Geometry — a paragon of mathematical deduction?

Joel David Hamkins, Professor of Logic, Oxford University

Lecture 4. Geometry

Classical Euclidean geometry is the archetype of a mathematical deductive process. Yet the impossibility of certain constructions by straightedge and compass, such as doubling the cube, trisecting the angle, or squaring the circle, hints at geometric realms beyond Euclid. The rise of non-Euclidean geometry, especially in light of scientific theories and observations suggesting that physical reality is not Euclidean, challenges previous accounts of what geometry is about. New formalizations, such as those of David Hilbert and Alfred Tarski, replace the old axiomatizations, augmenting and correcting Euclid with axioms on completeness and betweenness. Ultimately, Tarski’s decision procedure points to a tantalizing possibility of automation in geometrical reasoning.

[minch]

I'm surprised I'd never heard of the Tarski axioms before, what an amazing result!

[antoniolewis]

1:23:00 conic sections can definitely get you to cubic numbers, as there was work done on that during medieval times by middle eastern and european mathematicians.
As an example, the cube root of 2 can be made from the x-coordinate of the intersection of the parabola y = x^2 with the hyperbola xy = 2. Both curves are conic sections. It can also be shown that any algebraic number can be constructed using the intersection of a sufficient number of conic sections in a sufficient number of dimensions.

[jrshipley]

Just posting to nod to my man Poincare independently developing the modern transformation invariance viewpoint along with Lie and Klein (and shamelessly plugging my paper):

"There is a clear affinity between Poincar ́e’s view and that of Klein and Lie, ex-pressed in the Erlangen program. Indeed, there was interaction between Poincar ́eand Lie in 1882 when Lie was in Paris. Lie wrote to Klein that Poincar ́e held theconcept of a group to be the fundamental concept for all of mathematics (Hawkins, 2000, p. 182). Jeremy Gray notes that it is very likely that Poincar ́e’s use of groupsin his analysis of Fuchsian functions was independent of Klein’s Erlangen program(Gray, 2005, p. 551). It is not clear whether Klein or Lie ever indicated anything likePoincar ́e’s philosophical view that the group is prior in conception to the terms oc-curring in the axioms of geometry. Poincar ́e’s position is that group theory definesthose terms, that the objects to which they refer are constituted by invariant sub-groups, but an alternative view would be that the objects of geometry are presentedor constructed independently while groups afford a complete means of classificationwithout necessarily constituting the objects that comprise the spaces thus classified.This view is consistent with a mathematical interest in the use of groups to classifygeometries, but inconsistent with Poincar ́e’s fully developed philosophical position."

[joeldavidhamkins]

The puzzle occurs at 42:37. Can you solve it? I offer a proof that every triangle is isosceles (due originally to W. W. Rouse Ball). Can you spot the error? If so, commet here. If you've seen the puzzle before, kindly make only elliptical comments, so that others might enjoy figuring it out for themselves.

[johntavers]

56:30 so you're saying the earth is flat.