There are a number of issues raised in LCC that we will skip over:

1. Newton’s remarks about space being "as it were" the sensorium of God. See Query 28 of Newton’s Opticks: "…does it not appear from phenomena that there is a Being, incorporeal, living, intelligent, omnipresent, who in infinite space, as it were his sensory [sensorium], sees the things themselves intimately and thoroughly perceives them, and comprehends them wholly by their immediate presence to himself….?" Leibniz thinks this would make the universe independent of God if God needed an organ in order to perceive things.

2. Newton's God being required to intervene in order to prevent the fixed stars from coming together and to counteract the planetary irregularities that would result from gravitational attraction, and to replace the motion lost in hard body collisions between atoms. Leibniz criticizes this as imputing poor workmanship to God. See LCC 1 §4.

3. According to Leibniz, Newton's gravitational action at a distance is a perpetual miracle because it does not follow from the essence of matter. These passages are omitted from the Huggett book.

4. Leibniz 's arguments for the impossibility of a vacuum. (E.g., LCC 4 §46). Omitted from Huggett.

Why does Newton's mechanics requires absolute space and time? See the Scholium (on space and time) appended to Definition VIII in Book I of the Principia.

Newton's bucket and tied globe thought experiments.

Newton's Bucket and Tied Globe Argument for Absolute Space
 

                                                Shape of W            Rel. motion        Newton: Abs.        
                                                 surface                   of B & W           motion of W        
 
 

Initial state of rest                     flat                         none                   none

B begins rotating                     flat                             yes                     none

B continues rotating                 curved                     none                     yes

B suddenly stopped                 curved                     yes                         yes

Final state of rest                     flat                         none                          none

Probably the best historical discussion of what Newton was up to in the Scholium can be found in Ronald Laymon, "Newton's Bucket Experiment," Journal of the History of Philosophy 16 (1978) 399-413.

Laymon argues that:

• Newton's bucket experiment is different from the Nagel-Reichenbach embellishment of it; Newton does not stop the bucket.

• Newton's primary target was to refute Descartes's definition of motion as motion relative to contiguous bodies; especially important is Newton’s reliance on the principle, accepted by the Cartesians, that effects are proportional to their causes. Note that, as the velocity of the water relative to the sides of the bucket decreases, the curvature of the water surface increases.

• Newton's aim was not to demonstrate the existence of absolute space but to illustrate how, within his theory (which assumes the existence of absolute space) relative motion can be distinguished from absolute motion. Note that, in the tied globes thought experiment, the direction of rotation can also be determined.

Following tradition, we have interpreted Newton’s thought experiments as arguing for the existence of absolute space. Here is a crude version of that argument.

The curvature of the spinning water is a real, observable effect produced by a real (inertial) force acting on the water.as it rotates.

Rotational motion, like all other motion, is a relational concept: motion (whether velocity or acceleration) has to be with respect to something.

There is no other material thing (such as the sides of the bucket) with respect to which the water is rotating that is capable of explaining the changes in the curvature of the water.

Therefore, the bucket must be rotating with respect to some non-material thing, namely, absolute space.

Therefore, absolute space exists.

Leibniz's Relational Theory of Space and Time: Space (and time) is something ideal (not real).It is something merely relative (as time is). (LCC 3 §4, AG 324). Space is the order of "coexistences" (coexisting bodies; bodies existing simultaneously); time is an order of successions (nonsimultaneous events). (AG 324)

"...since space without matter is something imaginary, motion, in all mathematical rigor, is nothing but a change in the positions of bodies with respect to one another, and so, motion is not something absolute, but consists in a relation." "On Copernicanism and the Relativity of Motion" (1689) AG 91.

Leibniz’s most famous arguments against absolute space and time rest on two general metaphysical principles, the principle of sufficient reason (PSR) and the principle of the identity of indiscernibles (PII).

Principle of Sufficient Reason (PSR)

"Nothing happens without a reason why it should be so rather than otherwise" (2nd letter to Clarke, 1). Probably the most famous of all Leibniz's principles. See Monadology 32: for every truth and "existent" (positive) fact there is a reason (or explanation) why it is thus and not otherwise. In his 2nd letter to Clarke, Leibniz asserts that the PSR can be used to prove the existence of God, the rest of metaphysics, and the principles of natural philosophy (science) that go beyond mere geometry (extension and kinematics) and arithmetic by including the notion of force (dynamics). See Monadology 37-39 for Leibniz's cosmological argument for God's existence from the PSR. Put simply, the PSR says that there is an explanation (whether we know it or not) of (1) the existence of any object that exists, (2) why each existing object has each of the properties that it does, (3) every positive fact whatever (where positive facts are facts about existing objects and their properties).
 

