Out of Context: On Diagrams (Definitions)

 "Diagrams are of great utility for illustrating certain questions of vital statistics by conveying ideas on the subject through the eye, which cannot be so readily grasped when contained in figures." (Florence Nightingale, "Mortality of the British Army", 1857)

"Diagrams are sometimes used, not merely to convey several pieces of information such as several time series on one chart, but also to provide visual evidence of relationships between the series." (Alfred R Ilersic, "Statistics", 1959)

"Diagrams, whether representational or symbolic, are meaningless unless attached to some body of theory. On the other hand theories are in no need of diagrams save for psychological purposes. Let us then keep theoretical models apart from visual analogues."  (Mario Bunge, "Philosophy of Physics", 1973)

"Schematic diagrams are more abstract than pictorial drawings, showing symbolic elements and their interconnection to make clear the configuration and/or operation of a system." (Ernest O Doebelin, "Engineering experimentation: planning, execution, reporting", 1995)

"[...] (4) Diagrams are psychologically useful, but prove nothing; (5) Diagrams can even be misleading [...]" (James R Brown,"Philosophy of Mathematics", 1999)

"A model diagram declares some sets and binary relations, and imposes some basic constraints on them. A diagram is a good way to convey the outline of a model, but diagrams aren’t expressive enough to include detailed constraints." (Daniel Jackson, "Software Abstractions", 2006) 

"[...] diagrams are models, graphical in nature, that are used to illustrate structure (e.g., how components are physically interconnected); they do not capture functional behavior of a system. "  (Robbie T Nakatsu, "Diagrammatic Reasoning in AI", 2010)

On Diagrams: Definitions

"A diagram is a representamen [representation] which is predominantly an icon of relations and is aided to be so by conventions. Indices are also more or less used. It should be carried out upon a perfectly consistent system of representation, founded upon a simple and easily intelligible basic idea." (Charles S Peirce, 1903)

"A diagram is an icon or schematic image embodying the meaning of a general predicate; and from the observation of this icon we are supposed to construct a new general predicate." (Charles S Peirce, "New Elements" ["Kaina stoiceia"], 1904)

"[The diagram] is only an heuristic to prompt certain trains of inference; [...] it is dispensable as a proof-theoretic device; indeed, [...] it has no proper place in the proof as such. For the proof is a syntactic object consisting only of sentences arranged in a finite and inspectable array." (Neil Tennant, "The withering away of formal semantics", Mind and Language Vol. 1 (4), 1986)

"Diagrams are a means of communication and explanation, and they facilitate brainstorming. They serve these ends best if they are minimal. Comprehensive diagrams of the entire object model fail to communicate or explain; they overwhelm the reader with detail and they lack meaning." (Eric Evans, "Domain-Driven Design: Tackling complexity in the heart of software", 2003)

"A diagram is a graphic shorthand. Though it is an ideogram, it is not necessarily an abstraction. It is a representation of something in that it is not the thing itself. In this sense, it cannot help but be embodied. It can never be free of value or meaning, even when it attempts to express relationships of formation and their processes. At the same time, a diagram is neither a structure nor an abstraction of structure." (Peter Eisenman, "Written Into the Void: Selected Writings", 1990-2004, 2007)

"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)

"[...] diagrams are models, graphical in nature, that are used to illustrate structure (e.g., how components are physically interconnected); they do not capture functional behavior of a system. "  (Robbie T Nakatsu, "Diagrammatic Reasoning in AI", 2010)

On Construction V: Diagrams

"[…] deduction consists in constructing an icon or diagram the relations of whose parts shall present a complete analogy with those of the parts of the object of reasoning, of experimenting upon this image in the imagination, and of observing the result so as to discover unnoticed and hidden relations among the parts." (Charles S Peirce, 1885)

"A diagram is an icon or schematic image embodying the meaning of a general predicate; and from the observation of this icon we are supposed to construct a new general predicate." (Charles S Peirce, "New Elements" ["Kaina stoiceia"], 1904)

"A theorem […] is an inference obtained by constructing a diagram according to a general precept, and after modifying it as ingenuity may dictate, observing in it certain relations, and showing that they must subsist in every case, retranslating the proposition into general terms." (Charles S Peirce, "New Elements" ["Kaina stoiceia"], 1904)

"Exact figures have, in principle, the same role in geometry as exact measurements in physics; but, in practice, exact figures are less important than exact measurements because the theorems of geometry are much more extensively verified than the laws of physics. The beginner, however, should construct many figures as exactly as he can in order to acquire a good experimental basis; and exact figures may suggest geometric theorems also to the more advanced. Yet, for the purpose of reasoning, carefully drawn free-hand figures are usually good enough, and they are much more quickly done." (George Pólya, "How to solve it", 1945)

"The diagrams incorporate a large amount of information. Their use provides extensive savings in space and in mental effort. In the case of many theorems, the setting up of the correct diagram is the major part of the proof. We therefore urge that the reader stop at the end of each theorem and attempt to construct for himself the relevant diagram before examining the one which is given in the text. Once this is done, the subsequent demonstration can be followed more readily; in fact, the reader can usually supply it himself." (Samuel Eilenberg & Norman E. Steenrod, "Foundations of Algebraic Topology", 1952)

"When the correspondences on the plane can be established between: - all the divisions of one component - and all the divisions of another component, the construction is a DIAGRAM." (Jacques Bertin, "Semiology of graphics", 1967)

"The thinking person goes over the same ground many times. He looks at it from varying points of view - his own, his arch-enemy’s, others’. He diagrams it, verbalizes it, formulates equations, constructs visual images of the whole problem, or of troublesome parts, or of what is clearly known. But he does not keep a detailed record of all this mental work, indeed could not. […] Deep understanding of a domain of knowledge requires knowing it in various ways. This multiplicity of perspectives grows slowly through hard work and sets the state for the re-cognition we experience as a new insight." (Howard E Gruber, "Darwin on Man", 1981)

"According to mental model theory, human reasoning relies on the construction of integrated mental representations of the information that is given in the reasoning problem's premises. These integrated representations are the mental models. A mental model is a mental representation that captures what is common to all the different ways in which the premises can be interpreted. It represents in "small scale" how 'reality' could be - according to what is stated in the premises of a reasoning problem. Mental models, though, must not be confused with images. A mental model often forms the basis of one or more visual images, but some of them represent situations that cannot be visualized. Instead, mental models are often likened to diagrams since, as with diagrams, their structure is analogous to the structure of the states of affairs they represent." (Carsten Held et al, "Mental Models and the Mind", 2006)

Robbie T Nakatsu - Collected Quotes

"Why do people use mental models? First, they are used as inference tools to predict the behavior of a system under novel conditions. They enable us to predict system outcomes from system parameters: We may run our mental model by modifying the system parameters and observing how the behavior of the system changes. Second, mental models can be used to produce explanations and justifications. Such explanations may give us confidence in using e system and enable us to more readily trust the results of the system. Third, mental models can be used as mnemonic devices to facilitate remembering and long-term retention of information. Here, a mental model may provide one with a "cover story" to make the understanding of the system more memorable and easier to recall." (Robbie Nakatsu, "Diagrammatic Reasoning in AI", 1994)

"A hierarchy is a diagram that shows how various components of a system are related, often with a downward direction (or alternatively a left-to-right direction) that moves from more general to more specific. One way to envision a hierarchy is as an inverted tree: We start with a single component (referred to as the root node or topmost node), typically denoted by a square, and then we draw one or more paths leading from it to other nodes. Each of these nodes, in turn, may subdivide into additional subpaths to other nodes. This process may be repeated any number of times to arrive at a multitiered, tree-like structure." (Robbie T Nakatsu, "Diagrammatic Reasoning in AI", 2010)

"A mental model is the user's model of a target system; it is a model of a system that exists in a person's head. Through interaction with a complex system, it is a 'naturally evolving model. As a person develops more experience with a system, the model develops and becomes more refined. Hence, at any given point in time, the mental model, as seen through the eyes of the user, is a dynamic, usually incomplete specification of the target system. A conceptual model, on the other hand, is typically the designer's complete specification of a target system. As such, it is intended to be an accurate, consistent, and complete representation of a target system. Ideally, we would want the user's mental model to be the same as the system designer's conceptual model." (Robbie T Nakatsu, "Diagrammatic Reasoning in AI", 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)

