph had four such nodes, one could not find the desired path." (Albert-László Barabási, "Linked: How Everything Is Connected to Everything Else and What It Means for Business, Science, and Everyday Life", 2002)

"'There is an old debate', Erdos liked to say, 'about whether you create mathematics or just discover it. In other words, are the truths already there, even if we don't yet know them?' Erdos had a clear answer to this question: Mathematical truths are there among the list of absolute truths, and we just rediscover them. Random graph theory, so elegant and simple, seemed to him to belong to the eternal truths. Yet today we know that random networks played little role in assembling our universe. Instead, nature resorted to a few fundamental laws [...]. Erdos himself created mathematical truths and an alternative view of our world by developing random graph theory." (Albert-László Barabási, "Linked: How Everything Is Connected to Everything Else and What It Means for Business, Science, and Everyday Life", 2002)

"The important graphs are the ones where some things are not connected to some other things. When the unenlightened ones try to be profound, they draw endless verbal comparisons between this topic, and that topic, which is like this, which is like that; until their graph is fully connected and also totally useless." (Eliezer S Yudkowsky,  "Mysterious Answers to Mysterious Questions", 2007)

"The mathematical structure known as a graph has the valuable feature of helping us to visualize, to analyze, to generalize a situation or problem we may encounter and, in many cases, assisting us to understand it better and possibly find a solution." (Arthur Benjamin, "The fascinating world of graph theory", 2015)

"The theory of graphs is the fundamental study of relations in their purest, non-trivial form: binary connections between abstract points. And as so often in combinatorics, this simple assemblage of trivial objects results in a dazzlingly rich theory of seemingly endless depths." (Felix Reidl, "Structural Sparseness and Complex Networks", 2015)

"The theory of ramification is one of pure colligation, for it takes no account of magnitude or position; geometrical lines are used, but these have no more real bearing on the matter than those employed in genealogical tables have in explaining the laws of procreation." (James J Sylvester)