The Metaphysical Argument Based on PSR Against Absolute Space

 
Although the PSR and the PII arguments are stated together in the LCC (with versions for time as well as space), they are distinct arguments based on different principles. As Huggett remarks, empiricist philosophers in the 20^th century have frowned on Leibniz’s rationalist PSR while regarding the PII more favorably. (H 164) For Leibniz’s statement of his argument, see LCC 3 §5-7, 4 §13-19.

Suppose that absolute space exists. Thus, the world as a whole has some position in it. But, in that case, when God created the world, there could have been no reason for placing the world in one position as opposed to another. But since the PSR is true, there must be a reason for everything. Therefore, absolute space does not exist.

Or, alternatively: why does the world, today, have the spatial position it does? Answer: because it had the spatial position it had yesterday and, let us assume, no forces arose to move it. But, Leibniz would insist, however far we carry back this style of explanation, something will remain unexplained, thus violating the PSR. Therefore, absolute space does not exist.

How does Clarke respond to this argument? Does he reject the PSR?

Leibniz second group of arguments are based on another general principle that Leibniz regards as a truth of reason, known a priori: the PII.
 

Principle of the Identity of Indiscernibles (PII): If two things have exactly the same properties, then they are identical (i.e., they are numerically identical, they are one and the same thing). Or, equivalently, if two things are diverse (i.e., not numerically identical), then they must differ in at least one of their properties. It is important not to confuse this Leibnizian principle with the principle of the indiscernibility of identicals, viz., if two things are identical, then they must have exactly the same properties (numerical identity entails qualitative identity). The principle of the indiscernibility of identicals is widely accepted as a necessary truth. Leibniz's principle of the identity of indiscernibles is controversial. Many people think it is false. Among other things, Leibniz used the PII to argue that atomism must be false (because there cannot be two numerically distinct atoms that have all the same properties—5th letter to Clarke, 21-23). Leibniz sees a close connection between the PII and the principle of sufficient reason (PSR).

The Metaphysical Argument Based on PII Against Absolute Space
 

Huggett distiinguishes two versions of this argument, the static shift argument and the kinematic shift argument. Here is a version of the static shift argument.

Suppose that absolute space exists and that the absolute spatial positions of every body in the universe at a given time are P1, P2, P3,....Pn. Call this world, W1. If absolute space exists, then there is another possible world, W2, that is a different world from W1 (i.e., not identical with W1) in which each body is moved 10 feet to the east. Call this world, W2. But W1 and W2 are indiscernible (because they have all their properties in common). The PII is true. Therefore, W1 is identical with W2. But the existence of absolute space entails that W1 is not identical with W2. Therefore, absolute space does not exist.

  The PII appealed to here is not about individual objects but about possible worlds. But isn't it begging the question against Newton to insist that W1 and W2 have all their properties in common.

The Verificationist (Epistemic) Argument Against Absolute Space

 
Huggett, I think wrongly, includes this strand of reasoning as part of the PII argument (H 164). Though similar to the PII argument, it can be regarded as based on an entirely different general principle, the verificationist theory of meaning favored by hardline empiricists.

Leibniz LCC 5 §52 (AG 340; H 157): "motion indeed does not depend upon being observed, but it does depend upon being possible to be observed. There is no motion when there is no change that can be observed. And when there is no change that can be observed, there is no change at all." See also Berkeley, Principles 110-17 (W 66-70; omitted from Huggett),

Based on the verificationist theory of meaning. Claims about the world are meaningful (intelligible) only if they can, in principle, be verified. No possible observations, even in principle, could determine either position in absolute space or velocity wrt absolute space. Therefore, such assertions are meaningless. But if absolute space exists then such assertions are meaningful. Therefore, absolute space does not exist.

Comments: The verificationist theory of meaning is implausible. Besides, doesn't Newton's bucket argument satisfy the verificationist demand, at least with regard to absolute rotation and acceleration?

Another Leibnizian Argument Against Absolute Space

 
LCC 5 §36-46 (AG 336-37). This passage is omitted in Huggett’s book. It is a fascinating example of Leibniz showing that Clarke (and Newton) had failed to make sense of the concept of absolute space using the traditional categories of substance and attribute, categories that go all the way back to Aristotle.
 

Everything that exists is either a substance or an attribute (a property of a substance). Absolute space cannot be a substance (because otherwise there would be an infinite, eternal, unchangeable, thing that exists independently of God). If absolute space is an attribute, then either it is a property of something material or of something immaterial. It cannot be a property of anything material since material bodies occupy space and one material body cannot occupy another. It cannot be a property of anything immaterial (Clarke had suggested God) for a variety of reasons. (1) Space has parts, God does not. (2) If space is a property of God, then space (the predicate) is in God (the subject). But Clarke and others also want to say that God is in space and a subject cannot be in its predicate.