WHO’S AFRAID OF MATHEMATICAL DIAGRAMS?

Mathematical diagrams are frequently used in contemporary mathematics. They are, however, widely seen as not contributing to the justificatory force of proofs: they are considered to be either mere illustrations or shorthand for non-diagrammatic expressions.   Moreover, when they are used inferentially, they are seen as threatening the reliability of proofs.  In this paper, I examine certain examples of diagrams that resist this type of dismissive characterization. By presenting two diagrammatic proofs, one from topology and one from algebra, I show that diagrams form genuine notational systems, and I argue that this explains why they can play a role in the inferential structure of proofs without undermining their reliability.  I then consider whether diagrams can be essential to the proofs in which they appear.@font-face{font-family:"Cambria Math";panose-1:2 4 5 3 5 4 6 3 2 4;mso-font-charset:0;mso-generic-font-family:roman;mso-font-pitch:variable;mso-font-signature:-536870145 1107305727 0 0 415 0;}@font-face{font-family:Calibri;panose-1:2 15 5 2 2 2 4 3 2 4;mso-font-charset:0;mso-generic-font-family:swiss;mso-font-pitch:variable;mso-font-signature:-536859905 -1073697537 9 0 511 0;}p.MsoNormal, li.MsoNormal, div.MsoNormal{mso-style-unhide:no;mso-style-qformat:yes;mso-style-parent:"";margin:0in;line-height:200%;mso-pagination:widow-orphan;font-size:12.0pt;font-family:"Calibri",sans-serif;mso-fareast-font-family:Calibri;}.MsoChpDefault{mso-style-type:export-only;mso-default-props:yes;font-family:"Calibri",sans-serif;mso-ascii-font-family:Calibri;mso-fareast-font-family:Calibri;mso-hansi-font-family:Calibri;mso-bidi-font-family:Calibri;}.MsoPapDefault{mso-style-type:export-only;line-height:200%;}div.WordSection1{page:WordSection1;}

Establishment the program and mathematical diagrams of embankment stability analysis on soft soil reinforced by soil cement columns

In the calculation of improvement of embankments on soft soils, the geometrical parameters of soil cement columns such as the length L, diameter d, the distance between the columns D, greatly affect to the stability of embankments (settlement S, coefficient of stability Fs) and treatment efficiency. On the basis of the multivariate correlation equation predicting total settlement (S), residual settlement after treatment (DS) based on the unit weight of soil , the height of embankments H, the length L, diameter d, the distance between the columns D after statistical analysis on p_value index and R2 with the following models: Linear, Quadratic combined with the Bishop method on slope stability analysis, the authors have built the code of stability analysis for the embankment on soft ground reinforced with soil cement columns system. At the same time, we have developed mathematical diagrams for the analysis of the influence of each pair of column geometry factors on the settlement of the roadbed.

Why Were Greek Mathematical Diagrams Schematic?

This article presents the case for the claim that Greek mathematical diagrams were schematic. Following a deep dive into the practices and the process of transmission of those diagrams, the article situates Greek diagrammatic practices within the broader context of Greek scribal and readerly practices. The literary papyrus bookroll was produced and read as a tool for the projection of an imagined performance; so was the specialized type of bookroll containing mathematical proofs.

Constructing and Unpacking Diagrams in Geometry

By having students practice constructing diagrams for geometric theorems, teachers can develop students' understanding of mathematical claims, vocabulary, and notation methods. This practice can also strengthen students' ability to interpret mathematical diagrams and recognize their limitations.

Diagrammatic Reasoning from Reflections on Peircean Semiotics

The document illustrates some elements of reflection on Peirce's semiotics focused on reasoning through diagrams. The solution of a Euclidean geometry problem is taken as a reference in which mathematical diagrams are recognized as epistemological tools in the learning and teaching of geometry. This is how an interpreter, who systematically observes and experiments with a geometric diagram, generates different interpretants by means of abductive, inductive and deductive reasoning.

Visual Logic Maps (vLms)

This concluding chapter describes and analyses a concept map based, Visual Logic Maps (vLms). Essentially the vLms maps differ from the Thinking Maps discussed in the last chapter in that the Visual Logic Maps shifts its emphasis from map structure to map glyphs. In Thinking Maps it is the structure of the eight specific maps that determine how information is going to be organized and mentally process. The Visual Logic Maps reduces each of the maps to seven specific glyphs that operate as constant logical operators. In the conclusion of the chapter and the book it is argued that both the Thinking Maps and Visual Logic Maps are essentially non-verbal spatial maps that find the cognitive origins in the logical/mathematical diagrams of Venn and Euler. Unlike Venn an Euler Diagrams, however, Visual Logic and Thinking Maps are not domain specific nor mathematical in the numeric sense.