Greek geometry as constructive mathematics
Ancient Greek geometry is a "maker's knowledge." Euclid never proves a single theorem about objects he has not first carefully shown how to construct by ruler and compass. A large part of higher Greek geometry is similarly devoted to producing specific geometrical objects, such as duplicating a cube, trisecting an angle, or squaring a circle. Why this obsession with making? Shouldn't geometry be about proving theorems rather than giving recipes for how to draw things using mechanical tools?
Growing interest in constructive mathematics in recent decades has shed new light on this aspect of classical geometry. Many authoritative editions and interpretations of Greek mathematics from a century ago were arguably coloured by the Platonic philosophy of mathematics of the time---"Cantor's paradise," as Hilbert called it. This point of view played down the role of constructions, relegating it to minor subsidiary functions such as existence proofs. But renewed recognition of the value of constructive and operational modes of thought in modern mathematics has revealed rich foundational parallels with the Greek style of geometry. This suggests that the Greeks may well have focussed on constructions due to a philosophically sophisticated conception of mathematical method and foundations, rather than as quasi-applied problems pursued largely for reasons of tradition, as had previously been supposed.
In this seminar we read Greek geometrical works in this tradition and, informed by modern insights, try to reconstruct their conception of mathematics and its foundations. Since the classical Greek corpus is completely void of any explicit foundational reflection, we must study their technical works with an eye to try to extract the implicit assumptions they make in terms of what constitutes worthy research goals and legitimate mathematical method and rigor.
Attention to these questions are a longstanding focus of the Utrecht school in the history of mathematics, going back to the work of Henk Bos. It remains a burning question in recent scholarship, and one on which mathematically trained students have much to contribute to historical understanding.
The topic also affords ample connections to current research in logic and philosophy of mathematics. This includes formalisations of diagrammatic reasoning, constructive mathematics, and the philosophy of mathematical practice.
Schedule
In consultation with participants.
General mathematics at bachelor level. History of Mathematics is preferred but not strictly necessary. The seminar is suitable for Bachelor Math students in their last year as well as Master students, HPM students with sufficient mathematical background (depending on the subject, please contact in advance) and historically interested Physics students.
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