(credit: Wikipedia) 

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++, 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 for 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 79 to line 511 then back to line 79, 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 a "check" program would be difficult, but once it was written, it would save many people a great deal of time.

However, progress on the writing of this check program met with problem after problem. Eventually, Alan Turing published a proof showing that writing a check program was not possible. A foolproof algorithm for finding loops in 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. 

It confirms our deepest feelings about empiricism. It is doomed to remain incomplete. 

The possibilities for arguments and counter-arguments on this topic 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. Empiricism’s own methods have ruled out the possibility of an unshakable empiricist beginning point for epistemology. (What is the meaning of the word "meaning"?)