"Models are an important form of knowledge representation because expertise often lies in one's ability to reason about how the objects, or components, of a system are interconnected - whether physically, causally, relationally, or otherwise - in a domain of discourse." (Robbie T Nakatsu, "Diagrammatic Reasoning in AI", 2010)

"On the one hand, a conceptual model seeks to faithfully represent the components, the connections, the relations, and the processes that act on the components. On the other hand, a mental model that employs analogical representations is chosen to invite comparisons between two dissimilar domains, never to faithfully and completely represent the target domain." (Robbie T Nakatsu, "Diagrammatic Reasoning in AI", 2010)

"Prototyping is a method of developing systems rapidly by creating a quick-and-dirty mockup of a system, called a prototype. Once created, the prototype is given to end users so that they can provide their feedback and suggestions for improvement. Based on this feedback, you modify and enhance the prototype. It is an iterative process in that you can get feedback multiple times and enhance the prototype accordingly." (Robbie T Nakatsu, "Diagrammatic Reasoning in AI", 2010)

"Rule-based expert systems require that you preprogram all the rules that represent the knowledge in the domain. To create a realistic and complete solution. the knowledge engineer must be well versed in the domain and have a clear sense of what the decision procedures are. This understanding must take place beforehand, before the system is created: misunderstandings about the domain can be very costly later on because rule-based expert systems can be extremely difficult to modify and extend into other areas. Even when this is possible. the new rules must be created manually - the expert system does not learn how to tine-tune the rules on its own. Case-based reasoning and neural networks are two Al approaches that are more suitable when you want to create a system that 'learns' how to solve problems on its own." (Robbie T Nakatsu, "Diagrammatic Reasoning in AI", 2010)

"Semantic networks are used to illustrate how people organize information in their memories. Such representations have been used by cognitive psychologists to understand and theorize how one retrieves and processes information from long-term memory. In AI, semantic networks can also be used as a knowledge representation scheme that programs can use to retrieve information efficiently just like humans do." (Robbie T Nakatsu, "Diagrammatic Reasoning in AI", 2010)

"Structure is the way that the individual components of a system are interconnected (as given by a system's topology); Behavior refers to what each of these components is supposed to do. From this definition, we may distinguish three levels of system description. First, diagrams are models, graphical in nature, that are used to illustrate structure (e.g., how components are physically interconnected); they do not capture functional behavior of a system. Second, heuristics describe relationships between inputs and outputs, based on the way that experts describe how inputs are transformed into outputs. (Heuristics may be represented as IF-THEN rules). Heuristic knowledge, however, does not attempt to create an explicit representation of system structure. Model-based reasoning is a more complete representation system in the sense that it describes both structure and behavior. From this, three levels of system description can be distinguished, based on whether they describe structure, behavior, or both."  (Robbie T Nakatsu, "Diagrammatic Reasoning in AI", 2010)

"The development of a mental model, then, can be chronicled, much like the development of a cognitive skill. Three developmental processes8 seem to be at play when a mental model evolves. that mental model becomes more powerful because it works for a wider variety of situations. Second, discrimination means that a mental model is more sensitive to variations in a given situation so that a mental model may add an important new condition where previously it had been overlooked. Third, strengthening means that those aspects of a mental model that have been successfully applied in the past are strengthened and rendered more salient and significant." (Robbie T Nakatsu, "Diagrammatic Reasoning in AI", 2010)

"This is always the case in analogical reasoning: Relations between two dissimilar domains never map completely to one another. In fact, it is often the salient similarities between the base and target domains that provoke thought and increase the usefulness of an analogy as a problem-solving tool." (Robbie T Nakatsu, "Diagrammatic Reasoning in AI", 2010)

"Venn diagramming, it turns out, is a very effective technique for performing syllogistic reasoning. Its chief advantage (over the Euler graph in particular as we noted earlier) is the ability to incrementally add knowledge to the diagram. While an Euler graph has visual power in terms of representing the relations between sets very intuitively, it is impossible to combine more than one piece of information onto a Euler graph. A Venn diagram, on the other hand, easily lends itself to the representation of partial knowledge and can be manipulated to add successively more knowledge to the diagram. This means that when our knowledge of the relations between sets increases, we simply put in more symbols and shadings into the appropriate compartments of the Venn diagram. Thus we are able to accumulate knowledge in a Venn diagram. This capability turns out to be a powerful feature, one that endows Venn diagrams with a more dynamic quality that is sorely lacking in the Euler system." (Robbie T Nakatsu, "Diagrammatic Reasoning in AI", 2010)

"What advantages do diagrams have over verbal descriptions in promoting system understanding? First, by providing a diagram, massive amounts of information can be presented more efficiently. A diagram can strip down informational complexity to its core - in this sense, it can result in a parsimonious, minimalist description of a system. Second, a diagram can help us see patterns in information and data that may appear disordered otherwise. For example, a diagram can help us see mechanisms of cause and effect or can illustrate sequence and flow in a complex system. Third, a diagram can result in a less ambiguous description than a verbal description because it forces one to come up with a more structured description." (Robbie T Nakatsu, "Diagrammatic Reasoning in AI", 2010)

On Diagrams: Venn Diagrams

"[...] for merely theoretical purposes the rule of formation would be very simple. It would merely be to begin by drawing any closed figure, and then proceed [sic] to draw others, subject to the one condition that each is to intersect once and once only all the existing subdivisions produced by those which had gone before." (John Venn, "On the Diagrammatic and Mechanical Representation of Propositions and Reasonings", 1880)

"[…] it must be noticed that these diagrams do not naturally harmonize with the propositions of ordinary life or ordinary logic. […] The great bulk of the propositions which we commonly meet with are founded, and rightly founded, on an imperfect knowledge of the actual mutual relations of the implied classes to one another. […] one very marked characteristic about these circular diagrams is that they forbid the natural expression of such uncertainty, and are therefore only directly applicable to a very small number of such propositions as we commonly meet with." (John Venn, "On the Diagrammatic and Mechanical Representation of Propositions and Reasonings", 1880)

"[...] we can not readily break up a complicated problem into successive steps which can be taken independently. We have, in fact, to solve the problem first, by determining what are the actual mutual relations of the classes involved, and then to draw the circles to represent this final result; we cannot work step-by-step towards the conclusion by aid of our figures." (John Venn, "On the Diagrammatic and Mechanical Representation of Propositions and Reasonings", 1880)

"Whereas the Eulerian plan endeavoured at once and directly to represent propositions, or relations of class terms to one another, we shall find it best to begin by representing only classes, and then proceed to modify these in some way so as to make them indicate what our propositions have to say. How, then, shall we represent all the subclasses which two or more class terms can produce? Bear in mind that what we have to indicate is the successive duplication of the number of subdivisions produced by the introduction of each successive term. and we shall see our way to a very important departure from the Eulerian conception. All that we have to do is to draw our figures, say circles, so that each successive one which we introduce shall intersect once, and once only, all the subdivisions already existing, and we then have what may be called a general framework indicating every possible combination producible by the given class terms." (John Venn, "On the Diagrammatic and Mechanical Representation of Propositions and Reasonings", 1880)

"It will be found that there is a tendency for the resultant outlines thus successively drawn to assume a comb-like shape after the first four or five [...]The fifth-term figure will have two teeth, the sixth four, and so on. [...] There is no trouble in drawing such a diagram for any number of terms which our paper will find room for. But, as has already been repeatedly remarked, the visual aid for which mainly such diagrams exist is soon lost on such a path."  (John Venn, "Symbolic Logic", [footnote], 1881)

"There is no need here to exhibit such figures, as they would probably be distasteful to any but the mathematician, and he would see his way to drawing them readily enough for himself [...]" (John Venn, "Symbolic Logic", 1881)

"We endeavour to employ only symmetrical figures, such as should not only be an aid to reasoning, through the sense of sight, but should also be to some extent elegant in themselves." (John Venn, "Symbolic Logic", 1881)

