On Impossibility (1925-1949)

"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)

"The validity of demonstrably wrong law cannot conceivably be justified. However, any answer to the question of the purpose of law other than by enumerating the manifold partisan views about it has proved impossible - and it is precisely on that impossibility of any natural law, and on that alone, that the validity of positive law may be founded. At this point relativism, so far only the method of our approach, enters our system as a structural element." (Gustav Radbruch, "Rechtsphilosophie", 1932)

"While it is true that theory often sets difficult, if not impossible tasks for the experiment, it does, on the other hand, often lighten the work of the experimenter by disclosing cogent relationships which make possible the indirect determination of inaccessible quantities and thus render difficult measurements unnecessary." (Georg Joos, "Theoretical Physics", 1934)

"It is impossible to make a clear cut between science, religion, and art. The whole is never equal simply to the sum of its various parts." (Max Planck, "The Philosophy of Physics", 1936)

"Statements about impossibility in mathematics are of a wholly different character. A problem in mathematics which may not be solved for centuries to come is not always impossible. 'Impossible' in mathematics means theoretically impossible, and has nothing to do with the present state of our knowledge." (Edward Kasner & James R Newman, "Mathematics and the Imagination", 1940)

"When the number of factors coming into play in a phenomenological complex is too large, scientific method in most cases fails us. One need only think of the weather, in which case prediction even for a few days ahead is impossible. Nevertheless no one doubts that we are confronted with a causal connection whose causal components are in the main known to us. Occurrences in this domain are beyond the reach of exact prediction because of the variety of factors in operation, not because of any lack of order in nature." (Albert Einstein, "Science and Religion", 1941)

"Events with a sufficiently small probability never occur, or at least we must act, in all circumstances, as if they were impossible." (Félix E Borel, "Probabilities and Life", 1943)

"We have now to enquire how the neural mechanism, in producing numerical measurement and calculation, has managed to function in a way so much more universal and flexible than any other. Our question, to emphasize it once again, is not to ask what kind of thing a number is, but to think what kind of mechanism could represent so many physically possible or impossible, and yet self-consistent, processes as number does." (Kenneth Craik, "The Nature of Explanation", 1943)

"It is hard to have a good idea if we have little knowledge of the subject, and impossible to have it if we have no knowledge. Good ideas are based on past experience and formerly acquired knowledge." (George Pólya, "How to solve it", 1945)

"The rules of algebra show that the square of any number, whether positive or negative, is a positive number: therefore, to speak of the square root of a negative number is mere absurdity. Now, Cardan deliberately commits that absurdity and begins to calculate on such 'imaginary' quantities.     One would describe this as pure madness; and yet the whole development of algebra and analysis would have been impossible without that fundament - which, of course, was, in the nineteenth century, established on solid and rigorous bases. It has been written that the shortest and best way between two truths of the real domain often passes through the imaginary one." (Jacque S Hadamard, "An Essay on the Psychology of Invention in the Mathematical Field", 1945)