On Arguments and Arguants: Reason, Reasoning, and Reasoners
Helping repair the grave of Kurt and Adele Gödel
A Life In Deep Thought
Kurt Gödel was a 20th Century genius who is universally considered to be one of the greatest logicians of all time, and is colloquially legendary as Albert Einstein’s “best friend.”
In our common quest of asking questions and seeking answers, few rival the person nicknamed “Mr. Why” as a child.
Kurt (1906–1978) and his wife Adele (1899–1981) are buried together in the Princeton Cemetery, about 20 yards from John von Neumann, the first person to recognize the significance of Gödel’s work.
As von Neumann described KG’s famous incompleteness theorems (1930/31): “Kurt Gödel’s achievement in modern logic is singular and monumental — indeed it is more than a monument, it is a landmark which will remain visible far in space and time.” Perhaps fittingly, this result came a year after his PhD dissertation: Gödel’s completeness theorem (according to Aaronson’s QCSD, KG showed that 1st order logic is a sufficient sieve to reveal the consistency or inconsistency of one’s axioms).
On a visit to the memorial site, I noticed that the umlaut symbol (ö) on the Gödel gravestone was originally mismarked:
I contacted the Head of the Cemetery to inquire about the mistake, and was surprised to learn it hadn’t been brought to their attention before….
Caretakers: Old Friends & Colleagues
So I decided to reach out to someone who knew Gödel personally, the Turing Award winner Dana Scott. Professor Scott is the honorary President of the Kurt Gödel Society. He had met Gödel already as a graduate student in the late 1950s and made contact again when on the Princeton faculty in the early 1970s. Professor Scott confirmed my suspicion about the very obvious mistake and encouraged me to proceed with requesting the repair. In particular, he suggested I contact the Institute for Advanced Study (IAS) for advice.
The IAS is one of the greatest intellectual communities of all time, and was Gödel’s American home after leaving Europe. We believed they were the proper caretakers to remedy the matter, both legally and morally speaking, because of the lack of a next of kin and their management of Gödel’s work.
After reviewing the situation, the IAS decided to proceed with the remedy. But, after examining the wills of Kurt (’78) and Adele (’81) Gödel, they found an alternative executrix for Adele’s estate. Unfortunately, this person and her husband had since passed, but I was able to track down their daughter, a nice lawyer living in Boston (the executrix of her parents’ estate). She was excited to help and gave the requisite permission.
After interviewing various of the Cemetery’s authorized monument vendors — including learning about the specifics of the headstone material, the repair process, and how it would wear over the long term — and an umlaut design process that included ascertaining 12MM was too small, 13MM was too large, and 12.5MM was just right — I shared the design with the IAS for approval.
Although the Cemetery stated that none of the parties technically had the express legal authority to represent the Gödel estate, the combination of the IAS’s support and my diligence led the Cemetery to authorize the repair. Perhaps this was fitting: a Gödelian grey area of the law.
On December 3rd 2020, the job was completed:
To Truth and Beauty: Our Eternal Quest
Gödel proved that finite and recursive axiomatic formal systems of sufficient expressiveness — including those containing basic arithmetic — were incomplete IF they were consistent. From within such a system, one can’t prove only and all true statements (IT1), including the consistency of the system itself (IT2)! And while one’s first instinct might be to construct a more powerful system that circumvents the limitation, Gödel showed there is indeed no tower above reproach: an infinite ascent of more powerful languages continues to have ever-evasive (unprovable) true statements. Taken to its conclusion, for the finite, there is no final truth to output and no last axiom that will get us there.
While seemingly less emphasized, yet equally or even more beautiful in its participatory logos, Gödel also intimately understood that we ourselves build the castle brick-by-brick: as arguants making arguments, and as part of the System we seek to reason about, we must postulate and adjudicate the axioms (input). For the finite, even in a future that will one day include AI-mathematicians, our job will never be done!
In Budiansky’s wonderful new biography on Gödel [1], one feels his lifelong drive to bring logic and philosophy together, with a clear curiosity as to the nature of insight: that humans can see without seeing, intuiting the truth of the (good) axioms without them being proven (otherwise they'd be theorems... or false). To some, things are just proven relative to axioms: we postulate axioms as true to start, and the source of “truth” is the axioms. But to Gödel, more fundamentally, true things are concretely recognized through the axioms; where the good axioms “exist” atomically as a kind of periodic table of reason, “independent” of us.
Gödel was also inexorably drawn to the idea of an absolute, an ancient idea, which is equivalent ontologically to the question, “is there only 1 Original Possibility which is logically necessary as the exclusive 1st actual?” The concept of the absolute and the specificity of the good axioms– axioms not postulated arbitrarily as games with symbols, but that are instead properly recognized un-arbitrarily [1]– is deeply connected.
That’s because, insofar there is an absolute (an exclusive, specific, singular and necessary original existor), the “naturalness” of the good axioms flows by the Necessity of Nature’s orientation: there’s only 1 way the World could have been — and it needed to be.
I hope the memory of Kurt Gödel and his work will be increasingly enjoyed for generations, centuries, and even millennia to come. He is a wonderful example of the capacity of human achievement, and a platonic sherpa showing us that transcendent experience is available at the place where logical obsession meets playful composition — where Truth meets beauty.
I’m honored to contribute to the maintenance of his legacy, and I’d like to deeply thank the numerous parties involved in making this happen (to my knowledge): DS, SR, HH, RD, JP, LG, CG, JF, CB, MD, MF, and NB.
As you can see from the top of the headstone, many come to pay homage to the great thinker. If you are ever in Princeton, you should go visit!
Perhaps you too will be inspired.
Here is how it happened:
