Geography and Mathematics

“I have myself always thought of a mathematician as in the first instance an observer, a man who gazes at a distant range of mountains and notes down his observations. His object is simply to distinguish clearly and notify to others as many different peaks as he can.” (Godfrey H Hardy, “Mathematical Proof”, Mind 38, 1929)

“The study of mathematics is like climbing up a steep and craggy mountain; when once you reach the top, it fully recompenses your trouble, by opening a fine, clear, and extensive prospect.” (Tyron Edwards, “The New Dictionary of Thoughts: A Cyclopedia of Quotations”, 1948)

“Creating a new theory is not like destroying an old barn and erecting a skyscraper in its place. It is rather like climbing a mountain, gaining new and wider views, discovering unexpected connections between our starting point and its rich environment. But the point from which we started out still exists and can be seen, although it appears smaller and forms a tiny part of our broad view gained by the mastery of the obstacles on our adventurous way up.” (Leopold Infeld, “The Evolution of Physics”, 1961)

“The tantalizing and compelling pursuit of mathematical problems offers mental absorption, peace of mind amid endless challenges, repose in activity, battle without conflict, ‘refuge from the goading urgency of contingent happenings’, and the sort of beauty changeless mountains present to senses tried by the present-day kaleidoscope of events.” (Morris Kline, “Mathematics in Western Culture”, 1964)

“Nature’s beauty dies. The day dawns when the nautilus is no more. The rainbow passes, the flower fades away, the mountain crumbles, the star grows cold. But the beauty in mathematics — the divine proportion, the golden rectangle, spira mirabilis — endures for evermore.” (Henry E Huntley, “The Divine Proportion: A Study in Mathematical Beauty”, 1970)

"Mathematicians get a different kind of pleasure from the illumination of solving a problem, when what was once mysterious and obscure is made plain. Revealing the hidden connections in a situation is delightful - like reaching the top of a mountain after a hard climb, and seeing the landscape spread out before you. All of a sudden, everything is clear! If the result is extremely simple, so much the better . To start with confusing complexity and transform it into revealing simplicity is a marvellous reward for hard work. It really does give the mathematician a 'kick'!" (David Wells, "You Are a Mathematician: A wise and witty introduction to the joy of numbers", 1995)

“There is a strong parallel between mountain climbing and mathematics research. When first attempts on a summit are made, the struggle is to find any route. Once on the top, other possible routes up may be discerned and sometimes a safer or shorter route can be chosen for the descent or for subsequent ascents. In mathematics the challenge is finding a proof in the first place. Once found, almost any competent mathematician can usually find an alternative often much better and shorter proof. At least in mountaineering we know that the mountain is there and that, if we can find a way up and reach the summit, we shall triumph. In mathematics we do not always know that there is a result, or if the proposition is only a figment of the imagination, let alone whether a proof can be found.” (Kathleen Ollerenshaw, “To talk of many things: An autobiography”, 2004)

“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])

“It may be permissible to compare mathematical research with the opening up of a mountain range. There will always be the people whose principal interest it will be to try their ability in advanced mountaineering. They will go for the most difficult summits. Others will see their aim in making the mountain range accessible as a whole, by building convenient roads along the valleys and across the passes. They will also reach the summits eventually, but mainly for the sake of the beautiful views, and, if possible, by cable car.” (Hans Hermes)

“My approach to research consists in looking to the mathematical landscape, taking notice of the things I like and judge interesting and of those I don’t care about, and then trying to imagine what should be next. If you see a bridge across a river, you try to imagine what lies on the other shore. If you see a mountain pass between two high mountains, you try to imagine what is in the valley you don’t see yet but secretly know must be there.” (Enrico Bombieri)

"The scientific life of mathematicians can be pictured as an exploration of the geography of the 'mathematical reality' which they unveil gradually in their own private mental frame." (Alain Connes)

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)

Mental Models VIII (Limitations II)

“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)

