As von Mises sees it, applied mathematics is the field of central importance for every attempt at a philosophical picture of our world. In drawing such a picture, the central task is to understand the relation between the direct sense observation of the experimental physicist and the conceptual system of science, which consists of expressions such as "increase of entropy" or "principle of relativity." Most physicists are inclined to say that the picture drawn and the principles devised by our inductive ability are eventually checked by actual measurement of physical quantities like length, weight, electric charge, etc., but they use the expression "measurement of a length" in a perfunctory way, forgetting that no numerical value can ever be assigned to a length by a single measurement. In fact, a long series of measurements is needed from which eventually "the value of the length" can be computed.
In contrast to the procedure of the physicist, applied mathematics concentrates its efforts on the problem: how can "values of length" be computed from sets of different readings? And, in a general way, it has become the business of applied mathematics to investigate the connection between "direct pointer readings" and the abstract conceptions (as length, or electromagnetic field) that occur in all laws of science - in Newton's mechanics as well as in Maxwell's theory of the electromagnetic field. This problem of connection between sense observations and abstract principles has always been the critical point in the philosophy of science. As we see the problem, it is tackled quite directly by the methods of applied mathematics, and it is in this sense that von Mises has dealt with the tasks of "Applied Mathematics and Mechanics," building upon the ideas of the great Austrian scientist and philosopher Ernst Mach, who regarded both science and its philosophy as theories of sensations.
Investigating this problem of connection, von Mises discovered soon the all-important role that statistics plays in this task. He examined and presented this role in a precise and lucid way and removed the obscurity that bad been inherent in the traditional presentation of statistics and probability.
Thus a very rational line of thought connects von Mises' work in mechanical engineering (Theorie der Wasserräder, Fluglehre, etc.) with his investigations into the logical foundations of probability. If we study his work in fields of such complex structure as plasticity or turbulence, we never find smug contentment with rules of thumb or quick transitions from a vague assumption to a long row of figures, but meet everywhere the attempt to analyze these difficult problems in terms of rational mechanics and to examine critically "die bisherigen Ansätze." We see him, on the other hand, freeing probability theory from semi-mystical formulations, according to which the concept of probability is derivable from our "ignorance." To do this, he had to construct a system of statements, based, as is every physical theory, upon the combination of a formal system and the physical interpretation of its terms. In probability as well as in mechanical engineering, von Mises has investigated the complete range of problems that stretches from the construction of a suitable formal system to methods of numerical computation. Looking at the great variety of topics in his papers, we may marvel at the broad abilities of the author, but we must admire the work of a mind who is forever searching for the central problem hidden under the apparent variety.
Von Mises has summed up his ideas in several books, which are not the least known for the attractive presentation of topics that had suffered greatly in earlier presentations. Coherence in the large and precision in the small - both intimately connected with the nature of von Mises work - reappear in his style and give depth and clarity to his writing. With the Differential und Integralgleichungen der Mechanik und Physik he gave to those who wished to apply modern mathematics to physics and engineering problems, a lucid account of the mathematical fundamentals. This work, which first appeared in 1925, was reproduced (in German) in the United States during the Second World War for the benefit of all those working in defence research. In Probability, Statistics, and Truth, von Mises offered a brilliant presentation of his general ideas on probability to a wider class of readers; it is perhaps still the best book to make a general scientist or, for that matter, any well-educated person familiar with the conception of probability and its applications.
In Positivism, a Study in Human Understanding, von Mises gave us a summary of his views on many topics in science and life. In this book the word "positivism" is not meant to designate a sectarian doctrine of some philosophical school; von Mises uses it rather to characterize a way of presenting his views that takes its cue from the methods of science and should establish understanding among those willing to drop prejudice and accept what experience and reason suggest. Throughout the whole book von Mises does not fail to emphasize that the role played by human imagination is not less important in the invention of scientific theories than it is in the works of art and in religion. Perhaps it is best to characterize this book by the author's own words. "Positivism does not claim that all questions can be answered rationally, just as medicine is not based on the premise that all diseases are curable, or physics does not start out with the postulate that all phenomena are explicable. But the mere possibility that there may be no answers to some questions is no sufficient reason for not looking for answers, or for not using those that are attainable." He stresses the point that too many people interpret the present world struggle as a battle between two ideological systems of extremely metaphysical character. "If this goes on", writes von Mises, "the predictions of those who believe that the next step toward the solution of the basic sociological problems must come from physical annihilation of one of the two groups of people will be borne out".
"In our opinion, the only way out is less loose talk and more criticism of language, less emotional acting and more scientifically disciplined thinking, less metaphysics and more positivism."
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