## Monday, March 6, 2017

### A Paradoxical Life

Raymond Smullyan

It's been a month now since the death of Raymond Smullyan. Wikipedia calls Smullyan a "mathematician, concert pianist, logician, Taoist and philosopher" - for brevity they leave out writer of chess puzzles, amateur magician and I am sure much more besides. Smullyan belongs to a select group of writers - the masters of illogical logic. Others in this group are Lewis Carroll, G K Chesterton and Zhuangzi. Of these writers, if Carroll was the best pure writer and Zhuangzi the most poetic, it must be admitted that Smullyan was the brightest, the most gentle and had the strongest technical philosophical machinery. Nothing could be more pleasant to me than an admiring stroll through his life's work.

Raymond Smullyan

On a technical level, Smullyan's greatest achievement is probably his contributions to the interpretation of Gödel's Theorem. Working along the same path as Quine, Tarski and Rosser, Smullyan reworked the technical machinery of Gödel into a more elementary context.

Readers of Gödel's proof will note its difficulty. His proof involves quite complex operations on arithmetic and assumptions whose relevance to ordinary logic was deeply unclear ($$\omega$$-consistency being the most important). Gödel's arithmetization of logic was extremely inefficient - it involved nested exponentials and used factoring in interpretation - and inefficiency matters. Human beings who are not Kurt Gödel have difficulties working with at this level of abstraction.

While working out his Ph.D. thesis, Smullyan found he was able to work out Gödel's proof without inefficient number theory operations. The technical device that allowed him to do so was what he called a "norm" and what is now called a "quine".

What is a quine? Smullyan gifted us with a simple scenario in which a quine may occur. Imagine you are holding Smullyan's first book of logic puzzles. You may then think to yourself:

"What is the name of this book?" What Is The Name Of This Book?

A quine then is an expression preceded (or followed) by its quotation. The distinction between a sentence and it's quotation - analogous to that between an object and a representation of an object - is extremely deep and powerful.

Smullyan introduced the quine to the world in his 1957 paper, which is still worth reading. By replacing the exponential diagonalization function with the simpler norm function, Smullyan wrote Gödel's theorem in a complexity class that could be realistically worked out by a computer. Smullyan is able to arithmetize logic by working with strings of numbers directly instead of by working through exponentials.

Next to this, Smullyan's biggest techincal contribution is simplifying and formalizing "Analytic Tableaux", which are an alternative to Hilbert-style logic... well, that needs some clarification. In a Hilbert-style logic, one has many axioms and one rule for producing tautologies. A  Hilbert-style proof is more or less syntactical, of the form $$A \to B$$. In an Analytic Tableaux, one systemically generates the models in which $$A$$ and checks if $$B$$. If this happens, we get that $$A \vdash B$$. This is equivalent to $$A \to B$$ but found by examining models instead of using syntactical rules. These systems have their roots in the unfortunate Gerhard Gentzen's "Sequent Calculi" and Gödel's proof that a provable sentence was true in all models.

This method greatly simplified work with "Modal Logic", in which each model is identified with a possible world. If it turns out that in all models where it is raining I stayed indoors, then necessarily if it is raining then I stayed indoors. These sorts of things were not easy to work out with merely syntactical methods! Further, modal logic allows one to not just have a logic of possible worlds - and therefore a metaphysics - but also a logic of knowledge and provability! (This program also begun by Gödel - ol' Kurt really was the greatest logician in history.) Smullyan's informal introduction is a great classic.

Smullyan's literary brilliance extended to his technical work. The set of textbooks he wrote in the 90's are the greatest textbooks on mathematical logic and set theory ever written.

Raymond Smullyan

Unlike many famous mathematicians, Smullyan was not a child prodigy in mathematics. Smullyan did not publish until he was 38 and got his Ph.D. at 40. To compare, Gödel published the incompleteness theorems at 25 and essentially stopped publishing in his 40s.

Perhaps because he had a life before mathematics, Smullyan wrote very ably on subjects unrelated to these austere subjects. In fact, Goodreads lists Smullyan as "author of The Tao Is Silent" rather than as the man who sped up diagonalization. The Tao Is Silent is a book on philosophy in the old fashioned sense of worldview or whatever German words one likes to use. It's a very light book, a purposefully light popularization of the school of thought embodied by Lao Tse or Zhuangzi.

Smullyan concieves of that ancient school as not being so different from the so-called "ordinary language" philosophy popular in Oxford in the 60's. A typical trick of the Taoist-Rylean might be to proclaim

Thought does not exist!

The Taoist-Rylean is then questioned by the student or upbraided by a phony logician. How can one think that thought does not exist? The Taoist-Rylean can then explain (in their loopy way) that thought is not an action, but a collection of actions and a rather dubious collection at that. When one ceases to think of thought as a unitary action that one does separately from visualizing, speaking syllables to oneself, doing mental arithmetic, etc., one will find the world more comprehensible.

Smullyan's case for this type of deflation and relaxation is quite good. Like Wittgenstein, Smullyan often points to the quietist of this ordinariness but it is much more believable that Smullyan actually achieved this sort of inner peace. Smullyan even discusses morality in insightful ways, such as his famous dialog Is God A Taoist?.

However, there is much to complain about Smullyan's exposition of Taoism. Smullyan was much too peaceful and gentle to describe the life of the classical Chinese. As Laozi said

Heaven And Earth are not beneficent,
Neither is the sage beneficent.

Of course, Laozi believed that men would be good if they followed their nature and only illusion causes them to be ill. But Laozi would also admit the terrible force of illusion. Life is difficult, except for the sage, and in the context of that difficulty great wickedness can become justified. It is for this reason that the Japanese government prescribed Zen Buddhism - the branch most influenced by Daoism - for samurai.

Smullyan was a very capable philosopher. Not only did he do great work on technical logic, but also wrote some of the few Socratic dialogs worth reading in the 20th Century. Let's look at one of his dialogs as an example.

Did you finish reading it? Okay, I'll wait.

There, wasn't that interesting? The subject of this dialog is whether we have transcendental and unquestionable access to our stream of consciousness. Smullyan gives a common sense example that we are not - and it is common sense in spite of Smullyan's attempt to wake us up with a fantastical machine. This is an amazing feat of anti-phenomenology - even Daniel Dennett in Consciousness Explained admits of transcendental access!

I don't think anyone could have written this who was not a magician. Feeling out the distinction of what we think we think and what we think is easier to a serial trickster. Frank says that "[The book] seems red to me", his phrasing (as the psychologist - who is really just the thought reading machine in flesh and blood form - points out) reveals instantly that he doubts his own vision (for good reason as the introduction points out).

Frank sees red but has residual doubts about his own eyes, then grows frustrated when told he doesn't have transcendental access to his own seeing of red. The second situation with the psychologist - which, again, is identical to the machine - he finds that he can accept this.

It is the ordinaryness of the psychologist's language that makes Frank able to accept the results, not any change in situation. This captures, I think, both the difficulty people have with Dennett, despite his brilliant expositions and heterophenomenology's essential correctness.

Raymond Smullyan

I shall end this tribute with perhaps the deepest Smullyan of them all - Smullyan the concert pianist: