The Limits of Reason in Everyday Life

• We have described the great power of formal reason, but also identified a number of limits:
  • Godel's incompleteness theorems: for systems of reasoning of sufficient strength, there are unprovable truths
  • The Halting Problem: there is no effective procedure to find all the effective procedures (reason cannot find the precise limits of reason).
  • Complexity Incompleteness: generally, a theory of sufficient strength cannot decide questions of complexity substantially exceeding the complexity of that theory.
• Are final question is: Do these limits every have practical implications? Do we, or will we, ever run up against them?
• One caveat that needs to be considered is that it is not clear that we will notice when we hit such a limit: we may be blind to some limits because we see problems in terms of what we can do.
• But, we can plausibly describe some potential limits that could arise. Here are some speculative questions; these are not meant to be definite queries so much as illustrations of how we might start considering how the limits of reason could have practical effect:
  1. Have we blurred some distinctions between math and science?
  2. Does complexity tell us anything about consciousness?
  3. Do the limits of syntactic/formal methods reveal a flaw in the computational theory of mind?
  4. Must an ethical rules be comprehensible?

Mathematics as a (contingent) Science?

• What does this mean for reason, and for the special case of reason par excellence, mathematics?
• One way we can interpret the limitations that we've seen is that we will have to add information to our theories whenever we are trying to reason about things more complex than our theory.
• If you cannot prove P, but P looks likely or at least very useful, you may just need to assume P and work with it (perhaps abandoning P later if it turns out to lead to contradictions).
• But this blurs the lines now between pure reason and some of the fallible aspects of empirical science! Mathematics so practiced has strong analogies with science: we assume some things and work with them because of their power to explain, but are open to revision in the future! Our theories become pragmatic and fallible!

Applications to the Philosophy of Mind I: Consciousness and Complexity

Anyone interested in a slightly more rigorous version of the argument I gave (or will give) in class might want to read Physical Theory and the Complexity of Phenomenal Experience.

Applications to the Philosophy of Mind II: The Lucas Argument

• The philosopher Lucas first proposed an argument that has recently become widely known (in a differnt form) because of Roger Penrose. Below is something like Penrose's argument in The Emperor's New Mind.
• Recall Godel's First Incompleteness theorem.
  • Godel showed us that we could in arithmetic with multiplication create a well formed expression that can be inpreted to say "I am not provable." Call this G.
  • Such an expression is either provable or not.
  • If G is provable, then the system is inconsistent.
  • We typically like to assume arithmetic with multiplication is consistent. Hence, assuming that, G is unprovable.
  • If G is unprovable, G is true.
• Penrose makes a distinction between "formalist" truth, and truth (note: a formalist may deny this characterization). Formalist truth is what we would have in mathematics if we adopted one kind of "formalist" view. True would mean provable (in some system T).
• But G is true but not provable. If we adopt the formalist truth notion, then G is not true. But, Penrose observes, it's obviously true. So, we must abandoned the formalist truth notion. The result is that truth is something beyond the reach of mere formalism.
• Penrose claims we see in this case a mental capability that is beyond the reach of formal methods. If he is right, then the claim that we could make a thinking Turing-equivalent machine (or the claim that our minds are Turing-equivalent) is wrong.
• This extra ability, he says, is already evident in for example how we first go about setting up a mathematical system: we start with axioms that are obviously true, he says -- not true in some other formal system.
• Penrose also suggests that the difference here is one between syntax and meaning. He thinks there may be some analogy with Searle's Chinese Room argument.

Ethical Rules and Comprehensible Justice

• From Kafka's The Trial:
  "We are humble subordinates who can scarcely find our way through a legal document and have nothing to do with your case except to stand guard over you for ten hours a day and draw our pay for it. That's all we are, but we're quite capable of grasping the fact that the high authorities we serve, before they would order such an arrest as this, must be quite well informed about the reasons for the arrest and the person of the prisoner. There can be no mistake about that. Our officials, so far as I know them, and I know only the lowest grades among them, never go hunting for crime in the populace, but, as the Law decrees, are drawn toward the guilty and must then send out us warders. This is the Law. How could there be a mistake in that?"
  "I don't know this Law," said K.
  "All the worse for you," replied the warder....
  "You'll come up against it yet."
  Franz interrupted: "See, Willem, he admits that he doesn't know the Law and yet he claims he's innocent."
  "You're quiet right, but you'll never make a man like that see reason," replied the other.
  
• Must we be able to understand ethical rules, or the criteria that are used to make an ethical evaluation, in order for these to be just?
• Two plausible kinds of real world cases.
  1. Connectionist networks to recognize patterns in credit-worthiness.
  2. Connectionist networks to recognize potential security threats ("Total Information Awareness")
• It is important to understand that connectionist networks are:

  a. Trained, not directly programmed.
  b. Best understood using multidimensional state space models.
  

  Because of these features, the internal "rule" that a connectionist network utilizes might be irreducibly complex.
• Suppose now we have an accurate tool to recognize credit risky or security risky individuals, but that it is irreducibly complex. We know what the inputs are, but you cannot explain how it decides in any more simple way than just describing the high-dimensional rule.

  a. Could it ever be just to deny someone credit, or a space on a plane, using such a rule? How accurate could need such a rule be before we accept it?
  b. Should we allow rules that (because they are very Kolmogorov-Complex) we cannot understand, and to which we cannot mount any objections? (Consider Kafka's The Trial -- could that be a just situation?)

• Some preliminary thoughts

  a. If we are realists about ethics, and think that there really are such things independent of our understanding as good, evil, and so on, then perhaps we must allow that a situation like that described above could be both incomprehensible and morally accurate.
  
  b. If we decide we must restrict ethical decisions to what can be comprehended, this would be a very new kind of restriction on ethics. Many tasks arise as a result. For example, we need to define what can be comprehended (no small task!) and we need to explain how incomprehensible rules can be used, if at all.