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The Gödel Enigma: Unveiling a Hidden Logical Heritage

Periodic Reporting for period 2 - GODELIANA (The Gödel Enigma: Unveiling a Hidden Logical Heritage)

Reporting period: 2020-03-01 to 2021-08-31

Research in logic and foundations of mathematics received an enormous impact through the incompleteness theorems that Kurt Gödel published in 1931. They are among the most iconic scientific achievements of the 20th century. These results led to the development of true formal systems and to the notions of formal languages and algorithmic computability that are connected to such names as Alonzo Church and Alan Turing. The said notions are the direct basis on which the first programming languages and computers were built two decades later. Thus, the present information society owes a - well hidden - debt to the theoretically oriented foundational research that sprung off from Gödel's results.

Strangely enough, there are several thousand pages of notes by this foremost figure of logic that have remained almost completely untouched. Such a situation would be unthinkable in many other fields. Say, with modern physics, every effort would have been made if Einstein - even Gödel's colleague and friend at the Princeton Institute - had left behind such a patrimony!

With Gödel, the difficulty lies in part in the fact that the work is written down in an obsolete, forgotten old German stenographic script called Gabelsberger, a true enigma for those interested in the contents. A second difficulty is the intrinsic logical complexity of the work.

The central aim of the project is to make this work available to future generations of logicians and philosophers. The principal investigator is in a unique position of being able to read the Gabelsberger notes and to interpret their logical content: What they mean in a historical-foundational context, what their significance is for today's research problems in logic, and how they change the view of Gödel as one of the most original thinkers of a century.
The project's first 30 months have proceeded as planned, with two books published, two finished ones in press, a related fifth book to appear next month, and a sixth for which publisher is sought at present. Five of these books are mainly publications of Gödel's own writings, with editorial introductions and explanations. These books are detailed as follows:

1. "Can Mathematics Be Proved Consistent? Gödel's Shorthand Notes and Lectures on Incompleteness." With a transcription and English translation by Jan von Plato, Springer 2020. This book was published last August in the series "Sources and Studies in the History of Mathematics and Physical Science," the most prestigious series in the history of mathematics and exact science. Description: Kurt Gödel (1906–1978) shook the mathematical world in 1931 by a result that has become an icon of 20th century science: The search for rigour in proving mathematical theorems had led to the formalization of mathematical proofs, to the extent that such proving could be reduced to the application of a few mechanical rules. Gödel showed that whenever the part of mathematics under formalization contains elementary arithmetic, there will be arithmetical statements that should be formally provable but aren’t. The result is known as Gödel’s first incompleteness theorem, so called because there is a second incompleteness result, embodied in his answer to the question: Can mathematics be proved consistent? This book offers the first examination of Gödel’s preserved notebooks from 1930, written in a long-forgotten German shorthand, that show his way to the results: his first ideas, how they evolved, and how the jewel-like final presentation in his famous publication "On formally undecidable propositions" was composed. The book contains also the original version of Gödel’s incompleteness article, as handed in for publication with no mentioning of the second incompleteness theorem, as well as six contemporary lectures and seminars Gödel gave between 1931 and 1934 in Austria, Germany, and the United States. The lectures are masterpieces of accessible presentations of deep scientific results, readable even for those without special mathematical training, and published here for the first time.

2. "The Princeton Lectures on Intuitionism." Edited by Maria Hämeen-Anttila and Jan von Plato, in press with the same series as item 1. This book, Gödel's notes for lectures in Princeton in 1941, is preceded by an introduction by the first editor. Publication of this known series of lectures became possible after we found a gap of about ten pages hidden in a completely different place in the Gödel microfilm edition. The book's contents are a basis for further research on items 3-6 of our research agenda. Description: This book, Gödel’s lectures at the famous Princeton Institute for Advanced Study in 1941, shows to what point he had come with Hilbert’s second problem, namely to a theory of computable functionals of ‚€nite type and a proof of the consistency of ordinary arithmetic. It offers indispensable reading for logicians, mathematicians, and computer scientists interested in foundational questions. It will form a basis for further investigations into Gödel’s vast Nachlass of unpublished notes on how to extend the results of his lectures to the theory of real numbers. The book also gives insights into the conceptual and formal work that is needed for the solution of profound scientifi‚c questions, by one of the central fi‚€gures of 20th century science and philosophy.

3. "Gödel on Constructive Foundations and Intuitionism." This book is the doctoral thesis of Maria Hämeen-Anttila, defended in 2020 and published as an internal report of the University of Helsinki. It is the first study ever of Gödel's thinking on a specific topic, one that spans over a great number of years, that is based on a comprehensive usage of the shorthand materials available. Therefore it is a novelty, not just for the contents, but for its scientific methodology. Publication proper with an international publisher is planned, pending revision.

