In Pursuit of Patterns

``Computers make it easier to do a lot of things, but most of the things they make it easier to do don't need to be done."

Andy Rooney

Mathematics can be described as the ``science of patterns", pursuing patterns, relationships, generalized descriptions and recognizable structure in space, numbers, and other abstracted entities. Lynn Steen has observed [16],

Mathematical theories explain the relations among patterns; functions and maps, operators and morphisms bind one type of pattern to another to yield lasting mathematical structures. Application of mathematics use these patterns to ``explain" and predict natural phenomena that fit the patterns. Patterns suggest other patterns, often yielding patterns of patterns.

Science v240, 1988

This description conjures up images of cycloids, Sierpinski gaskets, ``cowboy hat" surfaces, and multi-colored graphs. However it isn't immediately apparent that this patently visual reference to patterns applies to all of mathematics. Many of the higher order relationships in fields like number theory defy pictorial representation or, at least, they don't immediately lend themselves intuitively to a graphic treatment. Much of what is ``pattern" in the knowledge of mathematics is instead encoded in a textual, sentential format born out of the logical formalist practices which currently dominate mathematics.

Within number theory, many problems offer large amounts of ``data" which the human mind has difficulty assimilating directly. These include classes of numbers which satisfy certain criteria (eg. primes), distributions of digits in expansions, finite and infinite series and summations, solutions to variable expressions (eg. zeroes of polynomials) and other relatively unmanageable masses of raw information. Typically real insight into such problems has come directly from the mind of mathematician who ferrets out their essence from formalized representations rather than from the data. Now computers are making it possible to ``enhance" the human perceptual/cognitive systems through visualization. Consequently, patterns of a new sort are beginning to appear in the morass of numbers.

However the epistemological role of computational visualization in mathematics is still not quite clear, certainly not any clearer than the role of intuition where mental visualization takes place. It can be seen to be fulfilling a number of particular functions in current day practice. These include inspiration and discovery, informal communication and demonstration, and teaching and learning. Lately though, forces in the area of experimental mathematics have been expanding its role to include exploration and experimentation and, prehaps more controversially, formal exposition and proof. Some carefully crafted questions have been posed about how experiment might contribute to mathematics [3]. Yet answers have been slow to come. This is in part due to the general resistance and, some cases, alarm [10] within the mathematical community and finds only conditional support from those who address the issue formally [5,7].

  

Figure 1: A simple ``visual proof" of $\sum_{n=1}^{\infty} (\frac{1}{2})^{2n} = \frac{1}{3}$

The value of visualization hardly seems to be in question. The real issue seems to be what it can be used for. Can it contribute directly to the body of mathematical knowledge? Can an image act as a form of ``visual proof"? There are a number of fine examples [5] (including in number theory), most of which are in the form of simplified, heuristic diagrams like Figure 1. They call into question what the epistemological criteria of an acceptable proof are. The full breadth of that issue is outside the scope of this paper. Rather it is suggested that three necessary, but prehaps not sufficient, conditions may be:

• reliability ; that the underlying means of arriving at the proof are reliable and that the result is unvarying with each inspection
• consistency ; that the means and end of the proof are consistent with other known facts, beliefs and proofs
• repeatibility ; that the proof may be confirmed by or demonstrated to others

Each requirement is difficult to satisfy in a single, static visual representation. Most criticisms of images as mathematical knowledge or tools make this clear[6,12].

It is clear that the nature of traditional exposition differs significantly from that of the visual. In the logical formal mode, proof is provided in linearly connected sentences composed of words that are carefully selected to infer unambiguous meaning. Each sentence follows the previous, specifying an unalterable path through the sequence of statements. Although error and misconception are still possible, the tolerances are extremely demanding and follow the strict conventions of deductivist presentation [11].

In graphical representations, the same facts and relationships are often presented in multiple modes and dimensions. For example, the path through the information is usually indeterminate, leaving the viewer to establish what is important (and what is not) and in what order the dependencies should be assessed. Further, unintended information and relationships may be perceived, either due to the unanticipated interaction of the complex array of details or due to the viewer's own perceptual and cognitive processes.

As a consequence, successful visual representations tend to be spartan in their detail. And the few examples of visual proof which withstand close inspection are limited in their scope and generalizability. In the effort to bring images closer to conformity with the prevailing logical modes of proof, they have subsequently lost the richness which is intrinsic to the visual.