### Consulo, ergo sum?

I used to use the movie AI in my Intro Philosophy course, to introduce the question of the self and the issue of the Cartesian split. The movie raises fascinating questions of what it means to be human, especially in its implicit acceptance that true "human" thought requires emotional responses as well as "intellectual" ones. The clear implications of the film are that rules alone do not define human intelligence. In fact, it is the lack of "emotional rules" (which the film assumes are a learned, not a defined, response) that sets the story in motion.

"Cogito, ergo sum," is the touchstone for modern philosophy thinking about what is the self, or even what is human, and I used the film to zero in on that issue. Descartes distinguished "rational" thought (which he considered the foundation of the "self") from emotions (which he considered "bodily" functions); but he was not just thinking about thought itself with his cogito. He was speaking of the consciousness of thought. Turing's famous "test" of AI is valid if the question is only the appearance of thought. Descartes' very statement, the one that turned his skepticism into certainty, is a statement of awareness of being aware. It is a "meta" statement about the process of thought itself. And it is on this point that the incompleteness theorem of Kurt Gödel enters the discussion.

Gödel's theorem, put simply, is that formal systems are incomplete, because the axioms upon which they are founded do not lead to a system powerful enough to eliminate the generation of propositions that are not both true and formally undecidable within the terms of the system. In other words, any formal system can generate a statement which it cannot establish as true within the terms of that system. The truth of the statement, if it can be established, must be established through recourse to terms from outside the system. That is, within a formal system such as mathematics, certain statements can be generated which cannot be established as true within mathematics. The truth of these statements can only be decided outside of mathematics, or meta-mathematically.

It's important to understand this fully before going further, because the easy answer here is: "Well, of course we can't decide all truths within any one system." The critical point here is that the system can generate the statement, so the statement itself does not come from wholly outside the system. It comes, in fact, from the axioms of the formal system itself. But the system, built upon those very axioms, cannot decide within the terms of that system, the validity or truth of the statement. It is, therefore, formally undecidable, and any decision on it must be made by appeal to source wholly outside the system.

As a matter of logic, in other words, no formal system of thought, being one founded on axioms which in turn generate the system of thought, is complete. It is, despite its rigor or complexity, essentially incomplete. Which brings us back to artifical intelligence.

Human thought is a complex subject. Is emotion, for example, a component of it? A complement to it? An obstruction that blocks it? Is thought only equivalent to reason? And does reason occur in spite of emotion? What, then, of curiousity? Is that an emotion? Could be build a curious machine? These are not idle questions: they challenge the notion that thought is simply a rules process, and that human thought can be replicated in computer software when enough of the rules are understood and anticipated to be reproducible as tools for a progam. (This idea is leveled as a criticism against Wittgenstein's concept of "language games," that he merely tries to reduce language to a set of acquired rules, and each "game" plays according to a different rule book.) But Godel's theorem challenges that very notion. Palle Yourgrau puts it this way:

Which reminds us that Descartes settled upon his 'cogito' because he placed an artifical limit to his thinking. Skepticism, as Kierkegaard after him also understood, is an acid that will destroy everything eventually, including itself.

All of which leaves us still with the question of the self, and the question of reason in pursuing the question of the self. Does reason, rigorously applied, lead us to the limits of our knowledge? Or the limits of our reason? Are the two coterminous? And is it because we think, that we know we have being? Or because we question, because we can ask, and discover answers?

"Cogito, ergo sum," is the touchstone for modern philosophy thinking about what is the self, or even what is human, and I used the film to zero in on that issue. Descartes distinguished "rational" thought (which he considered the foundation of the "self") from emotions (which he considered "bodily" functions); but he was not just thinking about thought itself with his cogito. He was speaking of the consciousness of thought. Turing's famous "test" of AI is valid if the question is only the appearance of thought. Descartes' very statement, the one that turned his skepticism into certainty, is a statement of awareness of being aware. It is a "meta" statement about the process of thought itself. And it is on this point that the incompleteness theorem of Kurt Gödel enters the discussion.

