Space from Zeno to Einstein; N. Huggett

Space and Time

Philosophy Course Index



Chapter 1:

Logic: All logic and reasoning is founded on the "law of non-contradiction": it is impossible for a claim to be both true and false in the same way at the same time (or for an object to have and not have that property in the same way at the same time). You are inconsistent when you deny the law of non-contradiction. If you are inconsistent, then you cannot say anything meaningful, in fact, you wouldn't convey any information at all.

Valid argument: if the premises are true, then the conclusion must be true. To reject the conclusion of a valid argument is to be inconsistent.

Indirect proof (reductio): To refute a claim, first assume that it is true, then deduce some unacceptable conclusion (i.e., something known to be false, or self-contradictory, or inconsistent). Since the conclusion must be rejected, so must the assumed premise.

Scientific Theories: What makes a "good" scientific theory?

A) It agrees with the facts/observations of empirical data. Does it accurately describe the world?

Hypothetico-Deductive Method: Form a hypothesis, derive a test of the hypothesis that can be empirically checked, and if the hypothesis passes the test, then it gains empirical support (and is a "good" theory, or better than it was, at least). However, other hypotheses might also pass the test.

B) Is the theory logically consistent? Is there a possible world where all the statements that comprise the theory are all true: if the answer is "no", then the theory is inconsistent.



3 Aspects of the Philosophical Study of Space

1) What kind of thing is space? (This is a question of metaphysics/ontology, which studies the nature of existing things, or "being".)

2) How can we come to know about space? (This is a question of epistemology, which is the study of knowledge)

3) In what ways do space and matter interact? (This is a question of both metaphysics and physics.)

Plato's 3 part distinction: (i) The material world, or that which "comes to be"; (ii) that in which the material objects "come to be", which is space, or the "Receptacle", as Plato classifies it; and (iii) the Forms, since all material objects are imitations, or copies, of the Forms. (That is, the Forms are eternally existing "beings" which are, essentially, perfect concepts that exist in a world outside of space and time (S+T) and are the perfect "models" which all objects are made to imitate, but are imperfect copies.)

The Receptacle is the "place" of all objects: it is a container in which all material objects exist. The reasoning seems to be: since everything that exists must exist somewhere, thus space (the Receptacle) is the collection of all "somewheres." Accordingly, "places" are important in science since, e.g., motion is defined as "change of position (place)".

What is space (the Receptacle) for Plato (a metaphysics question)? Answer: It cannot be a material object, because it would then need a place. (That is, the container would need a container, but then that container would also need a container, since it is also material, and this leads to an infinite regress of containers in containers, and this is an unacceptable state-of-affairs, since the "place" of the Receptacle would never be provided or established.)

Plato seems to suggest that matter is not separate from space: objects are "formed out of" the Receptacle (or space). Space is thus the "material" (substance) from which the copies of the Forms are made. But, Plato says the copies do not possess the properties of the Forms: it receives and makes the copies, but does not become like the Forms (yet, in what way?).

How do we know about space (epistemology)? Answer: by some kind of reason, and not from sense experience.

Do space and matter interact? Yes: The copies of the Forms cause motions in space, and these motion of space cause, in turn, motions of the copies. Space also sorts out the motions of the elements into their natural regions in the cosmos. (This reciprocal cause-effect relationship between space and matter is very similar to the General theory of Relativity.)

Chapter 2: Euclid First Axiomatic Treatment of Geometry (300 B.C.)

All geometric knowledge is deductive consequences of the 5 basic postulates (axioms) and a set of definitions. (The propositions and theorems are consequences of the axioms/definitions.)

Modern formulation of Euclid's system: point (has no parts, but is dimensionless); line (1-D, i.e., one-dimensional, breadthless length); lines are collections of points; lines are "dense" (a point lies between any 2 points); segment (a finite length of a line); lengths of segments can be compared; from any 2 segments, one can construct a third segment whose length is equal to the sum of the lengths of the other two.  The interior angles of a triangle equal 180 degrees (from prop. 32).

Does Euclidean space describe physical space? Most people have assumed that it does. Gauss's hypothetical experiment (measuring the angles formed by light reflected by mirrors on 3 mountain tops) seemed to verify that space was Euclidean; that is, Gauss judged that the interior angles of his light triangle would equal 180 degrees.

 Chapter 3: Zeno

Zeno (400 B.C.) was a follower of Parmenides (who believed that all things are one, and that there is no change, and thus no motion).

Zeno's arguments try to show motion is impossible, whether the units of space (and time) are considered to be discrete (i.e., having a smallest size, such that it cannot be divided any further) or continuous (i.e., there is no smallest unit, so that a spatial, or temporal, magnitude can be divided infinitely).

For our purposes, Zeno's arguments can be applied to the existence of space, and, in particular, whether or not space is Euclidean. Therefore, Zeno is trying to show that:

(P1) Space is Euclidean

(C) Therefore, motion is not possible.

(Here,  the first line, (P1), is the premise, and second line, (C), is the conclusion: this reconstruction of Zeno's argument uses an indirect proof, since the unacceptability of the conclusion is supposed to lead one to reject the truth of the premise): 3 ways to respond, here: (1) accept the conclusion as true; (2) accept the argument as valid, but deny the conclusion by rejecting the premise; or (3) the argument is invalid, so the premise is true but the conclusion is false. We will accept option (3): Space is Euclidean, but motion is possible.

Zeno's 1st argument (fr. 3,4,5,7): One can never traverse/pass a finite distance because you have to cover 1/2 the distance first, then 1/2 of the remaining distance, than 1/2 of that, etc., so you never can pass the distance. (This argument is known as the Dichotomy paradox.)

Aristotle's first response: one can't cover an infinitely big space, but one can, in the case of a finite distance, traverse an infinity of parts since there are an infinite number of parts of the finite temporal period in which it takes to traverse the finite distance. (Problem: This assumes that the infinity of temporal parts adds up to a finite sum. Second, Aristotle also states that the above response is week due to the fact that Zeno's paradox can also be applied to temporal units. Thus, appealing to time does not solve the problem because time also runs afoul of the paradox.)

Aristotle's second response: The infinity of the finite spatial distance is only potential and not actual, thus one can cover a potential infinity without problem (but not an actual infinity). This response seems to imply that physical space is not the same as mathematical/geometric space, since the infinite geometric divisions of the space are not real (actual), but only potential.

Another defense: One can solve the problem by using a different mathematical model of motion (p.50): v=d/t , so the distance is covered by plugging the values into the equation (t=d/v). Viewed in this new way, the finite distance is easily covered once one knows the speed of the object, and the finite spatial distance that has to be traversed. Problem: This response works, but it seems that the Euclidean theory of space is not addressed, or corrected, by this response. So, we still need to know if the Euclidean theory of space can handle Zeno's argument on its own? If not, then Euclidean space is somehow inconsistent (which is why the paradox arises).

A Euclidean version of Zeno's 1st argument (that is, a version of the argument intent on demonstrating that Euclidean geometry is inconsistent). (p.40)

(P1) All segments can be divided into segments without limit. (This premise is true because Euclidean space is "dense"; that is, since there always exists a third point between any two points, and two points mark the endpoints of a segment, it thus follows that any segment can be divided in to two smaller segments, and those two segments can each be divided again, and so on to infinity.)

(P2) All segment have finite length (This is true by the definition of a segment.)

(P3) The length of any segment = sum of the lengths of segments of which it is composed. (This is also true by definition in Euclidean space.)

(P4) All infinite sums of finite quantities are infinite. (This is a premise that we assume is true since it seems quite plausible.)

Conclusion: (C) All segments are infinitely long. (This conclusion follows from the premises because if each segment is of finite length, and there are an infinity of smaller segments in any larger finite segment, then the sum of this infinity of smaller segments must equal an infinite length. Yet, how can the sum of the parts of a finite length equal an infinite length? This is a contradiction; thus there must be something terribly wrong with the very concept of a Euclidean space. (or, as stated above, premise (P2) and the conclusion (C) contradict one another).

The best strategy to defeat Zeno's argument (and retain Euclidean space) is to deny premise (P4) i.e., that all infinite sums of finite quantities are infinite.

Modern developments of mathematics (Cauchy/Weirstrass) showed that the infinite sum of finite quantities can be finite. We can write an write an infinite series as: And, as a specific example of an infinite series: 1/2+1/4+1/8+1/16+...+1/2n+... Yet, as can be seen, the sum (or addition) of all the members of this infinite series does not equal an infinity, but a finite number; namely 1 (1/2+1/4+1/8+1/16+...+1/2n+... = 1), or, more precisely, the sum this infinite series converges to 1 as its (finite) limit. Thus, mathematics gives a precise, consistent definition of infinite sums, and how some infinite sums have finite limits. Accordingly, premise (P4) is false, and thus Euclidean space has not been proven to be inconsistent.

A Euclidean version of Zeno's 2nd argument (p.44: This argument is known as the Plurality paradox.)

(P1) All finite segments are composed of an infinity of points. (This premise is true by definition in Euclidean space.)

(P2) The points are either zero in length or finite in length. (This premise seems to exhaust all the possibilities of length for points.)

(P3) The length of any segment = sum of the lengths of the points of which it is composed. (This premise is an intuitively plausible assumption.)

Conclusion 1 (C1) If the points have 0 length, then total length of a segment is zero (since if all the points are zero in length then: 0+0+0+0+0+...=0)

Conclusion 2 (C2) If the points have a finite length, then the total length of a segment is infinite. (Here, it is important to remember that all points in Euclidean space are identical; so, if they have finite length, then they all have the same length, and thus an infinite sum should give an infinite total length: L+L+L+L+...= infinity).

Thus, this argument seems to lead to the absurd conclusion that all segments in Euclidean space (which are finite in length) are either of zero length or of infinite length! Thus, Euclidean space is inconsistent. Using the mathematical definition of an infinite series adding up to a finite number (length), as used in the 1st argument above, won't help here because we were concerned there with the sum of a series each of whose members decreased in size (1/2+1/4+1/8+...); but, here, we are concerned with points, all of which are the same size (e.g., finite or dimensionless).

Reply: The size of the set of points in a line is uncountably infinite: That is, it can't be put in 1-to-1 correspondence with the natural numbers (1,2,3,4...), as was proven in the late nineteenth century by Cantor. More precisely, if you pair off each point on a line with one of the natural numbers (1,2,3,4...), then there will always be some point that fails to be paired off to a number; thus, there are more points in a line then there are numbers in the set of all natural numbers. (The proof of this is very complex, but it is valid.) Accordingly, given that a set of points in a Euclidean line is larger than the set of natural numbers, the sum of the points on a line (called "nondenumerably infinite") may be different from the sum of our usual infinite series, such as the infinite series that comprise the natural numbers (called denumerably infinite). Yet, the fact that the two infinite series do not have an identical size does not really tell us any information on how their respective summations may, or may not, differ. Consequently, this observation really doesn't undermine the Zeno argument.

Nevertheless, the modern mathematical definition of infinite addition cannot be applied to a continuum of points (which is the size of the set of points in line) because modern mathematics has also shown that all line segments have the same number of points. So, the length of a segment is not determined by adding the lengths of its points, because each segment, no matter how long, has the exact same number of points! The proof of this is complex, but a simple diagram can show you how it can be visualized: In the diagram (p. 47) , segment CD is shorter than segment AB, yet each point in CD can be paired off with each point in AB without remainder (that is, each point in CD is paired off with each point in AB without any points being left over in AB). The dotted line is drawn from the top of this right triangle, and, as can be seen, it pairs off a point on CD with a point in AB: and a line can be drawn (from the top of this triangle) in the same manner that connects every single point in CD with every single point in AB. Conclusion: CD and AB have the exact same number of points despite the fact that CD is shorter than AB. (It is important in trying to understand the use of this diagram to bear in mind that lines have no thickness, and thus one can set-up an infinity of lines that start at the top of the triangle, X, and which then pair off each point in CD with each point in AB without remainder.)

Overall, this realization undermines premise (P3) in Zeno's second argument. Why?: Because the length of a segment has been proven not to be a function of the number of points that comprise the segment (since all segments have the same number of points no matter what their length happens to be). In fact, what this response to Zeno's argument demonstrates is that it is quite misleading to describe lines, segments, surfaces, etc. as "composed" of points: i.e., as if a "segment" is simply what you get when add a bunch of points together. On the contrary, some of the properties of lines, segments, surfaces, etc., have nothing to do with points: namely, "length" is such a property, since it is independent of the number of points in the segment (and where the term, metric, is often used to describe the structure of a space that determines length). Thus, maybe Aristotle's attempted solution really is the best answer to Zeno's paradoxes, since he argued that segments of lines could be "understood" to have points in them in the sense that these points were a mathematical/conceptual idealization of the parts of segments; and, consequently, that segments were not composed, or constructed, from points in an atom-like manner. Or, alternatively, the properties of the parts of a segment (the points) are not the same as the properties of the whole segment.

Conclusion, metrical notions (notions of length) are not an intrinsic property of a collection of points. Length is an independent metrical property (or, segments have length, but collections of points do not). Segments do not get their length from points: it is an independent metrical property of segments. So, Zeno's 2nd argument fails because premise (P3) is false.

Zeno's 3rd Argument. The "Arrow" argument.

An arrow is either motionless or in motion at each instant of its flight: (1) if it is in motion, then instants of time can be divided into earlier and later parts, because the motion must cover a series of points (and traversing each point takes time). But, this conclusion is a contradiction, since an instant (much like a point) has no parts; so then (2) the arrow is not moving at each instant of its flight: but then, how do you get motion from a collection of non-moving arrows? That is: How does a collection of stationary arrows (since they are stationary at each point) add up to a moving arrow?

Zeno's third argument seems to accept the following principle: if an arrow doesn't move during any instant, then it doesn't move at all. However, this assumption must be rejected.

