On Symbols (1950-1959)

"The most elementary communication is not possible without some degree of conformity to the 'conventions' of the symbolic system." (Talcott Parsons, "The social system", 1951) 

"We could compare mathematics so formalized to a game of chess in which the symbols correspond to the chessmen; the formulae, to definite positions of the men on the board; the axioms, to the initial positions of the chessmen; the directions for drawing conclusions, to the rules of movement; a proof, to a series of moves which leads from the initial position to a definite configuration of the men.” (Friedrich Waismann & Karl Menger, “Introduction to Mathematical Thinking: The Formation of Concepts in Modern Mathematics”, 1951)

"Culture consists of patterns, explicit and implicit, of and for behavior acquired and transmitted by symbols, constituting the distinctive achievement of human groups, including their embodiments in artifacts; the essential core of culture consists of traditional (i.e., historically derived and selected) ideas and especially their attached values; culture systems may, on the one hand, be considered as products of action, on the other as conditioning elements of further action." (Alfred L Kroeber & Clyde Kluckhohn, "Culture", 1952)

"A signal is comprehended if it serves to make us notice the object or situation it bespeaks. A symbol is understood when we conceive the idea it presents." (Susanne Langer, "Feeling and Form: A Theory of Art", 1953)

"Philosophy is in history, and is never independent of historical discourse. But for the tacit symbolism of life it substitutes, in principle, a conscious symbolism; for a latent meaning, one that is manifest. It is never content to accept its historical situation. It changes this situation by revealing it to itself." (Maurice Merleau-Ponty, "Éloge de la philosophie" ["In Praise of Philosophy"], 1953) 

"Beauty had been born, not, as we so often conceive it nowadays, as an ideal of humanity, but as measure, as the reduction of the chaos of appearances to the precision of linear symbols. Symmetry, balance, harmonic division, mated and mensurated intervals – such were its abstract characteristics." (Herbert Read, "Icon and Idea: The Function of Art in the Development of Human Consciousness", 1955)

"A symbol, therefore, may have no effect and indeed ordinarily will have no effect on the image of the immediate future around one. It does produce an effect, however, of what might be called the image of the image, on the image of the future, on the image of the past, on the image of the potential or even of the image of the possible."(Kenneth E Boulding, "The Image: Knowledge in life and society", 1956)

"The symbol and the metaphor are as necessary to science as to poetry." (Jacob Bronowski, "Science and Human Values", 1956)

"To be sure, mathematics can be extended to any branch of knowledge, including economics, provided the concepts are so clearly defined as to permit accurate symbolic representation. That is only another way of saying that in some branches of discourse it is desirable to know what you are talking about." (James R Newman, "The World of Mathematics", 1956)

"Behind these symbols lie the boldest, purest, coolest abstractions mankind has ever made. No schoolman speculating on essences and attributes ever approached anything like the abstractness of algebra." (Susanne K Langer, "Philosophy in a New Key", 1957)

"The function of mathematical logic is to reveal and codify the logical processes employed in mathematical reasoning and to clarify the concepts of mathematics; it is itself a branch of mathematics, employing mathematical symbolism and technique, a branch which has developed in its entirety during the past hundred years and which in its vigor and fecundity and the power and importance of its discoveries may well claim to be in the forefront of modern mathematics." (Reuben L Goodstein, "Mathematical Logic", 1957)

"Mathematics is neither a description of nature nor an explanation of its operation; it is not concerned with physical motion or with the metaphysical generation of quantities. It is merely the symbolic logic of possible relations, and as such is concerned with neither approximate nor absolute truth, but only with hypothetical truth. That is, mathematics determines what conclusions will follow logically from given premises. The conjunction of mathematics and philosophy, or of mathematics and science is frequently of great service in suggesting new problems and points of view." (Carl B Boyer, "The History of the Calculus and Its Conceptual Development", 1959)

"The mathematician, the statistician, and the philosopher do different things with a theory of probability. The mathematician develops its formal consequences, the statistician applies the work of the mathematician and the philosopher describes in general terms what this application consists in. The mathematician develops symbolic tools without worrying overmuch what the tools are for; the statistician uses them; the philosopher talks about them. Each does his job better if he knows something about the work of the other two." (Irving J Good, "Kinds of Probability", Science Vol. 129 (3347),  1959)