Project description

Had Kurt Gödel not thought very deeply about the foundations of mathematics around 1930, there would be no information society in the form we have it today.

Gödel's solitary work was the single most important factor in the development of precise theories of formal languages, which, through the coding that he invented, can be handled by a machine. His work led to precise notions of algorithmic computability from which a direct path leads to the first theoretical ideas of a computer in the work of Alan Turing in 1936 and John von Neumann some years later.

Intellectual background

The development of logic and foundational study has been known to us only in broad outlines until Jean van Heijenoort's pioneering work. He published in 1967 a collection of original papers under the title From Frege to Gödel: The Development of Mathematical Logic 1879-1931. There was, however, no attempt at organised historical work with the sources. Such detailed work is still in a very early state and crucial sources for the development of logic remain unexplored.

Deep intellectual history cannot be achieved if only published work is considered. Comparison to a parallel and partly overlapping development is illuminating. Quantum mechanics, as well as foundations of mathematics, received a decisive impetus around the year 1900, with Planck's discovery of the quantum of action, and the discovery of Russell's paradox and Hilbert's idea of formalisation. All of these developments led to a well-established theoretical account by the early 1930s, right before the Nazi takeover in Germany: quantum mechanics in the second part of the 1920s in the work of the Göttingers Heisenberg and Born, logic and foundations with the full development of first-order logic by the early 1930s by the Göttingers Hilbert, Bernays, and Gentzen, assisted mainly by Gödel's striking discoveries.

Historical work on the development of quantum mechanics began in the 1960s in a massive effort in which every conceivable source was unearthed and made available in several hundred microfilm reels, the Archive for the History of Quantum Physics.

Sadly, nothing of this kind was ever done with logic and foundations. Our present information society owes a part of its existence to a well-hidden but essential line of development. There are several thousand pages of notes by Gödel waiting to be studied, written in a long-forgotten, difficult stenographic script. Similarly, extensive notes for lecture courses during the crucial years, written by Paul Bernays, the central driving force in foundational study around 1930, as well as over five thousand contemporaneous letters exist in the Bernays archive. These have been exploited only sporadically. The fate of many other sources, such as the papers of Gerhard Gentzen and Rose Rand, has been similar.

Rediscovering Kurt Gödel

The main aims of our project are twofold:

  1. To obtain a full, objective picture of Kurt Gödel's crucial role as the logician behind the birth of present information society through a careful study of sources next to published work.
  2. To exploit Gödel's hidden discoveries to the full, an endeavour whose results can be partly foreseen, as with his 1942 proof of independence of the axiom of choice, but in some cases still shrouded in mystery, as with his suggested proof of the independence of the continuum hypothesis "through the Brouwerian method." It should be clear that so far completely unknown scientific developments will surface – one early finding is Gödel's discovery of linear natural deduction some thirty years before Fitch's published works .

Understanding Gödel's unpublished papers requires a combination of mastery of the logical details and the application of professional standards of historical-foundational research to the source material, which, in addition, is only accessible to a reader trained in stenographic writing. So far, much of this work remains to be done. There are two exceptions: the transcription of Gödel's so-called "Zilsel lecture" in Vienna in January 1938 and his Göttingen lecture on the consistency of the continuum hypothesis of December 1939, both published in the third volume of Gödel's Collected Works. Both of these texts were lecture notes, relatively finished in comparison to the private notes. More recently, much effort has been dedicated to producing transcriptions of Gödel's philosophical notebooks that he called "Max Phil," or Philosophical Maxims, in order to better understand Gödel the philosopher. Our project will provide ample material for future generations of philosophers and logicians to complement the picture given by the Philosophical Maxims as well as other works of Gödel's published so far.

Methods and sources

The project consists of theoretically oriented basic research. It uses special methods for the study of mostly unpublished primary sources, the contemporary methods of logic and foundational study to aid in their interpretation, up-to-date formal methods for the development of research themes suggested by the sources, and scholarly methodological standards of historical-foundational research.

The main sources comprise a number of series of notebooks, all written in shorthand and logical symbolism, left virtually untouched except for the Max Phil series:

  1. Resultate Grundlagen is a collection of four notebooks continuously numbered from page 1 to 368, with systematically presented results on logic and foundations that Gödel considered to be in a final form. These were written in a relatively short time span between 1942 and 1943.
  2. Arbeitshefte is an incredibly rich and complicated series of sixteen notebooks on various logical and mathematical topics, altogether some twelve hundred shorthand pages. Most of the notes have been written between the mid-1930s and mid-1940s.
  3. Logik und Grundlagen is a well-organized collection of six notebooks, continuously numbered from page 1 to 440, often referred to simply as Excerptenhefte. It records Gödel's own work, often in the form of concise surveys, as well as the work by others that Gödel had read.
  4. MaxPhil is a series of 15 notebooks (one lost) of philosophical and foundational observations, written between 1934-1955. Lately, the notebooks have been studied and transcriptions exist of most of the notebooks, although they have not been published yet.
  5. Earlier Excerptenhefte and isolated notebooks. There are many isolated notebooks and diaries, such as the "Amerika 1933/34" and "Amerika 1935" notebooks, as well as a series of earlier Excerptenhefte, preceding the Logik und Grundlagen books, the first entries of which are from the early 1940s. Many of these isolated notebooks call for a separate study.

Gödel's notebooks have some overlap in themes, ideas and results that are presented in Max Phil or Resultate Grundlagen often appearing in less finished form in the Arbeitshefte, to give an example. Gödel was meticulous about keeping his notebooks, and they contain detailed lists of contents from which we can infer that he did reread them. There is a separate list of "Progress" entries, in which he summed both his achievements and objectives in numbered points in the Arbeitshefte.

Current research

As of early Spring 2019, our current projects include the following:

  1. Jan von Plato has been occupied by transcribing and translating Gödel's notebooks on incompleteness, the Resultate Grundlagen as well as the later Arbeitshefte. The latter two notebooks are a summary of finished work that remained unpublished, a legacy to be. They turned out to be very readable, modulo the archaic Gabelsberger shorthand, with almost no cancellations or hesitation, and a good coverage that makes them largely self-contained for a trained reader. What were Gödel's central aims in logic and foundations at that time, and why did it all remain unpublished?
  2. Annika Kanckos and Tim Lethen have made remarkable discoveries about Gödel's's ontological argument and are preparing a definitive account of this topic. Tim has been digging into the theological notebooks that reveal surprising new aspects of Kurt Gödel. He has also transcribed a diary-type notebook from 1937-38 that gives a unique view of the last days of Viennese intellectual life.
  3. Maria Hämeen-Anttila has been working on Gödel's early results, views, and remarks on intuitionism and constructivity in mathematics, with materials collected from the Arbeitshefte and other early notebooks. She should be finished with her work, which constitutes her doctoral dissertation, in late spring.
  4. Jan and Maria have edited for publication Gödel's "Princeton Lectures on Intuitionism": These make up to more than a hundred pages. Maria's extensive introduction to the lecture series explains how the notes, written mostly in longhand English, relate to the shorthand Arbeitshefte of the same time, 1941.