But what’s a writer to do—this blog is the only semantic space in which I can discuss these issues, and I’ve been stimulated by two books I’ve just read—Rebecca Goldstein’s Incompleteness: The Proof and Paradox of Kurt Godel (Great Discoveries) and John Searle’s Mind, Language, and Society : Philosophy in the Real World. Both authors are philosophers who write for a larger public; Goldstein is also a novelist—evident in her vivid portrayal of Gödel as a person, and of his intellectual milieu.
Goldstein’s book stands out among treatments of Gödel’s ideas meant for broad audiences for two reasons. First, it doesn’t talk down to the intelligent layman and follows, step by step, the proofs of his theorems. The logical notation and equations she uses may look scary, but persist—it’s all explained very well. If you ever had a course in basic logic (which you probably had in high school math), you’ll be able to follow it.
Second, she presents the philosophical background to Gödel’s concept of incompleteness in a way I haven’t seen it discussed before. In part, Gödel’s theorems show that any formal logical system complex enough to include number theory (that is, the basic elements of mathematics) will contain statements that are true but which cannot be proved within the system itself. In a broader sense, it shows that there is a deficiency in formal logical systems sophisticated enough to describe the mathematical patterns we see in the world around us and which our minds can imagine. To show that these systems are self-consistent, we must always invoke axioms from outside the systems.
The concept has at times been invoked in support of postmodernism and relativism—if mathematics, the purest of sciences, can’t be grounded solidly in logic, such schools of thought claim, then there cannot be any absolute truths. Goldstein shows that Gödel not only rejected such interpretations of his work, but in fact believed that his theorems proved the opposite—that there are ultimate truths we can arrive at only through intuition, not through logic. Gödel was, Goldstein maintains, a Platonist—he believed that a world of truths exists outside of and independent of the human mind.
Searle is best known for his role in the philosophical debate over the nature of consciousness (see a summary of the field here; a good, basic, and readable book-length gateway into the subejct is Susan Blackmore’s Consciousness: An Introduction). Many, perhaps most, prominent figures in the field today maintain that our sense of consciousness (that sense of having a mind that is somehow distinct from and in control of our bodies, the sense of being a me) and our conscious sensations (for example, the “blueness” of blue—that part of the sensation that seems to lie beyond any physical description of how our brains receive and transmit the sense perception—philosophers call these sensations “qualia”) are simply illusions. The mind is no more than the physical functioning of our brains, and if we know enough about brains, we will be able to account for these sensations. That physiological account, these philosophers maintain, is the only proper explanation and description of the phenomenon of consciousness.
Searle accepts the premise that the mind is only the brain—that there is no ethereal mind or soul that exists independent of the physical body. But he rejects the claim that a physiological account of the brain can explain the mind. Our sensation of having being consciousness, our experience of the qualia, cannot be dismissed as illusions or artifacts of the physical brain. We experience them, and these experiences must be accepted at face value. Our intuition of having a mind may have no explanation within the formal scientific system we use to describe the world, but this does not mean they are false. On the contrary, we may have intuitions that are true but which cannot be accounted for by our formal systems.
While Searle does not adduce Gödel in Mind, Language, and Society, he’s clearly thinking along the same lines.
Gödel’s proofs are just that—proofs. Their philosophical implications can be debated, but their mathematical and logical truth is incontrovertible. Mathematics is by nature an incomplete system. Is physiology as well? Unfortunately (or not), we can’t transform physiological facts into logical symbols that we can manipulate in formal proofs. But Searle offers some intriguing lines of thought about the question of how we perceive and understand the world beyond our brains. If the way our brains work is incomplete in the Gödelian sense, we may have true intuitions of the world that can never be proven.
In short, Goldstein’s and Searle’s books make a good pair that are more than the sum of their parts. If you are interested in more than a blog post on this topic, read them together for a philosophical experience that transcends the formal limitations of each of them.