Shufflebrain

The prohibition against points on lines was an artifact of trying to approach points from a flattened-down, squared-off, cut-straight universe; and the same had been true about parallel lines. And where did we ever come up with the idea that the shortest distance has to be the straight path? The argument in favor of straight lines has had its epistemological justification in the peroration of my little sister's ontology: "Because the teacher said!"

But how can all this Riemann stuff really be? Why don't our bridges all fall down?

The irony is that Riemann's system is not anti-Euclidean. As I pointed out earlier, Euclid's geometry belongs to the realm of experience. It is the geometry we invent and use for distances neither very great nor very small; for uncomplicated planes; for simple spaces; for universes of non-complex dimensions; for standard forms of logic (like Aristotle's). With Riemann's system as a guide, Euclid's propositions even acquire a valid a priori foundation. [9] Their limitations no longer dangerously hidden from us, Euclid's rules can serve our needs without tripping us up. Infinitesimal regions on a straight line are beyond the Euclidean approach, and thus we must use a different method for handling them; parallel lines don't intersect on a Euclidean plane; we can safely lay railroad track in the conventional manner. As far as the shortest distances thing is concerned, we may keep the protractor, the ruler and the T-square for geometric operations within the ken of experience. For in a Riemannian universe, the shortest distances between points is the path of least curvature. In the Euclidean realm of experience, that least curvature approaches zero curvature, and it coincides with what looks like a straight line.

I once worked on a crew with a graduate student in philosophy we all called Al-the-Carpenter. (His thesis was "God is Love.") Older and wiser and more generous than the rest of us, Al let scarcely a day pass without an enlightening observation or an unforgettable anecdote. "I met a little boy this morning," he once told us as we gathered around the time clock, "while I was making picnic tables outside the recreation hall. 'Is the world really round?' the little boy asked me. 'To the best of my knowledge,' I answered. 'Then tell me why you cut all the legs the same length. Won't your tables wobble?'"

"Yes," his tables would wobble, Al had replied. But the wobble would be infinitesimally tiny. The smile he usually wore broadened as he lectured like a latter-day Aristotle. "If you want to hear philosophical questions, pay attention to the children."

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A theory constructed to fit the planes of experience will show its wobble when we invoke it on a very large, exceedingly small or extremely complex body of information.

Until now, we have viewed hologramic mind along Euclidean axes. But an understanding of hologramic mind requires the freedom only Riemann's simple but powerful ideas can give it.

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Riemann's name and influence pervade mathematics (e.g., Riemann surfaces, Riemann manifolds, Riemann integrals, Riemannian geomery) for reasons that are evident in his qualifying lecture. The seeds of his subsequent work are present there. The elementary relationships at points later became the means by which scientists learned to conceptualize invariance. Toward the completion of the lecture, Riemann even anticipated general relativity. (Einstein's 4-dimensional space-time continuum is Riemannian.) Where did Riemann's insight come from? What intuitive spark caused his genius to push against the outermost fringes of human intellectual capability? I'm not sure I have the answer and only raise the question because I think my personal suppositions (i.e., hunch) will help us in our quest of hologramic mind.

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Riemann, I believe, had a vivid concept of what I'll call "active" zero, the 0 between +1 and -1: the set we cross when we overdraw from the checking account--not the zero an erudite philosophy professor of mine, T. V. Smith used to call "what ain't." Judging from his own writings, Riemann seemed to have a crystal-clear intuitive idea of zero space--and even negative space![10] Let's try to develop a similar concept, ourselves. But so that what we say will have theoretical validity, let me make a few preliminary remarks.

There's a principle in logic known as Gödel's incompleteness theorem. The latter theorem tells us that we cannot prove every last proposition in a formal system. It's a sort of uncertainty principle of the abstract. We'll obey this tricky-- but powerful-- principle by always leaving at least one entire dimensional set beyond our reach. In fact, let's throw in two uncountable dimensions: negative and positive infinity which, by definition, are sets beyond our reach whether we admit our limitations or not. (Plus and minus infinity are by no means new ideas, incidentally.)