“It is by abstraction that one can separate movements, knowledge, and affectivity. And the analysis is, here, so far from being a real dismemberment that it is given only as probable. One can never effectively reduce an [mental] image to its elements, for the reason that an image, like all other psychic syntheses, is something more and different from the sum of its elements. […] We will always go from image to image. Comprehension is a movement which is never-ending, it is the reaction of the mind to an image by another image, to this one by another image and so on, in principle to infinity. “(Jean-Paul Sartre, “The Imaginary: A phenomenological psychology of the imagination”, 1940)

“Speaking without metaphor we have to declare that we are here faced with one of these typical antinomies caused by the fact that we have not yet succeeded in elaborating a fairly understandable outlook on the world without retiring our own mind, the producer of the world picture, from it, so that mind has no place in it. The attempt to press it into it, after all, necessarily produces some absurdities.” (Erwin Schrödinger, „Mind and Matter: the Tarner Lectures”, 1956)

“Mental models are fuzzy, incomplete, and imprecisely stated. Furthermore, within a single individual, mental models change with time, even during the flow of a single conversation. The human mind assembles a few relationships to fit the context of a discussion. As debate shifts, so do the mental models. Even when only a single topic is being discussed, each participant in a conversation employs a different mental model to interpret the subject. Fundamental assumptions differ but are never brought into the open. […] A mental model may be correct in structure and assumptions but, even so, the human mind - either individually or as a group consensus - is apt to draw the wrong implications for the future.” (Jay W Forrester, “Counterintuitive Behaviour of Social Systems”, Technology Review, 1971)

“At present, no complete account can be given - one may as well ask for an inventory of the entire products of the human imagination - and indeed such an account would be premature, since mental models are supposed to be in people's heads, and their exact constitution is an empirical question. Nevertheless, there are three immediate constraints on possible models. […] 1. The principle of computability: Mental models, and the machinery for constructing and interpreting them, are computable. […] 2. The principle of finitism: A mental model must be finite in size and cannot directly represent an infinite domain. […] 3. The principle of constructivism: A mental model is constructed from tokens arranged in a particular structure to represent a state of affairs.” (Philip Johnson-Laird, “Mental Models” 1983)

"Almost every aspect of our lives is shaped in some way by how we make sense of the world. Our thinking and our actions are affected by the mental models we hold. These models define our limits or open our opportunities. Despite their power and pervasiveness, these models are usually virtually invisible to us. We don't realize they are there at all.” (Robert Gunther et al, “The Power of Impossible Thinking: Transform the Business of Your Life and the Life of Your Business”, 2004)

“Humans have difficulty perceiving variables accurately […]. However, in general, they tend to have inaccurate perceptions of system states, including past, current, and future states. This is due, in part, to limited ‘mental models’ of the phenomena of interest in terms of both how things work and how to influence things. Consequently, people have difficulty determining the full implications of what is known, as well as considering future contingencies for potential systems states and the long-term value of addressing these contingencies. ” (William B. Rouse, “People and Organizations: Explorations of Human-Centered Design”, 2007)

“[…] we cannot accurately assess both what a mental model is and what it is becoming because the act of assessment affects the model.” (William B. Rouse, “People and Organizations: Explorations of Human-Centered Design”, 2007) [see Werner Heisenberg’s principle]

“Mental models are problematic because they typically operate unconsciously. This means that they influence behavior and structure thinking in ways that individuals do not recognize and therefore cannot easily articulate. This makes certain kinds of exploratory conversations difficult or impossible, and causes even reasonable propositions to be rejected out of hand.” (Kim Erwin, “Communicating The New: Methods to Shape and Accelerate Innovation”, 2013)

“A mental model is not necessarily founded on facts or complete understanding of reality. Let's be honest, most of our mental models are flawed in many ways, and that's perfectly normal. They work because they are fast and simple and not because they are a complete representation of the reality. […] The most important thing about a person's mental model is that it's simplified and very limited compared to what it models.” (Peter W Szabo, “User Experience Mapping”, 2017)

