4. Void
• What is the nature of space?
• A lecture about “nothing”—for 50 minutes!
• Universe is boundless (I, 960-970)
– 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, 1050-1070)
5. Void
• Universe is boundless (I, 960-970)
∀ ∴ Universe is infinite.
• Is this a valid argument?
• Inference works in ordinary
circumstances:
6.
7. Void
• Universe is boundless (I, 960-970)
∀ ∴ Universe is infinite.
• Is this a valid argument?
• Inference works in Euclidean space.
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.
12. Non-Euclidean Geometry
Georg Riemann
(1826-1866)
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 Equator & longitude
is a “great circle” lines are examples.
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
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.
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!
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!
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!
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, 1050-1070)
• 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.
