Euclid's Lost Masterpiece: The Porisms
This is a problem in restoration. Summarized with "horrible brevity" but in the most tantalizing of fashions by Pappus of Alexandria, this is the book that eluded the best efforts of Fermat, Newton and other early restorers of the lost art of Greek geometrical analysis. Neither the content of this book nor even the mode of reasoning it employed has ever been satisfactorily recovered. What was it about this book that it has remained so inscrutable to modern algebraic vision?
Euclid's lost book the Porisms treated of a third kind of mathematical proposition, somehow intermediate between a geometrical theorem (demonstration) and a geometrical problem (construction); our modern understanding of mathematical propositions makes it very hard for us even to imagine what such a tertium quid might be. It is also clear that the Porisms had some sort of fundamental connection with the Greek geometrical art of analysis & synthesis, the attempted restoration of which in the late Renaissance contributed greatly to the creation of modern algebra. It may then be highly significant that the Porisms was the one work in classical geometrical analysis that remained inaccessible to the early modern algebraists. Might it not be the case that what is most inaccessible to modern mathematical hindsight may have been most defining for Greek mathematics itself? That what seems familiar to us about ancient mathematics is due to anachronism? We wish to see whether the accurate restoration of the mode of reasoning underlying a porism - a bit of mathematical archaeology, if you will - can prompt a fresh reexamination of mathematical reasoning in general.
Mathematicians generally, especially geometers with a special interest in projective geometry and topology, as well as those interested in the history of Greek Mathematics; logicians generally, especially those interested in the logic of mathematical reasoning, and those interested in Aristotelian logic; those interested in "sacred geometry" and Plato's injunction to study mathematics in preparation for philosophy.
A. The Porisms in Hindsight
1) Restoration of the doctrine on the basis of the remarks by Pappus of Alexandria in his synopsis of the book; a careful explication of the term dedomena , which is central to poristic doctrine and the subject of another book by Euclid (somewhat misleadingly called the "Data"); and the actual employment of poristic reasoning in the method of analysis & synthesis. Outcome: poristic reasoning as the theoretical geometrization of the act of measurement.
Related Topics --An annotated translation of Marinus' Commentary on Euclid's "Data", showing the confusion over the central concept dedomena even in ancient times;" the relation between the "Data" and Euclid's Elements ; an inquiry into the reason why Aristotle called his logical writings "The Analytics," and a comparison of Aristotle's discussion of "intuitive induction" with the reasoning found in Euclid's "Data."
2) Differentiating poristic reasoning from modern algebraic reasoning, "analysis of deductive connections," "analysis of figures," hypothetical and heuristical reasoning.
Related Topics --The place of "poristic analysis" in Viète's analytic art; critique of the attempted restoration of the Porisms by Fermat & Newton; a recapitulization of the algebraization of geometry in Bonasoni and Descartes.
3) The problem of formalizing poristic or analytical reasoning in geometry so that it may be accommodated to modern mathematics without obscuring its distinctive character.
Related Topics --The analysis situs of Leibniz and his successors (particularly Grassmann), leading to topology.
B. Does poristic reasoning have applications beyond geometry; conversely, can modern logic address the kinds of inference found in porisms?
C. The esoteric implications of the doctrine of the porisms as a "sacred" formalization of the act of measurement; the hypothesis that it derives from Babylonian or Middle Eastern sources just as algebra did; ritualistic connection as the principle of "inference" in such argumentation; a comparison of the concept of the given (dedomena) with the concept of the fated or apportioned (heimarmena).