1. On the Nature of Things

3. Void
Movement requires void = empty space.

4. Void What is the nature of space?
A lecture about “nothing”—for 50 minutes!
Universe is boundless (I, )
If it did have a boundary, boundary has 2 sides. What’s on the other side?
 Universe is infinite
Center is what is equidistant from boundaries.
 Universe has no center (I, )

5. Void Universe is boundless (I, 960-970)  Universe is infinite.
Is this a valid argument?
Inference works in ordinary circumstances:

7. Void Universe is boundless (I, 960-970)  Universe is infinite.
Is this a valid argument?
Inference works in Euclidean space.

8. Euclid
( BC)
“Elements”
Treatise on
Math &
Geometry

9. Euclid’s Parallel Postulate:
Through a point not on a given line, there is one
and only one line that goes through that point
that is parallel to the given line.
Lines are “parallel” if they never intersect.

10. Void
Any space that satisfies Euclid’s Parallel Postulate is a Euclidean space.
Lucretius’ inference works in Euclidean spaces.
Are all spaces Euclidean? No
Non-Euclidean spaces
Non-Euclidean Geometry.

11. Non-Euclidean Geometry
Nicholai Lobachevski
Russian mathematician
( )
First proposed
non-Euclidean
geometry

12. Non-Euclidean Geometry
Georg Riemann
( )
German mathematician
First to formalize
non-Euclidean
geometry.

13. Surface of
a sphere is a non-euclidean space.
“Straight line”
is the shortest distance between two points.
On a sphere that is a “great circle”
Equator & longitude
lines are examples.

16. LA to
Jerusalem
A segment
of a great
circle.

17. On the surface of a sphere: Through a point
not on a given great circle, there is no great circle
that goes through that point that is parallel to
(never intersects) the given great circle.
Point not on the
given great circle
All great circles through
that point will intersect the equator somewhere.
Great circle

18. Continental U.S. is bounded & finite…. But, even without boundaries
surface is
finite.
1015 × m2 = surface area of the earth.

19. Non-Euclidean Space The surface of a sphere is a non-Euclidean space.
A non-Euclidean space can be boundless, and yet finite.
Our universe is a non-Euclidean space.

20. Albert Einstein
( )
Space is curved.

21. Space is curved
The surface of the earth looks flat over a small distance, but is curved.
Our space looks Euclidean over a small distance, but is curved.
If you shot an arrow that kept going, it would eventually hit you in the back!
It would never hit a boundary, but travel only a (long but) finite distance.

22. Space is curved Lucretius’ argument is invalid!
People who get outside the 2-dimensional surface of a sphere can see it is curved in 3 dimensions.
People who get outside our 3-dimensional space can see it is curved in 4 dimensions!

23. Space is curved
Positive curvature Negative curvature

24. Space is curved
The shortest way from one point to another on a non-Euclidean 2-dimensional surface is by leaving that surface and entering another dimension!

25. Shortest way
from here to
Hong Kong
is…
…through
The Earth!

26. Space is curved
The shortest way from one point to another in our non-Euclidean 3-dimensional space is by leaving that space and entering another dimension!

27. “We are entering a hole in the space-time
continuum.”

28. Void
Aristotle thought the universe had a center—the center of the earth.
Everything in the universe naturally moved toward the center.

29. Void Universe is boundless (I, 960-970)
Center is what is equidistant from boundaries.
 Universe has no center (I, )
Is that valid?
Surface of a sphere has a center, but it is not in the surface of the sphere!
Universe could have a center, which is not in the universe!

30. Void
If no center, toward which things naturally move, what accounts for movement?
Weight of atoms causes them to naturally move downward (II, 190).
But which direction is “down”?
“Down” only makes sense relative to a given frame of reference—like left/right.
Democritus rejected special direction.

31. Void Does space itself—the void—constitute a frame of reference?
Is there a difference between a universe with just one atom at rest, and a universe with just one atom moving at a constant speed in a constant direction?
Yes = Absolute Conception of Space.
No = Relative Conception of Space.

32. Isaac Newton (1642-1727) Absolute conception of space.
Lucretius agrees.

33. Gottfried Leibniz (1646-1716) Relative conception of space.
Democritus agrees.

34. Einstein’s
Theory of
Relativity.