A "map" takes points in one space (the source space) to certain points which the map identifies as the "corresponding points" in another space (the target space). Wittgenstein calls these "logical spaces." Symbolic structures which obey a system of rules for translation are isomorphisms, structural homologies. Thus the mapping amounts to a distorted image of the source space on the target space. Language maps thought on to sound. An input/output function can be understood as a mapping. Thus the toaster executes a function mapping from bread to toast, and the groove on a gramophone record maps to the sounds. The psycho-physiological problem in mechanistic psychology becomes a problem of point-to-point mapping of mental functions such as language and memory on to the brain. (see mind /brain )
In a computational theory of mapping, "the performance of a system is characterized as a mapping from one kind of information to another. The abstract properties of the mapping are precisely defined, and its appropriateness and adequacy for the task are demonstrated." (Marr) In his book on vision, David Marr proposes that three levels of understanding are required for the analysis of a complex information-processing system: At one extreme, the top level, is the abstract computational theory of the device. At an intermediate level is the choice of representation for the input and output and the algorithm to be used to transform one into the other. At the other extreme are the details of how the algorithm and representation are realized physically -- the detailed computer architecture, so to speak. (pp24-5) For Marr, "These three levels are coupled, but only loosely.
In the last chapter of On Growth and Form, D'Arcy Thompson's illustrates his "cartesian transformations" of animal forms. Thompson's mappings are referred to as "rubber sheet" mappings. They are the most general category of continuous geometric mappings that include identity, translation, rotation, reflection, scale, stretch, shear etc. (see Mitcell, Logic of Architecture, p. 121)
Continuous mapping is defined mathematically: F(X) is continuous at X=A when: 1) F is defined at A. 2) the limit x approaches A of F(X) exists. 3)the limit F(A) is the same when X approaches A from any and all directions. Such a map is " smooth" if the rate of change of F as X varies is also continuous. (see M. Shinbrot, "Fixed point theorems" Sci. Am. 214 (1) 105-110.) Arthur T. Winfree provides an introduction to the topology of various mappings in "The Geometry of Biological Time". As the book is primarily concerned with biological cycles, much of the mapping is to circles. Some of these mappings are necessarily discontinuous and create singularities. For example, mapping a finite linear interval onto a circle creates a point of discontinuity. Eg: 360° = 0°. Thus the necessity for the International Date Line and the impossibility of establishing time at the poles. (since the 24 time zones all meet)
Should we think of Thompson's transformations as the "same" form mapped into different spaces -- A class of figures equivalent under transformation? (see natural form) What is the relation between these mappings and the idea of type? If they are "identity preserving" transformations, they need a concept of type identity (and a distinction between the type and its instance, or token.) (see population / typological)
(cf also Deleuze's discussion of Leibniz and the fold)
"Of Exactitude in Science"
...In that Empire, the craft of Cartography attained such perfection that the Map of a Single province covered the space of an entire City, and the Map of the Empire itself an entire Province. In the course of Time, these Extensive maps were found somehow wanting, and so the College of Cartographers evolved a Map of the Empire that was of the same Scale as the Empire and that coincided with it point for point. Less attentive to the Study of Cartography, succeeding Generations came to judge a map of such Magnitude cumbersone, and, not without Irreverence, they abandoned it to the Rigors of sun and Rain. In the western Deserts, tattered Fragments of the Map are still to be found, Sheltering an occasional Beast or beggar; in the whole Nation, no other relic is left of the Discipline of Geography. from: Jorge Luis Borges, A Universal History of Infamy. (see ruins of representation text)
An extension of the concept of ground is in the relation between the map and the territory. According to Gregory Bateson, their progressive differentiation takes place through play, threat, histrionics, and allow for a vast variety of complications and inversions between communicative and metacommunicative meanings. Bateson claims that in primary process map and territory are equated, (see unconscious) in secondary process they can be discriminated, and in play they are both equated and discriminated. ("a Theory of Play and Fantasy" in Steps to an Ecology of Mind.)
In a computational sense, all systems of navigation answer the question "Where am I?" The mode in which the Western tradition of pilotage attempts to answer that question is in the establishment of the correspondence of map and territory. In Cognition in the Wild, Edwin Hutchins examines navigation as a form of cognitive computation that applies as much to the interaction of humans with artifacts and with other humans as it does to explicit symbol processing. For Hutchins, cognition is in a fundamental sense a cultural and social process. For Hutchins, navigational computation occurs through the propagation of representational state accross a series of representational media. (see pp 117 ff) In pilotage, the ship's situation is represented and re-represented until the answer to the navigator's question is transparent. For the navigator, the ship is where its lines of position intersect on the chart, the "common ground" of all the representations of its position.
Thongchai Winichakul describes the process of mapping Siam as a process of aligning map and power. "In terms of most communication theories and common sense, a map is a scientific abstraction of reality. A map merely represents something that exists objectively 'there.' In the history I have described , this relationship was reversed. A map anticipated spatial reality, not vice-versa. In other words, a map was a model for, rather than a model of, what it purported to represent....It had become a real instrument to concretize projections on the earth's surface. A map was now necessary for the new administrative mechanisms and for the troops to back up their claims...The discourse of mapping was the paradigm which both administrative and military operations worked within and served." (quoted in Benedict Anderson, Imagined Communities, pp 173-4) Benoit Mandelbrot cites maps as expressing power relations when a small country measures its shared border with a larger country and finds it longer than its neighbor does. (see fractals )
what is "cognitive mapping?" Frederic Jameson defines the aesthetic of the new cultural form of postmodern space as an aesthetic of cognitive mapping. He refers to Kevin Lynch's study The Image of the City, as showing that the alienated city is above all a space in which people are unable to map (in their minds) either their own positions or the urban totality in which they find themselves. "Disalienation in the traditional city then, involves the practical reconquest of a sense of place and the construction or reconstruction of an articulated ensemble which can be retained in memory and which the individual subject can map and remap along moments of mobile, alternative trajectories." (Postmodernism, p.51) (cf. Michel de Certeau's contrasts between the map and the tour -- De Certeau describes the "tour" as an everyday narration of movement and opposes it to the "map," a scientific representation that erases the itineraries that produced it, and whose history shows this process of disengagement. For de Certeau, who is interested in the tactics of poaching and consumption, everyday stories are guides to spatial practices. For Jameson, the kind of "tour" that De Certeau describes as "precartographic," diagrams organized around the still subject-centered or existential journey of the traveler. (see pp 51-52) For Jameson, cognitive mapping becomes more complex when it requires the coordination of existential data with unlived, abstract conceptions of the geographic totality.
Theory can be regarded as a kind of map. (or is it a tour?)