iCHSTM 2013 Programme • Version 5.3.6, 27 July 2013 • ONLINE (includes late changes)
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The imagery of gestalt and metaphor in mathematical proofs and its relevance to proof comparison
Arie Hinkis | Tel Aviv University, Israel

Many mathematical theorems have a number of proofs. Can the different proofs be methodically compared? We suggest that mathematical proofs can be described by a series of gestalts, with metaphors attached to each gestalt switch. The comparison of proofs can then be affected by comparing the proofs through their gestalt and metaphoric descriptors. To establish proof descriptors linguistic imagery, graphics and associations into art and poetry are applied. We will exemplify the suggested method for the case of the Cantor-Bernstein Theorem, an elementary but not trivial theorem of set theory. We will look at about half a dozen proofs of the theorem and compare them through their gestalt and metaphoric dimensions. The processing of proofs for their imagery descriptors may unravel internal links between proofs of different theorems, stemming seemingly from different contexts. We will exemplify this observation by relating Bernstein's proof of his division theorem, to one of the proofs of the Cantor-Bernstein Theorem. We will briefly further note how the imagery derived from another proof of the Cantor-Bernstein Theorem affected a research project which led from Bernstein's Division Theorem to important results in graph theory and topology. We will conclude with remarks on the possibility that the imagery of proof descriptors forms the arsenal of a mathematician, and is the touchstone for the semiformal discourse among peers. We will point out that to farther our knowledge on these subjects more study is required in the internal history of mathematics and the anthropology of the mathematicians' community.