Albert Einstein and Kurt Gödel

For years, the most optimistic of the empiricists were looking to AI for models of thinking that would work in the real world. Their position has been cut down in several ways since those early days. What exploded it for many was the proof found by Kurt Gödel, Einstein’s companion during his lunch hour walks at Princeton. Gödel showed that no rigorous system of symbols for expressing the most basic of human thinking routines can be a complete system. (In Gödel’s proof, the ideas he analyzed were basic axioms in arithmetic.) Gödel’s proof is difficult for laypersons to follow, but non-mathematicians don’t need to be able to do that formal logic in order to grasp what his proof implies about everyday thinking. (See Hofstadter for an accessible critique of Gödel.¹⁰)

 

       

                                                              Douglas Hofstadter

If we take what it says about arithmetic and extend that finding to all kinds of human thinking, Gödel’s proof says no symbol system exists for expressing our thoughts that will ever be good enough to allow us to express and discuss all the new ideas human minds can dream up. Furthermore, in principle, there can’t ever be any such system of expression.

What Gödel’s proof suggests is that no way of modelling the human mind will ever adequately explain what it does. Not in English, Logic, French, Russian, Chinese, Java, C++, music, or Martian. We will always be able to generate thoughts, questions, and statements that we can’t express in any one symbol system. If we find a system that can be used to encode some of our favorite ideas really well, we will only discover that no matter how well the system is designed, no matter how large or subtle it is, we will have other thoughts that we can’t express at all in that system. Yet we have to make statements that at least attempt, more or less adequately, to communicate our ideas. 

Science, like most human endeavors, is social. It has to be shared in order to advance.

Other theorems in computer science offer support to Gödel’s theorem. For example, in the early days of the development of computers, programmers were continually creating programs with loops in them. After a program had been written, when it was run it would sometimes become stuck in a subroutine that would repeat a sequence of steps from, say, line 193 to line 511 then back to line 193, again and again. Whenever a program contained this kind of flaw, a human being had to stop the computer, go over the program, find why the loop was occurring, then either rewrite the loop or write around it. The work was frustrating and time consuming.

Soon, a few programmers got the idea of writing a kind of meta-program they hoped would act as a check. It would scan other programs, find their loops, and fix them, or at least point them out to programmers so they could be fixed. The programmers knew that writing such a program would be difficult, but once it was written, it would save many people a great deal of time.

     Alan Turing winning a race ( He was a world class athlete as well as mathematician.)

However, progress on the writing of this check program encountered difficulty after difficulty. Eventually, Alan Turing, a man adept at computer languages, published a proof showing that writing a check program was, in principle, not possible. A foolproof algorithm for checking other algorithms is, in principle, not possible. (See “Halting Problem” in Wikipedia.¹¹)

 This finding in computer science, the science many people see as our bridge between the abstractness of thinking and the concreteness of material reality, is Gödel all over again. In another kind of proof, it confirms our deepest feelings about empiricism. It is doomed to remain incomplete. No completely effective check program has ever been found. Programs that are able to catch beginner programmers’ simpler mistakes have been written, but no foolproof one has ever been created in any of the many programming languages that have evolved in the field over the years.

The possibilities for arguments and counter-arguments on this topic in AI are fascinating, but for our purposes in trying to find a base for a philosophical system and a moral code, the conclusion is much simpler. The more we study both the theoretical points and the real-world evidence, including evidence from science itself, the more we’re driven to conclude that the empiricist way of seeing or understanding what thinking and knowing are will probably never be able to explain itself. If Gödel’s proof is right, and nearly every expert in math and computer science thinks it is, and if it is extended to human thinking in general, empiricism’s own methods have ruled out the possibility of an unshakeable empiricist beginning point for epistemology.

If I think I have found a way to describe what thinking is, then I will have to express what I want to say about the matter in a language of some kind—English, Russian, C++, or some other kind of language for encoding thoughts. But there is not, nor can there be, a code that is capable of capturing and communicating what the thinker is doing as she is thinking about her own thinking. It is a mental conundrum with no solution. (What is the meaning of the word meaning?)