On Abstraction (1950-1959)

"The first thing to realize about physics […] is its extraordinary indirectness. […] For physics is not about the real world, it is about 'abstractions' from the real world, and this is what makes it so scientific. […] Theoretical physics runs merrily along with these unreal abstractions, but its conclusions are checked, at every possible point, by experiments." (Anthony Standen, "Science is a Sacred Cow", 1950)

"In mathematics […] we find two tendencies present. On the one hand, the tendency towards abstraction seeks to crystallise the logical relations inherent in the maze of materials [….] being studied, and to correlate the material in a systematic and orderly manner. On the other hand, the tendency towards intuitive understanding fosters a more immediate grasp of the objects one studies, a live rapport with them, so to speak, which stresses the concrete meaning of their relations." (David Hilbert, "Geometry and the imagination", 1952)

"There is nothing mysterious, as some have tried to maintain, about the applicability of mathematics. What we get by abstraction from something can be returned. (Raymond L Wilder, Introduction to the Foundations of Mathematics, 1952)

"The theory of relativity is a fine example of the fundamental character of the modern development of theoretical science. The initial hypotheses become steadily more abstract and remote from experience. On the other hand, it gets nearer to the grand aim of all science, which is to cover the greatest possible number of empirical facts by logical deduction from the smallest possible number of hypotheses or axioms." (Albert Einstein, 1954)

"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)

"Abstractions are wonderfully clever tools for taking things apart and for arranging things in patterns but they are very little use in putting things together and no use at all when it comes to determining what things are for." (Archibald MacLeish, "Why Do We Teach Poetry?", The Atlantic Monthly Vol. 197 (3), 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)

"One great lesson that we can learn from its systematic absence in the work of the grand theorists is that every self-conscious thinker must at all times be aware of - and hence be able to control - the levels of abstraction on which he is working. The capacity to shuttle between levels of abstraction, with ease and with clarity, is a signal mark of the imaginative and systematic thinker." (C Wright Mills, "The Sociological Imagination", 1959)

"There is a logic of language and a logic of mathematics. The former is supple and lifelike, it follows our experience. The latter is abstract and rigid, more ideal. The latter is perfectly necessary, perfectly reliable: the former is only sometimes reliable and hardly ever systematic. But the logic of mathematics achieves necessity at the expense of living truth, it is less real than the other, although more certain. It achieves certainty by a flight from the concrete into abstraction." (Thomas Merton, "The Secular Journal of Thomas Merton", 1959)