"At the basis of our Symbolic Logic, however represented, whether by words by letters or by diagrams, we shall always find the same state of things. What we ultimately have to do is to break up the entire field before us into a definite number of classes or compartments which are mutually exclusive and collectively exhaustive." (John Venn, "Symbolic Logic" 2nd Ed., 1894)

"The best way of introducing this question will be to enquire a little more strictly whether it is really classes that we thus represent, or merely compartments into which classes may be put? […] The most accurate answer is that our diagrammatic subdivisions, or for that matter our symbols generally, stand for compartments and not for classes. We may doubtless regard them as representing the latter, but if we do so we should never fail to keep in mind the proviso, 'if there be such things in existence'. And when this condition is insisted upon, it seems as if we expressed our meaning best by saying that what our symbols stand for are compartments which may or may not happen to be occupied." (John Venn, "Symbolic Logic" 2nd Ed., 1894)

 "My Method of Diagrams resembles Mr. Venn's, in having separate Compartments assigned to the various Classes, and in marking these Compartments as occupied or as empty; but it differs from his Method, in assigning a closed area to the Universe of Discourse, so that the Class which, under Mr. Venn's liberal sway, has been ranging at will through Infinite Space, is suddenly dismayed to find itself "cabin'd, cribb'd, confined" in a limited Cell like any other Class! Also I use rectilinear, instead of curvilinear Figures" (Charles Dogson [Lews Carroll], 1896)

"This is why a 'web' of notes with links (like references) between them is far more useful than a fixed hierarchical system. When describing a complex system, many people resort to diagrams with circles and arrows. Circles and arrows leave one free to describe the interrelationships between things in a way that tables, for example, do not. The system we need is like a diagram of circles and arrows, where circles and arrows can stand for anything." (Tim Berners-Lee, "Information Management: A Proposal", 1989)

"Venn diagrams are widely used to solve problems in set theory and to test the validity of syllogisms in logic. […] However, it is a fact that Venn diagrams are not considered valid proofs, but heuristic tools for finding valid formal proofs." (Sun-Joo Shin, "Situation-Theoretic Account of Valid Reasoning with Venn Diagrams", [in "Logical Reasoning with Diagrams"], 1996)

"Venn diagrams provide us with a formalism that consists of a standardized system of representations, together with rules for manipulating them. In this regard, they could be considered a primitive visual analog of the formal systems of deduction developed in logic." (Jon Barwise & John Etchemendy, "Visual Information and Valid Reasoning", [in "Logical Reasoning with Diagrams"], 1996)

"A Venn diagram is a simple representation of the sample space, that is often helpful in seeing 'what is going on'. Usually the sample space is represented by a rectangle, with individual regions within the rectangle representing events. It is Often helpful to imagine that the actual areas Of the various regions in a Venn diagram are in proportion to the corresponding probabilities. However, there is no need to spend a long time drawing these diagrams - their use is simply as a reminder of what is happening." (Graham Upton & Ian Cook, "Introducing Statistics", 2001)

"Two types of graphic organizers are commonly used for comparison: the Venn diagram and the comparison matrix [...] the Venn diagram provides students with a visual display of the similarities and differences between two items. The similarities between elements are listed in the intersection between the two circles. The differences are listed in the parts of each circle that do not intersect. Ideally, a new Venn diagram should be completed for each characteristic so that students can easily see how similar and different the elements are for each characteristic used in the comparison." (Robert J. Marzano et al, "Classroom Instruction that Works: Research-based strategies for increasing student achievement, 2001)

"It is a curious fact that if you draw an endless line on a piece of paper so that it cuts itself any number of times (but never cuts itself more than once at the same point), then you can color the resulting regions using only two colors without any adjoining regions being the same color. [...] Venn diagrams also possess this property, but for a separate reason, which at first sight seems to be nicely demonstrated by induction." (Anthony W F Edwards, "Cogwheels of the mind: The story of Venn diagrams", 2004)

"The notion of outcomes covering a space is a very useful mental image, as it ties in strongly with the use of Venn diagrams and tables for clarifying the nature of possible events resulting from a trial. There are two important aspects to this. First, when enumerating the various outcomes that comprise an event, the number of (equally. likely) outcomes should correspond, visually, with the area of that part of the diagram represented by the event in question - the greater the probability, the larger the area. Secondly, where events overlap (for example, when rolling a die, consider the two events 'getting an even score' and 'getting a score greater than 2' ), the various regions in the Venn diagram help to clarify the various combinations of events that might occur." (Alan Graham, "Developing Thinking in Statistics", 2006)

"Venn diagrams visually ground symbolic logic and abstract set operations. They do not ground probability. Their common overuse in introducing probability, especially in teaching, can have undesirable consequences." (R W Oldford & W H Cherry, "Picturing Probability: the poverty of Venn diagrams, the richness of Eikosograms", 2006)

"Venn diagramming, it turns out, is a very effective technique for performing syllogistic reasoning. Its chief advantage (over the Euler graph in particular as we noted earlier) is the ability to incrementally add knowledge to the diagram. While an Euler graph has visual power in terms of representing the relations between sets very intuitively, it is impossible to combine more than one piece of information onto a Euler graph. A Venn diagram, on the other hand, easily lends itself to the representation of partial knowledge and can be manipulated to add successively more knowledge to the diagram. This means that when our knowledge of the relations between sets increases, we simply put in more symbols and shadings into the appropriate compartments of the Venn diagram. Thus we are able to accumulate knowledge in a Venn diagram. This capability turns out to be a powerful feature, one that endows Venn diagrams with a more dynamic quality that is sorely lacking in the Euler system." (Robbie T Nakatsu, "Diagrammatic Reasoning in AI", 2010)

On Heuristics I

"The materialistic point of view in psychology can claim, at best, only the value of an heuristic hypothesis." (Wilhelm Wundt, "Principles of Physiological Psychology", 1874)

"Heuristic reasoning is good in itself. What is bad is to mix up heuristic reasoning with rigorous proof. What is worse is to sell heuristic reasoning for rigorous proof." (George Pólya, "How to Solve It", 1945)

"Heuristic, or heuretic, or 'ars inveniendi' was the name of a certain branch of study, not very clearly circumscribed, belonging to logic, or to philosophy, or to psychology, often outlined, seldom presented in detail, and as good as forgotten today. The aim of heuristic is to study the methods and rules of discovery and invention. [...] Heuristic, as an adjective, means 'serving to discover'." (George Pólya, "How to Solve It", 1945)

"Heuristic reasoning is reasoning not regarded as final and strict but as provisional and plausible only, whose purpose is to discover the solution of the present problem. We are often obliged to use heuristic reasoning. We shall attain complete certainty when we shall have obtained the complete solution, but before obtaining certainty we must often be satisfied with a more or less plausible guess. We may need the provisional before we attain the final. We need heuristic reasoning when we construct a strict proof as we need scaffolding when we erect a building." (George Pólya, "How to Solve It", 1945)

"The attempt to characterize exactly models of an empirical theory almost inevitably yields a more precise and clearer understanding of the exact character of a theory. The emptiness and shallowness of many classical theories in the social sciences is well brought out by the attempt to formulate in any exact fashion what constitutes a model of the theory. The kind of theory which mainly consists of insightful remarks and heuristic slogans will not be amenable to this treatment. The effort to make it exact will at the same time reveal the weakness of the theory." (Patrick Suppes," A Comparison of the Meaning and Uses of Models in Mathematics and the Empirical Sciences", Synthese  Vol. 12 (2/3), 1960)

"Factoring big numbers is a strange kind of mathematics that closely resembles the experimental sciences, where nature has the last and definitive word. […] as with the experimental sciences, both rigorous and heuristic analyses can be valuable in understanding the subject and moving it forward. And, as with the experimental sciences, there is sometimes a tension between pure and applied practitioners." (Carl B Pomerance, "A Tale of Two Sieves", The Notices of the American Mathematical Society 43, 1996)

"[…] mathematics does not come to us written indelibly on Nature’s Tablets, but rather is the product of a controlled search governed by metaphorical considerations, the premier instance being the heuristics of the conservation principles." (Philip Mirowski, "More Heat than Light: Economics as Social Physics: Physics as Nature’s Economics", 1989)

