Physics & Metaphysics Handout 3

H.S. Hestevold Spring 2014

THE TOPOLOGY OF SPACE

I.Inspiration

A.Go to YouTube and watch “Dr. Quantum – Flatland” (5 mins):

B.Conceptual exercises. Assume that absolute space exists and that it is constituted by infinitely many continuous spatial points. (That is, between any two spatial points, there exists a third.) Conceive of Flatland as a 2D physical world that is located within 3D space. (That is, the 2D region of space occupied by Flatland is a sub-region of 3D space. Assume that Flatland's 2D region could include various flat-planets and flat-stars.)

1.Imagine that Flatland is infinite in the sense that Flatland has no edge. Isn’t there a sense in which Flatlanders are nonetheless spatially bounded? Imagine that there is absolute space that Flatland occupies. Isn’t there a spatial direction d1 and a spatial point p occupied by a part of Flatland such that no part of Flatland lies in direction d1 relative to that point? (In short, no part of Flatland occupies any spatial point above or below the surface of Flatland.) Now, aren’t you imagining a 2D universe that is spatially infinite and yet bounded – a universe in which there is a direction in which Flatlanders cannot move?

2.Imagine that Flatland is a square, which is a topological shape with buta single edge. (Does this cause you any particular conceptual difficulty?) Imagine two spatial points, p1 and p2, occupied by parts of Flatland. Assume that there is a direction d2 such that p1 bears d2 to p2. (Between p1 and p2, there are infinitely many spatial points occupied by Flatland. Right?) If Flatland is a square, then there are infinitely many spatial points that bear d2 to p1 and p2, and no part of Flatland occupies these points. Right? (These are spatial points that lie outside the region of space that Flatland occupies.)You are imagining a bounded universe that occupies a sub-region of unbounded space.

3.Imagine that there exist four spatial dimensions: left/right; forward/backward; up/down; and… let’s call the other dimension ana/kata.[1]Imagine that our 3D world is located within 4D space and that it extends infinitely in three dimensions in the same way that the edge-less Flatland extends infinitely within two dimensions. Wouldn’t there be a spatial direction d3(e.g. kata-16) and a spatial point p occupied by a part of our 3D universe such that no part of our3D universe bears direction d3 relative to that point? (In short, no part of our 3D universe occupies any of the infinitely many spatial points that would exist ana or kata.) Now, aren’t you imagining a universe that is spatially infinite and yet bounded – a universe in which there is a direction in which we cannot move?

4.Take a deep breath… Continue to imagine that our 3D universe occupies a 3D sub-region of 4D space that is, say, sphere—shaped or torus-shaped. Imagine two spatial points, p1 and p2, occupied by parts of physical objects in our 3D universe. Assume that there is a direction d4 such that p1 bears d4 to p2. (Between p1 and p2, there are infinitely many spatial points occupied by our 3D universe. Right?) If our 3D universeis a sphere, then there are infinitely many spatial points that bear d4 to p1 and p2, and no part of our 3D universe occupies these points. Right? (These are spatial points that lie outside the 3D region of space that our 3D universe occupies.)

5.A final exercise: What would it be like if space were indeed bounded? Re-think world (2) above – the one-edged square Flatland: imagine that all spatial points not occupied by a part of Flatland disappear. That is what finite space would be like if we were Flatlanders: if spatial point p1 bears d2 to p2, then there is some spatial point occupied by Flatland such that there exists no other spatial point that bears d2 to thatpoint.Now, re-think world (4) and imagine that all spatial points not occupied by a part of our 3D universe disappear. You are now conceiving of a non-infinite 3D universe: if p1 bears d4 to p2, then there is some spatial point occupied by our 3D universe such that there exists no other spatial point that bears d4 to that spatial point.

II.Absolute space: a sketch

A.Robin Le Poidevin:

Space contains objects, rather like a box, only in this case we suppose the box to have no sides. (Travels in Four Dimensions: The Enigmas of Space and Time (Oxford, 2003))

B.Issac Newton:

II. Absolute space, in its own nature, without relation to anything external, remains always similar and immovable. …

III. Place is a part of space which a body takes up…. …the place of the whole is the same as the sum of the places of the parts…. [Scholium to the Definitions, Mathematical Principles of Natural Philosophy]

C.Two ping-pong scenarios.

1.In a still room, a player serves the ball straight across the net with a certain type of paddle movement, and the ball bounces directly on the center line.

a.Study Question: Does the ball displace air molecules as it travels straight from one side of the net to the other?
b.Study Question: Does the ball displace points of absolute space as it travels straight from one side of the net to the other?

