History of Mathematics

1. History of Mathematics Five Problems and Their Solution

2. Five Problems • Geometry: The parallel postulate: axiom or derivable proposition? • Set Theory: How to understand the Infinite • Analysis: The rigorous foundations of the infinitesimal • Number Theory: Fermat’s Last Theorem • Algebra: Solving higher order equations

3. Euclid’s Elements 23 Definitions • A point is that which has no part. • A line is breadthless length. • A straight line is a line which lies evenly with the points on itself. • Parallel straight lines are straight lines which, being in the same plane and being produced indefinitely in both directions, do not meet one another in either direction.

4. Euclid’s Elements Five Axioms/Postulates. Let the following be postulated: • To draw a straight line from any point to any point. • To produce a finite straight line continuously in a straight line. • To describe a circle with any center and distance. • That all right angles are equal to one another. • That, if a straight line falling on two straight lines make the interior angle on the same side less than two right angles, the two straight lines, if produced indefinitely, meet on that side on which are the angles less than two right angles.

5. Euclid’s Elements A modern form of the axioms. • For every point P and every point Q not equal to P there exists a unique line that passes through P and Q. • For every segment AB and for every segment CD there exists a unique point E such that B is between A and E and segment CD is congruent to segment BE. • For every point O and every point A not equal to O there exists a circle with center O and radius OA. • All right angles are congruent to each other. • For every line l and for every point  P that does not lie on l there exists a unique line m through P that is parallel to l.

6. Euclid’s Elements Five Notions • Things which are equal to the same thing are also equal to one another. • If equals be added to equals, the wholes are equal. • If equals be subtracted from equals, the remainders are equal. • Things which coincide with one another are equal to one another. • The whole is greater than the part. In a modern reading what Euclid calls "equality" we today call "congruences".

7. Major contributors • Euclid (ca 300 BC): The Elements 13 books. First Axiomatization of Geometry • Girolamo Saccheri (1667-1733) • Hypothesis of right angle (HRA). • Hypothesis of Obtuse Angle (HOA) • Hypothesis of Acute Angle (HAA) • Johann Lambert (1728-1777) Derives theorems with HAA, but still believes it is necessarily false. • Pierre-Simon Laplace (1749-1827). Used Physics to show that space is homogenous. This implies PP. • Adrien-Marie Legendre (1752-1833). Writes Eléments de Géométrie. Tries to prove PP in several editions.

8. Non Euclidian Geometry • Carl Friedrich Gauß (1777-1855) First to believe in PP as additional axiom, but keeps his findings unpublished until after death • János Bolyai (1802-1860) Also studies non Euclidian geometry. Publishes as appendix in his father’s book (1831) • Nikolai Lobachevsky (1792-1856) Published on non Euclidian geometry On the principles of Geometry 1829. Geometrische Untersuchungen zu Theorie der Parallellinien (1842). Credited with the discovery of Hyperbolic Geometry

9. Models for Hyperbolic geometry • Eugenio Beltrami (1835-1900) • Felix Klein (1849-1925) • Henri Poincaré (1854-1912)

10. New Geometries • Bernhard Riemann (1826-1866) Studied what is now called Riemann surfaces. This generalized the geometries locally and is the basis for Einstein’s theory of general relativity • Felix Klein (1849-1925) with Sophus Lie (1842-1899) Studied geometries from the point of view of transformation groups.

11. Hilbert’s Axioms • Undefined Terms • Points • Lines • Planes • Lie on, contains • Between • Congruent • Axioms • Axioms of Incidence • Axioms of Congruence • Axioms of Order • Axiom of Parallels • Axioms of Continuity

12. Axioms of Incidence • Postulate I.1. • For every two points A, B there exists a line a that contains each of the points A, B. • Postulate I.2. • For every two points A, B there exists no more than one line that contains each of the points A, B. • Postulate I.3. • There exists at least two points on a line. There exist at least three points that do not lie on a line. • Postulate I.4. • For any three points A, B, C that do not lie on the same line there exists a plane α that contains each of the points A, B, C. For every plane there exists a point which it contains. • Postulate I.5. • For any three points A, B, C that do not lie on one and the same line there exists no more than one plane that contains each of the three points A, B, C. • Postulate I.6. • If two points A, B of a line a lie in a plane α then every point of a lies in the plane α. • Postulate I.7. • If two planes α, β have a point A in common then they have at least one more point B in common. • Postulate I.8. • There exist at least four points which do not lie in a plane.

13. Axioms of Congruence • Postulate III.1. • If A, B are two points on a line a, and A' is a point on the same or on another line a' then it is always possible to find a point B' on a given side of the line a' such that AB and A'B' are congruent. • Postulate III.2. • If a segment A'B' and a segment A"B" are congruent to the same segment AB, then segments A'B' and A"B" are congruent to each other. • Postulate III.3. • On a line a, let AB and BC be two segments which, except for B, have no points in common. Furthermore, on the same or another line a', let A'B' and B'C' be two segments which, except for B', have no points in common. In that case if AB≈A'B' and BC≈B'C', then AC≈A'C'. • Postulate III.4. • If ∠ABC is an angle and if B'C' is a ray, then there is exactly one ray B'A' on each "side" of line B'C' such that ∠A'B'C'≅∠ABC. Furthermore, every angle is congruent to itself. • Postulate III.5. (SAS) • If for two triangles ABC and A'B'C' the congruences AB≈A'B', AC≈A'C' and ∠BAC ≈ ∠B'A'C' are valid, then the congruence ∠ABC ≈ ∠A'B'C' is also satisfied.

14. Axioms of Order • Postulate II.1. • If a point B lies between a point A and a point C then the points A, B, C are three distinct points of a line, and B then also lies between C and A. • Postulate II.2. • For two points A and C, there always exists at least one point B on the line AC such that C lies between A and B. • Postulate II.3. • Of any three points on a line there exists no more than one that lies between the other two. • Postulate II.4. • Let A, B, C be three points that do not lie on a line and let a be a line in the plane ABC which does not meet any of the points A, B, C. If the line a passes through a point of the segment AB, it also passes through a point of the segment AC, or through a point of the segment BC.

15. Axiom of ParallelsAxioms of Continuity • Postulate IV.1. • Let a be any line and A a point not on it. Then there is at most one line in the plane that contains a and A that passes through A and does not intersect a. • Postulate V.1. (Archimedes Axiom) • If AB and CD are any segments, then there exists a number n such that n copies of CD constructed contiguously from A along the ray AB will pass beyond the point B. • Postulate V.2. (Line Completeness) • An extension of a set of points on a line with its order and congruence relations that would preserve the relations existing among the original elements as well as the fundamental properties of line order and congruence (Axioms I-III and V-1) is impossible.