Philosophy of Mathematics Seminar (Monday - Week 5, HT26)

Abstract: In this talk I will consider the claim that Gödel’s incompleteness theorems imply that “the mind cannot be mechanized,” where this statement shall be understood as the statement that “the mathematical outputs of the idealized human mind do not coincide with the mathematical outputs of any idealized finite machine,” and where here the informal notion of “an idealized finite machine” shall be understood in terms of the mathematically precise notion of a Turing machine (or, equivalently, a formal system).

I will begin with Gödel’s view as to the philosophical significance of his incompleteness theorems in connection with our question. In his Gibbs Lecture (1951), Gödel  argued that his incompleteness theorems imply that either “the mind cannot be mechanized” (as understood in the more precise sense described above) or “there exist absolutely undecidable propositions in mathematics” (in the sense that there exist mathematical propositions which can neither be proved nor refuted by “the idealized human mind”). He referred to this disjunction as a “mathematically established fact,” a fact which was “very decidedly opposed to materialist philosophy” (because it showed that either “mind surpasses matter” or “mathematics surpasses mind”). He believed the first disjunct (“the mind cannot be mechanized”), but he did not think he was in position to prove it. The obstacle, as he saw it, was the lack of an adequate resolution of the paradoxes involving self-applicable concepts, such as the concept of truth and the concept of knowledge, when understood in their full generality. But he held out hope that “[i]f one could clear up [these paradoxes], one would get a clear proof that mind is not machine.”

Over four decades later, in his book Shadows of the Mind, Roger Penrose gave a remarkable argument for the first disjunct. The argument, as distilled into final form, which appears in his 1996 paper “The Doubting of a Shadow”, and reappears in his 2011 paper “Gödel, the Mind, and the Laws of Physics,” runs as follows, where here F is a formal system, and where I have corrected what I take to be two misprints (for the second- and third-to-last instances of ‘F’ in the following quotation, the original has ‘F*’):

Though I do not know that I necessarily am F, I conclude that if I were, then the system F would have to be sound and, more to the point, F* would have to be sound, where F* is F supplemented by the further assertion ‘I am F.’ I perceive that it follows from the assumption that ‘I am F’ that the Gödel statement G(F*) would have to be true and, furthermore, that it would not be a consequence of F*. But I have just perceived that ‘if I happened to be F, then G(F*) would have to be true,’ and perceptions of this nature would be precisely what F is supposed to achieve. Since I am therefore capable of perceiving something beyond the powers of F, I deduce that I cannot be F after all.

This is a brilliant argument, one that is far more sophisticated than any of the earlier arguments for first disjunct. If you stare at it for a while you will find yourself convinced. If you stare at it a little longer you will find that something interesting emerges.

In the main part of this talk I will examine this argument in detail, step by step, by providing a charitable reconstruction in a framework where the underlying assumptions governing the central notions—most notably, the concept of truth and the concept of knowledge—are made explicit. It turns out, that in order to render the argument precise, one must treat the concepts of truth and knowledge in their full generality, as self-applicable concepts, just as Gödel conjectured, and this means that one must contend with the paradoxes of self-applicable concepts. Fortunately, now, many decades later, we have a greater understanding (if not the ultimate resolution) of how to treat such self-applicable concepts. And so, perhaps we have here exactly the sort of argument that Gödel had hoped we might one day have, when he conjectured that “[i]f one could clear up the [paradoxes of self-applicable concepts], one would get a clear proof that mind is not machine.”

In order to render the argument precise, building on work of Feferman, I will introduce a formal system, DTK, with axioms for a self-applicable concept of truth and a self-applicable concept of knowledge. It will be shown that the system is consistent and that within it one can both express and prove Gödel’s Disjunction. This raises the question: “Which disjunct holds?” I shall show that neither disjunct is provable or refutable in DTK, and, moreover, that this independence result is robust in the sense that (i) independence persists when one strengthens the principles governing knowledge and (ii) independence persists when one alters the underlying theory of truth to any of the presently available options. I shall conclude with a disjunctive conclusion of my own: Either the statements that “the mind cannot be mechanized” and “there are absolutely undecidable statements” are indefinite (as the philosophical critique maintains), or they are definite and the above results provide evidence that they are about as good examples of “absolutely undecidable” propositions as one might find

Registration: Please write to daniel.isaacson@philosophy.ox.ac.uk to request the Zoom link.

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