In Support of Proof

``Computers are useless. They can only give you answers."

Pablo Picasso (1881-1973)

In order to offer the reliability, consistency and repeatability of the written word and still provide the potential inherent in the medium, visualization needs to offer more than just the static image. It too must guide, define and relate the information presented. The logical formalist conventions for mathematics have evolved over many decades, resulting in a mode of discourse that is precise in its delivery. To wit, the order of presentation of ideas is critical with definitions preceding their usage, proofs separated from the general flow of the argument for modularity, and references to foundational material listed at the end.

To do the same, visualization must include additional mechanisms or conventions beyond the base image. It isn't appropriate to simply ape the logical conventions and find some visual metaphor or mapping that works similarly (this approach is what limits existing successful visual proofs to very simple diagrams). Instead, an effective visualization needs to offer several key features

• dynamic ; the representation should vary through some parameter(s) to demonstrate a range of behaviours (instead of the single instance of the static case)
• guidance ; to lead the viewer through the appropriate steps in the correct order, the representation should offer a ``path" through the information which builds the case for the proof
• flexibility ; it should support the viewer's own exploration of the ideas presented, including the search for counter examples or incompleteness
• openness ; the underlying algorithms, libraries and details of the programming languages and hardware should be available for inspection and confirmation

With these capabilities available in an interactive representation, the viewer could then follow the argument being made visually, explore all the ramifications, check for counter examples, special cases and incompleteness, and even confirm the correctness of the implementation. In fact, the viewer will be able to perform all of the same inspections on the visual representation as he would be able to on a traditional logical formal proof.

While this does not yet offer any conclusion as to how images and computational tools might impact on mathematical methodologies or the underlying epistemology, it does indicate the direction that subsequent work may take. Examples from recent work done at the CECM offer some insight into how emerging technologies may eventually provide an unambiguous role for visualization in mathematics.