2.Before the second serve, someone turns on a large box fan to the right of the server. Using the very same paddle movement with the ball in the same starting position, the player hits the ball across the net. Ordinarily, the ball would have bounced again on the center line; but, with the fan breeze blowing from the server's right to the server's left, the ball flies across the net and bounces to the left side of the center line.

a.Study Question: Does the ball displace air molecules as it travels straight from one side of the net to the other?
b.Study Question: Does the fan displace air molecules as it blows across the ping-pong table?
c.Study Question: Does the fan displace spatial points as it blows across the ping-pong table?
d.Study Question: Does the ball displace points of absolute space as it travels straight from one side of the net to the other?
e.Exactly, why doesn't the ball follow the same trajectory across the net, bouncing on the center line? Why does it follow a new path through absolute space instead of the original path?

Read/review Huggett, "The Shape of Space I: Topology" in SUP.

III.“Space can't be bounded:” 1stArchytasian argument

Read Robin LePoidevin, "The Edge of Space" in SUP; 89-90.2

A.Aristotle’s view of bounded space:Review 33.3 of Huggett’s essay on “Topology” for a description of Aristotle’s view, which implies that space is bounded, not infinite.

B.See 90.2 for LePoidevin’s first interpretation of Archytas’ reasoning for the view that space is infinite.

C.An interpretation:

1. If space is bounded [i.e. if space has an outermost limit], then there exists at least one spatial point p and direction d such that there exists no spatial point that bears d to p.

2. If there exists at least one spatial point p and direction d such that there exists no spatial point that bears d to p, then it would be physically impossible to extend one's arm beyond p in direction d.

3. If it is physically impossible to extend one's arm beyond p in direction d, then there exists a physical entity that blocks the arm from extending beyond p in direction d.

4. If there exists a physical entity that blocks the arm from extending beyond p in direction d, then that physical entity occupies a region of space that includes a spatial point that bears d to p.

5. If there exists a physical entity occupies a region of space that includes a spatial point that bears d to p, then there exists a spatial point that bears d to p.

6. Therefore, if space is bounded, then there exists a spatial point that bears d to p. (from 1,2,3,4,5)

7. Therefore, if space is bounded, then there both exists and does not exist a spatial point that bears d to p. (from 1&7)

8. Therefore, it is false that space is bounded. (from7; reduction to absurdity: since the claim that space is bounded implies a contradiction, the claim must itself be false)

D.Is this 1stargument sound?

1.What is LePoidevin's 1st objection to the argument above? See 90.2.

Study Question.LePoidevin’s first point may involve a mistake. Does it? Would it change the force of the argument if the barrier were presumed to be 2D instead of 3D?

2.What is LePoidevin's 2ndobjection to the argument above? See 90.2.

Study Question.Is LePoidevin’s second objection plausible?

3.Study Question: Is premise (3) is true? What can be said to strengthen or weaken premise (3)?

IV.“Space can't be bounded:” 2nd Archytasian argument

Read LePoidevin, "The Edge of Space" in SUP; 90.2

A.See 90.2 for LePoidevin’s second interpretation of Archytas’ reasoning for the view that space is infinite. (“Suppose, then, that there is a physical barrier…")

B.Exercise:Explicate this second argument informally in your words.

C.An interpretation of the 2nd argument:

1. For any two discrete physical entities X and Y and for any number N, if X and Y are N meter sticks apart at a time t, then it is possible that X and Y are more than N meter sticks apart at a time other than t.

2. If it is possible that any two discrete physical entities X and Y are more than N units apart at a time other than t, then there exists a spatial direction d and spatial point p (occupied by X or Y at t) such that infinitely many spatial points bear d to p.

3. If there exists there exists a spatial direction d and spatial point p such that infinitely many spatial points bear d to p, then space is not bounded there is a direction in which spatial points extend infinitely.

4. Therefore, space is not bounded. (from 1,2,3)

D.Study Question: Are the three premises above all obviously true?

V.“Space can't be bounded:” 3rdArchytasian argument

Read LePoidevin, "The Edge of Space" in SUP; 90.3-91.2

A.See 90.3-91.1 for LePoidevin’s third interpretation of Archytas’ reasoning for the view that space is infinite. (“Suppose, then, that there is a physical barrier…")

B.An interpretation:

1. If space is bounded [i.e. if space has an outermost limit], then there exists at least one spatial point p and direction d such that it is logically impossible that there exists any spatial point that bears d to p.

2. If there exists at least one spatial point p and direction d such that it is logically impossible that there exists any spatial point that bears d to p, then it would be impossible that there exists something that prevents a physical entity from moving in direction d beyond p.

