A Magnificent Structure: an Introduction to Geometry

What is geometry? And what does it do? For millennia, Euclid’s Elements has served as a model for mathematical inquiry—an epistemological model, in fact, for the exact sciences tout court: a logical axiomatic system, consisting of a few “self-evident truths” (axioms, or postulates), provides the basis from which additional truths (theorems, or propositions) may be derived by way of logical deduction. This “magnificent structure” (Einstein), which in its classical form aimed to model the physical world, has far more to do with logical connections amongst themselves than it does with objects of human experience. How, then, has this science of lines, curves, angles, planes, areas, volumes, symmetries, coordinates, and vectors resulted in practical achievements that have transformed our very experience of the world—from ocean navigation to space travel, from mapping to computer modeling, from architecture to the visual arts?

In this course, we will explore key paradigm shifts in the historical development of geometry and, in equal measure, their implications for the philosophy of mathematics. Beginning with Euclidean principles, we will ask: how can knowledge be systematically built from a set of assumptions through logical deduction? We will then delve into the sea change that came with the Cartesian coordinate plane and n-dimensional Euclidean space, by virtue of which geometry became increasingly algebraic, with linear algebra, vector spaces, and calculus emerging as tools for investigating geometric concepts. We will also consider how projective geometry—the geometry of visual perspective—opened new horizons for Renaissance art. Further, we will explore how non-Euclidean geometry—spherical, elliptic, hyperbolic—elaborated an entirely new space for geometric ideas and inaugurated an abstract turn in mathematics, with its attention tuned to axiomatic structures divorced from physical “reality.” What has been the impact, from antiquity to today, of geometry in all its forms—classical, Cartesian, projective, algebraic, differential, fractal, and topological?

Course Schedule

Wednesday, 6:30-9:30pm ET
October 16 — November 06, 2024
4 weeks

$335.00

This class is completed. Check out our upcoming courses.