In the course of his debate with Clarke, Leibniz presents an argument (a reduc-tio) directed against Newtonian substantivalism.66 In this case, as van Fraassen ([1990], p. 239) nicely expresses it, "Leibniz's God was Buridan's ass magnified" (see also Earman [1989], p. 118): given certain features about the nature of Newtonian space, it is impossible for God to make a reasoned decision regarding the placement of the world within it. Here, in its entirety, is the oft-quoted argument designed "to confute the fancy of those who take space to be a substance, or at least an absolute being":

[I]f space was an absolute being, there would something happen for which it would be impossible there should be a sufficient reason____Space is something absolutely uniform; and, without the things placed in it, one point of space does not absolutely differ in any respect from another point of

63 Cf. Ismael and van Fraassen [2003] and Belot [2003a].

64 See Marsden [1992] for a fairly elementary introduction to these ideas.

65 There are ways of making an extended space compatible with a deflationist interpretation of a theory in a certain restricted sense: we can impose conditions on measurement theory so that the symmetries in question are not distinguished by any measurement—i.e. observables are made insensitive to non-qualitative differences. However, I ignore this complication here (for details see Belot and Earman ([2001], p. 221) and §3.3 of this book).

66 As I have already intimated, in fact, it is Clarke—in the course of discussing the principle of sufficient reason—who initially suggests the possibility of different places for a system of matter. The reason he gives as to why a system of matter has the place it has is down to "the mere will of God" (C-II.1, [Alexander, 1956], p. 21).

space. Now from hence it follows, (supposing space to be something in itself, besides the order of bodies among themselves,) that 'tis impossible there should be a reason, why God, preserving the same situations of bodies among themselves, should have placed them in space after one certain particular manner, and not otherwise; why everything was not placed quite the contrary way, for instance by changing East into West. But if space is nothing else, but that order of relation; and is nothing at all without bodies, but the possibility of placing them; then those two states, the one such as it is now, the other supposed to be the quite contrary way, would not at all differ from one another. Their difference therefore is only to be found in the chimerical supposition of the reality of space in itself. But in truth the one would exactly be the same thing as the other, they being absolutely indiscernible; and consequently there is no room to enquire after a reason of the preference of the one to the other.67 (L-III.5, [Alexander, 1956], p. 26)

As Belot explains, "Newtonians multiply possibilities in a manner which violates the PSR" ([2001], p. 3):

In things absolutely indifferent, there is no [foundation for] choice; and consequently no election, nor will; since choice must be founded on some reason, or principle. (L-IV.1, [Alexander, 1956], p. 36)

For a relationalist like Leibniz, there is only one possibility corresponding to the entire set of possibilities brought about by the symmetry transformations of the shift argument. Hence, "PSR can be restored if we can assure ourselves that there are many absolutist possibilities for each genuine possibility. Where Clarke sees many possible arrangements of bits of matter, each placed differently in space but all satisfying the same spatial relations between bits, Leibniz sees only one" ([Belot, 2001], pp. 3-4):

To suppose two things indiscernible, is to suppose the same thing under two names. And therefore to suppose that the universe could have had at first another position of time and place, than that which it actually had; and yet that all the parts of the universe should have had the same situation among themselves, as that which they actually had; such a supposition, I say, is an impossible fiction. (L-IV.6, [Alexander, 1956], p. 37)

Let us attempt to couch this debate in the terms of the previous section. We see that the shift argument in the Leibniz-Clarke correspondence is constructed by using the symmetries of Newton's theory to generate new configurations for some specified system—in this case the material content of the universe! One starts out with a single instantaneous state representing a possible embedding of matter in space, and then acts on this state (configuration) with a symmetry to produce

67 Leibniz quickly follows up this argument against substantivalism about space with a similar one about time. The idea is that, on the assumption of absolute, substantival time, God could have created the universe sooner or later than He actually did. Hence, He has no sufficient reason to actualize it at one time as opposed to some other, since the points of time are exactly alike (i.e., indifferent). The details of the spatial and temporal arguments are so alike that we can take them as having the same form, and we may, therefore, restrict our discussion to the spatial scenario.

another equivalent state (configuration).68 The symmetries are such that one can produce infinitely many states in this way; specifically they are elements of the (six dimensional) Euclidean group £3 of transformations on R3: translations, rotations, and combinations (under the group composition law).69 The set of states related by the symmetries describe qualitatively identical situations: all qualitative monadic and relational properties are preserved.

