🪷On Mind: On Reasoning (1940-1949)

"It is difficult, however, to learn all these things from situations such as occur in everyday life. What we need is a series of abstract and quite impersonal situations to argue about in which one side is surely right and the other surely wrong. The best source of such situations for our purposes is geometry. Consequently we shall study geometric situations in order to get practice in straight thinking and logical argument, and in order to see how it is possible to arrange all the ideas associated with a given subject in a coherent, logical system that is free from contradictions. That is, we shall regard the proof of each proposition of geometry as an example of correct method in argumentation, and shall come to regard geometry as our ideal of an abstract logical system. Later, when we have acquired some skill in abstract reasoning, we shall try to see how much of this skill we can apply to problems from real life." (George D Birkhoff & Ralph Beately, "Basic Geometry", 1940)

"The first step in discussing a proposition with a view to proving it true or false is to discover the earlier propositions upon which it depends. A few of these earlier propositions must remain unproved and be taken for granted, for we must have some place from which to begin our chain of reasoning. We cannot prove everything. The propositions that we take for granted without proof are called assumptions. When we criticize an argument or assert that a certain proposition is true or false, we should first discover the assumptions upon which the argument or proposition depends." (George D Birkhoff & Ralph Beately, "Basic Geometry", 1940)

"The main thing geometry gives us is the ideal of a logical system and of precise thinking, and some acquaintance with the language in which logical arguments are usually expressed. The answer to a problem in actual life can often be obtained by further investigation of the actual facts, while in geometry it can always be obtained by reasoning alone." (George D Birkhoff & Ralph Beately, "Basic Geometry", 1940)

"It is a remarkable fact that complaints of the want of clearness and rigour in that part of Mathematics which respects calculation, - whether it be called 'Arithmetic', Universal Arithmetic', 'Mathematical Analysis', or aught else, - recur from time to time, now uttered by subordinate writers, now repeated by the most distinguished of the learned. One finds contradictions of the theory of 'opposed magnitudes'; - another is merely disquieted by 'imaginary quantities'; - a third finally meets difficulties in 'infinite series', either because Euler and other distinguished mathematicians have applied them with success in a divergent form, while the complainant thinks himself convinced that their convergence is a fundamental condition, - or because in general investigations general series occur, which, precisely because they are general, can be neither accounted divergent nor convergent. ... The [...] question: how may the paradoxes of calculation be most securely avoided? - obliges us to submit to a very exact examination of the subject of mathematical analysis, its first and simplest ideas, as also the methods of reasoning which are applied to it." (Martin Ohm, "The Spirit of Mathematical Analysis and its Relation to a Logical System", 1842)

“It is his intuition, his mystical insight into the nature of things, rather than his reasoning which makes a great scientist." (Karl R Popper,"The Open Society and Its Enemies", 1945) 

"Whenever we discuss a geometric statement in order to prove it, we refer to previously proved statements for 'evidence'. In this process of referring back to earlier statements that have already been established by proof, it is clear that some few of the earliest statements must of necessity remain unproved - that is, they have to be taken for granted, since we must begin our chain of reasoning somewhere. It is not possible to prove every statement. Those statements which are thus taken for granted without proof are called assumptions. Hence in all logical reasoning, two considerations are of prime importance: (1) the meanings of the terms used, and (2) the basic assumptions made." (William L Schaaf, "Plane and solid geometry for home study", 1944)

"Figures alone prove or disprove nothing. Only the conclusions drawn from the collected material can do this. And these are theoretical." (Ludwig von Mises) "Geometry is the science of correct reasoning on incorrect figures." (George Pólya, "How to Solve It", 1945)

"Heuristic reasoning is good in itself. What is bad is to mix up heuristic reasoning with rigorous proof. What is worse is to sell heuristic reasoning for rigorous proof." (George Pólya, "How to Solve It", 1945)

"Heuristic reasoning is reasoning not regarded as final and strict but as provisional and plausible only, whose purpose is to discover the solution of the present problem. We are often obliged to use heuristic reasoning. We shall attain complete certainty when we shall have obtained the complete solution, but before obtaining certainty we must often be satisfied with a more or less plausible guess. We may need the provisional before we attain the final. We need heuristic reasoning when we construct a strict proof as we need scaffolding when we erect a building." (George Pólya, "How to Solve It", 1945)

"If scientific reasoning were limited to the logical processes of arithmetic, we should not get very far in our understanding of the physical world. One might as well attempt to grasp the game of poker entirely by the use of the mathematics of probability." (Vannevar Bush, "As We May Think", Atlantic Monthly, 1945)

"It is his intuition, his mystical insight into the nature of things, rather than his reasoning which makes a great scientist." (Karl R Popper, "The Open Society and Its Enemies", 1945)

"The cookbook gives a detailed description of ingredients and procedures but no proofs for its prescriptions or reasons for its recipes; the proof of the pudding is in the eating. [...] Mathematics cannot be tested in exactly the same manner as a pudding; if all sorts of reasoning are debarred, a course of calculus may easily become an incoherent inventory of indigestible information." (George Pólya, "How to solve it", 1945)

"We need heuristic reasoning when we construct a strict proof as we need scaffolding when we erect a building." (George Pólya, "How to solve it", 1945)

"That deep emotional conviction of the presence of a superior reasoning power, which is revealed in the incomprehensible universe, forms my idea of God." (Albert Einstein [in  Lincoln Barnett's "The Universe and Dr. Einstein", 1948) 

"As everyone knows, it is not always easy to think. By thinking, we mean computing, reasoning, and other handling of information. By information we mean collections of ideas - physically, collections of marks that have meaning. By handling information, we mean proceeding logically from some ideas to other ideas - physically, changing from some marks to other marks in ways that have meaning." (Edmund C Berkeley, "Giant Brains or Machines that Think", 1949)

"The amount of human effort needed to handle information correctly depends very much on the properties of the physical equipment expressing the information, although the laws of correct reasoning are independent of the equipment." (Edmund C Berkeley, "Giant Brains or Machines that Think", 1949)