ENDING THE FOUNDATIONAL CRISIS OF MATHEMATICS

[GÖDEL] & CARNAP, RUDOLF + AREND HEYTING + JOHANN v. NEUMANN, etc.

[Bericht uber die 2. Tagung für Erkenntnislehre der exakten Wissenschaften Königsberg 1930]. Die Logizistische Grundlegung der Mathematik + Die intuitionistische Grundlegung der Mathematik + Die formalistische Grundlegung der Mathematik. [In: Erkenntnis. Zweiter Band - Heft 2-3 (Annalen der Philosophie - Band X Heft 2-3) herausgegeben von Rudolf Carnap und Hans Reichenbach.].

Leipzig, Felix Meiner, 1931. The entire volume present. 8vo. Orig. printed green wrappers. Sunning to spine, and a bit of soiling and minor wear to front wrapper w. minor loss of upper layer of paper at two pages, not gione through paper. A few leaves w. marginal markings, quite discreet. Library marking to inside of front wrapper, library stamp to title-page (Mathematical Institute of the University of Amsterdam). Overall a fine and nice copy. Pp. (91) - 105 + (106) - 115 + (116) - 121. The entire volume: (2) pp., Pp. (91) - 190.

First edition of the Erkenntnis-volume from the Königsberg congress of 1930, where Gödel introduced his incompleteness results and Carnap, Heyting and von Neumann held the seminal papers (here printed for the first time) that ended the "Grundlagenkrise der Mathematik" (foundational crisis of mathematics). It is also in this volume that the seminal discussions following Gödel's announcements of his results are printed for the first time ("Discussion on the Foundation of Mathematics", between Gödel, von Neumann, Carnap, Hahn, Reidemeister, Heyting, and Scholz) (Gödel, Collected Works, 1931a) as well as the article which inaugurated the logicist foundation of mathematics, in which the modern sense of "logicism" is introduced (Carnap's contribution).

In Königsberg in September 1930, Gödel presented his incompleteness results, a landmark in mathematical logic, at the second congress of scientific epistemology, -a congress which proved to be a turning point in the history of philosophical and mathematical logic. It is the papers presented at this congress which are printed in the present volume, apart from the contributions by Gödel and Scholtz (which were printed elsewhere) together with the seminal discussions that followed the presentation of the papers. The groundbreaking papers that are printed here include Carnap's "Die Logizistische Grundlegung der Mathematik", which furthermore introduced the modern sense of the term "logicism", Arend Heyting's "Die intuitionistische Grundlegung der Mathematik" and Johann von Neumann's "Die formalistische Grundlegung der Mathematik" as well as papers by Neugebauer, Reichenbach and Heisenberg.

The present papers, as well as the following discussion, mark a turning point in the history of logic and a cornerstone in the future development of the field. The so-called "Foundational Crisis of Mathematics" was a phase within mathematics begun in the early 20th century due to the search for proper foundations of mathematics and the uncertainty of this quest, which was supported by the many difficulties that philosophy of mathematics faced at the end of the 19th and beginning of the 20th century . The crisis took its actual beginning with the publication of Russell's "principles of Mathematics" of 1903, culminated in the 1920'ies with the main advocates of Formalism and Intuitionism respectively, Hilbert and Brouwer, in what is called the "foundational struggle of mathematics", and ended with the present volume in 1931, following the congress of 1930.

With the discovery of non-Euclidean geometry in the 18th century, it became evident that not only one sort of mathematics was possible, and even that some propositions could be true in one mathematical system, but false in another. This was the actual basis for the awareness of a mathematical foundation in the mathematical public, which again was the basis for the fact that the question of the foundation of mathematics could develop -and could develop into an actual crisis. During the first 30 years of the 20th century, almost all great mathematicians worked on their answer to the question of the correct foundation of mathematics, and thus it came to a crisis that developed into a struggle. It is this struggle and crisis that Carnap, Heyting and von Neumann break in 1930, where they present the three great positions of the struggling years: logicism (Carnap), intuitionism (Heyting) and formalism (von Neumann), and it is these three papars that pave the way for the discussion that follows, "Diskussion zur Grundlegung der Mathematik", between Gödel, Hahn, Carnap, Heyting, von Neumann, Reidemeister, and Scholz.

They all presented their positions in the most conciliatory manner, out of the comprehension that all parties who had contributed to the crisis had also contributed because they wanted to solve it, and because they were also searching for the best possible foundation. It is also this comprehension that Hilbert takes over, when he, in his program, sets out to prove the contradiction-freedom of infinite mathematics on the basis of finite arithmetic.
Thus, the seminal papers in the present volume once and for all ended the foundational crisis of mathematics and any fear of new antinomies.

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