Mental Models IX – Mathematics and Mental Models

“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])

“The words or the language, as they are written or spoken, do not seem to play any role in my mechanism of thought. The psychical entities which seem to serve as elements in thought are certain signs and more or less clear images which can be 'voluntarily' reproduced and combined.” (Albert Einstein [in a letter to Jaque Hadamard, in “The Psychology of Invention in the Mathematical Field”, 1949])

“Finally, students must learn to realize that mathematics is a science with a long history behind it, and that no true insight into the mathematics of the present day can be obtained without some acquaintance with its historical background. In the first-place time gives an additional dimension to one's mental picture both of mathematics as a whole, and of each individual branch.” (André Weil, “The Mathematical Curriculum", 1954)

“That is to say, intuition is not a direct perception of something existing externally and eternally. It is the effect in the mind of certain experiences of activity and manipulation of concrete objects (at a later stage, of marks on paper or even mental images). As a result of this experience, there is something (a trace, an effect) in the pupil's mind which is his representation of the integers. But his representation is equivalent to mine, in the sense that we both get die same answer to any question you ask - or if we get different answers, we can compare notes and figure out what's right. We do this, not because we have been taught a set of algebraic rules, but because our mental pictures match each other.” (Philip J Davis & Reuben Hersh, “The Mathematical Experience”, 1981)

“We are also limited by the fact that verbalization works best when mental model manipulation is an inherent element of the task of interest. Troubleshooting, computer programming, and mathematics are good examples of tasks where mental model manipulation is central and explicit. In contrast, the vast majority of tasks do not involve explicit manipulation of task representations. Thus, our access of mental models - and the access of people doing these tasks - is limited.” (William B. Rouse, “People and Organizations: Explorations of Human-Centered Design”, 2007)

“Where things get really interesting is when unexpected bridges emerge between parts of the mathematical world that were remote from each other in the mental picture that had been developed by previous generations of mathematicians. When this happens, one gets the feeling that a sudden wind has blown away the fog that was hiding parts of a beautiful landscape. In my own work this type of great surprise has come mostly from the interaction with physics.” (Alain Connes [in “The Princeton Companion to Mathematics” Ed. by Timothy Gowers et al, 2008])

“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 0 Cairbre, “The Importance of Being Beautiful in Mathematics”, IMTA Newsletter 109, 2009)

“Thus, when we speak of a mathematical concept, we speak not of a single isolated mental image, but rather of a family of mutually correcting mental images. They are privately owned, but publicly checked, examined, corrected, and accepted or rejected. This is the role of the mathematical research community, how it indoctrinates and certifies new members, how it reviews, accepts or rejects proposed publication, how it chooses directions of research to follow and develop, or to ignore and allow to die. All these social activities are based on a necessary condition: that the individual members have mental models that fit together, that yield the same answers to test questions. A new branch of mathematics is established when consensus is reached about the possible test questions and their answers. That collection of possible questions and answers (not necessarily explicit) becomes the means of accepting or rejecting proposed new members.” (Reuben Hersh, “Mathematical Intuition: Poincaré, Pólya, Dewey” [in “The Courant–Friedrichs–Lewy (CFL) Condition”, 2013])

“A mathematical entity is a concept, a shared thought. Once you have acquired it, you have it available, for inspection or manipulation. If you understand it correctly (as a student, or as a professional) your ‘mental model’ of that entity, your personal representative of it, matches those of others who understand it correctly. (As is verified by giving the same answers to test questions.) The concept, the cultural entity, is nothing other than the collection of the mutually congruent personal representatives, the ‘mental models’, possessed by those participating in the mathematical culture.” (Reuben Hersh, “Experiencing Mathematics: What Do We Do, when We Do Mathematics?”, 2014)