Let us call the space containing all of the possible configurations (worlds) for some system, including those related by symmetries, I (for inflated). A crucial fact, for Leibniz, is that the states related by the symmetries form equivalence classes— this accounts for the qualitative identity of the represented states of affairs. Thus, for any state x e I, there is a set [x] = {y: y ~ x} = {y: y = g(x)} (3g e £3) of isomorphic states (where denotes isomorphism, and g is a Euclidean symmetry transformation). We can factor out the symmetries of the six dimensional Euclidean group, yielding a reduced (relative) configuration space, with states represented by equivalence classes of states from the extended space. Let us call this space D (for deflated). Clearly D = I/£3.

Putting historical accuracy to one side, the shift argument goes as follows. Suppose we have a very simple Newtonian world W (considered at a single instant of time) containing three point particles qi (i = 1,2,3), each with mass m, living in three dimensional Euclidean space, E3. If we fix the masses mi = 1,2,3 for the particles (so distinguishing them and avoiding any initial identity problems), the (extended) configuration space for this system has nine degrees of freedom corresponding to the positions the particles relative to space (three coordinates per particle). W can be represented by the structure W = (E3, qi, Ri} (where Ri represent the relative distances between the particles, and masses are ignored). A Euclidean symmetry g acts on the qi, and so on W, so as to generate a new configuration g(qi) with corresponding structure Wg = (E3,g(qi), Ri) (representing world Wg). But the fact that g is a symmetry means that the relative distances stay the same: Vjk R(qj, qk) = R(g(qj),g(qk)). Consider the case where g is a rigid translation, so that g(qi) moves all of the particles five meters to the East of qi. Now, suppose that x is the spatial point lying at the center of the qi-system; suppose also that y is the spatial point lying five meters to the East of x. Then the center of mass of the system of particles is at point x in W but at the point y in Wg, where x,y e E3.

Now, the substantivalist's position can be characterized by the claim that the points (parts, for Newton) of space have their existence and identities fixed independently of any material objects occupying them—where, in our example, the material objects are just the three particles and space is three dimensional and Euclidean. Therefore, since any (non-trivial) group action will result in these particles having different positions in space, the substantivalist is committed to each such transformed state representing a distinct possibility: W and Wg are distinct

68 The specific kind of equivalence is determined by the kind of symmetry: some symmetries may lead to physical differences (these are symmetries in the strict sense) and others will lead to no discernible alteration whatsoever (these are the gauge redundancies).

69 I only consider the kinematical shift argument here, using only the symmetries of space. But note that Leibniz considers a dynamical shift argument too, where the symmetries of Newtonian space and time (i.e. the action of the ten dimensional Galilean group G on E3 x R) are used. See Maudlin [1993] for an explanation of the difference.

possible worlds because the center of mass point is played by x in the first and y in the second. However, since each configuration that results from a group action is equivalent, the possibilities will be indistinguishable. Thus, there could be no reason for one to be actualized over any other ^-related one. Leibniz, of course, used this feature as a reductio of substantivalism on the grounds that it violates PSR. To rule out such indistinguishable possibilities, Leibniz then invoked the principle of identity of indiscernibles [PII: VF, Vab | (Fa = Fb) ^ (a = b)].70 The latter move simply corresponds to reduction: each symmetrical configuration corresponds to one and the same physical reality.71 The folk wisdom is that relationalism and reduction (i.e. deflation) go hand in hand, as do substantivalism and inflation. The next section will introduce a number of key concepts from the metaphysics of modality. These concepts are then utilized to show that both alignments are false.

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