"Mathematicians, like the rest of us, cherish clever ideas; in particular they delight in an ingenious picture. But this appreciation does not overwhelm a prevailing skepticism. After all, a diagram is - at best - just a special case and so can't establish a general theorem. Even worse, it can be downright misleading. Though not universal, the prevailing attitude is that pictures are really no more than heuristic devices; they are psychologically suggestive and pedagogically important - but they prove nothing. I want to oppose this view and to make a case for pictures having a legitimate role to play as evidence and justification - a role well beyond the heuristic.  In short, pictures can prove theorems." (James R Brown, "Philosophy of Mathematics: An Introduction to the World of Proofs and Pictures", 1999)

"In the language of mental models, such past experience provided the default assumptions necessary to fill the gaps in the emerging and necessarily incomplete framework of a relativistic theory of gravitation. It was precisely the nature of these default assumptions that allowed them to be discarded again in the light of novel information - provided, for instance, by the further elaboration of the mathematical formalism - without, however, having to abandon the underlying mental models which could thus continue to function as heuristic orientations." (Jürgen Renn, "Before the Riemann Tensor: The Emergence of Einstein’s Double Strategy", [in "The Universe of General Relativity"] 2000)

"You can often hear from non-mathematicians, especially from philosophers, that mathematics consists exclusively in drawing conclusions from clearly stated premises; and that in this process, it makes no difference what these premises signify, whether they are true or fa1se, provided only that they do not contradict one another. But a per. son who has done productive mathematical work will talk quite differently. In fact these people [the non-mathematicians] are thinking only of the crystallized form into which finished mathematica1 theories are finally cast. However, the investigator himself, in mathematics as in every other science, does not work in this rigorous deductive fashion. On the contrary, he makes essential use of his imagination and proceeds inductively aided by heuristic expedients. One can give numerous examples of mathematicians who have discovered theorems of the greatest importance which they were unable to prove. Should one then refuse to recognize this as a great accomplishment and in deference to the above definition insist that this is not mathematics? After all it is an arbitrary thing how the word is to be used, but no judgment of value can deny that the inductive work of the person who first announces the theorem is at least as valuable as the deductive work. of the one who proves it. For both are equally necessary and the discovery is the presupposition of the later conclusion." (Felix Klein)

On Facts (1890-1899)

"The study of theory must go hand in hand with that of facts: and for dealing with most modern problems it is modern facts that are of the greatest use." (Alfred Marshall, "Principles of Economics", 1890)

"The graphical method has considerable superiority for the exposition of statistical facts over the tabular. A heavy bank of figures is grievously wearisome to the eye, and the popular mind is as incapable of drawing any useful lessons from it as of extracting sunbeams from cucumbers." (Arthur B Farquhar & Henry Farquhar, "Economic and Industrial Delusions", 1891)

"All great scientists have, in a certain sense, been great artists; the man with no imagination may collect facts, but he cannot make great discoveries." (Karl Pearson, "The Grammar of Science", 1892)

"It is a capital mistake to theorize before one has data. Insensibly one begins to twist facts to suit theories, instead of theories to suit facts." (Sir Arthur C Doyle, "The Adventures of Sherlock Holmes", 1892)

"The classification of facts, the recognition of their sequence and relative significance is the function of science, and the habit of forming a judgment upon these facts unbiased by personal feeling is characteristic of what may be termed the scientific frame of mind." (Karl Pearson, "The Grammar of Science", 1892)

"The true aim of the teacher must be to impart an appreciation of method and not a knowledge of facts." (Karl Pearson, "The Grammar of Science", 1892)

"Facts are not much use, considered as facts. They bewilder by their number and their apparent incoherency. Let them be digested into theory, however, and brought into mutual harmony, and it is another matter. Theory is of the essence of facts. Without theory scientific knowledge would be only worthy of the mad house." (Oliver Heaviside, "Electromagnetic Theory", 1893)

"Scientific facts accumulate rapidly, and give rise to theories with almost equal rapidity. These theories are often wonderfully enticing, and one is apt to pass from one to another, from theory to theory, without taking care to establish each before passing on to the next, without assuring oneself that the foundation on which one is building is secure. Then comes the crash; the last theory breaks down utterly, and on attempting to retrace our steps to firm ground and start anew, we may find too late that one of the cards, possibly at the very foundation of the pagoda, is either faultily placed or in itself defective, and that this blemish easily remedied if detected in time has, neglected, caused the collapse of the whole structure on whose erection so much skill and perseverance have been spent." (Arthur M Marshall, 1894)

"Without a theory all our knowledge of nature would be reduced to a mere inventory of the results of observation. Every scientific theory must be regarded as an effort of the human mind to grasp the truth, and as long as it is consistent with the facts, it forms a chain by which they are linked together and woven into harmony." (Thomas Preston, "The Theory of Heat", 1894)

"The first step, whenever a practical problem is set before a mathematician, is to form the mathematical hypothesis. It is neither needful nor practical that we should take account of the details of the structure as it will exist. We have to reason about a skeleton diagram in which bearings are reduced to points, pieces to lines, etc. and [in] which it is supposed that certain relations between motions are absolutely constrained, irrespective of forces. Some writers call such a hypothesis a fiction, and say that the mathematician does not solve the real problem, but only a fictitious one. That is one way of looking at the matter, to which I have no objection to make: only, I notice, that in precisely the same sense in which the mathematical hypothesis is 'false', so also is this statement 'false', that it is false. Namely, both representations are false in the sense that they omit subsidiary elements of the fact, provided that element of the case can be said to be subsidiary which those writers overlook, namely, that the skeleton diagram is true in the only sense in which from the nature of things any mental representation, or understanding, of the brute existent can be true. For every possible conception, by the very nature of thought, involves generalization; now generalization omits, means to omit, and professes to omit, the differences between the facts generalized." (Charles S Peirce, "Report on Live Loads", cca. 1895)

"The world is chiefly a mental fact. From mind it receives the forms of time and space, the principle of causality, color, warmth, and beauty. Were there no mind, there would be no world." (John L Spalding, "Means and Ends of Education", 1895)

"In scientific investigations, it is permitted to invent any hypothesis and, if it explains various large and independent classes of facts, it rises to the ranks of a well-grounded theory." (Charles Darwin, "The Variations of Animals and Plants Under Domestication" Vol. 1, 1896)

"Mathematics is the most abstract of all the sciences. For it makes no external observations, nor asserts anything as a real fact. When the mathematician deals with facts, they become for him mere ‘hypotheses’; for with their truth he refuses to concern himself. The whole science of mathematics is a science of hypotheses; so that nothing could be more completely abstracted from concrete reality." (Charles S Peirce, "The Regenerated Logic", The Monist Vol. 7 (1), 1896)

"Round about the accredited and orderly facts of every science there ever fl oats a sort of dust-cloud of exceptional observations, of occurrences minute and irregular and seldom met with, which it always proves more easy to ignore than to attend to […]" (William James, "The Will to Believe", 1896)

"Science like life feeds on its own decay. New facts burst old rules; then newly developed concepts bind old and new together into a reconciling law." (William James, "The Will to Believe and Other Essays in Popular Philosophy", 1896)

"The scientific value of truth is not, however, ultimate or absolute. It rests partly on practical, partly on aesthetic interests. As our ideas are gradually brought into conformity with the facts by the painful process of selection, - for intuition runs equally into truth and into error, and can settle nothing if not controlled by experience, - we gain vastly in our command over our environment. This is the fundamental value of natural science" (George Santayana, "The Sense of Beauty: Being the Outlines of Aesthetic Theory", 1896)

"To use an old analogy - and here we can hardly go except upon analogy - while the building of Nature is growing spontaneously from within, the model of it, which we seek to construct in our descriptive science, can only be constructed by means of scaffolding from without, a scaffolding of hypotheses. While in the real building all is continuous, in our model there are detached parts which must be connected with the rest by temporary ladders and passages, or which must be supported till we can see how to fill in the understructure. To give the hypotheses equal validity with facts is to confuse the temporary scaffolding with the building itself." (John H Poynting, 1899)

