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The universe (2)

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Hunter Cullen
Hunter Cullen

1. The document discusses Lucretius' argument that since the universe is boundless, it must be infinite, and examines whether this inference is valid. 2. It explains that Lucretius' inference only holds in Euclidean spaces, but the universe is a non-Euclidean space like the surface of a sphere, which can be boundless yet finite. 3. The key insight of Einstein's theory of relativity is that space is curved, so Lucretius' argument fails and the universe could be finite without boundaries.

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On the Nature of Things
The universe (2)
Void Movement requires void = empty space.
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)
Void Universe is boundless (I, 960-970) Universe is infinite. Is this a valid argument? Inference works in ordinary circumstances:
The universe (2)
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 on Math & Geometry
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.
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.
Non-Euclidean Geometry Nicholai Lobachevski Russian mathematician (1793-1856) First proposed non-Euclidean geometry
Non-Euclidean Geometry Georg Riemann (1826-1866) German mathematician First to formalize non-Euclidean geometry.
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.
The universe (2)
The universe (2)
LA to Jerusalem A segment of a great circle.
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
Continental U.S. is bounded & finite. But, even without boundaries surface 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 way from here to Hong Kong is 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-time continuum.
Void Aristotle thought the universe had a centerthe 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 referencelike left/right. Democritus rejected special direction.
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.
Isaac Newton (1642-1727) Absolute conception of space. Lucretius agrees.
Gottfried Leibniz (1646-1716) Relative conception of space. Democritus agrees.
Einsteins Theory of Relativity.

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The universe (2)

  • 1. On the Nature of Things
  • 3. Void Movement requires void = empty space.
  • 4. 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)
  • 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.
  • 9. 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.
  • 10. 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.
  • 11. Non-Euclidean Geometry Nicholai Lobachevski Russian mathematician (1793-1856) First proposed 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.
  • 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.
  • 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!
  • 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 centerthe 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 referencelike left/right. Democritus rejected special direction.
  • 31. 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.

Editor's Notes

  1. 10 15 0.510 m 2 = surface area of the earth.