3. If it is impossible that there exists something that prevents a physical entity from moving in direction d beyond p, then the claim that nothing can possibly move in direction d away from p is absurd.

4. If the claim that nothing can possibly move in direction d away from p is absurd, then it is possible that something moves in direction d away from p.

5. If it is possible that something moves in direction d away from p, then there exists a spatial point that bears d to p.

6. Therefore, if space is bounded, then there exists a spatial point that bears d to p. (from 1,2,3,4,5)

7. Therefore, if space is bounded, then there both exists and does not exist a spatial point that bears d to p. (from 1&7)

8. Therefore, it is false that space is bounded. (from7; reduction to absurdity: since the claim that space is bounded implies a contradiction, the claim must itself be false)

C.Study Question: Is the 3rd argument sound?

Study Question.Should one be tempted to reject premise (3)? See 91.2.

D.Summary

VI.A Kantian defense of finite space

Skip LePoidevin's sub-section on "Is There Space beyond the Universe?"; but do read "The Illusion of Infinity," SUP, 95.2-97.1.

A.See 95.3 for the Kantian defense of finite space.

B.An interpretation:

1. If it is possible that space is infinite, then it is possible that we acquire the concept of infinite space.

2. It is possible that we acquire the concept of infinite space only if we have the concept of the totality of the parts of infinite space.

3. It is possible that we have the concept of the totality of the parts of infinite space only if it is possible to count all the parts of infinite space.

4. It is impossible to count all the parts of infinite space an infinite number of entities isn't countable in its entirety.

5. Therefore, it is impossible to acquire the concept of infinite space. (from 2,3,4)

6. Therefore, it is impossible that space is infinite. (from 1,5)

C.1st objection

Study Question.See 96.2. What is LePoidevin’s first objection to this argument?

D.2nd objection

Study Question.See 96.3. What is LePoidevin’s second objection to this argument?

VII.An Aristotelian defense of finite space.

Read LePoidevin, 96.3-97.2

A.Aristotle on the actu infinitum. As LePoidevin notes in 96.3, Aristotle appears to be a finitist, rejecting the possibility that exists an actual infinitude of particular entities:

When we speak of the potential existence of a statue we mean that there will be an actual statue. It is not so with the infinite. There will not be an actual infinite. [Physics, III.6.206a18]

Franz Brentano apparently agreed with Aristotle:

…every part of the continuous whole is a real thing. And since parts are to be differentiated ad infinitum but do not become things merely in virtue of being differentiated, the continuous whole seems to consist of an infinity of things. [Psychology from an Empirical Standpoint, p. 353]

The truth is, however, that even if we assume the continuous whole, we can reject the inference to an actu infinite as invalid. Someone who assumes a continuous whole of one meter long can, of course, describe it as two entities ½ meter long or as a continuous whole of three entities 1/3 of a meter long instead of describing it as one entity. He can just as well describe it as any number of correspondingly small entities he pleases, but he cannot understand it as an infinite number of infinitely small entities… One can say of a continuous whole, then, only that it can be described as being as large a finite number of actual entities as you please, but not as an infinitely large number of actual entities. [Psychology from an Empirical Standpoint, p. 354]

If it is indeed impossible that an actu infinitum exists, then this would weaken the Kantian claim that we can acquire the concept of infinity only if we can conceive that there is an actual infinitude of entities. But the impossibility of an actual infinitude also suggests another argument against unbounded space…

B.An Aristotelian argument against unbounded space:

1. If it is possible that space is infinite, then there do exist infinitely many spatial points.

2. It is impossible that there exist infinitely many particulars of any kind.

3. Therefore, it is impossible that space is infinite. (from 1,2)

C.Study Question: Is the Aristotelian argument sound?

VIII.Poincaré’s diversion

Read LePoidevin, 98.1-99.3

A.Poincaré’s thought experiment. Review Huggett’s, LePoidevin’s, and LeClair’s treatments of this thought experiment.

B.The moral of the story.

Study Question.According to LePoidevin, what is the moral of the story?

C.Questions for physicists.

IX.Spacelessness

A.Our study of whether space could be bounded has presupposed Newton’s view that absolute space exists. Soon, we shall study Leibniz’s view that there is no such thing as space – that all talk of space can be reduced to talk of physical objects and spatial relations (e.g. (crudely) the tip of the pyramid is up relative to earth below it; and the tip is west of the Sphinx).

B.Study Question: if there is no such thing as space constituted by spatial points, then in what sense could “the universe” be bounded? Or unbounded?

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[1]To my knowledge, Charles Howard Hinton introduced the terms ‘ana’ and ‘kata’ (in A New Era of Thought (1888)) to refer to seventh and eight directions in 4D space.