4. "Chapters from Gödel's Unfinished Book on Foundational Research in Mathematics." Edited with an English translation by Jan von Plato. This little book was happily recovered from the enormous wealth of materials in Kurt Gödel's papers, in what turned out to be a real detective work. Gödel had agreed to write a concise introduction to foundational research in mathematics, right after his incompleteness theorems had made him famous at a young age in 1931. This was to be done in collaboration with Arend Heyting, but Gödel never delivered his part and it has been generally believed that he had not made progress with the project. The "detective work" mentioned consisted in recovering, against general belief, two complete chapters and sketches for a final third one in Gödel's hand. Publication of this book will be in the series of The Vienna Circle Institute hosted by the University of Vienna.

The next two books result from collaboration with Tim Lethen, whom the PI was able to employ in a smaller project financed by the Academy of Finland. (His existence was not known to the PI at the time of writing the ERC grant proposal.) Lethen has become the clearly most skilled reader of Gödel's forgotten shorthand and has been an inestimably valuable asset in our project. Even if formally not part of the ERC-funded Gödel project for those works published under his name only, the following must be listed, as they answer to items listed among the main objectives of the present project (items 7 and 9):

5. "Gespräche, Vorträge, Séancen: Kurt Gödels Wiener Protokolle 1937/38." Transkription and commentary by Tim Lethen. Veröffentlichungen des Instituts Wiener Kreis, Springer, May 2021. Description: This book provides detailed transcriptions of two notebooks written by Kurt Gödel in Vienna in 1937/38 in the nearly forgotten Gabelsberger shorthand system. The first of these notebooks, simply entitled as the Protokoll-book, contains notes on conversations Gödel had with people like Rudolf Carnap, Rose Rand, Friedrich Waismann, Else Frenkel-Brunswik, and many others who were—at least to some degree—connected to the Vienna Circle. It also covers detailed descriptions of the regular meetings organized by Edgar Zilsel. The second notebook includes notes on a series of lectures given at the Vienna Psychological Institute, which was led by Karl Bühler at the time. Both notebooks are part of Gödel’s huge Nachlass kept at the Institute for Advanced Study in Princeton, which consists of literally thousands of stenographic pages covering logic and the foundations of mathematics, philosophy, physics, and theology. The now transcribed and commented notes reveal a very personal side of Gödel which has—to a large extent—been unknown to a mainly scientific-oriented audience. The book is of interest to people wanting to learn about Gödel’s personal background in Vienna in the late 1930s as well as his keen interest in philosophy, psychology, and parapsychology.

6. "Kurt Gödel's Notizen zu Quantenmechanik." Edited by Oliver Passon and Tim Lethen. Book manuscript 171 p. The PI found out in 2018, at the project's start, that Gödel had written two systematic notebooks on foundational questions in quantum mechanics. These have been transcribed by Lethen and commented by Oliver Passon, a physics professor from Wuppertal University and expert on the topic with whom we sought collaboration in this difficult area. We quote from Prof. Wolfgang Schleich who generously offered to write a foreword to the book: "Der interessierte Leser und Kenner der Quantenmechanik wird in den Gedanken von Gödel die Kernaussagen der Quantenmechanik wiederfinden, aber auch vieles, das zum Nachdenken anregt. Die Antworten auf seine zahlreichen und auf den ersten Blick unverständlichen Fragen haben durchaus das Potential eine dritte Quantenrevolution auszulösen." To what revolutions Gödel's ideas in quantum mechanics lead remains to be seen.

Work finished but for various reasons not yet ripe for publication include transcription of Gödel's notebook series "Resultate Grundlagen," 368 pages, together with an English translation. This is work in collaboration with Hämeen-Anttila and von Plato, where the former perfected her skills in reading Gödel's shorthand. This book is a summary of Gödel's formal results from his sixteen "Arbeitshefte" workbooks. Of these latter, several hundred preliminary transcriptions have been made. Because of the very wide range of topics the universalist Gödel covers, we seek expert help in fields such as set theory, to assess the importance of Gödel's "Results on Foundations," and plan meetings to this purpose. Publication will be some time during the second half of the project.

These are the main achievements so far. There are in addition numerous articles that contribute to our research agenda as outlined in nine items.
The research group is unique in the world, and it the first one to to make Gödel's so far unknown work in logic available. It is to be foreseen that all the objectives of the project will be achieved, and much more.
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