Logic and completeness
This morning I sat outside at the Alliance Bakery, in unseasonably cool Chicago summer weather, and worked through the chapters my students will have to read in Patrick Hurley’s logic text. Coupled with my own reading and most recent paper attempt, it was a good reminder of the topic this blog has been flirting with: the limitations of logic. Below I’ll talk a little about Gödel’s ideas regarding the subject and move into Nagarjuna’s complementary thoughts. In another post I’ll discuss Hofstadter’s little story about an ant fugue and what the implications of Goedel and Nagarjuna could be for philosophy of science and philosophy of mind. At least, that’s what I’m aiming to do.
The First Incompleteness Theorem
What Gödel points out is that if a formal logical system contains a statement S, "This statement is unprovable" which can be proven true, then we have a contradiction. (Because the statement is provable, despite claims to the contrary.) On the other hand, if the statement is false, then the theory is incomplete–it contains a statement which is not able to be proved. This means no formal logical system can be both consistent and complete.
The Second Incompleteness Theorem
In this theorem, Gödel demonstrates that it is impossible to prove that number theory is consistent, from within number theory itself. The implications of this theorem are generally taken to extend beyond this one example, however, as with the first theorem. It means that no formal logical system can, within itself, contain a statement of its own consistency, on pain of being inconsistent.
The kind of system being discussed here is recursive, computably enumerable, one that can generate all of its axioms with reference to itself and can demonstrate truth-values through proofs containing those axioms. This means that Gödel’s theorems don’t apply to just any group of statements or necessarily make conclusions about religion, God or the Bible, although you can find people online applying them to just about anything.
Connection with Nagarjuna’s logical system
One of the consequences of Gödel’s proofs is that we have demonstrable limits to what humans can formalize. Recursive, computably enumerable systems are not able to 1) be both consistent and complete and 2) prove their own consistency. We can only go so far in our representation of the world around us. This doesn’t preclude rational thought, however, since Gödel used proofs to demonstrate these two theorems. Nagarjuna gets at this idea in another way, from analyzing the nature of language and truth. Mark Siderits, in summarizing Nagarjuna’s work, says "the ultimate truth is that there is no ultimate truth." This is known as a "limit paradox", and its most famous example is the Liar’s Paradox, which is similar to statement S above in Gödel’s first theorem (except it is expressed in terms of truth, not proveability). Can we assign a truth value to the statement "This sentence is false"?
Where Nagarjuna goes beyond Gödel, however, is in driving the limit paradox to the ontological, not merely semantic. The problem is not just that our language is inadequate for reality (as Gödel shows), but that reality itself is paradoxical:
Everything is real and is not real,
Both real and not real,
Neither real nor not real.
This is Lord Buddha’s teaching.
(Interestingly, the last of the "just about anything" links above alludes to this ‘contradictory Eastern’ thought, in the context of extrapolating Gödel to the necessity for an omniscient being to ground truth.)
What Jay Garfield and Graham Priest argue, in the paper I’ve discussed here before, "Nagarjuna and the Limits of Thought," is that these apparently nonsensical contradictions are limit contradictions or paradoxes, and that Nagarjuna does not merely toss logic out the window. As well, because of his ontological paradox, he collapses any dualisms between the ultimate and the conventional (or the noumenal/phenomenal, etc). All that we have is the conventional world of language, which is inadequate to describe the nature of reality. But the nature of reality is that it has no nature, so we are not lost in nihilism.
What does this mean, though (if anything), for my contentions about the mind being physical, all of reality being physical, and the power of science to explain our surroundings? Does Dennett’s program (for example) of explaining evolution as a series of self-generated scaffolds, fall short (this is the contention of the link I talked about above)? What about the human mind? It seems to be built upon a recursive system of some kind.
This is where Hofstadter’s essay comes in, and another paper by Jay Garfield on Nagarjuna and scientific explanation. In short: physicalism doesn’t require that we can have a Theory of Everything, based on quantum particles or Einsteinian relativity, to which we reduce psychology, sociology, history, etc. But unlike what some might contend, the limits of logic do not require a spiritual omniscient being to ground truth, or an immaterial soul to explain the mind.