Post on 09-Apr-2018
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On the Nature of Things
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g
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Void
Movement requires void = empty space.
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Void What is the nature of space?
A lecture about ³nothing´²for 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)
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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)
³E
lements´Treatise on
Math &
Geometry
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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 ³parallel´ if they never intersect.
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Void
Any space that satisfies Euclid¶s Parallel
Postulate is a Euclidean space.
Lucretius¶ inference works in Euclideanspaces.
Are all spaces Euclidean?
No
Non-Euclidean spaces
Non-Euclidean Geometry.
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Non-Euclidean Geometry
Nicholai LobachevskiRussian mathematician
(1793-1856)
First proposed
non-Euclidean
geometry
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Non-Euclidean Geometry
Georg Riemann(1826-1866)
German mathematician
First to formalize
non-Euclidean
geometry.
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Surface of
a sphere is a non-euclidean space.
³S
traight line´is the shortest
distance between
two points.
On a sphere that
is a ³great 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 surf ace o f a spher e: 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 through
that point will intersectthe equator somewhere.
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Continental
U.S. isbounded &
finite«.
But, even
without
boundaries
surface is
f inite.
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Non-E
uclideanS
pace 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-Euclideanspace.
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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 overa small distance, but is curved.
Our space looks Euclidean over a smalldistance, 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, buttravel only a (long but) finite distance.
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Space is curved
Lucretius¶ argument is invalid!
People who get outside the 2-dimensionalsurface of a sphere can see it is curved in
3 dimensions.
People who get outside our 3-dimensionalspace 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«
«throughThe 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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³We are entering a hole in the space-time
continuum.´
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Void Aristotle thought
the universe had a
center²the center
of the earth.
Everything in the
universe naturallymoved toward the
center.
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Void Universe is boundless (I, 960-970)
Center is what is equidistant fromboundaries.
@ Universe has no center (I, 1050-1070)
Is that valid?
Surface of a sphere has a center, but it isnot in the surface of the sphere!
Universe could have a center, which isnot 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 naturallymove downward (II, 190).
But which direction is ³down´?
³Down´ only makes sense relative to a
given frame of reference²like left/right.
Democritus rejected special direction.
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Void
Does space itself²the void²constitute a
frame of reference?
Is there a difference between a universewith 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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Einstein¶s
Theory of Relativity.