Suppose we dispense with the Mandelbrot set for the time being. Imagine that we can represent a true statement, for instance, “existence exists,” by a black pixel and a false statement as a white pixel. Knowledge is inexhaustible — we will never be able to know everything about everything, thus there are an infinite number of true statements.

I would say "a potentially infinite number of true statements"; there is always at any time a finite number of true statements, because nothing, including propositional knowledge, is infinite.

As well, the world is infinitely complex, as was succinctly expressed by a programming guru: “Everything is deeply intertwingled.”

The world is highly complex, but not infinitely so.

Thus, all true statements can be linked to one another and they must be mutually supportive.

All true statements can be derived from “existence exists.”

This graphically demonstrates Rand’s dictum, “If you discern contradictions, examine your premises.” There is no such thing as a grey area with respect to truth and falsehood — you just haven’t examined the logic at a deep enough level. Grey areas are called grey areas because contradictions exist.

If you don't want to accept my proposition that all knowledge can be derived from "Existence exists" then prove it. I'm not going to get involved in a long, time consuming discussion on what I thought was just an article on my ruminations.

Paul:

I love your essay: it combines recent computer and complexity theory to create a new mental model of how to visualize the long-standing mathematicians' dream of creating a firm logical foundation for the entire edifice of mathematics. The hope was that with the correct set of axioms, all mathematical truths could be demonstrated to stand on a rigorous and entirely logical foundation, thus creating a single magnificent and coherent structure that is a creation of the human mind and that in no way depends upon the physical world for its existence.

This was the motivation behind Russell & Whitehead's 3 volume opus, Principia Mathematica. The trouble is that they had barely got this monster written when Kurt Godel presented his proof that any sufficiently complex (useful & interesting) system of logical statements will contain true statements that cannot be proven from the system's axioms. Or to put it another way: no consistent system of primitive recursive arithmetic can, within itself, determine that every proposition that can be formulated within the system is decidable, i.e. can decide whether every proposition and its negation is provable within the system. This was a devastating blow to any attempt to create the structure that had motivated the Principia. I can't begin to imagine how it must have felt to have worked so hard to try to build such a structure, only to have a third party demonstrate that the entire opus was utterly doomed from the start! Russell maintained that the effort required to write the Principia was to great that his brain never fully recovered its intellectual powers.

My view is that their monumental project just took far too long and he was therefore really looking at the problems of an aging brain. I'm having this problem myself, in my own small way, and feel I'm in a desperate race to complete my work before I lose both the intellectual power and the energy required to do so.

A recent book by the mathematician Gregory Chaitin, "Meta Math: the Search for Omega" makes the interesting observation that (a) mathematics was, until the advent of the computer, treated (of necessity) as an entirely logical and philosophical enterprise concerned, at its foundation, with axioms and their inevitable and logically deduced theorems; but (b) since then, however, an entirely new branch of mathematics has evolved that uses the computer as an experimental tool. This new class of mathematical inquiry has a surprising aspect in common with the sciences, namely experimentation.

My artwork is a very visual example o this new paradigm - as, of course, is the Mandelbrot Set. Mathematica's Stephen Wolfram wrote a book a few years back (a monster) "A New Kind of Science" that is all about experimental computational mathematics. He thinks he invented it all and thus his self-published vanity press book and his egomaniacal attitude (he gives no one else any real credit) have deeply offended the extraordinarily many people who have preceded him in thinking along these lines. Despite his being seen by his colleagues as an offensive prick, the book is interesting. But the 'artwork' he raves on about and with which his book is filled, is pretty deadly dull stuff. This chap should really have left the keyboard a few times and perhaps gone to an art gallery or two! Since he's filthy rich, from his ingenious and much admired software package, Mathematica, he doesn't really need to care what others think of him. Chatin, however, seemed to find much to admire in Wolfram's book.

Your idea, in its delightful new suit of fractal clothing, sounds to me, at first blush, like an ingenious revisiting of Russell & Whitehead's ambitious program to demonstrate that all of mathematics is a coherent and unified structure. Godel has demonstrated that the world of mathematics and logic cannot be structured that way. Godel, if I remember correctly, once tried to use his new methods of theorem proving to demonstrate that there must be a god, or at least higher levels of nested intelligence. I seem to recall that other mathematicians who were conversant with his methods were unable to convince themselves his proof worked. Godel's new ideas about how to use numbers and number theory to reason about entirely different concepts were, I believe, partly inspired by some of Cantor's ingenious proofs in set theory and in particular by the way he demonstrated that infinity was not a single concept. Cantors work also helped inspire Mandelbrot and thus, in turn, it underlies my artwork.

It is possible however, that your approach does not have the Principia's flaw of expecting that at any given level all statements can be deduced from the axioms supporting  the structure at that level. Mathematics, as it now appears to stand, is a recursively layered structure in which the only way to prove the truth or falsehood of all true statements in any given system A is to create a higher or meta system A', using additional axioms, from which to look down on the incomplete system A and thereby acquire the power to prove the truth or falsehood of all statements in system A from A'. Clearly this implies an open ended system of systems. Perhaps, therefore, it can be reconciled with your approach? Your approach would then, to my mind, be a very fitting visual metaphor with which to view the endlessly intertwined, ever expanding, open ended system of systems of axioms and theorems that constitute mathematics.

Perhaps you might contact the math department at a local university - someone with the right mathematical bent might take a serious professional  interest. The problem is finding the right player: math is a such vast domain of specializations in which many players won't have the philosophical interests required to make good use of your idea.

Ken

I happen to think that it is essentially wrong, but wrong in a fruitful way. I agree with William (Bill) that it is a mistake to say that one can derive all try (sic) statements from existence exists.