On the Nature of Things

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On the Nature of Things
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Void

• Movement requires void empty space.

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Void

• What is the nature of space?
• A lecture about nothingfor 50 minutes!
• Universe is boundless (I, 960-970)
• If it did have a boundary, boundary has 2 sides.
  Whats on the other side?
• ? Universe is infinite
• Center is what is equidistant from boundaries.
• ? Universe has no center (I, 1050-1070)

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Void

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

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Void

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

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Euclid (325-265 BC) Elements Treatise on Math
Geometry
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Euclids 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.
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Void

• Any space that satisfies Euclids Parallel
  Postulate is a Euclidean space.
• Lucretius inference works in Euclidean spaces.
• Are all spaces Euclidean?
• No
• Non-Euclidean spaces
• Non-Euclidean Geometry.

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Non-Euclidean Geometry
Nicholai Lobachevski Russian mathematician (1793-1
856) First proposed non-Euclidean geometry
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Non-Euclidean Geometry
Georg Riemann (1826-1866) German
mathematician First to formalize non-Euclidean geo
metry.
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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.
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LA to Jerusalem A segment of a great circle.
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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
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Continental U.S. is bounded finite.
But, even without boundaries surface is finite.
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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.

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Albert Einstein (1879-1955) Space is curved.
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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.

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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!

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Space is curved
Positive curvature Negative curvature
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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!

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

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We are entering a hole in the space-time continuu
m.
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Void

• Aristotle thought the universe had a centerthe
  center of the earth.
• Everything in the universe naturally moved toward
  the center.

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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!

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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 referencelike left/right.
• Democritus rejected special direction.

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Void

• Does space itselfthe voidconstitute 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.

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Isaac Newton (1642-1727) Absolute conception
of space.
Lucretius agrees.
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Gottfried Leibniz (1646-1716) Relative
conception of space.
Democritus agrees.
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Einsteins Theory of Relativity.