The "At -At" theory of motion strives to undermine this assumption: The arrow moves when for every time unit (instant), that is every value of t, the function x(t) has a value such that there are no "jumps" (discontinuities) in x as t varies. So, at every t, the arrow is at some x. The position of the arrow, x, is a function x(t), for every value t of time spanned by the motion. So, this theory defeats Zeno because it denies his extra assumption: motion is not something that happens during and instant, but is rather just a matter of being at the right sequence of places at the right sequence of times. (Problem: Does this really solve the problem, or just raise the problem of motion all over again?) By the way, this "At-At" attempt to resolve the problem would seem to answer Zeno by replacing his view of time/motion with another view of time/motion (namely, (t=d/v), as mentioned above). Moreover, as also argued above with respect to space, the best counter-reply to the arrow argument is to claim (as we did above for the length of a segment) that adding together a set of instants does not give you a particular temporal duration, or temporal length (just as adding a set of points does not give you a segment with a certain spatial length). That is, Zeno seems to be assuming that adding together a set of instants gives you a finite temporal length, but we have seen how the same argument does not work in the case of space. Overall, Zeno's 3rd argument doesn't really seem to be a problem for Euclidean Space, but rather a problem for continuous/discrete notions of time.

 Chapter 4: Aristotle

Aristotle views of Science: science is derived from "first principles" which are believed to be the most basic (as obviously true claims); from these you deduce (logically) further theorems (theories of the world). The first principles are believed to be understandable by the average person (without technical knowledge).

Aristotle view of the heavens (p.74): 4 elements (earth, air, fire, water), 4 natural places for these elements, and a finite spherical universe. 2 aspects of "natural" motion of the elements: (i) elements move to places so Aristotle needs a theory of "place"; (ii) elements move in order to reach their proper places, so elements have a "built-in" (teleological) goal as an intrinsic, basic, internal property.

Place: Basic Beliefs (1) A place can be separated from the objects it contains (208B6). This is a common-sense based belief about place (as well as definitional).

(2) Places do not have places (209A24). If places needed places, then an infinite regress of places inside of places would occur (which is impossible to ever complete, like Zeno's paradoxes).

(3) The difference between "up" and "down" (left and right) is not just relative to peoples' views, but is absolute (208B17). In nature, directions/places of the elements are distinct, and the same for all observers.

4 Theories of "place"

(A) "shape": the outer surface of an object. This does not work because it is not compatible with basic belief (1), since the place would be attached to an object (211B10-14), which is counter-intuitive.

(B) "matter": the matter of an object is the place. This theory also violates BB(1), for the same reason as (A). (211B30-212A2)

(C) "extension between the extremities"; that is, the extension, or spatial distance, between the containing surface of a containing body (e.g., the inside surface of a coke bottle which surrounds the body, the coke). Aristotle rejects this theory because it conflicts with BB(2), since it leads to an infinity of places in the same place. In short, the spatial distance that constitutes the place (e.g., the three-dimensional volume or distance between the sides of the coke bottle) can divided into an infinity of smaller places inside that original spatial distance. Another explanation: if a balloon is filled out completely, and the air is slowly released, then at each instant, the balloon will have a new inner containing surface, and thus a new place; so, conclusion, there are many places in one place (in fact, an infinite amount).(211B19-29) (Reply: Modern mathematics can tolerate infinities, so it is not necessarily a problem.)

(D) "the boundary of the containing body at which it is in contact with the contained body"; that is, place is the common boundary between the contained and containing bodies (e.g., the boundary between the outermost surface of the coke and inner surface of the coke bottle). Aristotle rejects a vacuum, so there is no space between contained and containing body. Place is thus 2-Dimensional, with no thickness (It is a 2-D limit of a 3-D space). Thus, space (place) becomes a property of material objects (place is dependent on bodies). This theory is consisted with BB(1) since place is in the contained and containing bodies only in the sense "as the limit is in the limited" (212B29): or, in other words, place is a mathematical/conceptual limit between the two material surfaces (thus it is not in the bodies). BB(2) also appears to be met since the 2-D surface which constitutes "place" cannot be divided without dividing the surface into incomplete parts (which thus destroys the original place). That is, a 2-D surface has no thickness, so no other places can be inside that surface (place). BB(3) is also satisfied, since the common surface between the earth and the next celestial sphere, as well as the common surface of the last sphere and the one directly inside it, serve as absolute positions in Aristotle's universe (see the discussion of motion/universe below).

Motion: Motion is a form of change: "actualization of a potential in so far as it a potential". Motion is change of place. (208A31)

Elements move to a natural places because of the "substantial" form that they possess (not Platonic "Form"). The "place" doesn't move the object, but the object moves naturally (due to an internal principle of motion) to the place (268B23). (On the Heavens, B1, C2) Consequently, the absolute places of the universe explains/satisfies BB(3) Why are absolute "up" and "down" needed?: If place moved relatively with the object (as a boat in a moving stream) then there would be no place to identify as the initial place of the motion, and thus no motion could be assigned to a motion (or no motion could ever be identified/recognized), which is a contradiction. So, the need for an unchanging place is crucial, and there are 2 such places: earth, and the outermost ring of the universe. (212A 14-28)

Kinematic/Dynamic distinction: ("kinematics" is the study of motion per se, without regard to forces; whereas "dynamics" is the study of the motion of bodies under the action of forces) Aristotle views the motion of "free fall" on the surface of the earth as kinematic, and not dynamic (as Newton believed). Aristotle was the first to see the need for a kin./dyn. distinction. "Unnatural" motions are forced motion of a body away from its natural place (or natural motion): e.g., shooting an arrow into the air forces the arrow, which is made of the element earth (in part) away from its natural place (which is the earth, or center of the universe). Therefore, Aristotle explains "unnatural" forced motions as dynamic (caused by outside forces, such as the bow which acts on the arrow), and explains the natural motion of the elements towards their proper place in the universe as kinematic motion. (The modern General Theory of Relativity also favors a kinematic theory of free-fall: the curvature of space-time is responsible for the motion of bodies near large masses, since the mass curves the space-time, and this motion can be considered as a kinematic, inertial motion, at least for infinitesimal spatial and temporal distances.)

The Universe

Space is 3-D. (On Heav. B1, C1) Why? Because there is no further spatial direction to point to! Problem: if there was a 4-D space, then there would be a 4th direction, although (like the flat-landers) we would not be able to point to it. So, not being able to point at it is not conclusive proof that there is no 4-D space.

There is only one world, and no others. (On Heav. B1, C8) There can be no other worlds because it would have to be made up of the element earth (thus the 4 elements are assumed to be the same throughout the universe), but if it is made up of the element earth, then it would move to the center of the universe, where our world is located. Aristotle considers the following problem: the element earth must fall to the center, but if there are two worlds, then there are two centers. So, earth will either go up in one world (which violates the natural motion of elemental earth) or it falls down in its world, but then it does not move to our center (which also violates the definition of the element earth). Could there be 2 centers of the universe? No, because then the elements have 2 natural places, and thus 2 separate goals (i.e., two teleological internal principles of motion), and thus they don't have the same (substantial) forms, and thus they are not the same element (which is a contradiction, of course). But, the form is an internal property (not dependent on external situation), and this would be violated since the form is relative to the particular local center-of-universe.

What is the "place" of the World? (212A31- B21) The "world" does not have a place because "place" is the common boundary between contain and containing bodies (surfaces); and since the universe/world is the whole, thus there are no bodies that are outside the world which could contain it (since all bodies are part of the one whole). Although there are no further bodies on the outer portion of the final sphere, this outer celestial sphere has "linear" containment, since any part of the outer sphere is contained by the parts on its sides (but not on its extreme outer part, of course). In this way, Aristotle seems to want to provide a place for the final sphere of the universe (but this answer is not very convincing, since there are no bodily part on the outer surface of the final sphere; and so many parts on the outer sphere will not be completely surrounded by adjacent bodily parts, and will thus lack a containing surface, and thus place). Rotational motion is acceptable for the outer sphere because this form of motion never leaves the same place (the place of the motion remains constant). That is, rotational motion does not move a body forward, backward, left, right, etc., in space, but merely stays in the same place and rotates.

Why is the universe spherical? A plane sphere, or solid sphere, is the most "simple" since the area/volume is bounded by only one line, rather than several (as with, e.g., a square). Since what is one is prior to what is many, the circle is the primary plane/solid figure. Also, the circle is complete, since nothing can be added to it (as a line can be constantly added to, since it has infinite length). So, what is complete is prior to what is incomplete, thus the circle is the primary figure. For Aristotle, the primary figure belongs to the primary body, which is the outermost sphere of the universe. But, the sphere below this one must be spherical with it, since it is contiguous with it. But, then all the spheres in the sequence below will also be spherical. Thus, the whole world is spherical. The motion of the universe must be spherical. Since there is no empty space outside the outermost sphere, there is no place for the universe to move into. But circular motion does not change place (as described above), so this must be the motion of the universe. The motion of the outermost sphere is the quickest motion, which means it takes the least time. Yet, since the smallest is the means of measuring all the others of the same kind, thus the motion of the outermost sphere is the measure of all worldly motions. Also, the motion of the outermost sphere is the most regular and everlasting (most perfect). For Aristotle, the quickest movement is circular, since it is the shortest in going from, and returning to, the same point. So, the motion of the outermost sphere is circular.

Chapter 5: The Aristotelian Tradition

Many Philosophers tried to resolve the internal problems with Aristotle's theory.

John Philoponus (6th cent. A.D.): The place of the earth is not stationary as Aristotle assumed (since the spheres do not move out of their place, Aristotle assumes that the inner surfaces of the spheres can count as motionless places). But, the parts that make up the spheres are constantly changing place relative to the parts of the sphere contiguous with it (since they rotate relative to one another). Thus, there are no motionless places. Also, since the outermost sphere has no contiguous body outside of it, it can have no place, and thus no motion. But, if the outermost sphere has no place, then the world has no place either, nor does the earth (which is a contradiction). Conclusion: Aristotle's theory of place/motion is deficient. Philoponus accepted the theory (C) rejected by Aristotle: "space is a the extension between the extremities"; place is a 3-D extension that exists distinct or separate from the bodies that occupy it (much like an empty space, or vacuum).

Crescas (15th century) argued that Aristotle was wrong to assume that objects are the same size as their places (211A1). If one removes a thin cylinder from a sphere, the holed-out sphere has more surface area than the sphere before the cylinder was removed, and thus (according to Aristole) the holed-out sphere has a larger place than the whole sphere (which seems to be contradictory). So, if the size of an object is equal to the size of its place, that is, size of object = surface area of object (or surface area of containing surface), then a part of the sphere must be larger than the whole sphere, which is a contradiction (since no part can be greater than the whole). (It should be noted that a body’s surface area does not equal its volume.)

Jean Buridan (14th century): Aristotle believed that the motion of an object after it has left the source of the "unnatural" motion (e.g., arrow after it has left the bow) is due to either: (i) the air continued to be pushed by the bow string, and the arrow is carried along with the air; (ii) or the tip of the air pushed the air into a loop that circled around to the back of the arrow and pushed the arrow forward. Aristotle needed to resolve this problem since a body returns to its natural place (and its natural motion) once the outside force (violent motion) no longer acts on the object. But, then why does the arrow not fall directly towards the earth once it leaves the bow? This is the problem of "inertia": Buridan argued that Aristotle's theory could not work because a javelin/arrow with a sharp tail moves just as far as a javelin with a flat end; but, the sharp end should make it more difficult for the air to push the arrow forward (since there is less surface to push the arrow directly forward, as opposed to directions askew). So, Aristotle's theory fails, because experimental evidence demonstrates that all arrows fly the same length regardless of having or lacking a sharp end.

Buridan favored an "impetus" theory: Bodies not only have natural motions, but a "force" or property that a body can temporarily possess that allows the object to maintain a certain "violent" motion (until this property is used up or destroyed by air resistance, and the body then resumes its natural motion entirely). Thus, 2 types of motion (natural and forced) account for the observed trajectory of the arrow (much as Galileo argued). Also, these 2 types of motion can be applied to all bodies, both terrestrial and celestial, so that there is no difference between the motion of celestial or terrestrial bodies. (Aristotle assumed circular motion was the natural motion of the 5th, celestial element, but not for the other 4 terrestrial elements). In fact, Buridan's theory seems to indicate that maybe all matter is governed by the same laws of motion: a "universal mechanics".

Galileo and Copernicus (16th/17th centuries) argued further for the idea of inertial motions, since they believed the earth rotated. It was often argued that if the earth rotated, then an arrow shot straight up in the air would not fall back to the same place, which is where the arrow does return. (Why? Because the earth rotates while the arrow is up in the air, the arrow does not fall back to the same place.) (296B23) But, Galileo argued that the arrow has a "circular impetus" alongside its motion up and down, and this explains why it falls back to the same place: e.g., as the arrow is going up or down, it is also partaking in a circular motion which is in the same direction and speed an the rotating earth. (The "circular impetus" is the same for all terrestrial objects.) Thus, for Galileo and Copernicus, it seems that impetus (inertia) can be circular as well as rectilinear (as opposed to Newton and Descartes, for whom it is only rectilinear).


Chapter 6: Descartes

"A priori" knowledge (using reason alone, and not experience) provides the understanding of the laws of nature, space, and motion. Descartes is a "Dualist": i.e., there are 2 substances: mind and matter (where a "substance" is an independently existing thing, and a "property", or mode, can only exist inside substance). All substances have a defining property, or "essence", as it is called: What is the essence of matter? Answer: Extension (in 3-D, or "length, breadth, and width"), since all other properties can be removed from matter and it would still be matter (e.g., "hardness" can be removed because we can imagine a body which we can never touch, yet this would not undermine the concept of the body's matter). But, if you remove extension, then you eliminate the matter (since a non-extended body is an impossible object), so extension is the essence of matter (I. 48, II. 4, 11).