Gödel's theorem, put simply, is that formal systems are incomplete, because the axioms upon which they are founded do not lead to a system powerful enough to eliminate the generation of propositions that are not both true and formally undecidable within the terms of the system. In other words, any formal system can generate a statement which it cannot establish as true within the terms of that system. The truth of the statement, if it can be established, must be established through recourse to terms from outside the system. That is, within a formal system such as mathematics, certain statements can be generated which cannot be established as true within mathematics. The truth of these statements can only be decided outside of mathematics, or meta-mathematically.

It's important to understand this fully before going further, because the easy answer here is: "Well, of course we can't decide all truths within any one system." The critical point here is that the system can generate the statement, so the statement itself does not come from wholly outside the system. It comes, in fact, from the axioms of the formal system itself. But the system, built upon those very axioms, cannot decide within the terms of that system, the validity or truth of the statement. It is, therefore, formally undecidable, and any decision on it must be made by appeal to source wholly outside the system.

As a matter of logic, in other words, no formal system of thought, being one founded on axioms which in turn generate the system of thought, is complete. It is, despite its rigor or complexity, essentially incomplete. Which brings us back to artifical intelligence.

Human thought is a complex subject. Is emotion, for example, a component of it? A complement to it? An obstruction that blocks it? Is thought only equivalent to reason? And does reason occur in spite of emotion? What, then, of curiousity? Is that an emotion? Could be build a curious machine? These are not idle questions: they challenge the notion that thought is simply a rules process, and that human thought can be replicated in computer software when enough of the rules are understood and anticipated to be reproducible as tools for a progam. (This idea is leveled as a criticism against Wittgenstein's concept of "language games," that he merely tries to reduce language to a set of acquired rules, and each "game" plays according to a different rule book.) But Godel's theorem challenges that very notion. Palle Yourgrau puts it this way:

The complete set of mathematical truths will never be captured by any finite or recursive list of axioms that is fully formla. Thus, no mechanical device, no computer, will ever be able to exhaust the truths of mathematics. It follows immediately, as Godel was quick to point out, that if we are able somehow to grasp the complete truth of this domain, then we, or our minds, ar enot machines or computers. (Enthusiasts of artificial intelligence were not amused.) A World Without Time, Palle Yourgrau (New York: Basic Books 2005) p. 3However, Rebecca Goldstein brings us to a slightly different conclusion:

According to Wang, Godel believed that what had been rig-orously proved, presumably on the basis of the incomplete-ness theorem, is: "Either the human mind surpasses all machines (to be more precise it can decide more number the-oretical questions than any machine) or else there exist num-ber theoretical questions undecidable for the human mind."

What exactly did Gödel have in mind with this second dis-junct? I think that what he is considering here is the possibil-ity that we are indeed machines-that is, that all of our thinking is mechanical, determined by hard-wired rules-but that we are under the delusion that we have access to unfor-malizable mathematical truth. We could possibly be machines who suffer from delusions of mathematical grandeur. What follows from his theorem, he seems to be suggesting, is that just so long as we are not delusional as regards our grasp of mathematical truths, just so long as we do have the intuitions that we think we have, then we are not machines. If indeed we truly have the intuitions that we do, then it is impossible for us to formalize (or mechanize) all of our mathematical intu-itions, which means that we truly are not machines. Of course there is no proof that we know all that we think we know, since all that we think we know can't be formalized; that, after all, is incompleteness. This is why we can't rigorously prove that we're not machines. The incompleteness theorem, by showing the limits of formalization, both suggests that our minds transcend machines and makes it impossible to prove that our minds transcend machines. Again, an almost-paradox. Incompleteness, Rebecca Goldstein (New York: W.W. Norton 2005), p. 203

Which reminds us that Descartes settled upon his 'cogito' because he placed an artifical limit to his thinking. Skepticism, as Kierkegaard after him also understood, is an acid that will destroy everything eventually, including itself.

All of which leaves us still with the question of the self, and the question of reason in pursuing the question of the self. Does reason, rigorously applied, lead us to the limits of our knowledge? Or the limits of our reason? Are the two coterminous? And is it because we think, that we know we have being? Or because we question, because we can ask, and discover answers?

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