On Imagination (1900-1924)

"This is the greatest degree of impoverishment; the [mental] image, deprived little by little of its own characteristics, is nothing more than a shadow. […] Being dependent on the state of the brain, the image undergoes change like all living substance, - it is subject to gains and losses, especially losses. But each of the foregoing three classes has its use for the inventor. They serve as material for different kinds of imagination - in their concrete form, for the mechanic and the artist; in their schematic form, for the scientist and for others." (Théodule-Armand Ribot, "Essay on the Creative Imagination", 1900)

"This means that it is not a dead thing; it is not at all like a photographic plate with which one may reproduce copies indefinitely. Being dependent on the state of the brain, the image undergoes change like all living substance, - it is subject to gains and losses, especially losses. But each of the foregoing three classes has its use for the inventor. They serve as material for different kinds of imagination - in their concrete form, for the mechanic and the artist; in their schematic form, for the scientist and for others." (Théodule-Armand Ribot, "Essay on the Creative Imagination" , 1900)

"We form in the imagination some sort of diagrammatic, that is, iconic, representation of the facts, as skeletonized as possible. The impression of the present writer is that with ordinary persons this is always a visual image, or mixed visual and muscular; but this is an opinion not founded on any systematic examination." (Charles S Peirce, "Notes on Ampliative Reasoning", 1901)

"Imagination is as vital to any advance in science as learning and precision are essential for starting points." (Percival Lowell, "The Solar System", 1903)

"Nature talks in symbols; he who lacks imagination cannot understand her." (Abraham Miller, "Unmoral Maxims", 1906)

"Mathematics makes constant demands upon the imagination, calls for picturing in space (of one, two, three dimensions), and no considerable success can be attained without a growing ability to imagine all the various possibilities of a given case, and to make them defile before the mind's eye." (Jacob W A Young, "The Teaching of Mathematics", 1907)

"The motive for the study of mathematics is insight into the nature of the universe. Stars and strata, heat and electricity, the laws and processes of becoming and being, incorporate mathematical truths. If language imitates the voice of the Creator, revealing His heart, mathematics discloses His intellect, repeating the story of how things came into being. And the value of mathematics, appealing as it does to our energy and to our honor, to our desire to know the truth and thereby to live as of right in the household of God, is that it establishes us in larger and larger certainties. As literature develops emotion, understanding, and sympathy, so mathematics develops observation, imagination, and reason." (William E Chancellor, "A Theory of Motives, Ideals and Values in Education" 1907)

"The beautiful has its place in mathematics as elsewhere. The prose of ordinary intercourse and of business correspondence might be held to be the most practical use to which language is put, but we should be poor indeed without the literature of imagination. Mathematics too has its triumphs of the Creative imagination, its beautiful theorems, its proofs and processes whose perfection of form has made them classic. He must be a 'practical' man who can see no poetry in mathematics." (Wiliam F White, "A Scrap-book of Elementary Mathematics: Notes, Recreations, Essays", 1908)

"No system would have ever been framed if people had been simply interested in knowing what is true, whatever it may be. What produces systems is the interest in maintaining against all comers that some favourite or inherited idea of ours is sufficient and right. A system may contain an account of many things which, in detail, are true enough; but as a system, covering infinite possibilities that neither our experience nor our logic can prejudge, it must be a work of imagination and a piece of human soliloquy: It may be expressive of human experience, it may be poetical; but how should anyone who really coveted truth suppose that it was true?" (George Santayana, "The Genteel Tradition in American Philosophy", 1911)

"Only in men’s imagination does every truth find an effective and undeniable existence." (Joseph Conrad, "Some Reminiscences", 1912)

"What is the imagination? Only an arm or weapon of the interior energy; only the precursor of the reason." (Ralph W Emerson, "Miscellanies, Natural history of intellect", 1912)

"The concept of an independent system is a pure creation of the imagination. For no material system is or can ever be perfectly isolated from the rest of the world. Nevertheless it completes the mathematician’s ‘blank form of a universe’ without which his investigations are impossible. It enables him to introduce into his geometrical space, not only masses and configurations, but also physical structure and chemical composition." (Lawrence J Henderson, "The Order of Nature: An Essay", 1917)

"[…] because mathematics contains truth, it extends its validity to the whole domain of art and the creatures of the constructive imagination." (James B Shaw, "Lectures on the Philosophy of Mathematics", 1918)

"Nature uses human imagination to lift her work of creation to even higher levels." (Luigi Pirandello, "Six Characters in Search of an Author", 1921)

"The story of scientific discovery has its own epic unity - a unity of purpose and endeavour - the single torch passing from hand to hand through the centuries; and the great moments of science when, after long labour, the pioneers saw their accumulated facts falling into a significant order - sometimes in the form of a law that revolutionised the whole world of thought - have an intense human interest, and belong essentially to the creative imagination of poetry." (Alfred Noyes, "Watchers of the Sky", 1922)

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 (1890-1899)

"Judged by the only standards which are admissible in a pure doctrine of numbers i is imaginary in the same sense as the negative, the fraction, and the irrational, but in no other sense; all are alike mere symbols devised for the sake of representing the results of operations even when these results are not numbers (positive integers)." (Henry B Fine, "The Number-System of Algebra", 1890)

"The mathematician, carried along on his flood of symbols, dealing apparently with purely formal truths, may still reach results of endless importance for our description of the physical universe." (Karl Pearson, “The Grammar of Science”, 1892)

"The mechanism of thought consists in combinations, separations, and recombinations of representative images or symbols […] the object of thought is adaptation to environment." (Paul Carus, “Le probeme de la conscience du moi", 1893)

"At the basis of our Symbolic Logic, however represented, whether by words by letters or by diagrams, we shall always find the same state of things. What we ultimately have to do is to break up the entire field before us into a definite number of classes or compartments which are mutually exclusive and collectively exhaustive." (John Venn, "Symbolic Logic" 2nd Ed., 1894)

"The best way of introducing this question will be to enquire a little more strictly whether it is really classes that we thus represent, or merely compartments into which classes may be put? […] The most accurate answer is that our diagrammatic subdivisions, or for that matter our symbols generally, stand for compartments and not for classes. We may doubtless regard them as representing the latter, but if we do so we should never fail to keep in mind the proviso, 'if there be such things in existence'. And when this condition is insisted upon, it seems as if we expressed our meaning best by saying that what our symbols stand for are compartments which may or may not happen to be occupied." (John Venn, "Symbolic Logic" 2nd Ed., 1894)

“We form ourselves images or symbols of external objects; and the form which we give them is such that the necessary consequents of the images in thought are always the images of the necessary consequents in nature of the things pictured." (Heinrich Hertz, 1894)

"Art is a human activity consisting in this, that one consciously, by means of certain external symbols, conveys to others the feelings one has experienced, whereby people so infected by these feelings, also experience them." (Leo Tolstoy, "What is Art?", 1897)

"The elements of plane geometry should precede algebra for every reason known to sound educational theory. It is more fundamental, it is more concrete, and it deals with things and their relations rather than with symbols." (Nicholas M Butler, "The Meaning of Education, and Other Essays and Addresses", 1898)

On Symbols (1910-1919)

"Pure mathematics is a collection of hypothetical, deductive theories, each consisting of a definite system of primitive, undefined, concepts or symbols and primitive, unproved, but self-consistent assumptions (commonly called axioms) together with their logically deducible consequences following by rigidly deductive processes without appeal to intuition." (Graham Fitch, "The Fourth Dimension simply Explained", 1910)

"Things and events explain themselves, and the business of thought is to brush aside the verbal and conceptual impediments which prevent them from doing so. Start with the notion that it is you who explain the Object, and not the Object that explains itself, and you are bound to end in explaining it away. It ceases to exist, its place being taken by a parcel of concepts, a string of symbols, a form of words, and you find yourself contemplating, not the thing, but your theory of the thing." (Lawrence P Jacks, "The Usurpation Of Language", 1910)

