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
.