Shufflebrain

In its present form, though, hologramic theory will not serve our needs. Fourier transform space is too cramped for us to appreciate, for instance, the similarities and differences between our own mental cosmos and the minds of other creatures. Fourier transform space is too linear to explain the nonlinear relationship between the time we measure by the clock and the intervals that elapse in dreams, for example. Nor in Fourier and kindred transform spaces can we readily envisage the smooth, continuous movement of information from sensations to perceptions to memories to behavior to whatever. How can the information be the same while the events retain their obvious differences? Described only in the language of Fourier theorem, mind seems more like the inner workings of a compact disc player than the subjective universe of a living organism. If the reader has already felt that something must be wrong with our picture, it's because he or she knows very well that we mortals aren't squared-off, smoothed-down, case-hardened, linearly perfected gadgets.

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Thus far, the constraints on our understanding have two main causes. First, our theory is now anchored to the axioms and postulates of Euclid's geometry. Second, we haven't yet given sufficient theoretical identity to the particular, having been too preoccupied with escaping from the particular to give the attention it really deserves. In this chapter, I will first reassemble hologramic theory free of assumptions in previous chapters. Towards the end, I will draw the realm of the particular into our discussion. Ready? Take a deep breath!

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June 10, 1854 is an noteworthy date in the history of thought. It was also a momentous day in the brief life of mathematician Georg Friedrich Bernhard Riemann whose discoveries were crucial to Albert Einstein.[2] It was when Riemann stood before the distinguished professors of Göttingen's celebrated university to deliver his formal trial lecture as a probationary member of the faculty. Entitled, "On the Hypotheses Which Lie at the Foundations of Geometry," the lecture attacked a dogma that had ruled rational belief, "From Euclid to Legrendre," asserted the twenty-eight year old Privat-dozent. [3]

"It is well known," he continued, "that geometry presupposes not only the concept of space but also the first fundamental notions for constructions in space as given in advance. It gives only nominal definitions for them, while the essential means of determining them appear in the form of axioms. The relation [logic] of these presuppositions [the postulates of geometry] is left in the dark; one sees neither whether and in how far their connection [cause-effect] is necessary, nor a priori whether it is possible."

In the detached bon ton of scholars then and now, Riemann in effect was telling his august audience (including no less that his mentor, Karl Friedrich Gauss) that mathematicians and philosophers had flat-footedly assumed that space is just there. Like the gods in The Iliad who had an external view of the mortal realm, mathematicians and philosophers had inspected space in toto, had immediately brought rectilinear order to the nullity, with the flat planes of length, width and height, and thereby had known at a glance how every journey on a line, across a surface or into a volume must start, progress and stop. Geometric magnitudes--distance, area, volume--plopped inexorably out like an egg from laying hen's cloaca.

Ever since Newton and Leibnitz had invented calculus, infinitely small regions of curves had been open to mathematical exploration; but points on a straight line had eluded mathematician and philosopher, alike. Riemann doubted that the same fundamental elements of geometry--points--could obey fundamentally different mathematical laws in curves versus lines. He believed that the point in flat figures had become an enigma because geometry had been constructed from the top down, instead of from the bottom up: "Accordingly, I have proposed to myself at first the problem of constructing a multiply extended [many dimensional] magnitude [space] out of the general notions of quantity." He would begin with infinitely small relationships and then reason out the primitive, elemental rules attending them, instead of assuming those rules in advance. Then, not taking for granted that he knew the course before his journey into the unknown had begun, he would, by parts, follow the trail he had picked up, and he'd let the facts dictate the way.