"A symbol which has not been properly defined is not a symbol at all. It is merely a blot of ink on paper which has an easily recognized shape. Nothing can be proved by a succession of blot, except the existence of a bad pen or a careless writer." (Alfred N Whitehead, "An Introduction to Mathematics", 1911)

"The symbols organized by knowledge, or concepts, themselves belong to social nature as its ideological elements. Therefore, by operating upon them, knowledge is able to expand its organizing function much more broadly than labour in its technological operation of real things; and as we have already seen that many things, which are not organized in practice, can be organized by knowledge, i.e. in symbols: where the ingression of things is absent, the ingression of their concepts is still possible." (Alexander A Bogdanov, "Tektology: The Universal Organizational Science" Vol. I, 1913)

"This diagrammatic method has, however, serious inconveniences as a method for solving logical problems. It does not show how the data are exhibited by cancelling certain constituents, nor does it show how to combine the remaining constituents so as to obtain the consequences sought. In short, it serves only to exhibit one single step in the argument, namely the equation of the problem; it dispenses neither with the previous steps, i.e., 'throwing of the problem into an equation' and the transformation of the premises, nor with the subsequent steps, i.e., the combinations that lead to the various consequences. Hence it is of very little use, inasmuch as the constituents can be represented by algebraic symbols quite as well as by plane regions, and are much easier to deal with in this form." (Louis Couturat, "The Algebra of Logic", 1914)

"The rigor of mathematics is not absolute - absolute rigor is an ideal, to be, like other ideals, aspired unto, forever approached, but never quite attained, for such attainment would mean that every possibility of error or indetermination, however slight, had been eliminated from idea, from symbol, and from argumentation." (Cassius J Keyser, "The Human Worth of Rigorous Thinking: Essays and Addresses", 1916)

"In obedience to the feeling of reality, we shall insist that, in the analysis of propositions, nothing 'unreal' is to be admitted. But, after all, if there is nothing unreal, how, it may be asked, could we admit anything unreal? The reply is that, in dealing with propositions, we are dealing in the first instance with symbols, and if we attribute significance to groups of symbols which have no significance, we shall fall into the error of admitting unrealities, in the only sense in which this is possible, namely, as objects described." (Bertrand Russell, "Introduction to Mathematical Philosophy" , 1919)

On Symbols (1980-1989)

"[…] mathematics is not just a symbolism, a set of conventions for the use of special, formal vocabularies, but is intimately connected with the structure of rational thought, with reasoning practices. [...] mathematics is not just a language, and of refusing the foundationalist move of trying to reduce mathematics to logic, instead seeing mathematics as providing rational frameworks for science, is to set science against a background of rational structures and rational methods which itself has a built-in dynamics. The rational framework of science is itself historically conditioned, for it changes with developments in mathematics." (Mary Tiles, "Bachelard: Science and Objectivity", 1984)

"Scientific laws give algorithms, or procedures, for determining how systems behave. The computer program is a medium in which the algorithms can be expressed and applied. Physical objects and mathematical structures can be represented as numbers and symbols in a computer, and a program can be written to manipulate them according to the algorithms. When the computer program is executed, it causes the numbers and symbols to be modified in the way specified by the scientific laws. It thereby allows the consequences of the laws to be deduced." (Stephen Wolfram, "Computer Software in Science and Mathematics", 1984)

"A computer is an interpreted automatic formal system - that is to say, a symbol-manipulating machine." (John Haugeland, "Artificial intelligence: The very idea", 1985)

"We who are heirs to three recent centuries of scientific development can hardly imagine a state of mind in which many mathematical objects were regarded as symbols of spiritual truths or episodes in sacred history. Yet, unless we make this effort of imagination, a fraction of the history of mathematics is incomprehensible.” (Philip J Davis & Rueben Hersh, “The Mathematical Experience”, 1985)

"When a graph is constructed, quantitative and categorical information is encoded, chiefly through position, size, symbols, and color. When a person looks at a graph, the information is visually decoded by the person's visual system. A graphical method is successful only if the decoding process is effective. No matter how clever and how technologically impressive the encoding, it is a failure if the decoding process is a failure. Informed decisions about how to encode data can be achieved only through an understanding of the visual decoding process, which is called graphical perception." (William S Cleveland, "The Elements of Graphing Data", 1985)

"Artificial intelligence is based on the assumption that the mind can be described as some kind of formal system manipulating symbols that stand for things in the world. Thus it doesn't matter what the brain is made of, or what it uses for tokens in the great game of thinking. Using an equivalent set of tokens and rules, we can do thinking with a digital computer, just as we can play chess using cups, salt and pepper shakers, knives, forks, and spoons. Using the right software, one system (the mind) can be mapped onto the other (the computer)." (George Johnson, "Machinery of the Mind: Inside the New Science of Artificial Intelligence", 1986)

"Meaning does not reside in the mathematical symbols. It resides in the cloud of thought enveloping these symbols. It is conveyed in words; these assign meaning to the symbols." (Marvin Chester, "Primer of Quantum Mechanics", 1987)

"[…] the chain of possible combinations of the encounter can be studied as such, as an order which subsists in its rigor, independently of all subjectivity. Through cybernetics, the symbol is embodied in the apparatus - with which it is not to be confused, the apparatus being just its support. And it is embodied in it in a literally trans-subjective way." (Jacques Lacan, 1988)

"Western culture’s world-view appears to be dominated by material objects. […] One of the ways mathematics has gained its power is through the activity of objectivising the abstractions from reality. Through its symbols (letters, numerals, figures) mathematics has taught people how to deal with abstract entities, as if they were objects." (Alan J Bishop, "Mathematics education in its cultural context", Educational Studies in Mathematics 19, 1988)

"People who have a casual interest in mathematics may get the idea that a topologist is a mathematical playboy who spends his time making Möbius bands and other diverting topological models. If they were to open any recent textbook in topology, they would be surprised. They would find page after page of symbols, seldom relieved by a picture or diagram." (Martin Gardner, "Hexaflexagons and Other Mathematical Diversions", 1988)

On Symbols (1990-1999)

"When a person has learned a symbolic system well enough to use it, she has established a portable self-contained world within the mind." (Mihaly Csikszentmihalyi, "Flow", 1990)

"Mathematics […] is mired in a language of symbols foreign to most of us, [it] explores regions of the infinitesimally small and the infinitely large that elude words, much less understanding." (Robert Kanigel, "The Man Who Knew Infinity", 1991)

"Great mathematics seldom comes from idle speculation about abstract spaces and symbols. More often than not it is motivated by definite questions arising in the worlds of nature and humans." (John L Casti, "Reality Rules: Picturing the world in mathematics", 1992)

"Mathematical modeling is about rules - the rules of reality. What distinguishes a mathematical model from, say, a poem, a song, a portrait or any other kind of ‘model’, is that the mathematical model is an image or picture of reality painted with logical symbols instead of with words, sounds or watercolors." (John Casti, "Reality Rules", 1992)

"Scientific claims or statements are inexact and provisional. They depend on dozens of simplifying assumptions and on a particular choice of words and symbols and on 'all other things being equal'." (Bart Kosko, "Fuzzy Thinking: The new science of fuzzy logic", 1993)

"The insight at the root of artificial intelligence was that these 'bits' (manipulated by computers) could just as well stand as symbols for concepts that the machine would combine by the strict rules of logic or the looser associations of psychology." (Daniel Crevier, "AI: The tumultuous history of the search for artificial intelligence", 1993)

"Above all, words must be recognized as symbolic pointers to truth, not objective containers of truth." (John S Spong, "Resurrection: Myth or Reality?", 1994) 

"[...] images are probably the main content of our thoughts, regardless of the sensory modality in which they are generated and regardless of whether they are about a thing or a process involving things; or about words or other symbols, in a given language, which correspond to a thing or process. Hidden behind those images, never or rarely knowable by us, there are indeed numerous processes that guide the generation and deployment of those images in space and time. Those processes utilize rules and strategies embodied in dispositional representations. They are essential for our thinking but are not a content of our thoughts.” (Antonio R Damasio, “Descartes' Error. Emotion, Reason, and the Human Brain”, 1994)

