Abstract
MENTAL IMAGERY IN auditory, sensual, and visual modes has played a central role in creative thought. Wolfgang Amadeus Mozart’s auditory imagery permitted him to hear a new symphony “tout ensemble.” The great French mathematician and philosopher Henri Poincaré’s “sensual imagery” led him to sense a mathematical proof in its entirety “at a glance.” Albert Einstein’s creative thinking occurred in visual imagery, and words were “sought after laboriously only in a secondary stage” (Hadamard, 1954).
Logic . . . remains barren unless it is fertilized by intuition.
H. Poincaré (1908a)
The very fact that the totality of our sense experience is such that by means of thinking (operations with concepts, and the creation and use of definite functional relations between them, and the coordination of sense experiences to these concepts) it can be put in order, this fact is one which leaves us in awe, but which we shall never understand. One may say “the eternal mystery of the world is its comprehensibility.”
A. Einstein (1936)
[Cognitive science] shows how one can construct a normative theory—a “logic” if you will—of discovery processes [and] shows how one can treat operationally and formally phenomena that have usually been dismissed with fuzzy labels like “intuition” and “creativity.”
H.A. Simon (1973)
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Notes
Here there is a deep problem facing computer science that had been realized by Poincaré in the context of mathematical logic; namely, that “pure logic cannot give us [a] view of the whole; it is to intuition that we must look for it” (Poincaré, 1904b).
Among the sharpest critics of computer science is John R. Searle (1981, 1982) who has accused them of “simply play acting at science.” According to Searle computers are merely useful tools for cognitive psychology, a view that has been dismissed by computer science (see Newell and Simon, 1981).
For example, the interplay between depictive and long-term descriptive information was illustrated by an experiment in which subjects viewed a 3 × 6 array of letters. After the array was removed, the subjects were told that the array would be referred to either as 6 columns of 3 or 3 rows of 6. Subjects took longer to image the array when it was described in the first way. For other experiments see Kosslyn (1979, 1981).
For example, the propositional account of Shepard’s data is that the image of an object can be encoded with a network of propositions describing its shape and orientation, and containing a rotation operator. Kosslyn’s reply to the propositional account is that the imagery account seems “somewhat plausible and relatively straightforward” (Kosslyn, 1979). The reason is, continues Kosslyn, that the propositional account requires the imposition of ad hoc constraints to account for why images are rotated continuously; for, after all, a rotation of 180° should be accomplished quicker than one of 45° because it can be propositionalized as a spatial reflection.
For example, in order to translate from a code 1 to a code 2, it would be necessary to translate code 1 into a code 3 and then code 3 to code 2. But then a new code is required to translate code 1 into code 3, and so on (Anderson, 1978). Paivio has proposed that there is a dual-code theory for image and verbal modes, asserting that long-term memory representaion for images is separate from that of verbal behavior (see Kosslyn (1981) for criticism).
Kosslyn (1979, 1981) writes that theories relying heavily on computer encodings are so powerful that they are not easily operationalizable, and so a host of theories can be formulated post hoc for each experiment; these theories usually lack predictive power and are “primarily metatheoretical commitments to the form that a theory and model will ultimately take” (Kosslyn, 1981). Furthermore, emphasizes Kosslyn using an argument due to Putnam (1973), it is unnecessary to assert that a phenomenon is explained only if it is reduced completely to a formal system—for example, to explain why a round peg does not fit into a square hole requires only notions of form and not elementaryparticle physics.
Anderson (1978) continues by emphasizing that at issue is also the important point that one cannot probe a representation in the abstract, but the “representation in combination with certain assumptions about the processes that use the representation.” For example, how the letter R is internally represented (imaginal as a display on a two-dimensional grid or propositional with a list of propositions describing its shape), and how the letter is processed (rotated by computer subroutines for rotating a matrix representation in small steps or through the propositional calculus).
Their other tack has been to appeal to Thomas S. Kuhn’s scenario for the advance of science. So, for example, we find that after 532 pages of almost nothing but the computer science approach, the concluding sentence in the Lachman (1979) tome is: “It is our final hope that our treatment of the information processing paradigm has been persuasive.”
