"We must have a logical intuition of the probable relations between propositions. Once the existence of this relation between evidence and conclusion, the latter becomes the subject of the degree of belief." (John M Keynes, "Treatise on Probability", 1921)
"If a fact is to be a picture, it must have something in common with what it depicts. […] What a picture must have in common with reality, in order to be able to depict it correctly or incorrectly - in the way it does, is its pictorial form. […] What any picture, of whatever form, must have in common with reality, in order to be able to depict it - correctly or incorrectly in any way at all, is logical form, i.e., the form of reality. […] Logical pictures can depict the world." (Ludwig Wittgenstein, "Tractatus Logico-Philosophicus", 1922)
"Mathematics, or the science of magnitudes, is that system which studies the quantitative relations between things; logic, or the science of concepts, is that system which studies the qualitative (categorical) relations between things." (Peter D Ouspensky, "Tertium Organum: The Third Canon of Thought; a Key to the Enigmas of the World", 1922)
"Mere deductive logic, whether you clothe it in mathematical symbols and phraseology or whether you enlarge its scope into a more general symbolic technique, can never take the place of clear relevant initial concepts of the meaning of your symbols, and among symbols I include words. If you are dealing with nature, your meanings must directly relate to the immediate facts of observation. We have to analyse first the most general characteristics of things observed, and then the more casual contingent occurrences. There can be no true physical science which looks first to mathematics for the provision of a conceptual model. Such a procedure is to repeat the errors of the logicians of the middle-ages." (Alfred N Whitehead, "Principle of Relativity", 1922)
"The logical picture of the facts is the thought. […] A picture is a model of reality. In a picture objects have the elements of the picture corresponding to them. The fact that the elements of a picture are related to one another in a determinate way represents that things are related to one another in the same way." (Ludwig Wittgenstein, "Tractatus Logico-Philosophicus", 1922)
"All traditional logic habitually assumes that precise symbols are being employed. It is therefore not applicable to this terrestrial life but only to an imagined celestial existence." (Bertrand Russell, 1923)
"The logic of things, i.e., of the material concepts and relations on which the structure of a science rests, cannot be separated by the logic of signs. For the sign is no mere accidental cloak of the idea, but its necessary and essential organ. It serves not merely to communicate a complete and given thought content, but is an instrument, by means of which this content develops and fully defines itself. […] Consequently, all truly strict and exact thought is sustained by the symbolic and semiotics on which it is based." (Ernst Cassirer, "The Philosophy of Symbolic Forms", 1923)
"These symbols are so constituted that the necessary logical consequences of the image are always images of the necessary natural consequences of the imagined objects." (Ernst Cassirer, "The Philosophy of Symbolic Forms", 1923)
"[…] mathematics, accessible in its full depth only to the very few, holds a quite peculiar position amongst the creation of the mind. It is a science of the most rigorous kind, like logic but more comprehensive and very much fuller; it is a true art, along with sculpture and music, as needing the guidance of inspiration and as developing under great conventions of form […]" (Oswald Spengler, "The Decline of the West" Vol. 1, 1926)
"I have now come to believe that the order of words in time or space is an ineradicable part of much of their significance – in fact, that the reason they can express space-time occurrences is that they are space-time occurrences, so that a logic independent of the accidental nature of spacetime becomes an idle dream. These conclusions are unpleasant to my vanity, but pleasant to my love of philosophical activity: until vitality fails, there is no reason to be wedded to one's past theories." (Bertrand Russell," Review of The Meaning of Meaning", 1926)
"Once a statement is cast into mathematical form it may be manipulated in accordance with [mathematical] rules and every configuration of the symbols will represent facts in harmony with and dependent on those contained in the original statement. Now this comes very close to what we conceive the action of the brain structures to be in performing intellectual acts with the symbols of ordinary language. In a sense, therefore, the mathematician has been able to perfect a device through which a part of the labor of logical thought is carried on outside the central nervous system with only that supervision which is requisite to manipulate the symbols in accordance with the rules." (Horatio B Williams, "Mathematics and the Biological Sciences", Bulletin of the American Mathematical Society Vol. 38, 1927)
"In the presence of certain objects of thought or of certain affirmations the child, in virtue of previous experiences, adopts a certain way of reacting and thinking which is always the same, and which might be called a schema of reasoning. Such schemas are the functional equivalents of general propositions, but since the child is not conscious of these schemas before discussion and a desire for proof have laid them bare and at the same time changed their character, they cannot be said to constitute implicit general propositions. They simply constitute certain unconscious tendencies which live their own life but are submitted to no general systematization and consequently lead to no logical exactitude. To put it in another way, they form a logic of action but not yet a logic of thought." (Jean Piaget, "Judgement and Reasoning in the Child", 1928)
"The rule is derived inductively from experience, therefore does not have any inner necessity, is always valid only for special cases and can anytime be refuted by opposite facts. On the contrary, the law is a logical relation between conceptual constructions; it is therefore deductible from upper laws and enables the derivation of lower laws; it has as such a logical necessity in concordance with its upper premises; it is not a mere statement of probability, but has a compelling, apodictic logical value once its premises are accepted."(Ludwig von Bertalanffy, "Kritische Theorie der Formbildung", 1928)
"Metaphysics may be, after all, only the art of being sure of something that is not so and logic only the art of going wrong with confidence." (Joseph W Krutch, "The Modern Temper", 1929)
"The art of discovery is confused with the logic of proof and an artificial simplification of the deeper movements of thought results. We forget that we invent by intuition though we prove by logic." (Sarvepalli Radhakrishnan, "An Idealist View of Life", 1929)
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