"Just as music comes alive in the performance of it, the same is true of mathematics. The symbols on the page have no more to do with mathematics than the notes on a page of music. They simply represent the experience." (Keith Devlin, "Mathematics: The Science of Patterns", 1994)

"Every phenomenon on earth is symbolic, and each symbol is an open gate through which the soul, if it is ready, can enter into the inner part of the world, where you and I and day and night are all one." (Hermann Hesse, "The Fairy Tales of Hermann Hesse", 1995)

“Mathematics is not the study of an ideal, preexisting nontemporal reality. Neither is it a chess-like game with made-up symbols and formulas. Rather, it is the part of human studies which is capable of achieving a science-like consensus, capable of establishing reproducible results. The existence of the subject called mathematics is a fact, not a question. This fact means no more and no less than the existence of modes of reasoning and argument about ideas which are compelling an conclusive, ‘noncontroversial when once understood’." (Philip J Davis & Rueben Hersh, “The Mathematical Experience”, 1995)

"Schematic diagrams are more abstract than pictorial drawings, showing symbolic elements and their interconnection to make clear the configuration and/or operation of a system." (Ernest O Doebelin, "Engineering experimentation: planning, execution, reporting", 1995)

"The logic of the emotional mind is associative; it takes elements that symbolize a reality, or trigger a memory of it, to be the same as that reality. That is why similes, metaphors and images speak directly to the emotional mind." (Daniel Goleman, "Emotional Intelligence", 1996)

“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.” (Clifford A Pickover, "The Loom of God: Mathematical Tapestries at the Edge of Time", 1997)

"Reality contains not only evidence, but also the means (such as our minds, and our artefacts) of understanding it. There are mathematical symbols in physical reality. The fact that it is we who put them there does not make them any less physical." (David Deutsch, "The Fabric of Reality", 1997)

"Meaning is conferred not by a one-to-one correspondence of a symbol with some external concept or object, but by the relationships between the structural components of the system itself." (Paul Cilliers, "Complexity and Postmodernism: Understanding Complex Systems", 1998)

"A formal system consists of a number of tokens or symbols, like pieces in a game. These symbols can be combined into patterns by means of a set of rules which defines what is or is not permissible (e.g. the rules of chess). These rules are strictly formal, i.e. they conform to a precise logic. The configuration of the symbols at any specific moment constitutes a ‘state’ of the system. A specific state will activate the applicable rules which then transform the system from one state to another. If the set of rules governing the behaviour of the system are exact and complete, one could test whether various possible states of the system are or are not permissible." (Paul Cilliers, "Complexity and Postmodernism: Understanding Complex Systems", 1998)

“Cultural archetypes are the unconscious models that help us make sense of the world: they are the myths, narratives, images, symbols, and files into which we organize the data of our life experience” (Clotaire Rapaille, “Cultural Imprints”, Executive Excellence Vol. 16 (10), 1999)

"In broad terms, a mental model is to be understood as a dynamic symbolic representation of external objects or events on the part of some natural or artificial cognitive system. Mental models are thought to have certain properties which make them stand out against other forms of symbolic representations." (Gert Rickheit & Lorenz Sichelschmidt, "Mental Models: Some Answers, Some Questions, Some Suggestions", 1999)

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 Diagrams (-1899)

"It hath been an old remark, that Geometry is an excellent Logic. And it must be owned that when the definitions are clear; when the postulata cannot be refused, nor the axioms denied; when from the distinct contemplation and comparison of figures, their properties are derived, by a perpetual well-connected chain of consequences, the objects being still kept in view, and the attention ever fixed upon them; there is acquired a habit of reasoning, close and exact and methodical; which habit strengthens and sharpens the mind, and being transferred to other subjects is of general use in the inquiry after truth." (George Berkeley, "The Analyst; Or, A Discourse Addressed to an Infidel Mathematician", 1734)

"In mathematics, such processes [of reasoning] are much longer than in any other science; and hence the study of it is peculiarly calculated to strengthen the power of steady and concatenated thinking, - a power which, in all the pursuits of life, whether speculative or active, is one of the most valuable endowments we can possess. This command of attention, however, it may be proper to add, is to be acquired, not by the practice of modern methods, but by the study of Greek geometry, more particularly, by accustoming ourselves to pursue long trains of demonstration, without availing ourselves of the aid of any sensible diagrams; the thoughts being directed solely by those ideal delineations which the powers of conception and of memory enable us to form." (Dugald Stewart, "Elements of the Philosophy of the Human Mind", 1792)

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

"There are, undoubtedly, the most ample reasons for stating both the principles and theorems [of geometry] in their general form […] But, that an unpractised learner, even in making use of one theorem to demonstrate another, reasons rather from particular to particular than from the general proposition, is manifest from the difficulty he finds in applying a theorem to a case in which the configuration of the diagram is extremely unlike that of the diagram by which the original theorem was demonstrated. A difficulty which, except in cases of unusual mental powers, long practice can alone remove, and removes chiefly by rendering us familiar with all the configurations consistent with the general conditions of the theorem." (John S Mill, "A System of Logic", 1843)

"Diagrams are of great utility for illustrating certain questions of vital statistics by conveying ideas on the subject through the eye, which cannot be so readily grasped when contained in figures." (Florence Nightingale, "Mortality of the British Army", 1857)

"Whenever I am infuriated, I revenge myself with a new Diagram." (Florence Nightingale, [letter to Sidney Herbert] 1857)

"They [diagrams] are designed not so much to allow of reference to particular numbers, which can be better had from printed tables of figures, as to exhibit to the eye the general results of large masses of figures which it is hopeless to attack in any other way than by graphical representation." (William S Jevons, [letter to Richard Hutton] 1863)

"Every process of what has been called Universal Geometry - the great creation of Descartes and his successors, in which a single train of reasoning solves whole classes of problems at once, and others common to large groups of them - is a practical lesson in the management of wide generalizations, and abstraction of the points of agreement from those of difference among objects of great and confusing diversity, to which the purely inductive sciences cannot furnish many superior. Even so elementary an operation as that of abstracting from the particular configuration of the triangles or other figures, and the relative situation of the particular lines or points, in the diagram which aids the apprehension of a common geometrical demonstration, is a very useful, and far from being always an easy, exercise of the faculty of generalization so strangely imagined to have no place or part in the processes of mathematics." (John S Mill, "An Examination of Sir William Hamilton’s Philosophy", 1865)

"[…] it must be noticed that these diagrams do not naturally harmonize with the propositions of ordinary life or ordinary logic. […] The great bulk of the propositions which we commonly meet with are founded, and rightly founded, on an imperfect knowledge of the actual mutual relations of the implied classes to one another. […] one very marked characteristic about these circular diagrams is that they forbid the natural expression of such uncertainty, and are therefore only directly applicable to a very small number of such propositions as we commonly meet with." (John Venn, "On the Diagrammatic and Mechanical Representation of Propositions and Reasonings", 1880)

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

"At the basis of our Symbolic Logic, however represented, whether by words by letters or by diagrams, we shall always find the same state of things. What we ultimately have to do is to break up the entire field before us into a definite number of classes or compartments which are mutually exclusive and collectively exhaustive." (John Venn, "Symbolic Logic" 2nd Ed., 1894)

"The best way of introducing this question will be to enquire a little more strictly whether it is really classes that we thus represent, or merely compartments into which classes may be put? […] The most accurate answer is that our diagrammatic subdivisions, or for that matter our symbols generally, stand for compartments and not for classes. We may doubtless regard them as representing the latter, but if we do so we should never fail to keep in mind the proviso, 'if there be such things in existence'. And when this condition is insisted upon, it seems as if we expressed our meaning best by saying that what our symbols stand for are compartments which may or may not happen to be occupied." (John Venn, "Symbolic Logic" 2nd Ed., 1894)

"Deduction is that mode of reasoning which examines the state of things asserted in the premises, forms a diagram of that state of things, perceives in the parts of the diagram relations not explicitly mentioned in the premises, satisfies itself by mental experiments upon the diagram that these relations would always subsist, or at least would do so in a certain proportion of cases, and concludes their necessary, or probable, truth." (Charles S Peirce, "Kinds of Reasoning", cca. 1896)

