Download OLLI: History of Mathematics for Everyone Day 2: Geometry

OLLI: History of Mathematics for Everyone
Day 2: Geometry
instructor: Jill Faudree
February 22, 2017
I
Questions about Geometry Facts and Experience
I
Experimental Geometry
I
Cut-and-Paste Geometry
I
Perfection: the Axiomatic Geometry of Euclid
I
Brave New Worlds: the Extraordinary Tale of Euclid’s 5th Axiom
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Foundations Split Asunder: Gödel’s Incompleteness Theorems
Experimental Geometry
Experimental Geometry:
the volume of a cone is one-third the volume of the cylinder that contains
it
Vcone =
Vcylinder
3
or
Vcone =
πr 2 h
3
Experimental Geometry:
the volume of a cone is one-third the volume of the cylinder that contains
it
Vcone =
Vcylinder
3
or
Vcone =
πr 2 h
3
Experimental Geometry
the ratio of the circumference to the diameter is about 3
d =2
0
1
2
3
4
5
C
≈3
d
or
C
2πr
=
=π
d
d
6
7
Experimental Geometry
the ratio of the circumference to the diameter is about 3
d =2
0
1
2
3
4
5
C
≈3
d
or
C
2πr
=
=π
d
d
6
7
Cut-and-Paste Geometry
the Pythagorean Theorem
a2 + b 2 = c 2
c
b
a
Cut-and-Paste Geometry
the Pythagorean Theorem
a2 + b 2 = c 2
c
b
a
Cut-and-Paste Geometry
the Pythagorean Theorem
a2 + b 2 = c 2
c
b
a
a
the Pythagorean Theorem :a2 + b 2 = c 2
c
b
aa
Timeline:
I
(1850 BC) Egyptians correctly calculated the volume of a truncated
pyramid.
I
(1650 BC) Egyptians found area of circle with π ≈ 3.1604 · · · .
I
(600 BC) Greeks knew the Pythagorean Theorem. It was certainly
known much much earlier by Egyptians, Babylonians, Chinese, and
Indians.
I
(213 BC) Chinese mathematicians formula to find the sector of a
circle (using π ≈ 3.)
No formulas and no proofs. Just worked out examples.
Timeline:
I
(1850 BC) Egyptians correctly calculated the volume of a truncated
pyramid.
I
(1650 BC) Egyptians found area of circle with π ≈ 3.1604 · · · .
I
(600 BC) Greeks knew the Pythagorean Theorem. It was certainly
known much much earlier by Egyptians, Babylonians, Chinese, and
Indians.
I
(213 BC) Chinese mathematicians formula to find the sector of a
circle (using π ≈ 3.)
No formulas and no proofs. Just worked out examples.
Euclid’s Elements, 300 BC
I
By about 600 BC, Greeks had established the importance of logic in
the study of mathematics.
I
The theorems in the Elements were not new. It was the organization
that was new.
I
It consists of 13 books covering plane geometry, solid geometry,
number theory, and ratios.
I
Each book begins with the “ground rules:" common notions,
definitions, postulates (axioms) then proves one theorem after
another without motivation.
I
Gold-standard of logical thought and mathematical organization.
Elements: Book I
Postulates
1. A straight line segment can be drawn between any pair of points.
2. A straight line segment can be extended continuously in a line.
3. A circle can be drawn with any center and radius.
4. All right angles are equal.
5. If a straight line falling on two straight lines makes the sum of the
interior angles on one side less than the sum of two right angles, then
the two straight lines if extended indefinitely meet on that side on
which the angle sum is less than the two right angles.
Hmmmmm.....which axiom is not like the other ones???
Elements: Book I – Postulate 5
If a straight line falling on two straight lines makes the sum of the interior
angles on one side less than the sum of two right angles, then the two
straight lines if extended indefinitely meet on that side on which the angle
sum is less than the two right angles.
Euclid proves 28 propositions, including ones characterizing parallel lines,
prior to using Postulate 5!
The controversy begins:
Is Postulate 5 really a Proposition?
Elements: Book I – Postulate 5
If a straight line falling on two straight lines makes the sum of the interior
angles on one side less than the sum of two right angles, then the two
straight lines if extended indefinitely meet on that side on which the angle
sum is less than the two right angles.
Euclid proves 28 propositions, including ones characterizing parallel lines,
prior to using Postulate 5!
The controversy begins:
Is Postulate 5 really a Proposition?
I
( 250 CE) Ptolemy gives a proof of the 5th Postulate using earlier
results.
I
(5th Century) Proculus records Ptolemy’s proof is shown to be
incorrect and gives a proof of his own. (Also, wrong.)
I
(8th and 9th Century) Multiple Arab scholars try to prove the 5th
Postulate. All wrong.
I
(11th Century) Omar Khayyam
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(1663) John Wallis inspired by Al-Tusi
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(18th Century) John Playfair (Scottish) proposes new axiom:
Given any line L and point P not on L, there exists a unique line
through P parallel to L.
I
(1800) AM Legendre
(It turns out to be equivalent to Euclid’s postulate!)
