Periodic Reporting for period 4 - GODELIANA (The Gödel Enigma: Unveiling a Hidden Logical Heritage)
Berichtszeitraum: 2023-03-01 bis 2024-08-31
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 was to make this work available to future generations of logicians and philosophers. The principal investigator was 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.
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 was to make this work available to future generations of logicians and philosophers. The principal investigator was 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 has exceeded the initial expectations by a wide margin. The work performed is best reflected in the published products of the project, five books published and a sixth in press, and close to twenty articles in the very best journals and conference proceedings appeared or in print. The books are mainly publications of Gödel's own writings, transcribed from his forgotten Gabelsberger shorthand and translated into English with editorial introductions and explanations. They are detailed as follows:
1. "Can Mathematics Be Proved Consistent? Gödel's Shorthand Notes and Lectures on Incompleteness." Jan von Plato, Springer 2020. This book was published 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. Gödel 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.
2. "The Princeton Lectures on Intuitionism." Edited by Maria Hämeen-Anttila and Jan von Plato, 2021. Gödel's lectures at the famous Princeton Institute for Advanced Study in 1941 show to what point he had come with a theory of computable functionals of finite type. 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 scientific questions, by one of the central figures of 20th century science and philosophy.
3. "Chapters from Gödel's Unfinished Book on Foundational Research in Mathematics." By Jan von Plato, 2022. 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 is in the series Vienna Circle Institute Library, hosted by the University of Vienna.
4. "Results on Foundations." Edited by Maria Hämeen-Anttila and Jan von Plato, 2023, in the same series as items 1 and 2. This book is a 368-page summary of Gödel's finished results on logic and foundations, an enormously rich source of mostly unknown results that have already been used in contemporary work on logic.
5. "Portrait of Young Gödel." By Jan von Plato, 2024, in the same series as item 3. This book displays Gödel's education and development, through sources from Gödel's last year at high-school through sources on his university studies to his first result, the completeness theorem of quantificational logic.
6. "Mathematical Logic in Vienna." Edited by Jan von Plato, in press for 2024. This book is the record of a seminar Gödel and his professor Hans Hahn gave in Vienna in 1931-32. It gives a unique view of mathematical logic right after Gödel had published his incompleteness theorem. The book also contains a surprise discovery in the Gödel papers, a lecture on intuitionistic logic Gödel gave in February 1933, amply explaining his numerous results he had published in very short, laconic publications of the time.
Results of the project have been presented in numerous talks and conferences. In particular, there was a meeting for specialists in Helsinki in 2019, a one-day Gödel program in a conference in Vienna also 2019. Next, a Gödel conference was organized in collaboration with the University of Tübingen in 2021, amidst great difficulties caused by the Covid epidemic. Another such meeting was organized in 2023, in Tübingen. Of the single talks, of particular mention is a Nordic Logic Seminar presentation by the PI in September 2023 that was attended, by Zoom, by over 700 persons from five continents.
1. "Can Mathematics Be Proved Consistent? Gödel's Shorthand Notes and Lectures on Incompleteness." Jan von Plato, Springer 2020. This book was published 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. Gödel 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.
2. "The Princeton Lectures on Intuitionism." Edited by Maria Hämeen-Anttila and Jan von Plato, 2021. Gödel's lectures at the famous Princeton Institute for Advanced Study in 1941 show to what point he had come with a theory of computable functionals of finite type. 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 scientific questions, by one of the central figures of 20th century science and philosophy.
3. "Chapters from Gödel's Unfinished Book on Foundational Research in Mathematics." By Jan von Plato, 2022. 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 is in the series Vienna Circle Institute Library, hosted by the University of Vienna.
4. "Results on Foundations." Edited by Maria Hämeen-Anttila and Jan von Plato, 2023, in the same series as items 1 and 2. This book is a 368-page summary of Gödel's finished results on logic and foundations, an enormously rich source of mostly unknown results that have already been used in contemporary work on logic.
5. "Portrait of Young Gödel." By Jan von Plato, 2024, in the same series as item 3. This book displays Gödel's education and development, through sources from Gödel's last year at high-school through sources on his university studies to his first result, the completeness theorem of quantificational logic.
6. "Mathematical Logic in Vienna." Edited by Jan von Plato, in press for 2024. This book is the record of a seminar Gödel and his professor Hans Hahn gave in Vienna in 1931-32. It gives a unique view of mathematical logic right after Gödel had published his incompleteness theorem. The book also contains a surprise discovery in the Gödel papers, a lecture on intuitionistic logic Gödel gave in February 1933, amply explaining his numerous results he had published in very short, laconic publications of the time.
Results of the project have been presented in numerous talks and conferences. In particular, there was a meeting for specialists in Helsinki in 2019, a one-day Gödel program in a conference in Vienna also 2019. Next, a Gödel conference was organized in collaboration with the University of Tübingen in 2021, amidst great difficulties caused by the Covid epidemic. Another such meeting was organized in 2023, in Tübingen. Of the single talks, of particular mention is a Nordic Logic Seminar presentation by the PI in September 2023 that was attended, by Zoom, by over 700 persons from five continents.
The research group was unique in the world with its aim to make Gödel's so far unknown work in logic available. This task has been performed splendidly. The results are described in the previous section. To these can be added the results of a smaller project of the PI, financed by the Academy of Finland, in which Tim Lethen has produced two books with Gödel materials and close to twenty articles. Of particular mention is the "Notizen zur Quantenmachenik" (Remarks on quantum mechanics), a book-length set of notes by Gödel on foundational problems of quantum mechanics, published by Springer in 2021.