On Diagrams (1975-1999)

"Pencil and paper for construction of distributions, scatter diagrams, and run-charts to compare small groups and to detect trends are more efficient methods of estimation than statistical inference that depends on variances and standard errors, as the simple techniques preserve the information in the original data." (W Edwards Deming, "On Probability as Basis for Action", American Statistician, Volume 29, Number 4, November 1975)

"The formalist makes a distinction between geometry as a deductive structure and geometry as a descriptive science. Only the first is regarded as mathematical. The use of pictures or diagrams, or even mental imagery, all are non- mathematical. In principle, they should be unnecessary. Consequently. he regards them as inappropriate in a mathematics text, perhaps even in a mathematics class." (Philip J Davis & Reuben Hersh, "The Mathematical Experience", 1981)

"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)

"[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)

"We distinguish diagrammatic from sentential paper-and-pencil representations of information by developing alternative models of information-processing systems that are informationally equivalent and that can be characterized as sentential or diagrammatic. Sentential representations are sequential, like the propositions in a text. Diagrammatic representations are indexed by location in a plane. Diagrammatic representations also typically display information that is only implicit in sentential representations and that therefore has to be computed, sometimes at great cost, to make it explicit for use. We then contrast the computational efficiency of these representations for solving several. illustrative problems in mathematics and physics." (Herbert A Simon, "Why a diagram is (sometimes) worth ten thousand words", 1987) 

"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)

"The value of diagram techniques even at this rudimentary level should be clear by now: it is easier to visualize where simplifications may be found in a complicated network by searching for a reducible linkage than by examining a complicated algebraic expression."(Geoffrey E Stedman, "Diagram Techniques in Group Theory", 1990)

"Diagrams are physical situations. They must be, since we can see them. As such, they obey their own set of constraints. […] By choosing a representational scheme appropriately, so that the constraints on the diagrams have a good match with the constraints on the described situation, the diagram can generate a lot of information that the user never need infer. Rather, the user can simply read off facts from the diagram as needed." (Jon Barwise & John Etchemendy, "Visual information and valid reasoning", [in "Visualization in Teaching and Learning Mathematics"], 1991)

"It has been said that the art of geometry is to reason well from false diagrams." (Jean Dieudonné, "Mathematics - The Music of Reason", 1992)

"A mental model is not normally based on formal definitions but rather on concrete properties that have been drawn from life experience. Mental models are typically analogs, and they comprise specific contents, but this does not necessarily restrict their power to deal with abstract concepts, as we will see. The important thing about mental models, especially in the context of mathematics, is the relations they represent. We will use diagrams to depict mental models for a variety of concepts, and it is important to keep in mind that any diagram, or even a non-diagrammatic representation that represents the same essential relations would be equally effective." (Lyn D English & Graeme S Halford," "Mathematics Education: Models and Processes", 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)

"Given particular tasks of reasoning, different types of diagrams show different degrees of suitedness. For example, Euler diagrams are superior in handling certain problems concerning inclusion and membership among classes and individuals, but they cannot be generally applied to such problems without special provisos. Diagrams make many proofs in geometry shorter and more intuitive, while they take certain precautions of the reasoner's to be used validly. […] Mathematicians experience that coming up with the 'right' sorts of diagrams is more than half-way to the solution of most complicated problems." (Atsushi Shimojima, "Operational Constraints in Diagrammatic Reasoning" , [in "Logical Reasoning with Diagrams"], 1996)

"Making a good choice of representational conventions is always important in solving a problem, but especially true of charts. This sensitivity of type of chart to the particularities of the task at hand makes a very general logic of charts useless." (Jon Barwise & Eric Hammer, "Diagrams and the Concept of Logical System", [in "Logical Reasoning with Diagrams"], 1996)

"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)