Space (extension) is thus identical to matter: they are ontologically identical (i.e., the same being), although a conceptual distinction can be drawn between them. So, space is not independent of matter (as held by the "container" view). Problem: But, as Aristotle argued, space/place and matter are separable (since objects occupy places, and move out of them, etc., as they move). Reply by Descartes: There are 2 senses of "place" which often get confused (II. 12-13). (i) Internal place is place considered in the general sense, and is picked out as staying the same over time. Internal place is picked out relative to any bodies which retain the same position with respect to one another. Descartes says that internal place is the normal view of "space" since different bodies can occupy internal places at different times. Thus, internal place is separable from a body's matter. (ii) External place is place in the true, philosophical sense, and is considered to be intrinsic (not separable) from the body. This place is uniquely defined as the common surface between the contained and containing bodies (and is identical to Aristotle's notion of place). Motion: Each body has many different motions viewed from its generic place (internal place) since there are many perspectives from which to judge the body's motion (For example: Is the watch on the ship in motion or stationary? Answer: It depends on your viewpoint: if viewed from the shore, the watch moves; if viewed from the deck of the ship, the watch is stationary).

Yet, each body has only one distinct motion when conceived as change of external place, since this is motion understood as "transfer, or change, of contiguous bodies". When the neighborhood of bodies does change, there is motion (and all observers will agree that there has been a change of external place). However, the change of external place is reciprocal, so one can conclude that either the body moved and the surrounding contiguous bodies were at rest, or that the body was at rest and the neighborhood of surrounding bodies moved (or a combination of both). Thus, for Descartes (and Aristotle) motion is relational (it is relative to a viewpoint, but only in a limited sense, as defined above for external place). This allows Descartes to say that the Earth is properly stationary (since it does not change its external place in the vortex of contiguous particles surrounding the earth), but that the entire vortex is in motion (but this would be only in the generic sense of change of internal place).

Descartes proposes first fully developed theory of inertial motion: bodies when moved continue to move in the same direction and at the same speed (which is velocity, as understood in the modern sense). Bodies do not have a natural motion to a natural place (as Aristotle believed). There are no natural places; rather, all bodies retain their same state of motion unless acted upon by an outside force. (II. 23, 26, 37-39) Problems: Descartes thinks rest and motion are opposite states, so they do not naturally change from one state to the other (e.g., a body in motion does not, unless acted by an outside force, change into its opposite state, which is rest). Yet, the modern concept of inertia holds that they are identical states, and not opposite, or distinct states.

Descartes believes the universe is infinite (since there is no limit to extension), that all matter is infinitely divisible, and that all matter is the same (i.e., no internal difference in the elements of matter, as held by Aristotle). Problem with relational motion: Once the motion of a body, X, occurs, which is a change of X's neighborhood (external place), the neighborhood of bodies surrounding the moving body X will be changed/altered as they reshuffle and assume new positions (i.e., these neighboring bodies will need to fill in the now empty (internal) place of the moved body X). But, if this disturbance of the neighborhood of bodies does occur, then how can the motion of body X then be determined? Without an enduring external place, there is no way to determine X's state of motion (i.e., the places that X has passed over), and thus X's velocity cannot be determined, which is defined as "distance (places) divided by time". (This is Newton's "De Grav." argument). If one judges motion from internal place, however, then any given motion can be judged from many different perspectives, leading to many different assignments of (contradictory) values of motion to X.

Definitions: "inertial" motion is motion of constant velocity (which is motion of constant speed and direction), and "noninertial" motion is accelerated motion (which is motion not of constant velocity). Non-inertial motions generate "inertial forces", also called "non-inertial forces", which can be observed by all. (So, non-inertial force = inertial force) Also, "relationalism" is not the same as "relativity".

Chapter 7: Newton

In his unpublished paper, "De Grav" (early paper c.1666), Newton tries to disprove Descartes' relational theory of motion. There are 2 arguments (generally):

(1) External places (contiguous bodies) disperse after a body's motion, so external place cannot be used to measure the distance traversed by a body (i.e., the places it has occupied), and thus its velocity cannot be ascertained (this is the same problem as described at the end of the Descartes' notes). Likewise, one cannot use Descartes' concept of internal place to determine distance and velocity, since any outside reference frame that one chooses will not remain fixed over time. According to Descartes, all bodies, and thus all reference frames, are constantly changing their relative positions (due to the flux of matter in the Cartesian plenum). So, no body or reference frame, can serve as a coherent, or unambiguous, means of precisely fixing the successive places occupied by a body over the course of its motion. Consequently, both internal place and external place cannot serve as the foundation for a coherent notion of velocity for a relationalist. (This is an indirect proof of the falsity of relationalism, since any theory of physics/science must make sense of velocity.) Newton assumes, in this argument, that any meaningful concept of velocity must be able to fix precisely the many places that an object occupies over the course of its motion. In other words, since "velocity" is defined as "distance (places) divided by time", one must be able to meaningfully identify these places over time. Consequently, space must be endowed with "enduring" places: i.e., places that remain the same over time (and never change), thus allowing the places (that make up a distance) to be meaningfully identified.

(2) The earth exhibits a tendency to recede, so it must be accelerating. Yet this non-inertial force (from the acceleration) cannot be due to a "true" motion (i.e., a motion with respect to external place, which is a change of contiguous bodies) since the earth is at rest relative to its contiguous bodies (or so Descartes believes). And, the earth cannot be accelerating in the "common" sense, either (i.e., with respect to internal place, which is motion relative to some arbitrary body or reference frame). Why?: Because the earth is moving relative to many bodies and reference frames, and the motion ascribed to the earth will be different in most of these different reference frames (e.g., some frames will judge the earth to be at rest, while some observe it rotating, etc.), and so the earth will have many different, conflicting assignments of acceleration (and thus inertial forces) at the same time (which is a physical impossibility, and thus contradicts relational motion, once again). In short, the earth should have a different inertial force for every "common" relational motion relative to each different reference frames; but this is false (and impossible), since the earth only has one inertial force which is relative to one frame.

What is this frame? Descartes could say that it is the Sun, but Descartes also states that the Sun rotates and has inertial effects too (i.e., the Sun's particles have a tendency to recede, as well), so what does the Sun rotate relative to?

Newton's resolution to this problem is to provide the following answer: Absolute space. Inertial forces (also called non-inertial forces) are caused by motion relative to absolute space (since all motions must be relative to something). Newton believes that the only way to make sense of our experience of inertial forces (which are due to accelerated motion), and his laws of motion, is to postulate absolute space.

Absolute space is a "container" in which objects (material objects) exist. Each material objects has a unique position (X,Y,Z coordinate) in absolute space. These positions or coordinates in absolute space remain the same over time, thus fulfilling Newton's demand for the "enduring" places required to make sense of the concept of velocity (as argued in the De Grav paper). Absolute space is Euclidean, is 3-D, and: (i) locations exist in absolute space before they are occupied by bodies (p. 111); (ii) space would still exist even if there was no matter (p.113) (Philoponus might accepted (i), but not (ii).)

De Gravitatione: Absolute space is "an effect arising form the first existence of being" (p.112); i.e., space is a consequence of God's existence (as a co-eternal fact or property of "being qua being"). Absolute space is not a substance because it depends upon God (or another being) for its existence. And, since a "substance" is commonly understood as a being whose existence does not depend on anything else, absolute space thus cannot be a substance. (Problem: Newton says absolute space also does not count as a substance because it doesn't affect any other bodies. But this seems false because the inertial effects of accelerated motions are apparently caused by absolute space.) Absolute space is not a property of matter because it can exist without matter, since it is easy to imagine empty space without matter (Problem: just because you can imagine empty space, that does not mean that it is physically possible. Maybe empty space is physically impossible although logically/conceptually possible? Also, absolute space can still be a property of God, or another being, even if it is not a property of matter. In fact is "disposition of being qua being" identical to a property? It would seem so; and thus maybe absolute space is really a property, after all.)

Newton's conclusion: Absolute space is neither a substance nor property; but it is something, even though it is closer in spirit to substance. This view (space as substance, or quasi-substance) is called "substantivalism" (In short, absolute space is as real as matter, it is not dependent on matter, and it is not identical to matter.) Absolute space is immutable: it is not affected by matter (p. 113). p.112 (#3): Newton also suggests that the points of space get their individual identity from their position in the whole of space. So, absolute space is a "holistic" entity. (That is, if a part of space, A, could occupy the position of another part of space, B, then A would become numerically the same as that other part, B; and visa versa, if B were to take the place of A.) Accordingly, the parts of space seem to get their properties/qualities, etc. from the whole of space, and do not possess them individually, or intrinsically, apart from one another. Newton also plays with the notion that space is the only substance, and that "material bodies" are just collections of bodily properties, such as "impenetrability", that move around in space in organized ways. This view is known as "supersubstantivalism" (since space is the only physical substance, and bodies are merely properties that move around in the one substance, absolute space). But, Newton never really adopts this view (which is similar to Plato's).

Absolute motion: all bodies have a place, and thus have one real motion (relative to absolute space). Accelerations, and thus inertial effects, are due to motion relative to absolute space. Relative space is position, motion, etc., relative to some object (which we call a "reference frame". A body is at rest in a reference frame if it remains at a fixed distance to the origin (center) of the reference frame; it is in constant motion if it moves with a uniform, non-changing speed and direction, and is in accelerated motion if its motion is non-uniform (i.e., it either changes speed or direction, or both). Newton's laws demand a unique and principled way of separating inertial motions from non-inertial motions, because the non-inertial forces have to be identified/coupled to the non-inertial motions for all observers; but, relational motion/space cannot provide this necessary feature. (Huggett gives a good example; 2nd paragraph, p.133.) Rotation is a non-inertial motion exhibiting non-inertial effects. Example: imagine a rock being twirled around in a sling. Since the rock wants to continue to move inertially, and thus move along the tangent to its circular path, a center-directed pull must be applied to the rock to keep it rotating. But, Newton's 3rd law of motion says that "for every action there is an equal and opposite reaction"; hence there is a non-inertial force directed outward on the sling as one pulls on the sling in the inward direction. Overall, any linear acceleration produces a non-inertial force.

Newton's "bucket experiment" (an observational experiment designed to undermine relational motion by showing that it cannot explain non-inertial forces).

When the water in a bucket rotates, the water rises due to the non-inertial forces that are generated by the water particles hitting the side of the bucket. Since the water particles want to move on a straight line (that is, on the tangent line at there position on their circular path inside the bucket), the inner surface of the bucket causes the particles to rise up the side of the bucket when those particles hit its sides, and this event (i.e., the water rising up the inner side of the bucket to form a curved, or concave surface) is a non-inertial effect/phenomenon. Once again, Newton's "bucket experiment" is an attempt to show that relational motion (Descartes' theory, in particular) cannot explain non-inertial forces. Take a bucket of water suspended on a rope; twist the rope very slowly, and then release it. The following events will occur: (i) before the bucket is released, the bucket is still, and the water is still; (ii) after the bucket is released, the bucket rotates on its axis (the rope), but the water is still, and surface flat; (iii) eventually, the bucket and water rotate together in unison at the same rate (since the friction between bucket and water, as the bucket rotates, starts to make the water rotate, too). The water's surface is concave. (iv) Grab the bucket so it stops rotating: the water still rotates and is concave shaped (for a short while, at least).

Problems for relational motion/space due to Newton's "bucket experiment".

(A) In stage (ii), there is relative rotation (relative motion) between the water and bucket, but the water is flat (not curved), and thus has no non-inertial force is acting on it. But in stage (iii), there is a non-inertial force acting on the water, since its surface is curved (concave), but there is no relative motion between the bucket and water. Thus, relative motion between the water and the bucket does not explain the presence of the non-inertial force; since the force was only manifest in event (iii), when there was no (or little) motion relatively between the water and the bucket.

(B) Looked at another way, the problem for relational motion can be put thus: events (i) and (iii) both manifest the same relational state of motion between the water and bucket, namely, they are at rest relative to one another, but there is a non-inertial force in (iii) and not in (i); also, events (ii) and (iv) have the same relational state of motion between the bucket and water, namely, there is a rotational motion manifest between the water and bucket, but there is only a non-inertial force evident in (iv) and not in (ii). Thus, the existence of the non-inertial force is not due to motion of the water relative to the side of the bucket (since we have cases of non-inertial forces both when it is, and is not, in motion relative to the bucket).

Could the non-inertial forces be due to motion relative to the room? If the room were rotated around a fixed bucket (say, the bucket is suspended from a crane with a hole cut in the roof, while a big turntable spun the room around), it seems quite implausible to think that the water would become curved/concave (i.e., have a non-inertial effect). The non-inertial curved surface of the water seems independent of what outside, or external, objects are doing (i.e., moving or not moving relative to the water). There is no suitable interaction between the water and room to explain its curved shape, thus the motion of other, separate objects cannot explain the non-inertial effect. So, all other outside objects, and their motions, would also seem to fail in accounting for the non-inertial curved surface of the water; in particular, using the Earth (as in, motion relative to the Earth causes the non-inertial effects).

Newton's "rotating globes" example also can be used against relational motion.