See also:
Mental Models VIII – More on their Limits
Mental Models I, II, III, V, VI, VII

Mental Models VII

“It is obvious then, that memory belongs to that part of the soul to which imagination belongs. […] Just as the picture painted on the panel is at once a picture and a portrait, and though one and the same, is both, yet the essence of the two is not the same, and it is possible to think of it both as a picture and as a portrait, so in the same way we must regard the mental picture within us both as an object of contemplation in itself and as a mental picture of something else […]. Insofar as we consider it in relation to something else, e.g. as a likeness, it is also an aid to memory.” (Aristotle, “De Memoria et Reminiscentia” [On Memory and Recollection], 4th century BC)

“In the same way as regards the soul, when that kind of thing in us which mirrors the images of thought and intellect is undisturbed, we see them and know them in a way parallel to sense-perception, along with the prior knowledge that it is intellect and thought that are active. But when this is broken because the harmony of the body is upset, thought and intellect operate without an image, and then intellectual activity takes place without a mind-picture.” (Plotinus, “Enneads”, cca. 270 AD)

“Part of the functions of the imaginative faculty is, as you well know, to retain impressions by the senses, to combine them, and chiefly to form images." (Moses Maimonides, “Guide of the Perplexed”, 1190 [translated by Michael Friedländer, 1904])

”[…] the painter cannot produce any form or figure […] if first this form or figure is not imagined and reduced into a mental image (idea) by the inward wits. And to paint, one needs acute senses and a good imagination with which one can get to know the things one sees in such a way that, once these things are not present anymore and transformed into mental images (fantasmi), they can be presented to the intellect. In the second stage, the intellect by means of its judgement puts these things together and, finally, in the third stage the intellect turns these mental images […] into a finished composition which it afterwards represents in painting by means of its ability to cause movement in the body.” (Romano Alberti, “Della nobiltà della Pittura”, 1585)

“The conception of lines of force was introduced by Faraday to form a mental picture of the processes going on in the electric field. To him these lines were not mere mathematical abstractions. He ascribed to them properties that gave them a real physical significance.” (Hendrik van der Bijl, “The Thermionic Vacuum Tube and Its Applications”, 1920)

“The continuing effects of mental images and ideas that, later, emerge in memory actually take place in the sphere of our feelings. […] Our life of feelings – with its joys, pains, pleasures, displeasures, tensions, and relaxations – is the actual vehicle for the enduring qualities of the ideas and mental images that we can recall at a later stage.” (Rudolf Steiner, “Education for Adolescents”, 1921)

“While the stuff from which our world picture is built is yielded exclusively from the sense organs as organs of the mind, so that every man's world picture is and always remains a construct of his mind and cannot be proved to have any other existence, yet the conscious mind itself remains a stranger within that construct, it has no living space in it, you can spot it nowhere in space.” (Erwin Schrödinger, „Mind and Matter: the Tarner Lectures”, 1956)

“A mental image occurs when a representation of the type created during the initial phases of perception is present but the stimulus is not actually being perceived; such representations preserve the perceptible properties of the stimulus and ultimately give rise to the subjective experience of perception.” (Stephen Kosslyn, “Image and Mind”, 1980)

“[…] a mental model is a mapping from a domain into a mental representation which contains the main characteristics of the domain; a model can be ‘run’ to generate explanations and expectations with respect to potential states. Mental models have been proposed in particular as the kind of knowledge structures that people use to understand a specific domain […]” (Helmut Jungermann, Holger Schütz & Manfred Thuering, “Mental models in risk assessment: Informing people about drugs”, Risk Analysis 8 (1), 1988)

“Mental models are the mechanisms whereby humans are able to generate descriptions of system purpose and form, explanations of system functioning and observed system states, and predictions of future system states.” (William B Rouse & Nancy M Morris, “On looking into the black box: Prospects and limits in the search for mental models”, Psychological Bulletin (3), 1986)

See also:
Mental Models I, II, III, IV, V, VI, VIII