![]() ![]() He leaves the question to the philosopher. Whether this knowledge springs from an ability of the human mind or from experience, from some cooperation of the two or from some other source, is not for the mathematician to decide. How is this axiom to be interpreted in the older sense and in the more modern sense? The older interpretation: everyone knows what a straight line is, and what a point is. Let us for a moment consider from this point of view any axiom of geometry, for instance, the following: through two points in space there always passes one and only one straight line. It seems to me that complete clarity as to this state of things became common property only through that trend in mathematics, which is known by the name of “axiomatics.” The progress achieved by axiomatics consists in its having neatly separated the logical-formal from its objective or intuitive content according to axiomatics the logical-formal alone forms the subject matter of mathematics, which is not concerned with the intuitive or other content associated with the logical-formal. ![]() In my opinion, the answer to this question is, briefly, this: as far as the propositions of mathematics refer to reality, they are not certain and as far as they are certain, they do not refer to reality. How can it be that mathematics, being after all a product of human thought which is independent of experience, is so admirably appropriate to the objects of reality? Is human reason, then, without experience, merely by taking thought, able to fathom the properties of real things? But there is another reason for the high repute of mathematics, in that it is mathematics, which affords the exact natural sciences a certain measure of certainty, to which without mathematics they could not attain.Īt this point, an enigma presents itself, which in all ages has agitated inquiring minds. For it cannot occasion surprise that different persons should arrive at the same logical conclusions when they have already agreed upon the fundamental propositions (axioms), as well as the methods by which other propositions are to be deduced therefrom. In spite of this, the investigator in another department of science would not need to envy the mathematician if the propositions of mathematics referred to objects of our mere imagination, and not to objects of reality. One reason why mathematics enjoys special esteem, above all other sciences, is that its propositions are absolutely certain and indisputable, while those of all other sciences are to some extent debatable and in constant danger of being overthrown by newly discovered facts. Volume 7: The Berlin Years: Writings, 1918-1921 (English translation supplement) © The Collected Papers of Albert Einstein Translation: from Einstein, Ideas and Opinions Published: 1921 by Julius Springer (Berlin) ![]() Lecture before the Prussian Academy of Sciences As far as the propositions of mathematics refer to reality, they are not certain and as far as they are certain, they do not refer to reality. ![]()
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