Suppose there are 2 globes in empty space, with a chord/rod connecting them. Since there are no outside objects to serve as a reference frame, there can be no motion of these 2 globes for a relationalist since there is no relational motion in such a world. That is, for a relationalist about motion/space, a situation where the 2 globes rotate is the same situation as one where the 2 globes are at rest. Why?: Because the two states are indistinguishable in an empty space: with no other material bodies to provide a reference frame, there is no possible way to determine if the globes are rotating or not (or, put differently, there is no reference frame for the globes to rotate relative to). But, Newton says you could measure the tension on the rod connecting the 2 globes to determine if they are rotating (a tension on the chord would indicate a rotation due to the non-inertial, centrifugal, forces acting on them as they rotate; i.e., the globes want to recede from the center of the rotation, which places a tension on the chord). If there is no tension on the chord, then the globes are not rotating. So, a relationalist cannot make sense of this "absolute rotation" of the globes, as we may call it (i.e., their rotation relative to absolute space); thus, relational motion is not an adequate account of motion/space since it cannot account for this simple fact about the 2 globes. This "globes" argument is more general than the other "bucket" argument, since it seems to entail that all relational theories of motion/space are not adequate to explain (accelerating/rotational) non-inertial forces (i.e., they cannot explain absolute rotation). (Huggett has a nice paragraph about this, middle of page, 139.)

"Rotating" globes example: Newton also says that if you use the fixed stars as a reference frame to determine motion, then the relationalist must say that 2 physical situations are possible: (i) the stars rotate, and the globes are at rest, or (ii) the globes rotate, and the stars are at rest (or both rotate). These 2 situations are identical for a relationalist, since both manifest the same motion relative to each other (i.e., globes relative to stars); yet, by measuring the tension on the chord, one can determine which of the pair is really rotating (an absolute rotation). Hence, relational motion is inadequate to explain the existence of inertial forces because relational space cannot provide a reference frame to distinguish inertial motion (no inertial forces) from non-inertial motion (inertial forces). Only absolute space can do this.

Problems for Newton's Absolute space: (1) no one can ever detect what their state of inertial motion is relative to absolute space. That is, it seems impossible to determine if you are at rest, or moving with a uniform velocity, with respect to absolute space. So, if one cannot detect, or measure, the absolute velocity of an object relative to absolute space, why should this property/feature of the Newton's theory be accepted? (Why not eliminate it, since it can never be verified?) Newton believes one can approximately detect such absolute spatial positions, but we need to use the relative motions among material bodies to do this (p. 120). Unfortunately, this response admits that absolute velocity cannot be observed, since it is dependent on relational motion to acquire an approximate measure. (2) Absolute Space affects bodies, but it cannot be affected, in turn, by bodies. Does this violate Newton's 3rd law of motion? How can something, absolute space, influence something else causally (material bodies), but not be influenced causally in return? This seems contrary to our experience of the world.


Chaper 8: Leibniz and Clarke (1715-1716)

Clarke: philosopher/friend and prophet of Newton (Newton helped him in this correspondence); Leibniz: co-inventor of calculus, philosopher. This is the most famous debate in the history of philosophy on the topic of space. Leibniz was a relationalist about space and motion. He was also a rationalist, who believe that the most important knowledge of the world comes from reason, not experience.

2 facts about absolute space that Leibniz relies upon in his arguments: (i) all points are identical, and space is Euclidean, so there is no way to distinguish one absolute place from another; (ii) consequently from (i), there is no way to distinguish, or determine, a body’s velocity relative to absolute space (i.e., it is not possible to determine the body's absolute velocity). Why?: Because physics is the same for all inertial reference frames: i.e., the behavior of bodies in a reference frame at rest is the same at a constant velocity (uniform speed and direction). For example, the behavior of bodies is the same when car is at rest or moving at a constant speed; so, if you could not see out of the windows, and the car rode very smoothly, you would never know if you were at rest or moving at a constant speed.

Leibniz's 1st argument; Static shift: If the world (or a different world) were located in a different position in absolute space, since the two places are exactly alike and relations among the bodies are exactly the same, it follows that no one could ever determine if the world really was in a different absolute place. So, absolute space leads/allows for situations which are different, but indistinguishable with respect to empirical evidence (which is not a feature of a successful theory in science). Liebniz uses fact (i) above in this argument (III.5). Leibniz's argument relies upon the "Principle of Sufficient Reason" (PSR): "there ought to be some sufficient reason why things should be so, and not otherwise" (LII. 1); i.e., all facts or states-of-affairs need a reason why they are as they are. But, God would have no reason for placing the world in one position in absolute space as opposed to another position in absolute space, so since there can be no reason for this decision (because all spatial absolute positions are identical), absolute space violates the PSR. Leibniz's conclusion: reject absolute space in favor of relational space. For a relationalist, both situations are the identical (i.e., numerically identical), since the relations among the bodies are identical in both cases. That is, whether the universe occupies absolute position X1 or absolute position X2 makes no difference to the relations among the bodies, and since space is just the relations among the bodies for a relationalist, both scenarios are identical for the relationalist.

Clarke’s reply: God can make such decisions based on His mere "will" (CII.1); so, even if we cannot discover the exact reason why the universe is in position X1 or X2, God can still make the choice (thus preserving absolute space). Thus, for Leibniz’s argument to work, one must accept the PSR, but it is unclear if this principle really is true (e.g., modern quantum mechanics may violate it, because the outcome of some quantum events are not determined, and any violation of determinism is a violation of the PSR).

Leibniz's 2nd argument; Kinematic shift. Imagine our universe (or a different universe) that is exactly the same except that the bodies have a different absolute velocity (but all the relations among bodies are exactly the same, i.e., all the bodies have the exact same absolute velocity). But, since no one could distinguish these two different absolute velocities of the world form each other (they are observably indistinguishable, as mentioned above, point (ii) accepted by Leibniz), absolute space, once again, possesses concepts, or theoretical entities, which are not observable (LIV.13). Leibniz uses the "Principle of the Identity of the Indiscernibles" in this argument (PII), which, he claims, is based on the PSR (LIV.3): If 2 things are alike (identical) in every way, then they are really the same thing; or, 2 things with identical properties are really one thing. Therefore, all individual objects in the universe must possess some property which is different from (non-identical to) all the other properties in the world. By the means of unique property, each object gains its "individuality". (Leibniz rejects the ancient/medieval concept of "bare numerical difference", or "haccaety", which holds that two bodies could possess identical properties but still be different bodies; i.e., they are differentiated from one another due to the possession, by each body, of a quasi-"property" called "bare numerical difference".) So, Leibniz concludes that a universe U1 with one absolute velocity, V1, is identical with a universe U2 with a different absolute velocity, V2. In other words, U1 is really the same universe as U2 (since they have no real difference in properties, as mandated by PII). But, this argument only works if you already assume that absolute space does not exist. If absolute space does exist, then the 2 universes do have different properties; namely, V1 and V2 (and V1 does not equal V2, as we assumed in the beginning). So, Leibniz cannot reject the existence of absolute space using PII without "begging the question" (assuming what he was supposed to prove). However, it is best to view Leibniz's argument as using "Ockham's Razor" (which holds that you should always choose the simplest theory which explains the observed phenomena): since a theory that introduces unobservable, untestable entities (i.e., absolute velocity) is more complex than a theory which does not introduce unobservable, untestable entities (such as relational space/motion), one should accept the theory which does not use such unobservable entities (i.e., one should accept relational space, and not absolute space, since relational space is a simpler theory). In fact, modern science is very skeptical, and usually outright rejects, theories which have unobservable consequences, or have conceptual components which cannot be verified, tested, or play no function in the observable consequences of the theory. (If I said I had a law of gravity, G*, which was identical in every way to Newton's law of gravity, G, but that G* included "ghosts" which make no difference at all in any observations of gravitational phenomena, then most people would either reject G*, or say G=G*.)

Clarke responds by saying that motions can be different although they are not detected (as in our example of the car with covered windows) (C.IV.13). Yet, Leibniz argues that 2 different states of absolute velocity are in principle unobservable, not that they have simply not been detected (e.g., you can open the car door to see if the car is really moving; but 2 different absolute velocity states of the universe can never be detected at all.) (L.V.52) However, although absolute velocity cannot be in principle observed, absolute acceleration can be observed (as argued in chap 6 & 7) because absolute accelerations manifest non-inertial forces (whereas absolute velocities do not manifest non-inertial forces). Thus, since absolute acceleration is just change in absolute velocity, and since absolute acceleration is a real, observable state of a body, it follows that absolute velocity must be real (because absolute acceleration is dependent on absolute velocity: i.e., change in absolute velocity is absolute acceleration). In fact, Clarke seems to introduce this argument (CIII.4) because he says that if a universe would experience a non-inertial effect, then this non-inertial effect would provide an observable feature which could distinguish this universe form all other identical universes possessing different states of absolute acceleration (thus defeating Leibniz’s 2nd argument based on PII). So, absolute space has observable consequence, after all. One can call the above reply the "dynamic shift": If 2 universes are identical, and always maintain the same relative positions among the bodies, yet one of the universes undergoes an absolute acceleration and the other does not, then the one with the absolute acceleration will manifest a non-inertial force effect which will make it distinguishable (observationally) from the other universe with no absolute acceleration (but only as long as it is not a uniform acceleration which accelerates all of the bodies, and the parts of the bodies, simultaneously). The relationalist cannot account for the non-inertial forces experienced by all the bodies in such an accelerating universe because the only meaningful concept of acceleration for a relationalist is a relative change in acceleration between individual bodies, and not an identical change of the state of acceleration of all bodies in unison (since such a change is not relative among bodies at all). Problem’s with Clarke’s response: (1) He doesn’t introduce the "dynamic shift" argument with enough emphasis. (2) He declares that space is a property, but this allows Leibniz to argue "what is space a property of?" Since both Clarke/Newton believe it is somehow connected with God’s existence (and not matter), this means that God is in space (as the "sensorium" of God), and thus God is divisible since space is divisible (but God cannot be divided, since God is a whole, indivisible thing). (3) Clarke’s argument that space can be divided in our thought (i.e., conceptually), but cannot be divided physically, is a pretty good argument. But, Leibniz can still argue that a theory that allows conceptual divisions of God is contradictory (since God is not divisible, and so our concept of God should not be divisible); thus, Clarke’s/Newton's theory of God "Spread Out" (diffused) throughout all space (space as the sensorium of God") is very problematic. Unfortunately, Leibniz never responds to the "dynamic shift" argument. Yet, he needs to give a response if relational space/motion is to be considered a successful and consistent alternative to absolute space. One possible response, which seems to work, is to claim that we can only tell that our universe is undergoing a non-inertial force when we compare our universe with a non-accelerating universe (which is not undergoing a non-inertial force), as mentioned above; but, we can only observe our universe, and can never compare our universe with another universe, so the existence of these non-inertial forces due to the alleged acceleration can never be observed by us, and thus they really are not observable (and this entails that a dynamic shift is undetectable, and hence meaningless). 

For Leibniz, "space" is "the order of co-existences" of material bodies. Space is merely the relations (co-existences) of bodies among themselves (where "co-existence means the relations of bodies that exist at an instant of time). (LIII.4&5) Thus, if there are no material bodies in existence, then there is no "situation of objects", and thus no space (since space is just these relations). This view is called "relational space". Problem: Leibniz's view of space is very unclear. At times, he seems to say that bodies can have motions which are not merely relative to other bodies, but are "real", because the bodies manifest a "real" force of motion (as he comes close to stating at LIV. 53, but comes out clearer in other works, such as the "Discourse on Metaphysics"). For instance, in the Leibniz/Clarke correspondence, he seems to state that every time a body has a "real" motion (whatever that means), there will be a change in position among the co-existing bodies, but that it is also true that the "real" motion is in the one body, and not in the co-existing bodies. Thus, Leibniz is apparently rejecting the view that motion is a reciprocal change of relative position of bodies (since one body has it, and not the others). A "strong" relationalist, on the other hand, would deny this claim (since any questions about the individual states of bodily motion are meaningless, as discussed in chapters 6 & 7: i.e., there is only a relative motion among bodies, thus there are no individual states of motion for the bodies involved in this relative motion). So, Leibniz seems to be backing away from a "strong" relational theory of motion/space (where, once again, a "strong" relational theory holds that motion is purely reciprocal, and thus no individual bodies have a determinate state of motion). Rather, Leibniz seems to be accepting a much "weaker" relational theory of motion; namely, that bodies can have individual states of motion given a relative motion between several bodies, but that these individual states of motion do not require absolute space. That is, these individual states of bodily motion can be consistently utilized and discussed by a relationalist without requiring the introduction of absolute space (as will be discussed in more length in later chapters), thus Clarke's/Newton's arguments for absolute space fail.


Chapter 9: Mach & Berkeley: (Berkeley: 17th cent./Mach: late 19th, early 20th cent.)

Both Berkeley and Mach were empiricists (all knowledge is ultimately based on experience)

Both argue that absolute space, absolute location, absolute velocity, are unobservable, unmeasurable; thus these concepts should be rejected. (Mach favored the "verifiability" criterion of meaningfulness: a theoretical/scientific concept is only meaningful if it can be verified (as true) by observable experience.)

Berkeley argues that one can only conceive a motion if it is relative to another body (sec. 58). Consequently, without a natural reference frame (i.e., using material bodies as a means of determining motion), the motion of the globes in Newton's "globes experiment" (where no bodies exist except the globes) cannot be conceived, thus there is no difference between a resting globe or rotating globe (sec. 59). Mach argues similarly: all motion is relative: one can’t say how K behaves in the absence of reference frame bodies "A, B, C...". Thus: (i) we can’t know how K would act in the absence of A, B, C...; and (ii) without the reference frames (constituted by A, B, C...), it is impossible to determine how K would behave (Sec 4). Both of these points undermine Newton’s use of absolute space to explain the inertial effects of absolute rotation. Why?: Because Newton assumes that absolute rotation will produce non-inertial effects even if all other bodies in the universe are removed. Yet, we can never know if this belief is true.

Newtonian response: But doesn’t the existence of non-inertial effects give indirect support for the existence of absolute space (as argued in chap 7 & 8)? We use indirect evidence to prove the existence of many unobservable, theoretical entities (such as electrons, etc.)? The relationalists (Mach, Berkeley, etc.) need to find some relationist means of accounting for the existence of non-inertial forces (i.e., using relative motion alone, and not absolute space): Thus the relationalist needs to fill-in the blank: "absolute acceleration is acceleration relative to____."

