"And so it was that the complex number, which had its origin in a symbol for a fiction, ended by becoming an indispensable tool for the formulation of mathematical ideas, a powerful instrument for the solution of intricate problems, a means for tracing kinships between remote mathematical disciplines." (Tobias Dantzig, "Number: The Language of Science", 1930)
"But how can we avoid the use of human language? The [....]symbol. Only by using a symbolic language not yet usurped by those vague ideas of space, time, continuity which have their origin in intuition and tend to obscure pure reason - only thus may we hope to build mathematics on the solid foundation of logic." (Tobias Dantzig, "Number: The Language of Science", 1930)
"Greek thought was essentially non-algebraic, because it was so concrete. The abstract operations of algebra, which deal with objects that have been purposely stripped of their physical content, could not occur to minds which were so intently interested in the objects themselves. The symbol is not a mere formality; it is the very essence of algebra. Without the symbol the object is a human perception and reflects all the phases under which the human senses grasp it; replaced by a symbol the object becomes a complete abstraction, a mere operand subject to certain indicated operations." (Tobias Dantzig, "Number: The Language of Science", 1930)
"I recall my own emotions: I had just been initiated into the mysteries of the complex number. I remember my bewilderment: here were magnitudes patently impossible and yet susceptible of manipulations which lead to concrete results. It was a feeling of dissatisfaction, of restlessness, a desire to fill these illusory creatures, these empty symbols, with substance. Then I was taught to interpret these beings in a concrete geometrical way. There came then an immediate feeling of relief, as though I had solved an enigma, as though a ghost which had been causing me apprehension turned out to be no ghost at all, but a familiar part of my environment." (Tobias Dantzig, "The Two Realities", 1930)
"Symbolism is the foundation of all sublimation and of every talent, since it is by way of symbolic equation that things, activities and interests become the subject of libidinal phantasies." (Melanie Klein, "The Importance of Symbol-formation in the Development of the Ego", The International Journal of Psychoanalysis 11,1930)
"Most mistakes in philosophy and logic occur because the human mind is apt to take the symbol for the reality." (Albert Einstein, "Cosmic Religion: With Other Opinions and Aphorisms", 1931)
"Mathematics is the science of the infinite, its goal the symbolic comprehension of the infinite with human, that is finite, means." ( Hermann Weyl, "The Open World: Three Lectures In the Metaphysical Implications of Science", 1932)
"This assimilation of every fresh object to already existing motor schemas may be conceived of as the starting point of ritual acts and symbols, at any rate from the moment that assimilation becomes stronger than actual accommodation itself." (Jean Piaget, "The Moral Judgment of the Child", 1932)
"By the logical syntax of a language, we mean the formal theory of the linguistic forms of that language - the systematic statement of the formal rules which govern it together with the development of the consequences which follow from these rules. A theory, a rule, a definition, or the like is to be called formal when no reference is made in it either to the meaning of the symbols (for examples, the words) or to the sense of the expressions (e. g. the sentences), but simply and solely to the kinds and order of the symbols from which the expressions are constructed." (Rudolf Carnap, "Logical Syntax of Language", 1934)
"Mathematics is the science of the infinite, its goal the symbolic comprehension of the infinite with human, tha”[…] the merit of mathematics, in all its forms, consists in its truth; truth conveyed to the understanding, not directly by words but by symbols which serve as the world’s only universal written language.” (David Eugene Smith, “The Poetry of Mathematics and Other Essays”, 1934)
"[...] our knowledge of the external world must always consist of numbers, and our picture of the universe - the synthesis of our knowledge - must necessarily be mathematical in form. All the concrete details of the picture, the apples, the pears and bananas, the ether and atoms and electrons, are mere clothing that we ourselves drape over our mathematical symbols - they do not belong to Nature, but to the parables by which we try to make Nature comprehensible." (Sir James H Jeans, "The New World-Picture of Modern Physics", Supplement to Nature, Vol. 134 (3384), 1934)
"What is the inner secret of mathematical power? Briefly stated, it is that mathematics discloses the skeletal outlines of all closely articulated relational systems. For this purpose mathematics uses the language of pure logic with its score or so of symbolic words, which, in its important forms of expression, enables the mind to comprehend systems of relations otherwise completely beyond its power. These forms are creative discoveries which, once made, remain permanently at our disposal. By means of them the scientific imagination is enabled to penetrate ever more deeply into the rationale of the universe about us." (George D Birkhoff, "Mathematics: Quantity and Order", 1934)
"Whenever we pride ourselves upon finding a newer, stricter way of thought or exposition; whenever we start insisting too hard upon 'operationalism' or symbolic logic or any other of these very essential systems of tramlines, we lose something of the ability to think new thoughts. And equally, of course, whenever we rebel against the sterile rigidity of formal thought and exposition and let our ideas run wild, we likewise lose. As I see it, the advances in scientific thought come from a combination of loose and strict thinking, and this combination is the most precious tool of science." (Gregory Bateson, "Culture Contact and Schismogenesis", 1935)
"Given any domain of thought in which the fundamental objective is a knowledge that transcends mere induction or mere empiricism, it seems quite inevitable that its processes should be made to conform closely to the pattern of a system free of ambiguous terms, symbols, operations, deductions; a system whose implications and assumptions are unique and consistent; a system whose logic confounds not the necessary with the sufficient where these are distinct; a system whose materials are abstract elements interpretable as reality or unreality in any forms whatsoever provided only that these forms mirror a thought that is pure. To such a system is universally given the name Mathematics." (Samuel T. Sanders, "Mathematics", National Mathematics Magazine, 1937)
"We can now return to the distinction between language and symbolism. A symbol is language and yet not language. A mathematical or logical or any other kind of symbol is invented to serve a purpose purely scientific; it is supposed to have no emotional expressiveness whatever. But when once a particular symbolism has been taken into use and mastered, it reacquires the emotional expressiveness of language proper. Every mathematician knows this. At the same time, the emotions which mathematicians find expressed in their symbols are not emotions in general, they are the peculiar emotions belonging to mathematical thinking." (Robin G Collingwood, "The Principles of Art", 1938)
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