The great epistemological revolution math was based

  • McGill University
  • MATH 338
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The Great Epistemological Revolution Math was based on geometry; geometry was based on Euclid’s five axioms. How do you know the axioms are true? Lagrange showed the parallel postulate (Axiom 5) is equ ivalent to Δ = 180° and to the Pythagorean Theorem. How do you if any of those are true? Lecture 30 Monday, November 18, 2019 The second crisis: geometry As I said, Lagrange (and others) went back to Euclid’s axioms. The question wa s Axiom 5 (Parallel postulate). It says, Given a line and a point, there is a unique parallel line through the point . Euclid did not like the idea. He does not know how to avoid it. Lagrange said that if this were not true, then triangles do not add up to 180°. So he thought this must be true.
©2019 An Li. All rights reserved. 43 Around 1830, Lobachevsky, Bolyai, Gauss decided to prove the parallel postulate by reduction ad absurdum. What would happen if axiom S were false? Suppose there are many parallels, But all these parallels were curved . They got no contradictions (in our universe). This leads to a different “New Euclidean” geometry. Logically speaking, there could be an alternate universe, where Euclid’s parallel postulate is false! In this alternate universe, they found Δ < 1 80° and the Pythagorean theorem would say for a right triangle that a 2 + b 2 < c 2 . One parallel Multiple parallels No parallels The bugs follow the surface and realize that the surface is curving ! They seem to go parallel, but they get farther from each other… A triangle would have looked like this in that universe: Δ = 180° Δ < 180° Δ > 180°, a 2 + b 2 > c 2 It could be there are many parallels. It could be there are no parallels. We all know that the real universe obeys Euclid’s laws! However, according to Euclid’s relativity rules, our universe is not Euclidean!
44 ©2019 An Li. All rights reserved. Lecture 31 Wednesday, November 20, 2019 The first epistemological crisis: Non-Euclidean geometry There are three possible geometric universes: Euclidean world Elliptic world Hyperbolic world Unique tangents Δ = 180°, a 2 + b 2 = c 2 No tangents Δ > 180°, a 2 + b 2 > c 2 Many tangents Δ < 180°, a 2 + b 2 < c 2 A line goes on forever The entire universe is finite, but unbounded A line goes on forever In each of these worlds, the first four axioms are true. Only the fifth axiom breaks down. So, what is mathematical truth based on? Algebra! But algebra is based on numbers. This leads the ontological question: What is a number ? Types of numbers Integers (5) Fractional (rational) numbers ( 5 9 ) Algebraic numbers (roots of rational polynomial) (5 √14 3 ) Transcendental real numbers (.1010010001…, e , π ) o How many are there? You cannot write them all down! o In fact, most numbers are transcendental. Complex numbers ( 1 2 + √3 2 i ) Set theory ( Georg Cantor** , ~1875) Set theory became the foundation of all math. What does an “infinite set” mean?
©2019 An Li. All rights reserved. 45 According to Cantor, a set is any bunch of things you can think of. Can I count an “infinite set”? Obviously, I cannot count them all. To compare the size of two large sets M and W , I try to pair them off one-to-one. If no element is left over in either set, then # M = # W .

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