Both Berkeley and Mach argue that absolute acceleration (and thus non-inertial forces) are due to motion relative to the fixed stars (Berkeley: Sec. 64; Mach: Sec. 5 & 6). Mach states that it is relative to the fixed stars, but that the reference frame is not the distant stars themselves, but the "center of mass" of the entire universe, which is defined as the position that gives the average of all bodies according to their mass (weight): e.g., the center-of-mass (com) of a 1 lb. and 2 lb. object 3 feet apart from each other is at a position on a line between them that is 2 feet from the 1 lb. object and 1 foot from the 2 lb. object (so that the average "mass times distance" of each body is the same numerical value: i.e., 2=2, in the case provided above). Essentially, the com is the point on a see-saw where the fulcrum needs to be put in order to balance the weight of the 2 objects.

According to Mach, there is no difference between: (i) a non-rotating bucket and rotating universe; and (ii) a rotating bucket and non-rotating universe. Why?: Because the relative motion is exactly the same. And since the relative motion is exactly the same, (i) and (ii) are really just 2 different descriptions of the same state-of-affairs (sec. 5). Mach believes, in fact, that Newton's argument has only proven that a relational motion of the water in the bucket cannot be with respect to the sides of the bucket; but Newton has not shown that the water's rotation cannot be with respect to the "fixed stars", or, more accurately, the average mass of the universe (as represented by the com reference frame of the universe). For Mach, the com reference frame provides the means of determining accelerated motions, and thus non-inertial force effects: i.e., the water rotates relative to the com reference frame of the world. Moreover, being a relationalist, Mach argues that if the sides of the bucket were to grow to a very large size (possibly as large as the size of the galaxy), then the rotation of this large bucket might actually cause the water in the bucket to exhibit a non-inertial force (a concave shape) even though the water is at rest.


(1) Is Mach's theory a violation of his own "verifiability" criterion? Can one "observe" all the mass in the universe? And, should subatomic particles be included in the com reference frame (even though Mach did not believe in atoms)? Also, since one must rotate a significant part of the universe to test Mach's theory (that a very large rotating bucket can cause a non-inertial force effect on the resting water in Newton's bucket), is this theory any more observable, or verifiable, than Newton’s?

(2) Newton believed that any use of the fixed stars/universe as the cause of non-inertial forces (in absolute acc) was crazy. In short, "How can a spinning universe cause the water to curve in the bucket?!" (Yet, how can absolute space provide the same function without being equally mysterious/ridiculous?)

(3) Modern attempts to introduce Machian formulations of classical/relativistic space-times have not been successful (so far). In fact, Mach's brand of relational motion does not coincide well with dynamical theories; and this is the case despite the fact that modern General Relativity appears to vindicate the idea that a very large rotating bucket would cause the resting water to exhibit a non-inertial effect! In the General Theory of Relativity (GTR), spacetime itself is curved or warped by large bodies, so it is conceivable that a very large body could warp the spacetime in the bucket to such a degree that the water takes on a curved, concave shape. Yet, this facet of GTR does not make Machian relational concepts of motion and space amenable to GTR overall. Why?: (i) in GTR, mass/matter warps a spacetime that has its own metrical properties (i.e., distance, shape, etc.) even in the absence of matter (which is usually considered to be flat). In fact, the story of curvature of the resting water in the large rotating bucket is explained by appealing to the warping of spacetime by that bucket; and this seems to imply that spacetime has its own form of existence separate from the matter in spacetime (but more on this in later chapters). More importantly, (ii) Mach's theory of relational motion still fails because it cannot separate the two following cases: Assume the observer is situated at, or near, the resting bucket, and that the bucket is surrounded by two large spheres or cylinders of matter (one inside the other), with one sphere being very large and the other very thin. Next, assume that there exists a rotational motion between the two spheres as observed from the bucket (thus, the observer can apparently see through the first sphere/cylinder and observe the relative rotation). According to Mach's form of relational motion, it is not a meaningful question to ask if it is really the large sphere that rotates and the thin sphere at rest, or visa versa, since these two possible situations cannot be observationally distinguished from one another. Thus, both possibilities are really the same state-of affairs (i.e., rotating thick sphere and resting thin sphere, or rotating thin sphere and resting thick sphere). Unfortunately, GTR holds that these two situations are really different, since the rotating thick sphere will cause the spacetime to warp, and thus cause the water to curve in the bucket, whereas the rotation of the thin bucket will not cause this outcome (or at least no to the same large degree). Accordingly, Mach's brand of relational motion is still incompatible with modern theories of physics.

(Note: Maybe Mach would disagree with this conclusion? Since the rotation of the thick sphere causes more warping than the rotation of the thin sphere, Mach could claim that they are in principle observationally distinguishable, and thus not identical states-of-affairs. If this is true, then maybe Mach doesn't accept a "strong" relational theory of motion after all (rather, maybe he simply rejects absolute space, which we've dubbed a "weaker" form of relationism)? Mach’s confused statements on relational motion might have only been intended to (1) reject the strong from of relationism (which is the claim that all motion is the relative motion of bodies), while (2) accepting weak relationism (that is, rejecting the theory that space is an independently existing entity, called substantival or absolute space). His verificationist theory of meaning is perfectly compatible with a weak relationism, as his com reference frame method for determining motion (without the need of substantival space).

Berkeley's attempts to solve the "bucket" experiment seems confused: He seems to think that Newton argued that a centrifugal (center of axis fleeing) force arises due to rotational motion, but Berkeley argues that the effect of the water rising is due to the tangential tendency of inertially moving particles (without a sufficient centripetal, or inward directed, force). Berkeley is correct, but this misses the point of Newton's argument: Newton argued that the curvature of the water is due to an absolute acceleration that only occurs when the bucket rotates with respect to absolute space. So, Berkeley's explanation assumes that such non-inertial effects (of absolute rotation) really do occur only in rotational motion; thus, he has unwittingly accepted Newton's account of the phenomena, and thus accepted absolute acceleration and absolute rotation. In fact., Berkeley's account only makes sense if one accepts absolute acceleration and absolute rotation This suggests Berkeley did not understand Newton's argument. That is, he focuses only on an axis-fleeing force rather than on the non-inertial force’s existence, and its manifestation only in certain motions (such as rotation).

Berkeley's other argument: Since the bucket moves with the earth, and the earth is engaged in numerous other motions (spinning on its axis, motion around the sun, etc.), it is not the case that the bucket is involved in a true circular motion. In other words, the bucket's motion is not really circular with respect to absolute space (rather, it is some form of complex curve). Accordingly, Berkeley seems to believe that an object must rotate around a fixed point of absolute space (and thus not translate) to count as a true rotational motion. Problem: the rotational motion of the earth (around axis, sun, etc.) does produce a non-inertial force (but it is counter-acted by gravity), so many non-inertial forces can exist simultaneously (Coriolus Force, etc). Also, a fixed rotation (around a fixed spatial location in absolute space) is not needed to cause non-inertial effects of absolute rotation/acceleration All that is needed is a rotation relative to absolute space as a whole (and not relative to just one position, which would entail that an object can only rotate about that fixed position and cannot partake in any other motions).

Overall, Berkeley/Mach did help to promote the idea that inertial forces and motions are affected by matter, which is an important development in the road to modern STR/GTR.

Huggett's Problems: (2) Hugget wants to show (apparently) that displacement acceleration is needed relative to all bodies, and not just to those bodies near an object. In some cases, a body and its neighborhood of bodies, may all be spiraling out of the universe together, so the displacement acceleration of the bodies with respect to the (local) com reference frame of this small group of bodies is equal to zero (i.e., d2R/dt2 of the local R = 0), and so our body is at rest relative to this local com reference frame. Yet, this is a problem because this small group of bodies is really accelerating relative to rest of material bodies in the universe. One can (hypothetically) correct for this problem if all the bodies in the universe are considered, such that the displacement acceleration of our body (d2R/dt2) must include all universal bodies. If all bodies are taken into consideration, then our original body is spiraling (accelerating) relative to this universal com reference frame R (and not merely the local R). Thus, d2R/dt2 does not equal zero, and is thus not at rest, but accelerating (as Mach's theory should claim, if it is going to be a successful theory, that is).

Problem (3): This is Sklar’s notion of "absolute acc" as a primitive, non-relational, and intrinsic property of bodies. That is, acceleration is not "relative to ____", but is merely an internal property of a body (like such non-relational properties as "circle", "square", etc., and unlike such relational properties as "mother of___", or "taller than ____"). This theory of acceleration denies that non-inertial effects are due to motion relative to something (Mach's com reference frame, Newton's absolute space, etc.). Some bodies just have an intrinsic property of "acceleration", just as some bodies have the intrinsic property of "square", or "six feet tall", etc. For these "monadic" (as opposed to "dyadic") types of properties, no other body, object, item, concept, etc. is needed to ground (or base/secure) the existence of the property. 


Chapter 10: Space-Time

Spacetime (ST) is the collection of all possible events. It is 4-D: 3 space/1 time (but ST diagrams are 3-D, with 2-space/1 time). Each temporal instant is a slice of the ST into a "spatial slice". Points in ST picks out a certain place at a specific time (and can be called events).

Newton would have been a "substantivalist", which holds that spacetime is some sort of "substance" (but ST is one enduring entity, not a collection of "space slices").

Newtonian ST: This ST must preserve Newton's belief in static shifts, kinematic shifts, and inertial effects (i.e., absolute place, absolute velocity, absolute acceleration).

Newtonian ST: Between any two points (events) P and Q, there is a definite Euclidean spatial distance R(P,Q), and temporal interval T(P,Q), which gives the distance between the points in space, R, or time, T. The ST "riggings" mark the "same place" over time, so it makes sense to talk about the an absolute place in this ST (which thus allows "static" shifts from an absolute place A to an absolute place B). In Newton ST, it makes sense to talk about the speed of an object as it changes from absolute place to another absolute place over time, so "kinematic" shifts are also possible because a universe that is moving uniformly through space may retain the same relative positions among each of its bodies but will be occupying different absolute places at each instant. It also makes sense to talk about absolute acceleration because this ST allows one to measure motions relative to straight line inertial motions (i.e., even if no material objects exist, or are at rest, the same ST position marks such inertial paths over time). Every world line has a definite slope (which is determined relative to the ST rigging). World lines with constant slope/velocity are inertial, but become curved if the slope/velocity changes (which is acceleration, of course). This substantival version of Newtonian ST can be named, "Sub. Newton ST".

Galilean ST: Between any two points (events) P and Q, (i) let there be a definite temporal interval, T(P,Q); (ii) if they are simultaneous (T(P,Q)=0), let there be a definite spatial distance, D(P,Q); (iii) given any curve C, through P, let C have a definite curvature at P, S(C,P), such that a curve C is straight if S(C,P)=0 for all points in the infinitesimal neighborhood of P. Overall, in Galilean ST, it is not possible/meaningful to ask about the spatial distance between non-simultaneous event/points: since there is no concept of absolute position, it is not possible to determine the spatial separation between events that occur at other temporal moments (but this is possible in Newtonian ST, of course). This spacetime can tell if a trajectory (line) passing through a given point is straight or curved (inertial or accelerating) because the ST has a special feature (the covariant derivative), which we have labeled S(C,P). In Galilean ST, since there are no ST riggings which mark off/determine the "same absolute spatial place over time", any straight line (constant slope line) is just as privileged as any other straight line. Any line (straight line) can be transformed into another straight line (where straight lines are inertial, of course) by "skewing" the ST spatial slices (sliding the slices a fixed distance over one another, while preserving the relations of points/objects on each spatial slice; or, alternatively, just skewing the lines and points on each slice a fixed distance, while preserving their relations among one another on each slice).

Substantival (Sub.) Galilean ST (also known as Neo-Newtonian ST) eliminates both static and kinematic shifts because one cannot determine absolute place (i.e., there is no such property in this ST), but it does allow absolute acceleration because the ST has a property, namely, the function S(C,P), which can determine which paths/lines through ST are straight (non-accelerating/inertial) or curved (accelerating/non-inertial). This ST is the modern (or preferred) version of a Sub. Newtonian ST because it eliminates the unobservable absolute place and absolute velocity concepts, but retains the absolute acceleration concept (which is observable due to inertial force affects).

NOTE: Huggett’s discussion of substantival/relational Galilean ST seems confused to me: he believes a Sub. Galilean ST allows static shifts, but not kinematic shifts. Yet, if you can discern a universe in a different ST position (i.e., having a different absolute place), then you can also determine the "rate" of the universe's change of absolute position, which is the notion of absolute velocity. Therefore, the Sub. Galilean ST should also allow kinematic shifts (i.e., absolute velocity), contrary to Huggett's assertions (as well as every other ST theorist's assertions). For all other ST philosophers whose work I am familiar with, Sub. (and Relationist) Galilean ST eliminates both static and kinematic shifts. However, maybe Huggett thinks that static shifts, unlike kinematic shifts, only occur instantaneously; or, in other words, on one "spatial slice", which does not involve a sequence of spatial slices over time. If time is not involved in a static shift, and one cannot refer back to the previous absolute positions prior to the static shift, then one cannot appeal to these previous absolute places, that endure over time, to measure the universes absolute velocity. But, this seems to me to be a strange notion of absolute position.

Relationalism (Rel.): (i) ST is not independent of matter; (ii) ST is the collection of ST relations among material objects.

Rel. Newtonian ST: this ST is the same as Sub. Newtonian ST, although one can only use material events, or points, to identify positions. One cannot use points in ST separate/independent of matter (since this is a feature of Sub. Newtonian ST).

(i) This universe cannot be statically shifted because the material points maintain the same relations among one another in the two universes (or different static shift). In other words, since the points/events of this ST are just the material bodies/points of the universe (and these material points/event are identical in both worlds), both universes are identical (which eliminates the static shift problem).

