1.Introduction

Gauss once said "Mathematics is the queen of the sciences and number-theory the queen of mathematics". That sets the mathematical tradition. One cannot cite a single instant in number theory where an idea is borrowed from another science. All mathematical objects are by implication the creation of the mind, not borrowed from any external phenomena, visual or otherwise. On the other hand, empirical sciences have borrowed heavily from mathematics and have greatly benefitted from it. Chaitin remarked that the usefulness of a hypothesis, and not necessarily its "self-evident truth", is the key criterion by which to judge whether it should be regarded as the basis for a theory [1]. Godel expressed in a discussion of Russell's mathematical logic [2,3] that "...axioms need not be evident in themselves, but rather their justification lies (exactly as in physics) in the fact that they make it possible for these sense perceptions to be deduced... ". Here is a remark from Weyl in the same vein (1949): "A truly realistic mathemtics should be conceived, in line with physics, as a branch of the theoretical construction of the one real world, and should adopt the same sober and cautious attitude toward hypothetic extensions of its foundations as is exhibited by physics." [4]. To these remarks, the author adds: "We should not set number theory apart ... it should be treated as just another science amongst sciences".

The author postulated back in 1988 that the point is divisible [11]. He pointed out that the axiom of the indivisibility of the point was the cause of our inability to visualise beyond 3-dimensions. To break this deadlock, the axiom itself must be changed. This is because visualisation is not just an external phenomenon since it has a neurophysiological linkage from the external sceme through our optic receptors to the visual cortex at the back of the brain. This is the basis of visualisation the principle of which has been perfected by evolution some 65 million years ago when warm blooded mammals played second strings to Dinosaurs by hunting at night with frontal eyes and colour visions. Let me take the liberty to quote an excerpt from Geoffrey Montgomery's The Mind's Eye from Discover [5]:

"....In 1968, ....two young neurobiologists at the University of Wisconsin found, to their surprise, that the visual cortex was only one map among many. By probing the brain of the South American owl monkey...... John Allman and Jon Kaas discovered that the back half of the primate brain contains more than a dozen different maps of the visual field. Each one exists within an area of the size of a postage stamp, and each displays the same scene, like television sets arrayed in a department store, all turned to the same channel. .... they encountered a lot of resistance and even hostility because people just couldn't see what purpose these multple maps might be serving..... Today, however, Allman, Kaas, and others have begun to figure out the purpose. As they have monitored the reactions of neurons in the brains of monkeys, they have learned that while these maps all show the same picture, they don't all accentuate the same aspects of it. Cells in one map are particularly sensitive to motion. Cells in another respond strongly to particular shapes. Those in a third respond to color. ... ".

Allman and Kaas' discovery pointed to an important geometrical principle -- the brain processes a point as multiple points. In other word, our mind treat a point as divisible which is contrary to the first definition established unchallenged by Euclid around 350 BC [6]. This fact was not known to Euclid and could hardly be considered self-evident even in modern times. If Euclid had defined the point as divisible, the course of scientific developments could have evolved differently. The dice was cast and mathematics and sciences had marched on for more than twenty centuaries since Euclid. Like the QWERTY keyboard invented more than a hundred years ago during the industrial revolution, no one since has succeeded in replacing it by modern keyboards.

Allman and Kaas' discovery is very relevant to geometry. Plane geometry was extended to analytical solid geometry but that is a theoretical cul-de-sac due the assumption of the indivisibility of points. Information since the advent of digital computers have become multidimensional requiring innovative visualisation techniques. But a practical multidimensional visualisation theory does not exist until the discovery of CAH [11]. CAH visualisation techniques and point geometry have a nine years' history which eventually led to the development of sequence algebra as a new offshoot in 1994 [7 to 15]. Sequence algebra is also called Visual algebra although the former name is currently being used. This paper explains how number theory could go multidimensional using sequence algebra. Left out entirely is the Hgram graph since its plotting requires elaborate computer graphics in gif or jpg formats which creates bloated HTML files [14]. This paper advocates the use of sequence algebraic graphs for the visualisation of number theoretic functions. It should benefit mathematical communications via the Internet.

2. Multidimensional Sequence Space

In conventional number theory, the natural number sequence is considered 1-dimensional. In sequence algebra, sequences can be multidimensional. Given in table 1 are classifications of dimensionalities of some well known number theoretic sequences: