Positivism Quotes

“The theory of descent presents biological science with a task that it attempts to fulfill by diligent and complicated experimental investigations. Mendel's law, verified by all observations, offers a basic statistical proposition about the facts of heredity. Further progress will most likely be connected with a precise delineation of the concepts involved and with the construction of axiomatic systems in the sense of the exact natural sciences. The short popular formulas and slogans and the philosophical and political generalizations based upon them are of no significance in this structure.”
― Richard von Mises, Positivism: A Study in Human Understanding

“The second half of the nineteenth century brought two new forms of historical theory which are methodologically oriented toward positivism but undoubtedly include also elements of the absolutistic ideology of Hegel. The so-called materialistic conception of history conceived by Karl Marx and Friedrich Engels regards as the cause of all historical events the defects of the social conditions in which men live and which they strive to improve.
In contrast to this, a more naturalistic movement, which is based upon Darwin and founded upon the modern theory of heredity, asserts that all history consists of the struggle and interplay of the different, essentially unchanging human races. A favorite antithesis in connection with these ideas is whether single "great men" or national groups "make" history, i.e., are the actual bearers of the historical development. In the same way one could ask in physics whether physical phenomena depend more upon electric or upon magnetic forces, or one could build up a theory according to which, e.g., the causes of all events are mechanical (as the philosopher Wilhelm Wundt did in 1866).”
― Richard von Mises, Positivism: A Study in Human Understanding

“All exact science starts with observations, which at the outset are formulated in ordinary language; then these inexact formulations are made more and more precise and are finally replaced by axiomatic assumptions, which at the same time yield definitions of the basic concepts; tautological transformations are then the means of deducing from the axioms conclusions, which after retranslation into common language are tested by new observations. Of no scientific statement or basic concept do we know what changes it will experience in the future.—All of these propositions are supposed to be merely descriptions of the hitherto observed ways of research.”
― Richard von Mises, Positivism: A Study in Human Understanding

“Every physical theory begins with observations and conjectures derived from them, which in the course of the process of testing grow into physical laws and—after the possible addition of suitable propositions regulating linguistic usage—become an axiomatic system (induction). From a sufficiently complete axiomatic system one can draw conclusions by tautological transformations (deduction).”
― Richard von Mises, Positivism: A Study in Human Understanding

“Brouwer, the founder of intuitionist mathematics, has shown that in certain mathematical problems dealing with infinite sets of numbers the elementary rule of the excluded middle is not admissible, without an additional arbitrary assumption. Statements like: there is a number . . . and: there is no number . . . , in this case only seemingly, by virtue of their abbreviated linguistic formulations, have the form of contradictory opposites.”
― Richard von Mises, Positivism: A Study in Human Understanding

“According to L. E. J . Brouwer, the founder of the intuitionist school, the simplest mathematical ideas are implied in the customary lines of thought of everyday life and all sciences make use of them; the mathematician is distinguished by the fact that he is conscious of these ideas, points them out clearly, and completes them. The only source of mathematical knowledge is, in Brouwer's opinion, the intuition that makes us recognize certain concepts and conclusions as absolutely evident, clear, and indubitable.
However, he does not assume that it is possible to list in a precise and complete way all basic fundamental concepts and elementary methods of deduction, which in this sense are to serve as a basis of mathematical derivations. It should always be possible to supplement the once fixed set of assumptions by accepting new ones, if a further intuition leads that way.”
― Richard von Mises, Positivism: A Study in Human Understanding

“It is not possible to state a special act of "understanding" in the humanities which would consist of anything else but reducing to earlier experiences, and subsumption under the known, habitual, and repeatedly observed. It is only the desire to introduce an irrational element into science that leads to the assumption of such a basic concept”
― Richard von Mises, Positivism: A Study in Human Understanding

“None of the three forms of the foundation of mathematics, the intuitionist, the formalistic, or the logicistic, is capable of completely rationalizing the relation between tautological systems and (extramathematical) experiences, which is its very purpose, i.e., to make this relation a part of the mathematical system itself.”
― Richard von Mises, Positivism: A Study in Human Understanding

“Dividing mathematics into a formal system, which progresses according to mechanical rules, and a metamathematics, which is supposed to lead to the justification of the formal system, does not exclude the difficulties that intuitionism has pointed out. The coordination between mathematics (its tautological side) and reality cannot be reached by a mathematicized doctrine and certainly cannot be settled by a consistency proof.”
― Richard von Mises, Positivism: A Study in Human Understanding

“The construction of a "problem calculus" in the sense of Heyting and Kolmogoroff yields a model of logic in which the theorem of the excluded middle does not appear among the basic formulas. The study of such a logic widens our insight into the basic elements of mathematics and, in particular, points out the special position of the so-called indirect proofs within mathematics.”
― Richard von Mises, Positivism: A Study in Human Understanding

“Brouwer, the founder of intuitionist mathematics, has shown that in certain mathematical problems dealing with infinite sets of numbers the elementary rule of the excluded middle is not admissible, without an additional arbitrary assumption. Statements like: there is a number . . and: there is no number . . . , in this case only seemingly, by virtue of their abbreviated linguistic formulations, hm·e the forrn of contradictory opposites.”
― Richard von Mises, Positivism: A Study in Human Understanding

“In agreement with the empiristic conception of science, intuitionism holds that the source of mathematics is the insight which we intuitively comprehend from experience of the external world, but which cannot once and for all be collected in a closed system of axioms.”
― Richard von Mises, Positivism: A Study in Human Understanding

“[d]isregarding certain rather mystic formulations that Brouwer gave to his doctrine, one recognizes his point of view as very close to a radical empiricism. The thesis that the fundamental assumptions of mathematics cannot be formulated in a definitely fixed and completed form, but are subject to continued examination and possible supplementation by intuition (we should prefer to say, by experience […]) corresponds exactly to our conception.”
― Richard von Mises, Positivism: A Study in Human Understanding