(ii) This universe can be kinematically shifted because the ST preserves absolute velocity once a material event is allowed to pick out an absolute place: i.e., once a material event is introduced, then the ST rigging is introduced, which means absolute velocity can be determined (and thus the whole universe can have an absolute velocity).

(iii) This universe allows absolute acceleration because of the concept of absolute velocity introduced in (ii), and absolute acceleration is just the change of absolute velocity.

Leibnizian ST: Between any 2 points P and Q, there is a definite time, T(P,Q), but only between simultaneous points is there a distance, D(P,Q). Also, there is no ability to determine the straight/curved nature of trajectories; i.e., there is no S(C,P). There is both a Sub. Leibnizian ST and a Rel. Leibnizian ST.

(i) Static shifts: "no" for Sub. Leibnizian ST (for the same reason described above with respect to Sub. Galilean ST); and "no" for Rel. Leibnizian ST (for the same reason as for Rel. Newtonian ST).

(ii) Kinematic shifts: "no" for Sub. Leibnizian ST and Rel. Leibnizian ST (both for the same reason as described above with respect to Sub. Galilean ST).

(iii) Absolute acceleration/inertia: In Leibnizian ST, both the Sub. and Rel. versions, only relative accelerations among bodies are meaningful/determinable, so the absolute acceleration/inertia of a single body—as a property that is intrinsic to that body—is not meaningful in either the Sub. or Rel. versions of Leibnizian ST. Both of these spacetimes can make sense of "relative accelerations", however; but this will not allow a unique class of bodily trajectories (paths through ST) to be picked out as the "inertial" paths of the ST—and the ST must provide an "invariant" determination of a class of such inertial trajectories (for all observers) if it is to secure a coherent account of inertial motion. In short, in a (Rel. or Sub.) Leibnizian ST, whether a path through ST is straight (inertial/non-acceleration) or curved (non-inertial/acceleration) depends on what reference frame you happen to choose to measure/determine the path of the body: some frames will see the path as straight, and some will see the path as curved (depending on whether that reference frame is, or is not, accelerating itself); so, it is impossible to uniquely define a set of paths through ST that everyone agrees are "inertial" (straight). Yet, if no one can agree on what paths really are inertial, then it will be impossible to establish a meaningful concept of inertial (straight) motion in the ST. This is the reason why Leibnizian ST is so problematic for our modern theories of dynamics (which require a coherent means of determining inertial from non-inertial trajectories).

Rel. Galilean ST: This ST eliminates static and kinematic shifts (for the same reason as Sub. Galilean ST), but it retains a measure of absolute acceleration (also for the same reason as Sub. Galilean ST); thus this ST has the ability to determine whether it is the bucket/globes or stars that are really moving, or at rest, in Newton's examples. Likewise this ST can (contrary to Mach's estimate) determine whether the globes are, or are not, rotating in an empty universe, since S(C,P) is introduced with these material objects/points (i.e., the matter that comprises the globes). But, it cannot make sense of: (a) an empty universe with gravity waves running through it, which is a fully consistent solution to the field equations of GTR; or (b) a rotating universe as a whole, which is also a consistent solution to the field equations of GTR (Godel's solutions). Thus, since GTR can make sense of these distinct scenarios, (a) and (b), but Rel. Galilean ST cannot, it follows that Rel Galilean ST is not necessarily compatible with GTR, which is our best current theory of the physics/dynamics of space and time.

Problem: On our "weaker" formulation of Rel. ST (such as provided in Earman’s version of relationalism, dubbed (R2) in his text), the conclusion reached above can (possibly) be avoided. But, the criterion of relationalism labeled (ii) above (i.e., that ST is a relation among objects alone) must be abandoned. On thus "weaker" Relational theory of ST (Weak Rel.): (i) ST is not independent of matter, but it seems to be more than our (ii) above (i.e., just the collection of ST relations among material objects). Why?: On the "strong" theory of relational motion (with respect to problem (b) above), a stationary and rotating universe have identical relations, and so must be equivalent descriptions of the same ST. But, Weak Rel. ST will rejects problem (b) because the S(C,P) can distinguish between fixed/rotating universes. Once a material object exists in the ST, the S(C,P) is automatically introduced, and this feature of the ST can distinguish between a stationary or rotating universe (since the matter that constitutes the former ST traces out a straight trajectory, while the matter that comprises the latter ST follows a curved trajectory—and the S(C,P) determines the straight/curved trajectories of all material bodies, including the straight/curved path of the whole material world moving in unison). In short, a Weak Rel. ST is the collection of ST relations among material objects and among the inertial trajectories provided by the matter-dependent S(C,P) (also called the covariant derivative). Unfortunately, a Weak Galilean Rel. ST does not seem to be able to solve problem (a) above, since there is no matter in the ST, at all. (Yet, if gravity waves are understood to carry mass/energy, as GTR does hold, then such waves provide a basis for the introduction of the matter-dependent S(C,P). Why?: Because mass/energy is a "state", or form, of matter according to GTR (i.e., E=MC2).


Chapter 11: Kant and Handedness

I. Kant (German philosopher, late seventeenth century)

Some objects and their mirror images are "congruent counterparts": i.e., a rigid motion of the object in space (equal in dimension to the dimensions of the object) will make the object completely cover its mirror-image (that is, both can occupy the same place, which is what makes them "congruent"). E.g., the letter "E" can be moved around in a space and will always coincide with a backwards-E . Some objects and their mirror images are "incongruent counterparts": i.e., a rigid motion of the object in space (equal in dimension to the dimensions of the object) will not make the object completely cover its mirror-image (that is, both cannot occupy the same place, which is what makes them "incongruent"). E.g., the letter "F" can be moved around in a space

The Kantian argument: left-handed (L-hand) and right-handed (R-hand) are different objects such that they are incongruent counterparts (incon. count.), but they do not differ on intrinsic spatial relations. That is, all the internal spatial relations of a L-hand and R-hand are equal: e.g., distance of thumb to forefinger, width of palm to width of middle finger, etc. So, according to Relationism, all of the relations between the parts of a R-hand are identical to the relations between the parts of a L-hand. Accordingly, a Relationist cannot distinguish a L-hand from a R-hand (since they can only appeal to the differences among material objects, in this case, the parts of a hand). But, since a R-hand is intrinsically different from a L-hand in reality, and the Relationist can’t account for this fact, then Relationism is not an adequate theory of space. Since absolute space can account for this difficulty (because L-hands are congruent to L-handed-shaped regions of absolute space, and similarly R-hands are congruent to R-handed-shaped parts of absolute space), it follows that absolute space is preferable to relational space.

Response by a Relationist: a left-hand has a certain spatial relation to other objects in space (called "fitness") and this accounts for its handedness without the need for absolute space. For example, a L-hand is closer to the heart of the body than the right hand, thus establishing a difference purely in terms of other bodies (as mandated by Relationism). So, other material reference frames can account for incongruent counterparts.

Absolute space counter-reply: (1) The relationalist's response assumes that one already understands that the heart is on the left-side of the body. But what if the body can form its own mirror image (i.e., it an incongruent counterpart)? In this case, one cannot tell on relationalist grounds if it is a left or right hand. Reply by Relationist: a universe which is the mirror image of the normal universe is unobservable and cannot be verified by any possible observation, so does it make sense to talk about such possibilities? (It is much like saying: "Could you detect if the whole universe doubled in size?") Thus, this Relationist reply is much like the "static shift" argument. (2) In a universe that is empty except for the existence of only one hand, the Relationist would not be able to state whether the hand was L- or R-handed. In fact, handedness does not make any sense to a Relationist in this case (since there exists no material reference frame, or "fit", to determine handedness). This is like the indistinguishability of the resting/rotating globe in Newton’s argument (Huggett seems to miss the point, here). Kant says that the Relationist must conclude that our solitary hand can fit either arm of the body (p.202). Yet, if the Relationist is committed to this outcome, then Relationism has a big problem: How can the introduction of a body instantaneously change a hand that, when alone in the world, could fit either arm (side of the body) into a hand that can only fit one arm (side of the body)?! How can the introduction of a material body (or reference frame) into space change/affect an intrinsic feature of the hand (L- or R-handedness, since this seems to be an internal feature of the hand)? Of course, the relationist might simply reply that our intuitions are wrong in this case: handedness is a relation among bodies, so if there is no other objects or reference frames in space, then the object just does not possess a left- or right-handed orientation.

Another absolutist argument: Not only do rigid motions in space determine if an object is a congruent counterpart or incongruent counterpart, but also whether the space is orientable (all rigid motions preserve incongruent counterparts) or nonorientable (some rigid motions do not preserve incongruent counterparts) plays a role. That is, in nonorientable spaces (like the 2-D Mobius strip for 2-D objects), an object can be changed into its counterpart if it moves through certain regions of the space (e.g., "F" into a "backwards-F", or a R-hand into a L-hand, in 3-D nonorientable space). So, the shape of the space as a whole determines the congruence/incongruence of counterparts (e.g., handedness, etc.). But, a Relationist cannot appeal to the shape/structure of space to account for this fact: they must simply say that it is a basic fact that some rigid motions in space preserve incongruent counterparts, and some rigid motions do not. Yet, this does not seem a satisfactory resolution of the problem because they cannot explain why certain motions do, or do not, change the incongruent counterparts. Absolute space can explain this, so it gains in (abductive) inductive support by having more explanatory power/unification than Relational theories of space.

Also, quite importantly, certain physical events/process are incongruent counterparts (have a L- or R-handed orientation, such as certain particle decay events). So, can a Relationist consistently account for these processes?

(On a Mobius strip: there is no front-side/back-side distinction because the space is only 2-D, and sidedness demands a thickness, which is 3-D.)


Chapter 12: Kant and Geometry

Summary of Kant's metaphysical philosophy in his later work (especially the Critique of Pure Reason). (These notes are borrowed from my Early Modern Philosophy web page).

First of all, Kant argued that previous philosophers had not gone far enough in categorizing the different types of statements with respect to their epistemological content (i.e., how we know that they are true, etc.). Kant accepts Leibniz’ distinction between what we he (Kant) calls "analytic", and "synthetic": analytic statements are such that they are true by definition, i.e., (as in "all bachelors are unmarried males", and "x=x"; here, the concept "unmarried males" is contained in the definition of the concept "bachelors", etc.). Synthetic statements, however, are not true by definition, since the concept of their predicate is not contained in their subject (such as, "this cup is hot", which is not analytic since the predicate "hot" is not contained in the definition of the concept "cup"). Kant, furthermore, accepts the distinction between "a priori", which he defines as a necessarily true statement (i.e., true of all possible experiences) known by reason, and "a posteriori" statements, which are not true of all possible experience and are thus known only through individual experiences. Kant then considers the four conjunctions of these four types of statements.

1) Analytic A Priori: These types of statements are true by definition and true of all possible experience. Hence, these are just the statements usually dubbed "a priori" by most of Kant's predecessors (although Hume called them "relations of ideas"). Examples: "all bachelors are unmarried males", "x=x" , "triangles have three sides".

2) Analytic A Posteriori: These types of statements are impossible because they would be true by definition but known only through experience (i.e., not true of all possible experience); but how could something known to be true only through the meanings of its own words also be known to be true only through experience!

3) Synthetic A Posteriori: These statements are not true by definition and are only known to be true by experience; thus, most of the contingent facts of the world would fit this category (such as "today is cold", "my coffee rules", etc.)

4) Synthetic A Priori: These statements are not true by definition, but are necessarily true of all possible experiences. This is Kant's contribution to the problem, since no one before him had ever made this distinction (as far as I can tell). Kant places most mathematical and geometrical knowledge in this category: While "1=1" is analytic a priori, because the definition of the two concepts are identical, he wants to claim that statements such as "7+5=12" are synthetic a posteriori because the concept "7+5" is a different concept than "12" (i.e., the concept of the one is not contained in the concept of the other). Kant reasons that the denial of "7+5=12" does not lead to a contradiction in the way that the denial of "1=1" does lead to contradiction (remembering, of course, that Kant also defines an "analytic" statement as one whose denial leads to a contradiction). If all math statements were analytic, he argues, then many very difficult mathematical theorems and equations (which were eventually proven to be true) should have been easily recognized as true (by definition) by just thinking about the meaning of the concepts involved.

Kant worries, of course, how we can know that synthetic a priori statements are true. This provides the reason for his introduction of the "forms of intuition". (Kant uses the term 'intuition' to designate sense experience.) Kant claims that all of our possible sense experiences (intuitions) are filtered through a "conceptual framework" which our minds impose upon those sense experiences. These "forms of intuition" are the categories of space and time: all sense data is represented, or manifested, to us as possessing spatiotemporal properties (i.e., all of our sense experiences, say, of a coffee cup, "represent" the coffee cup as taking up space and enduring through time). Here, it is important to remember that the "content" of a sense experience is "what the experience is about" whereas the "form" of a sense experience is "how the experience is presented/represented". For example, if I see a table, a desk next to it, and some papers on top of it, then the desk, table, and papers are the content of this experience; while the spatial relationships ("next to", "on top off") are the form of this experience since they concern the way in which the table, desk, and papers are represented.