The authenticity is disputed of the Mozart letter from which these passages were taken. Nevertheless, the contents square with other of Mozart’s descriptions of his creative style. For example, in a letter of 30 December 1780 to his father he wrote of work on Idomeneo that “everything is composed, just not copied out yet” (Hildesheimer, 1983).
This deep sentiment on thinking was meant no doubt to be read by logicians who were attempting to reduce mathematics to an axiomatic basis from which in turn could be generated a cookbook procedure to construct new theorems. As Poincaré wrote elsewhere in 1908, logicians claim “to have shown that mathematics is entirely reducible to logic, and that intuition plays no part in it whatever.” And in reply to one logician, Bertrand Russell, in 1909 Poincaré wrote that “there is no logic and epistemology independent of psychology.” No doubt he would have written similarly of computer science. Poincarés antireductionist stance in mathematics carried over into the life sciences. For example, in essays of (1900) and (1904c) he wrote of the impossibility to reconstruct the “unity of the individual” having analyzed the atomic structure of its cells: “Would a naturalist imagine that he had an adequate knowledge of the elephant if he had never studied the animal except through a microscope?” (Precisely this antireductionist argument has been rediscovered by others, e.g., Polanyi (1962) who used a frog.) Poincaré’s argument that the content of living organisms is greater than the sum of its parts is applicable also to the interpretation problem in mathematical logic, where the interesting interpretations are the sensible ones, and we make this judgment from our ability to recognize them by examining the completed structure. Haugeland (1981) suggests avoiding the problem of complete reduction by redefining reductionism in cognitive science to be “systematic reductions.” This is the process of explaining a technological layer by means of those in which it is instantiated, and so on. The very bottom layer is explained in the traditional sense, that is, by the laws of physics.
For example, the mathematician G.H. Hardy wrote: “The mathematical patterns, like the painter’s or the poet’s, must be beautiful; the ideas, like the colours or the words, must fit together in a harmonious way. Beauty is the first test: there is no permanent place in the world for ugly mathematics . . . . It may be very hard to define mathematical beauty, but that is just as true of beauty of any kind” (Hardy (1940) italics in original).
Poincaré’s view of the creative process contains what we may construe to be his attempt to exclude the homunculus from thinking. Neisser (1967) has succinctly described this agent: “Who does the turning, the trying and the erring? Is there a little man in the head, a homunculus, who acts the part of paleontologist vis-avis dinosaur?” Poincaré assumed that the mathematician’s “special aesthetic sensibility” served as the agent of selection. By interaction with the subliminal ego whose “automatic actions . . . blindly forms” a large number of combinations of facts this agent selects out only the most beautiful ones which somehow find their way into the conscious. In computer simulation of thinking, be it the purely descriptive encodings, or the mixed descriptive-depictive encodings, the homunculus problem is declared to be resolved through the definition of thinking or perception as information processing. Thus the computer functions that permit access to various subroutines are operationally the homunculi or mind’s eye; these guiding routines are referred to as “executive routines,” and they are “in no sense a programmulus or miniature of the entire program” (Neisser, 1967). For example, an executive routine can divert the flow of a calculation according to whether a certain variable is greater or less than zero.
Poincaré’s hypnagogic imagery was the typical sort. For further discussion of hypnagogic imagery that is pertinent to this essay see Richardson (1969) and Shepard (1978a).
Hadamard (1954) had proposed an analogous sequence in which Simon’s selective forgetting is referred to as the “forgetting hypothesis”; needless to say, there was no attempt on Hadamard’s part to suggest automation of this process.
