On the Nature of Things

Void• Movement requires void = empty space.

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)

Void• Universe is boundless (I, 960-970)

∀∴ Universe is infinite.

• Is this a valid argument?

• Inference works in ordinary circumstances:

Void• Universe is boundless (I, 960-970)

∀∴ Universe is infinite.

• Is this a valid argument?

• Inference works in Euclidean space.

Euclid(325-265 BC)

“Elements”Treatise onMath & Geometry

Euclid’s Parallel Postulate:Through a point not on a given line, there is one and only one line that goes through that pointthat is parallel to the given line.

Lines are “parallel” if they never intersect.

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.

Non-Euclidean Geometry

Nicholai LobachevskiRussian mathematician(1793-1856)First proposednon-Euclideangeometry

Non-Euclidean Geometry

Georg Riemann(1826-1866)German mathematicianFirst to formalizenon-Euclideangeometry.

Surface ofa sphere is a non-euclidean space.

“Straight line”is the shortest distance between two points.

On a sphere that is a “great circle”

Equator & longitudelines are examples.

LA toJerusalem

A segment of a greatcircle.

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.

Great circle

Point not on thegiven great circle

All great circles throughthat point will intersect the equator somewhere.

ContinentalU.S. isbounded &finite….

But, evenwithoutboundariessurface is finite.

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.

Albert Einstein(1879-1955)

Space is curved.

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.

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!

Space is curved

Positive curvature Negative curvature

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!

Shortest wayfrom here toHong Kongis…

…through The Earth!

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!

“We are entering a hole in the space-timecontinuum.”

Void• Aristotle thought

the universe had a center—the center of the earth.

• Everything in the universe naturally moved toward the center.

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!

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.

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.

Isaac Newton(1642-1727)

Absolute conception ofspace.

Lucretius agrees.

Gottfried Leibniz(1646-1716)Relative conception ofspace.

Democritus agrees.

Einstein’sTheory ofRelativity.