Girolamo Saccheri: Euclid Freed of Every Flaw (pub. 1733, read 1800’s)
I
Proof by contradiction
I
the negation of Playfair’s Postulate:
Playfair:
Given any line L and point P not on L, there exists a unique line
through P parallel to L.
Negation:
Given any line L and point P not on L, there exists
(1) no line through P parallel to L or
(2) multiple lines through P parallel to L.
What do you think? Are these obviously impossible???!!!
Recall Euclid proved the existence of parallel lines!
Girolamo Saccheri: Euclid Freed of Every Flaw (pub. 1733, read 1800’s)
I
Proof by contradiction
I
the negation of Playfair’s Postulate:
Playfair:
Given any line L and point P not on L, there exists a unique line
through P parallel to L.
Negation:
Given any line L and point P not on L, there exists
(1) no line through P parallel to L or
(2) multiple lines through P parallel to L.
What do you think? Are these obviously impossible???!!!
Recall Euclid proved the existence of parallel lines!
Girolamo Saccheri: Euclid Freed of Every Flaw (pub. 1733, read 1800’s)
I
Proof by contradiction
I
the negation of Playfair’s Postulate:
Playfair:
Given any line L and point P not on L, there exists a unique line
through P parallel to L.
Negation:
Given any line L and point P not on L, there exists
(1) no line through P parallel to L or
(2) multiple lines through P parallel to L.
What do you think? Are these obviously impossible???!!!
Recall Euclid proved the existence of parallel lines!
Girolamo Saccheri: Euclid Freed of Every Flaw (pub. 1733, read 1800’s)
I
Proof by contradiction
I
the negation of Playfair’s Postulate:
Playfair:
Given any line L and point P not on L, there exists a unique line
through P parallel to L.
Negation:
Given any line L and point P not on L, there exists
(1) no line through P parallel to L or
(2) multiple lines through P parallel to L.
Saccheri correctly proves hypothesis (1) produces a contradiction. He
hallucinates a proof that hypothesis (2) produces a contradiction.
Brave New Worlds:
I
Carl Friedrich Gauss (1810), Nikolai Ivanovich Lobachevsky (1829),
Janos Bolyai (1832) realized there exists a geometry in which multiple
parallel lines are possible!
Escher’s 1959 Circle Limit III
Brave New Worlds:
I
Carl Friedrich Gauss (1810), Nikolai Ivanovich Lobachevsky (1829),
Janos Bolyai (1832) realized there exists a geometry in which multiple
parallel lines are possible!
Escher’s 1959 Circle Limit III
Brave New Worlds:
I
Bernhard Riemann (1854) discovered mathematicians were too quick
to believe Euclid had eliminated the existence of geometries with no
parallel lines.
Brave New Worlds:
I
Bernhard Riemann (1854) discovered mathematicians were too quick
to believe Euclid had eliminated the existence of geometries with no
parallel lines.
Gödel’s Incompleteness Theorems
I
Kurt Gödel (1906-1978), published work in 1931
I
Destroyed David Hilbert’s (1862-1943) goal to find a complete and
consistent set of axioms for all of mathematics.
I
Any consistent axiomatic system complicated enough to define basic
arithmetic will contain true statements that are nevertheless
unprovable.
I
Further, such a system cannot be shown to be consistent.
Gödel’s Incompleteness Theorems
I
Kurt Gödel (1906-1978), published work in 1931
I
Destroyed David Hilbert’s (1862-1943) goal to find a complete and
consistent set of axioms for all of mathematics.
I
Any consistent axiomatic system complicated enough to define basic
arithmetic will contain true statements that are nevertheless
unprovable.
I
Further, such a system cannot be shown to be consistent.
Gödel’s Incompleteness Theorems
I
Kurt Gödel (1906-1978), published work in 1931
I
Destroyed David Hilbert’s (1862-1943) goal to find a complete and
consistent set of axioms for all of mathematics.
I
Any consistent axiomatic system complicated enough to define basic
arithmetic will contain true statements that are nevertheless
unprovable.
I
Further, such a system cannot be shown to be consistent.
Gödel’s Incompleteness Theorems
I
Kurt Gödel (1906-1978), published work in 1931
I
Destroyed David Hilbert’s (1862-1943) goal to find a complete and
consistent set of axioms for all of mathematics.
I
Any consistent axiomatic system complicated enough to define basic
arithmetic will contain true statements that are nevertheless
unprovable.
I
Further, such a system cannot be shown to be consistent.
Gödel’s Incompleteness Theorems
I
Kurt Gödel (1906-1978), published work in 1931
I
Destroyed David Hilbert’s (1862-1943) goal to find a complete and
consistent set of axioms for all of mathematics.
I
Any consistent axiomatic system complicated enough to define basic
arithmetic will contain true statements that are nevertheless
unprovable.
I
Further, such a system cannot be shown to be consistent.
Summary
Euclid’s Elements
I
became the benchmark of mathematics organization, exposition and
thought
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drove 2000 years of research ending in the discovery of alternate
geometries (hyperbolic and elliptic).
NOTE: This research proved Euclid right in his choice of axiom!
Geometry drove 2000 years of research including research into the nature
of axiomatic systems ultimately undermining the structure of the work
that inspired it!