Incomplete, Uncomputable & Inconsistent

A Mathematical & Philosophical Exploration of Mechanical Minds & Conscious Computers

Philosophy of mind is a broad and rich area of study investigating fascinating topics such as:

  1. The Mind: What is the nature of the human mind? Is it best thought of as a computer?
  2. Consciousness: How are we conscious to begin with? What is consciousness?
  3. Artificial Intelligence: Can AI systems become conscious? Can they really “think”?

Engaging with contemporary philosophical works on consciousness or AI requires a familiarity with mathematical principles and methods originating from the “Foundations Crisis” of mathematics in the 18 century and the flurry of mathematical discovery which ensued. Some of the most important results are the Incompleteness Theorems produced by Kurt Gödel. This seminar provides a conceptual and methodological introduction to the Incompleteness Theorems and explores the ways in which they are (mis-)used in contemporary philosophy of mind and philosophy of artificial intelligence.

Taking the Incompleteness Theorems as the focal point, part 1 of this seminar covers a broad set of fundamental mathematical results and techniques with which any contemporary analytic philosopher should be familiar:

  1. Formal Logic & Formal Systems: what they are, what they are not, and what properties they ought to have
  2. The Finite & The Infinite: What is infinity, anyway, and how do we count collections of infinite objects?
  3. Computation: What *is* computation, actually? Are there things no computer could do, no matter how much “power” it had? What is “power” in this context, anyway?

The second part of the seminar explores some contemporary philosophical positions about the mind and about AI, using incompleteness as a lens through which to evaluate the arguments put forth for and against such positions. The seminar concludes on a cautionary note, providing guidance on the many ways incompleteness is (often unknowingly) mis-applied and misappropriated in philosophical discourse.

Core Readings

Primary Text

Godel Without Too Many Tears

by Peter Smith

Secondary Texts

  1. Introduction to Incompleteness by Serafim Batzoglou (chapter 6)
  2. Are LLM’s Sentient by Chalmers
  3. Mind and Machines by Putnam
  4. Minds, Machines and Godel by Penrose
  5. The Emperor’s New Mind by Penrose
  6. Minds, Brains and Programs by Searle
  7. The Hard Problem of Consciousness by Chalmers
  8. Minds, Machines and Mathematics by Chalmers
  9. Escaping the Chinese Room by Margaret Boden
  10. Theoretical Impediments to Machine Learning by Judea Pearl

Part 1: Formal Systems & Their Limits

In part 1, we review Hilbert's program and its many responses. We get comfortable with formal systems, the formal definition of computability, as well as the concepts of infinity and countability. Finally, we learn how to use Cantor's diagonalization method to prove the incompleteness theorems.

1

A Formal System for All Mathematics

Hilbert's Program and the search for consistency and completeness.

2

Logic and Computability

Gödel Numbers and Turing Machines

3

The First Theorem

How to construct a sentence we know is true, yet cannot be proved

4

The Second Theorem

How to use logic to prove logic is consistent (or: is this even possible?)

Part 2: The Philosophical Consequences

With the mathematical fundamentals out of the way, we turn to some "big questions" in philosophy that invoke (either implicitly or explicitly) incompleteness such as: Is the human mind more than a machine? What are the true limits of AI? We look at the major positions taken by contemporary philosophers in response to these questions, their arguments, and their critiques.

The Argument: Minds are Not Machines

Week 5: The Lucas-Penrose Argument

The claim that Gödel's theorems prove human minds are not computers.

The Critique: The AI Debate

Week 6: The Critics

Counterarguments suggesting Gödel's theorems do not imply that an AI system cannot develop a mind of its own

Week 7: Misinterpretations, Abuse, and Final Thoughts

Examples of misapplications of the incompleteness theorems, how to recognize them and how to avoid them