Pestalozzi (1801) wrote (italics in original): “What have I especially done for the very being of education? I find I have fixed the highest, supreme principle of instruction in recognition of Anschauung as the absolute foundation of all knowledge.” A recent biographer of Pestalozzi has emphasized that whereas the ideas in Pestalozzi’s most important book, How Gertrude Teaches Her Children, have been widely accepted, the book itself is “now hardly ever read”: the prose is difficult because even as he wrote the book Pestalozzi was still shaping his educational system; and the central term in his system is virtually untranslatable into English, and even in German the word Anschauung possesses a multitude of philosophically weighted meanings through which Pestalozzi shifted with little warning (Silber, 1960). For example, we find: “Nothing is more difficult to grasp in Pestalozzi’s doctrine than what exactly is meant by the untranslatable word Anschauung” (Green, 1913). In fact, a painstaking 1894 translation of How Gertrude Teaches Her Children, contains the word Anschauung in parenthesis whenever it is rendered other than “sense-impression.” Detailed notes to this translation contain over two pages devoted to Anschauung and stress that Pestalozzi’s own usage covers the spectrum from sense-impressions (in the infant’s mind) to observation, perception, appreception, and intuition (”knowledge obtained by contemplation of ideas already in the mind, which have not necessarily been derived from the observation of external objects”). Sometimes Pestalozzi shifted through all of the above meanings in a single paragraph. Kantian shades of interpretation color all of Pestalozzi’s uses for Anschauung, coinciding exactly with “intuition.” Certain of Pestalozzi’s and Kant’s notions of the construction of knowledge are similar, but were conceived of independently. Pestalozzi first aired his thoughts in the book Leonard and Gertrude that was published in 1781, the same year as Kant’s Critique of Pure Reason. That fervent representative of Kantian philosophy, Johann Gottlieb Fichte, found in Pestalozzi’s 1781 book “many of the same results as in Kant.” Pestalozzi’s influence on Fichte can be seen in Fichte’s addresses of 1807–1808 in Berlin, “Reden an die deutschen Nation.” Fichte transferred Pestalozzi’s notion of the fundamental power of the human mind, i.e., Anschauung, to an innate characteristic of the German people (Merz, 1965; Hoffding, 1955). Friedrich Nietzsche would elaborate further on this hypothesis with unforeseen consequences. This darker side of Anschauung I do not discuss here, except to mention that the meaning of anschaulich in the relativity theory as compared to classical physics was a central point in Philipp Lenard’s criticisms of Einstein (see Beyerchen, 1977). Although Pestalozzi was aware of the works of such philosophers as Rousseau, Voltaire, Hume, and Kant, among others, he always claimed that recent developments in philosophy were “beyond his capacity” (Silber, 1960). Gratified to be compared with such “crystal-clear thinkers” as Kant, Pestalozzi preferred “feeling philosophers” such as Johann Gottfried Herder and Friedrich Heinrich Jacobi. Although the “organization of American technology in the first half of the nineteenth century tended naturally to follow the pattern set by the world of art” (Ferguson, 1977), the introduction of Pestalozzi’s teachings into mid-nineteenth century America no longer found fertile ground. For since the 1860s engineering education in the United States has stressed analytical over nonverbal thinking. Ferguson (1977) writes that “in engineering school a course in ’visual thinking’ is regarded as an aberration rather than as a discipline that should be incorporated into an engineer’s repertoire of skills . . . and the course in which it occurs is picked up by the New York Times.” See Hindle (1981) and Ferguson (1977) for discussion of engineering education in the United States during the first half of the nineteenth century.
Although offering Gedanken experiments was not uncommon in the literature, by comparison with Einstein’s those offered, for example, by Ernst Mach in his (1905b) “Über Gedankenexperimente,” pale. Mach defined a Gedanken experiment as an “idealization or abstraction” of existing physical conditions.” Among the examples he gave were Gustav Kirchhoff’s description of the perfect black body and the mathematical process of integration on a line as the limiting case of summing up many line elements. Clearly, Mach’s Gedanken experiments were the usual sorts of abstractions from direct experience that throughout were linked with sense perceptions: “Experience produces a thought experiment which is then spun further to be compared with experience and modified.”
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Miller, A.I. (1984). On the Limits of the IMAGination. In: Imagery in Scientific Thought Creating 20th-Century Physics. Birkhäuser, Boston, MA. https://doi.org/10.1007/978-1-4684-0545-3_7
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