Hi Dick,

You're probably still enjoying beautiful Vienna at this time. When you're back, and if you feel in the mood, I'd like to know your thoughts on this.

If I understand you correctly, no one can untie the Gordian knot of definitions because the whole thing has no exposed ends. Your approach is to slash a key point ("reality") and then the whole knot is untied. As with Alexander, I suppose this is a valid approach since freeing the wagon is what really matters. But I find the issue of how the knot was put together in the first place just as interesting, because I'm convinced the ends must exist, even if they cannot be clearly seen. So I came up with what seems to me a good analogy.

If you look at the code for any computer program, you will see that it is nothing but a long list of definitions! It's all mishmashed, that's for sure, but the open ends can be clearly seen: they are the basic words of the compiler. Stuff such as "IF", "THEN", "GOTO", and so on. Those terms cannot be defined using the language of the compiler itself. You can't write, say, a piece of Basic code to emulate the function of the Basic "IF" statement. The language just doesn't support it.

Of course I'm skipping something important: the compiler itself is a computer program, so how did they write it? Isn't a compiler required to build a compiler? Those are the kinds of pseudo-paradoxes that keep philosophers in business for millennia, but fortunately in this case we already know the answer. If you don't have any compiler around, you can still create computer programs, including compilers, using something other than a compiler: you can directly manipulate individual bits directly in the computer's hardware. That is exactly how the first computers were programmed. But one can still argue that "directly manipulating bits" is not essentially different from using a compiler. I just wanted to make sure I've covered this potential objection.

The really interesting question for me is: if, no matter how you create it, a program is just an ellaborate, complex set of definitions, are there "exposed ends" or not? In other words, are there basic, essential definitions which must be in place even before the first bit is manipulated? I'm of the opinion that there are exposed ends, and those would be the few basic definitions which make up boolean logic. Without "1", "0", "NOT", and "AND" (or "OR") being given a priori, there can be no software. But the more important thing is, if you understood the four basic definitions of boolean logic, you understand how every single piece of software works, without exception.

So the language of computer programs can be reduced to the four "exposed ends" which comprise the definitions of boolean logic. Everything can be explained in terms of those, and boolean logic itself is a closed, self-consistent set of very formal definitions. No wonder computers are so easy to program and so reliable!

If you think in terms of human language, the same thing applies, although obviously it's not boolean logic that is at the bottom. But there must be some exposed ends in the form of a small set of words which can be defined in terms of each other, to create a closed, self-consistent formal system from which everything else can be derived. And that closed, self-consistent system would be known as "mathematics". So it's not as much that the Gordian knot must be cut as it is that you have to find some exposed ends, and there are good indications that mathematics happens to be just that.

Well, that was harder to write than I first thought when I started. I'm now reluctant to post... I think I'll give it a try, maybe you'll offer some comments.

See you,

Aurino