On Diagrams (1925-1949)

"The preliminary examination of most data is facilitated by the use of diagrams. Diagrams prove nothing, but bring outstanding features readily to the eye; they are therefore no substitutes for such critical tests as may be applied to the data, but are valuable in suggesting such tests, and in explaining the conclusions founded upon them." (Sir Ronald A Fisher, "Statistical Methods for Research Workers", 1925)

"What the diagram has in common with the symbolic schema is the fact that the diagram spatially represents an abstract and unextended object. But there is here nothing other than a determinate location in space. This location serves as a mooring, an attachment, an orientation for our memory, but does not play any role in our thought." (Jean-Paul Sartre, "The Imaginary: A phenomenological psychology of the imagination", 1940)

"[…] statistical literacy. That is, the ability to read diagrams and maps; a 'consumer' understanding of common statistical terms, as average, per cent, dispersion, correlation, and index number." (Douglas Scates, “Statistics: The Mathematics for Social Problems”, 1943)

"When I undertake some geometrical research, I have generally a mental view of the diagram itself, though generally an inadequate or incomplete one, in spite of which it affords the necessary synthesis - a tendency which, it would appear, results from a training which goes back to my very earliest childhood." (Jacques Hadamard, "The Psychology of Invention in the Mathematical Field”, 1949)

"I believe, that the decisive idea which brings the solution of a problem is rather often connected with a well-turned word or sentence. The word or the sentence enlightens the situation, gives things, as you say, a physiognomy. It can precede by little the decisive idea or follow on it immediately; perhaps, it arises at the same time as the decisive idea. […] The right word, the subtly appropriate word, helps us to recall the mathematical idea, perhaps less completely and less objectively than a diagram or a mathematical notation, but in an analogous way. […] It may contribute to fix it in the mind." (George Polya [in a letter to Jaque Hadamard, "The Psychology of Invention in the Mathematical Field", 1949])

On Networks (1960-1969)

"Any pattern of activity in a network, regarded as consistent by some observer, is a system, Certain groups of observers, who share a common body of knowledge, and subscribe to a particular discipline, like 'physics' or 'biology' (in terms of which they pose hypotheses about the network), will pick out substantially the same systems. On the other hand, observers belonging to different groups will not agree about the activity which is a system." (Gordon Pask, The Natural History of Networks, 1960)

"Clearly, if the state of the system is coupled to parameters of an environment and the state of the environment is made to modify parameters of the system, a learning process will occur. Such an arrangement will be called a Finite Learning Machine, since it has a definite capacity. It is, of course, an active learning mechanism which trades with its surroundings. Indeed it is the limit case of a self-organizing system which will appear in the network if the currency supply is generalized." (Gordon Pask, "The Natural History of Networks", 1960)

"I am using the term 'network' in a general sense, to imply any set of interconnected and measurably active physical entities. Naturally occurring networks, of interest because they have a, self-organizing character, are, for example, a marsh, a colony of microorganisms, a research team, and a man." (Gordon Pask, "The Natural History of Networks", 1960)

"A NETWORK is a collection of connected lines, each of which indicates the movement of some quantity between two locations. Generally, entrance to a network is via a source (the starting point) and exit from a network is via a sink (the finishing point); the lines which form the network are called links (or arcs), and the points at which two or more links meet are called nodes." (Cecil W Lowe, "Critical Path Analysis by Bar Chart", 1966)

"In a network, one can plot the figures on a plane which has no meaning, and then look for the arrangement which produces the minimum number of intersections, or the simplest figure. After this transformation, the graphic will yield maximum efficiency, based on the discovery of a meaningful order expressed by the plane." (Jacques Bertin, "Semiology of graphics", 1967)

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

Mental Models LVIII

"We form in the imagination some sort of diagrammatic, that is, iconic, representation of the facts, as skeletonized as possible. The impression of the present writer is that with ordinary persons this is always a visual image, or mixed visual and muscular; but this is an opinion not founded on any systematic examination." (Charles S Peirce, "Notes on Ampliative Reasoning", 1901)

"A mental model is conceived here as a knowledge structure possessing slots that can be filled not only with empirically gained information but also with 'default assumptions' resulting from prior experience. These default assumptions can be substituted by updated information so that inferences based on the model can be corrected without abandoning the model as a whole. Information is assimilated to the slots of a mental model in the form of 'frames' which are understood here as 'chunks' of knowledge with a well-defined meaning anchored in a given body of shared knowledge." (Jürgen Renn, "Before the Riemann Tensor: The Emergence of Einstein’s Double Strategy", [in "The Universe of General Relativity"] 2000)

"In the language of mental models, such past experience provided the default assumptions necessary to fill the gaps in the emerging and necessarily incomplete framework of a relativistic theory of gravitation. It was precisely the nature of these default assumptions that allowed them to be discarded again in the light of novel information - provided, for instance, by the further elaboration of the mathematical formalism - without, however, having to abandon the underlying mental models which could thus continue to function as heuristic orientations." (Jürgen Renn, "Before the Riemann Tensor: The Emergence of Einstein’s Double Strategy", [in "The Universe of General Relativity"] 2000)

"A mental model represents a possibility, or, to be precise, the structure and content of the model capture what is common to the different ways in which the possibility could occur [...]" (Philip N Johnson-Laird, Mental Models, Sentential Reasoning, and Illusory Inferences, [in "Mental Models and the Mind"], 2006)

"According to mental model theory, human reasoning relies on the construction of integrated mental representations of the information that is given in the reasoning problem's premises. These integrated representations are the mental models. A mental model is a mental representation that captures what is common to all the different ways in which the premises can be interpreted. It represents in "small scale" how "reality" could be— according to what is stated in the premises of a reasoning problem. Mental models, though, must not be confused with images. A mental model often forms the basis of one or more visual images, but some of them represent situations that cannot be visualized. Instead, mental models are often likened to diagrams since, as with diagrams, their structure is analogous to the structure of the states of affairs they represent." (Carsten Held et al, "Mental Models and the Mind", 2006)

"Mental models are mental representations of a certain type. The main problem in the philosophy of mental representation is to characterize the relation between a mental representation and the represented object. Naively speaking, a mental representation is an entity that 'stands for' another—the represented object - , but here 'stands for' is just a metaphoric place-holder for 'represents', thus requires further explanation." (Carsten Held et al, "Mental Models and the Mind", 2006)

"Prom the processing view, the model theory distinguishes between three different operations. In the construction phase, reasoners construct the mental model that reflects the information from the premises. In the inspection phase, this model is inspected to find new information that is not explicitly given in the premises. In most variants of the model theory, the inspection process is conceptualized as a spatial focus that scans the model to find new information not given in the premises.. In the variation phase, reasoners try to construct alternative models from the premises that refute the putative conclusion. If no such model is found, the putative conclusion is considered true." (Carsten Held et al, "Mental Models and the Mind", 2006)

"The model theory postulates that mental models are parsimonious. They represent what is possible, but not what is impossible, according to assertions. This principle of parsimony minimizes the load on working memory, and so it applies unless something exceptional occurs to overrule it." (Philip N Johnson-Laird, Mental Models, Sentential Reasoning, and Illusory Inferences, [in "Mental Models and the Mind"], 2006)

"Just as physicists have created models of the atom based on observed data and intuitive synthesis of the patterns in their data, so must designers create models of users based on observed behaviors and intuitive synthesis of the patterns in the data. Only after we formalize such patterns can we hope to systematically construct patterns of interaction that smoothly match the behavior patterns, mental models, and goals of users. Personas provide this formalization." (Alan Cooper et al, "About Face 3: The Essentials of Interaction Design", 2007)

"We tend to form mental models that are simpler than reality; so if we create represented models that are simpler than the actual implementation model, we help the user achieve a better understanding. […] Understanding how software actually works always helps someone to use it, but this understanding usually comes at a significant cost. One of the most significant ways in which computers can assist human beings is by putting a simple face on complex processes and situations. As a result, user interfaces that are consistent with users’ mental models are vastly superior to those that are merely reflections of the implementation model." (Alan Cooper et al,  "About Face 3: The Essentials of Interaction Design", 2007)