The really interesting (and controversial) claim that Kant makes is that the form of the mind’s experiences are contributed by the mind itself. The mind imposes the "forms of intuition" on all sense experience; so it is just a brute fact that all possible sense experience will have spatiotemporal properties. But, Kant often says that these forms of intuition are preconditions for having experiences; that is, in order to have any sensory experience at all, such as "seeing the coffee cup", you must be equipped with the space and time mental framework necessary for the very perception of the cup. (Since the cup is a spatiotemporal object, if you remove space and time from your experience of the cup and you have nothing left!!) Yet, Kant also claims that these forms of intuition are means by which our minds acquire knowledge of synthetic a priori statements. Our minds are able to arrive at synthetic a priori truths by applying our space and time categories of understanding (the forms of intuition) to analytic a priori concepts: e.g., "a line is the shortest distance between two points" cannot be known by just examining the meaning of the concept "line", since this concept does not incorporate any information about distance (or how distances are determined, etc.); but, if the line is embedded (placed) in a (Euclidean) space, then it becomes quite obvious that a line is the shortest distance between two points. Hence, our synthetic a priori claim, "a line is the shortest distance between two points", is revealed to us as true by means of the spatial "form of intuition". (Kant says that arithmetical statements, which are also synthetic a priori, are known through the application of time's "form of intuition", since our experience of the succession of temporal moments provides us with our understanding of arithmetical progression; i.e., addition.). Moreover, since the "forms of intuition" are applied to all possible experiences, we thus know that synthetic statements such as, "a line is the shortest distance between two points" will be true a priori (i.e., true of all possible experience). By this clever means, we acquire knowledge of synthetic a priori statements.

Problems for Kant, or, more accurately, for the neo-Kantians, began to surface with the development of non-Euclidean geometries in the later 19th century. In many of these theories, such claims as, "a line is the shortest distance between two points", no longer hold true (since the shortest distance could be a curve, etc.)—and, in addition, it is possible that a posteriori experience of the world might be able to determine if space is either Euclidean or non-Euclidean. Thus, Kant's claim that geometrical (and mathematical) statements are a priori synthetic is apparently undermined, since now it appears that our experience of the world (as regards content, and not form) can reveal the truth or falsity of many of these claims (such as "a line is the shortest distance between two points"), and so they would appear to be synthetic a posteriori, after all. In other words, they were not necessarily true of all possible experiences, which is what Kant would need if they were to qualify as a priori.

Kant's synthetic a priori also demonstrates how he attempted to synthesize the Empiricists with the Rationalists. Kant was a big fan of Hume, and wanted to preserve Hume's doctrine that we can only know what is revealed through experience. However, Kant wanted also to argue, along with the Rationalists, that we have a priori, or necessary, knowledge about the world. The synthetic a priori accomplishes this daunting task in the following manner: Kant claims that we can never know the world separate, or apart, from our experience, since how can you claim to know anything that transcends all possible experience? Thus, the world as it exists apart from intuition (experience), which he calls "noumena", is forever unknowable. So, this conclusion would appear to agree with Hume's philosophy. However, since we know that all possible experience will partake of the forms of intuition (space and time), then any knowledge that we acquire from investigating the properties/features of these forms of intuition (such as synthetic a priori mathematical and geometrical propositions, as above) must be true a priori! Conclusion: we can have certain, a priori knowledge of many facets of the world of experience, which he dubs "phenomena"; and this a priori knowledge is in keeping with the spirit of Rationalism, of course.

But, if our synthetic a priori knowledge is only limited to phenomena, and we can never know noumena (things-in-themselves, as he calls them), does that mean he is an idealist like Berkeley (where "idealism" means that the only existing things are minds and God)? Kant vehemently rejects idealism, and wants to claim that we have a direct access to, or experience of, the outside, external world. In fact, he wants to reject the whole "representationalist" theory of perception (which holds that we gain knowledge of the outside world by means of ideas or sense perceptions in our minds, and that these sense perceptions are like "little pictures" in our minds which "represent" the world by resembling it—this is, by the way, how photographs "represent" their content). Overall, Kant says we directly experience the world (or so some passages seem to suggest) since it is not the case that our sense experiences function as some sort of go-between, or middle-man, betwixt the external world and our minds (which is how they would need to function, moreover, if sense experiences really were like little pictures in our minds). However, if we can never know the world as it exists outside all possible experiences, doesn't that entail idealism all the same? Its not quite clear how to resolve this problem entirely, but Kant also claims that noumena (things-in-themselves) are not really things, but only the limitations of our perceptual faculties: i.e., it is the limit of all our possible experiences. Thus, Kant's noumena could just be another way of saying that we can only talk about those features/properties of the world that we can ultimately experience, and thus it is fruitless to try to describe the properties of the world which will forever be outside any possible experience. On this interpretation of Kant, his concept of noumena seems compatible with his rejection of idealism. (But is it what he meant?)

Besides the forms of intuition, our minds also are equipped with innate principles of understanding that likewise hold true (or, more correctly are applied to) all possible experience, and thus they are synthetic a priori, too. Causation, and the Laws of Nature, are among these a priori categories of "understanding", hence causal relationships hold true of all possible experiences. In short, Kant held that the way in which "the future resembles the past", as Hume would put it, is a part of our experience that is contributed by our minds. That is, causation is something we can believe with justification because our minds automatically work in such a way that future experiences resemble past ones. Kant claims, by the way, that the existence of the synthetic a priori categories of understanding are not derived from experience, but are presupposed by experience: in other words, you need to have the a priori forms of intuition and categories of understanding operating/functioning in your mind before you can even have an experience.

(Here, I begin the summary of Huggett's discussion.)

For Kant, every experience of the physical world has 2 components: sensations of the things, and the framework in which the sensations are organized. Space and time are the framework in which the sensations are organized. From this Kant draws 2 conclusions: (1) the framework is required for us to have experiences, so it cannot come from experience but is provided, in advance, by us. Therefore, Space and time is not something we learn form experience, as the Relationist believe, but is something presupposed by experience (p.216). So, the form of intuition of space and time is "a priori" since it is built-into us as the framework by which experience is revealed. (2) Since the framework is provided prior to experience, our experiences will always be spatial, an we can be certain of our knowledge of space (it is "a priori", since it is true of all possible experience, or, put differently, it is necessarily true of all experience). For Kant, space is Euclidean, 3-D, and this is an "a priori" necessary truth. But, since space is just a framework in our minds (a pure intuition), we do not know if space in the world outside of our minds—i.e., in the world outside of all possible experience—is Euclidean, or anything, at all. That is, we can’t know about space and time as it pertains to the world outside of our experiences (where experiences are mental events in our minds, of course).

Kant’s argument that space is Euclidean, and knowable as a synthetic a priori claim was undermined by the development of non-Euclidean geometries in the later 19th century (as described above). So, if there are other geometries which are possible, which geometry is the geometry of space? It would seem that experiments/observations need to be conducted (such as Gauss’s hypothetical light experiments on mountain tops) to determine which geometry is the real geometry of physical space. Therefore, maybe space is "synthetic a posteriori" after all (since observations are required to determine which geometry is true of space, and it is thereby not true by reason alone). In short. Kant did not know about the possibility of other geometries, so he probably assumed Euclidean was correct without any reservations. Yet, other geometries are possible. For example, on a sphere, a great circle can connect any 2 points, and is the shortest distance between two points, thus it satisfies the definition of "line" in Euclidean geometry. Yet, in the "spherical" geometry, as it is called, all lines intersect, which is different from Euclidean geometry (Alternatively, through a point outside a line, no lines can be drawn which do not intersect the given line; whereas in Euclidean geometry, only one line can be drawn which doesn’t intersect a given line.) Also, in Euclidean geometry, all triangles=180 degrees, whereas on spherical surfaces triangles don’t equal 180 degrees, but equal a sum greater than 180 degrees. Example: a triangle formed by the equator and 2 longitudinal line coming from the North Pole can form 3 angles which are each 90 degrees, so the triangle equals 270 degrees.

The Neo-Kantians philosophers (late nineteenth/early twentieth century) tried to salvage Kant by saying that he was wrong to pick out Euclidean geometry as the geometry of the space provided by our mind's "pure intuition". Yet, the Neo-Kantians insisted that Kant's idea that space is a "pure intuition" was correct: i.e., space as an a priori synthetic truth. What the mind does supply as the a priori synthetic spatial framework, however, is the metric tensor Gij(dxidxj) (or simply, Gij). In modern differential geometric treatments of spacetime, the Gij allows for all different metrics (i.e., means of measuring distance, which determine the geometry of the space), so the Gij can allow Euclidean, Spherical Non-Euclidean, or Hyperbolic Non-Euclidean geometries. Therefore, if the mind provides the same synthetic a priori Gij function for all humans, then many different geometrical structures can be constructed from the identical Gij geometric function that all humans possess. Consequently, Kant did not need to declare that space must be Euclidean, 3-D; rather he needed only to declare that the mind has a built-in capacity (the Gij) for constructing a geometry that, in turn, serves as the framework for all possible sensory experiences. By this means, the Neo-Kantians tried to preserve the main idea behind Kant's theory of space (as a "pure intuition", a priori synthetic). Alternatively, a devoted Kantian could claim that all local perceptions of space do indeed uphold Euclidean 3-D geometry, but once we try to form a global (or non-local) theory of space that incorporates the local perceptions of many observers (such as the perceptions of space from each person on the three mountain tops of Gauss’ experiment), the geometry constructed from each separate, local Euclidean perception os space need not itself be Euclidean: in short, space is Euclidean locally for each observer (as Kant claims), but a theory of spatial geometry that transcends the local view of any one observer need not be Euclidean.


Chapter 13: Poincare (Physicist, Mathematician, Philosopher: late nineteenth/early twentieth century)

Hyperbolic Non-Euclidean, or Bolyai-Lobachevsky (BL) Geometry: This geometry is similar to non-Euclidean spherical, but instead of there being "no lines parallel to a given line through a point", BL geometry holds that "through a point outside a given line there are an infinity of lines that can be drawn which do not intersect the given line." Also, in BL geometry, the sum of the angles of any triangle equal less than 180 degrees.

In addition, in our three geometries, there are differences in the relationship of the circumference, c, of a circle to its diameter, d, which will be of use in our discussion: in Euc: c=pd; in BL: c>pd; and in non-Euclidean spherical, or Riemannian (R) geometry: c<pd. (Here, p =3.1416..., of course.)

Poincare wanted to show that we can imagine a world where the inhabitants would conduct measurements/observations that would convince them that they live in non-Euclidean BL space, although from our perspective it is Euclidean. Suppose there exists a force that shrinks all material objects as they move to the edge of a finite, flat disk: i.e., the space for the inhabitants of this world is just the flat, finite disk. Since the people/objects continuously shrink as they move to the circumference (outer-most edge) of the disk, they can never reach the edge (since they will continue to shrink indefinitely as they get closer to the edge, and thus can never reach it). So, from their perspective, the world is infinite (but we see it as finite, since we are examining their world from outside of their space). What other properties of the geometry of their world do they measure? Well, if they measure the circumference, c, and diameter, d, they would judges that c>pd, thus proving that their world is non-Euclidean BL: Poinc. says that objects shrink as they leave the center of the circle by the formula (R2–r2/R2; where "R" is the Euclidean radius of the entire disk, and "r" is the distance from the center point of the world to the object). So an object L meters long at the center is L(R2–r2/R2) at the distance r form the center. Since all objects shrink as they approach the circumference of the disk, all measuring apparatus (and people) will also shrink in the same manner. Let "d" be the diameter of a circle on the disk measured by our people (located in the space), let "c" be its circumference, and suppose the measuring apparatus has shrunk to 1/2 its size as originally possessed at the center point (P). Now, the circumference measured by our people on the disk is 2 times the total that we would observe/measure (i.e., since we do not live on the disk, our measuring sticks have not shrunk, thus we judge their meter sticks to be 1/2 smaller than ours). So, whereas we would measure the circumference of the circle, "C", to be 100m (meters), or C=100m, they would measure 200m (c=200m), since their meter sticks are 1/2 the size of ours. So, c=2C. Now, if their meter sticks remained at 1/2 the size of ours as they measured the diameter, d, of their disk world, then D=2d, as well (where "D" is the diameter as measures by us). Yet, since their meter sticks grow in size as they move back towards the center of the circle (P), the overall measurement of their diameter, d, will be smaller than 2D (because the increased size of their meter sticks will mean that they will not have to lay down 2 meter sticks for every 1 meter sticks that we lay down). So, d<2D. Plugging this into the equation: "circumference/diameter=p ", we see that c/d>2C/2D=C/D=p . That is, c=2C, but since d<2D, d will divide into c more times than 2D divides into 2C (or D divides into C). So, the ratio c/d is greater than the ratio 2C/2D (which is equal to c/d, and is equal to p). Thus, c/d>p , or c>pd, which is the reason why the inhabitants of the disk world judge that their world is Non-Euclidean BL.

Conventionalism: Poincare argues that any geometry can be made consistent with the observations/measurements of physical space if certain extra assumptions are made. Thus, in our example above, some people can come to the same conclusion that we did (i.e., the people not living on the disk, who believe the geometry of the disk is Euclidean). In other words, some of the "disk-landers" (who live on the disk) may believe that their disk is a flat, infinite, Euclidean plane with a "universal force" that shrinks all objects as they move outward from the center towards the circumference (according to the formula L(R2–r2/R2) provided above). Or, the disk-landers could interpret the measurements conducted in their world to reveal the their space is non-Euclidean BL. So, which Geometry, or interpretation of the measurements, is correct? Answer: neither, or both. For Poincare, geometry is "conventional": it is a conventional stipulation in that we have a freedom to postulate any geometry consistent with the following rule:

Geometry of Space + Behavior of Measuring Apparatus = Survey Measurements

We have examined two cases:

(1) Non-Euclidean BL + apparatus remain the same = "c>pd"

(2) Euclidean + apparatus shrink as they approach the circumference = "c>pd" (but it is assumed that space is such that "c=pd" in the absence of the shrinking force)

Both of these geometries are consistent with the overall observed facts (of the measurements using the meter sticks, etc.), so neither can be pick out as the real geometry of space, although some geometries may be more convenient (such as non-Euclidean BL, since one does not need to invoke a "mysterious" shrinking force). In fact, "Ockham's Razor" would easily favor case (1), because there is no need to introduce an additional assumption, i.e., the shrinking force, in addition to the geometry of the space ("Ockham's Razor" is the thesis that states "always choose the theory that explains the phenomena while using the fewest assumptions or theoretical entities.")

