On Reality (1960-1969)

"Science manipulates things and gives up living in them. It makes its own limited models of things; operating upon these indices or variables to effect whatever transformations are permitted by their definition, it comes face to face with the real world only at rare intervals. Science is and always will be that admirably active, ingenious, and bold way of thinking whose fundamental bias is to treat everything as though it were an object-in-general - as though it meant nothing to us and yet was predestined for our own use." (Maurice Merleau-Ponty, "L'Œil et l'Esprit", 1960)

"It is always extremely difficult to express thoughts. Words and phrases are so many fretters by which our spirit is bound. Words are mere symbols of reality, and the written word is not more than a one-dimensional fl ow across the two-dimensional page of a three-dimensional book." (Charles-Noël Martin, "The Role of Perception in Science", 1963)

"The probability concept used in probability theory has exactly the same structure as have the fundamental concepts in any field in which mathematical analysis is applied to describe and represent reality." (Richard von Mises, "Mathematical Theory of Probability and Statistics", 1964)

"A more problematic example is the parallel between the increasingly abstract and insubstantial picture of the physical universe which modern physics has given us and the popularity of abstract and non-representational forms of art and poetry. In each case the representation of reality is increasingly removed from the picture which is immediately presented to us by our senses." (Harvey Brooks, "Scientific Concepts and Cultural Change", 1965)

"It is paradoxical that while mathematics has the reputation of being the one subject that brooks no contradictions, in reality it has a long history of successful living with contradictions. This is best seen in the extensions of the notion of number that have been made over a period of 2500 years. From limited sets of integers, to infinite sets of integers, to fractions, negative numbers, irrational numbers, complex numbers, transfinite numbers, each extension, in its way, overcame a contradictory set of demands." (Philip J Davis, "The Mathematics of Matrices", 1965)

"The concept of reality (in the sense of independence from the cognizing consciousness) does not belong with (rational) science, but within metaphysics." (Rudolf Carnap, "The Logical Structure of the World", 1967)

"Knowing reality means constructing systems of transformations that correspond, more or less adequately, to reality. They are more or less isomorphic to transformations of reality. The transformational structures of which knowledge consists are not copies of the transformations in reality; they are simply possible isomorphic models among which experience can enable us to choose. Knowledge, then, is a system of transformations that become progressively adequate." (Jean Piaget, "Genetic Epistemology", 1968)

"We realize, however, that all scientific laws merely represent abstractions and idealizations expressing certain aspects of reality. Every science means a schematized picture of reality, in the sense that a certain conceptual construct is unequivocally related to certain features of order in reality […]" (Ludwig von Bertalanffy, "General System Theory", 1968)

"[...] one cannot describe reality; only give metaphors that indicate it. All human modes of description (photographic, mathematical, and literary) are metaphorical. Even the most precise scientific description of an object or movement is a tissue of metaphors." (John Fowles, "'Notes on an Unfinished Novel", 1969)

"Models are, for the most part, caricatures of reality, but if they are good, then like good caricatures, they portray, though perhaps in a distorted manner, some of the features of the real world." (Mark Kac, "Some mathematical models in science" Science, Vol. 166 (3906), 1969)

"Pure mathematics are concerned only with abstract propositions, and have nothing to do with the realities of nature. There is no such thing in actual existence as a mathematical point, line or surface. There is no such thing as a circle or square. But that is of no consequence. We can define them in words, and reason about them. We can draw a diagram, and suppose that line to be straight which is not really straight, and that figure to be a circle which is not strictly a circle. It is conceived therefore by the generality of observers, that mathematics is the science of certainty." (William Godwin, "Thoughts on Man", 1969)

"The advantages of models are, on one hand, that they force us to present a 'complete' theory by which I mean a theory taking into account all relevant phenomena and relations and, on the other hand, the confrontation with observation, that is, reality." (Jan Tinbergen, "The Use of Models: Experience," 1969)