Does this mean that the metric of space (i.e., distance-function) is indeterminate and totally relative to any choice? No, because there is a notion of "objectivity" at work, here. (The following discussion is largely dependent upon the discussion in Kosso's book, chapter 5 and 8.) Regardless of which geometric hypothesis, or the behavior of measuring devices, one eventually decides to adopt, there are objective constraints that the real world places on us. In particular, there are two particular scenarios associated with the example provided above:

(i) if the geometry is Euclidean, then you must invoke a universal force that shrinks the measuring apparatus.

(ii) if the measuring apparatus remain constant, then you will measure a geometry which is non-Euclidean BL.

So, objectivity is saved because the world constrains or determines the conclusions you can reach once you make an assumption (or a conceptual reference frame is introduced). If the world, and geometry, were entirely subjective/relative, then any geometry, and all assumptions about measuring apparatus, should all equally work. But it is not the case that any, and all, assumptions work with respect to space-time metrics (as described in (i) and (ii) above). For example, our 2 outcomes, (i) and (ii), cannot be mixed together to get, say: "if Euclidean geometry is assumed, then you cannot invoke forces which distort the meaning apparatus." This option/scenario does not work because it is falsified by observational experience (since, assuming no universal forces, it follows that c=pd; but observations showed that c>pd, thus demonstrating that this third option is contradictory/falsified). So, relativism/subjectivism is not upheld by conventionalism.

Put differently, in case (1), as you move away from the center position of the space, P, the meter sticks shrink in size according to the function L(R2–r2/R2); whereas in case (2), the meter sticks remain the same size anywhere in space, but the metric grows larger the farther you move away from P. (A metric is the function for measuring the distances, "ds", between coordinate positions, "dx", in a particular coordinate system/reference frame: i.e., "ds" measures the distance between coordinate positions "x" and "x+dx".) So, contrary to case (1), where a coordinate difference between two positions is provided by the standard Euclidean function, which is the square root of dx2+dy2; in case (2), the metric must take a different form to account for the swelling of space-time the further you travel from P. (By the way, the choice of P is arbitrary in both cases (1) and (2), so no matter what position you chooses as P, either the meter stick shrinks or space-time increases in metric distance the further you move away from that arbitrarily-chosen P. This would seem to be a feature of both cases (1) and (2), but especially case (2), that favor a relationalist interpretation of space and time. Why?: Because, as in case (1), objects shrink in size relative to a particular observer or coordinate position. Yet, if you allow the point P to be an unoccupied spatial position—that is, not occupied by a material body—then relationalism would not be upheld by these cases, of course.) Accordingly, in case (2), as you move away from P, the same coordinate difference, "x+dx", conducted from a coordinate position situated far away from P, say, x1, will result in an increase in the distance between these two coordinate points "x1+dx" that is much larger than the same coordinate difference "x+dx" conducted at the point P. (Why?: Because, in case (2), we assume that the meter stick remains the same length, and this assumption thus necessitates that it must be the space itself which increases in size (i.e., the distance between coordinate positions) to account for our measurement "c>pd", which is non-Euclidean BL). The increase in the distance between the coordinate points as you move farther away from P can be captured (I think) by the following new metric function: 1/(R2–r2/R2) times (the square root of dx2+dy2). In short, this new distance function makes the space (i.e., distance between coordinate points) increase in size in an exponential and opposite fashion as compared with the analogous shrinking force that shrinks the meter sticks in case (1). Therefore, notice that in both cases (1) and (2), the function R2–r2/R2 plays a role: more carefully, that function is in both L(R2–r2/R2), from case (1), and "1/(R2–r2/R2) times (the square root of dx2+dy2)", from case (2). Therefore, as discussed above, the function R2–r2/R2 is an "invariant" of both cases (1) and (2), and so it has a definite claim to count as an objective feature of all the geometric/physical theories of our disk world. (As used in this context, an "invariant" is a feature, or object, of a space-time theory that remains the same in all the different formulations of that space-time theory; and, in this respect, I am assuming that both cases (1) and (2) can be understood as two different formulations of the same general theory of space and time.)

Also, returning to options (i) and (ii), maybe it is the case that options (i) and (ii) are just 2 different descriptions of the same theory but with different definitions of what it means to be 1 meter long: (i) says that "1m" equals an increasing number of standard meter sticks as you move away from the center; while (ii) says that a standard meter stick always equals 1m no matter where it is in space. So, (i) and (ii) are one and the same geometry (i.e., slightly different alternatives or versions of the same geometry/theory), or, at least, (i) and (ii) are one and the same state-of-affairs/situation.

How does this discussion relate to the issues of realism and anti-realism (where a "realist" believes that theories describe/portray real, independent facts about the world; whereas an anti-realist rejects this view)? Given the previous debate surrounding the issue of objectivity and invariants, one might ask if there are any prospects or strategies for developing a viable realist theory of space-time that does not fall afoul of the many anti-realist objections to realism (see my notes to the philosophy of science course, PHIL 240, for a thorough discussion of these anti-realist objection to realism). While all philosophical theories are subject to criticism (which is the first conclusion all philosophy students reach, I might add), a recent trend towards a more sophisticated form of realism, sometimes called "structural" realism, appears promising (at least to me). In Kosso's book, Appearance and Reality, an analogous view of realism is dubbed "realistic realism"; so I will briefly discuss Kosso's theory. First, as has been noted often, it is certainly the case that falsified theories can make successful predictions, and thus generally continue to function as useful theories of science. Why? And, more importantly, How? Well, one way to approach this problem is to recognize that even a false scientific theory can have "truthful parts". That is, false theories can continue to provide useful information because various aspects of the theory accurately resemble some aspect of reality. This insight can be used, accordingly, to argue for a limited form of realism about scientific theories. To quote Kosso, "We cannot know everything [through our scientific theories], but we can know a lot." (p.182) More specifically, our best theories give us a partial picture of some ultimate reality out there in the physical world, and this partial picture provides real information of that world; but, of course, it is not the complete picture, and thus there are many facets of reality missed by our best theories.

In addition, there exist many different ways of gaining information on the world, and the content of the information we acquire is partially, but not entirely, dependent on the choice of framework (i.e., concepts and experimental apparatus). For instance (using Kosso's example), we can decide to measure the temperature of a body using either a thermometer scaled in Fahrenheit or Celsius, so that a reading of, say, 32 degrees F for one thermometer is matched by a reading of 0 degrees C on the other thermometer. Does this prove that the temperature of the body is purely relative to the choice of measuring equipment (which is merely a conventional choice, by the way)? Put differently, is there no objective aspect of reality that corresponds to our measurement (and which caused the readings on our respective thermometers)? Of course not! While the choice of measurement regarding the temperature scale (either Fahrenheit or Celsius) was a mere convention, once we choose a measuring convention, objective reality determines the outcome of the experiment! If we choose Fahrenheit, then water will freeze at 32 degrees (and not at, say, 50 or 70 degrees F). If we choose Celsius, then water will freeze at 0 degrees (and not at 10 or 32 degrees C). In short, the choice of measuring convention, or concepts, or framework, is up to us; but once we choose a particular framework, etc., objective reality determines the outcome. To quote Kosso, "Once the initial choice of language [i.e., theory, concepts, framework, measuring convention, etc.] is made, then nature dictates what is true and what is false. We choose how to say it, but we cannot choose what to say" (p. 108).

This sort of realization is at work in the discussion of the invariant R2–r2/R2, and the discussion of options (i) and (ii), above. The analysis of Einstein's General Theory of Relativity seemed to allow for different interpretations of the geometry of physical space: either Euclidean or non-Euclidean (where, roughly, a Euclidean space is flat while a non-Euclidean space is curved). As Poincare had earlier pointed out, a person on the surface of a curved space could accept one of two conclusions concerning the geometry of that space, our options (i) and (ii), of course: (ii) the space is actually curved, since certain measurement of length, with "meter sticks", reveal that various properties of the space are not Euclidean (e.g., the circumference of a (very large) circle in the space would not equal pi times the diameter of the circle, as it does in Euclidean space); (i) the space is flat, since the disconfirming measurements with the meter stick (which seem to prove that the space is not Euclidean) can be "explained away" by merely claiming that some "strange force" exists in the space which distorts the meter sticks used in the measurement. That is, if the strange force did not distort the meter stick, then the measurements of the space would reveal that it was actually flat (so that, in the absence of the force, the circumference of a large circle would equal pi times the diameter of the circle, as mandated by Euclidean space). Poincare claimed that both options (i) and (ii) are equally compatible with the person's experience of that space, so the observable evidence cannot determine which theory is preferable. (In fact, maybe the person really does live on a flat surface with strange forces, so that the "curved space" interpretation is actually false.) What does this example prove?: Is the geometry of space merely a convention, so that we could accept either a flat or curved theory space? Of course, there may be good reasons to accept that the curved space is the "real" geometry of the space. (For instance, the curved space theory is simpler, for it does not need to postulate "strange" forces, whose presence in the flat space theory may also conflict with our other beliefs about forces, moreover—that is, the Scientific Method may favor one interpretation over the other). Regardless of this debate, one thing remains true: not all aspects of the choice between (i) and (ii) are conventional. If one decides to retain a flat space, then one must postulate strange forces to distort the meter sticks. On the other hand, if one decides to directly accept the measurements conducted with the meter stick, then one must conclude that the space is curved (and not flat). Consequently, as in the case with our alternative thermometer measurements, once you conventionally choose a framework or a set of beliefs about the world (respectively, that the space is flat or that the meter sticks provide accurate measurements), nature determines what you can further say, or what conclusions you reach, about the world. In short, the constraints that nature places on our conclusions about the world, once a framework or language is chosen, is what "objectivity" means, and this constitutes a form of "realism"! However, it should be noted that the realism that is being upheld in this argument is not of the "robust" variety that has a clear and definitive answer about the nature of theory's existing entities (which is the "ontology" of a theory: i.e., "what are the existing things in the theory"): whether space is "really" flat (Euclidean) or "really" curved (non-Euclidean) can not be answered on this more minimal interpretation of realism (perspectival realism), since the answer given depends on a prior acceptance of framework or perspective, and this admission seems to undercut any strong type of realism about the geometry of space. Hence many traditional realists may not be satisfied with the story provided above.


Chapter 14: Einstein (Twentieth century scientist/philosopher)

Einstein argues that the concept of space arose when it was realized that 2 different solid objects could both fit in the same place (or "interval") between 2 other bodies. So, space was conceived as "something" separate form any particular object, and this lead to a concept of space as the collection of all such "places". (Poincare made the same argument in Chap.13.)

The Michelson-Morley experiments (1880s-1890s) showed that one could not detect the "ether" responsible for the propagation of light. This ether was believed to be at rest in absolute space (or directly play the role of absolute space). Consequently, absolute place (and absolute rest) could not be observationally/experimentally verified.

Two developments led to the Special and General Theories of Relativity (STR/GTR): (i) Mich./Mor. "null" results in finding the ether (as above); (ii) the development of non-Euclidean geometries, especially Reimann’s work on a general metric "ds" (Gij metric tenor) which can describe all kinds of metrics (and geometries, since geometries are determined by the distance function, or metric).

Einstein thought he was fulfilling Poincare's conventionalism by adopting "general covariance", which holds that "the laws of nature [should be] so constituted that they are not materially simplified through the choice of any one particular set of coordinates" (p. 259).

For Einstein (and Poincare), since each measuring apparatus, and hence system of coordinates, is only conventional (and could easily have been a different measuring apparatus or system of coordinates), it follows that the laws of nature should not be dependent upon them. (That is, the laws of physics should not be dependent on how we label space-time points, or whether we believe light rays move straight or in curved lines, or whether we believe our reference frame is, or is not, at absolute rest, etc.) So, STR/GTR was developed to show how many different coordinate systems can equally well describe the laws of nature, which is the notion of general covariance.

In GTR, the metric, and hence geometry, of space-time depends on how matter is distributed in space. But, since space-time determines which motions are inertial, there is a symmetrical action and reaction between matter and space. "Matter tells space how to curve (because matter curves space-time), but space tells matter how to move (i.e., because the inertial paths are a feature of the geometry/metric of space-time)."

Yet, GTR does not uphold Machian relationalist theories of space-time, for the following two reasons: (1) In GTR, one must arbitrarily choose four coordinates of the metric tensor which determine the so-called "background geometry" of the space: e.g., a flat background space, or a curved background space, can be chosen for the geometry of space prior to the discussion, or introduction, of the effects of matter on that space-time. Consequently, there can be two worlds with identical distributions of matter but with different background geometries. For example, there can be a world where inertial paths move forward in space (infinitely in any direction) without ever meeting (because the background space is flat), and there can be a world where the inertial paths of bodies come back to their starting points (since the background space is spherical or cylindrical); and both of these universes can have the same distribution of matter. Consequently, contrary to Mach, matter does not determine all the properties of space, since it is the geometry alone which accounts for some differences (such as the difference in long-range inertial paths, as described above). (It should also be noted, here, that GTR allows for solutions of its field equations which contain no matter, but where the geometry/space still has a determinate structure.) (2) GTR also allows one to talk meaningfully about the state of motion (i.e., acceleration) of a single object in a universe that contains only that single object. That is, a situation where the object rotates, or does not rotate, can be determined and measured by the equations/structure of GTR. This violates relationalism since, as so often discussed, since there are no other objects, or reference frames, to determine/measure its motion. Consequently, modern GTR is not very favorable to relationalism (but maybe a theory of GTR favorable to relationism can be developed in the future?). Also, a weaker form of relationism, as discussed above in earlier chapters, need not accept Machianism, and thus can escape the problems as raised above (as also noted in earlier chapters).

  Notes Completed up to this point: 3/24/04