BJUP-41-Spring-2011.46-52.pdf

8/14/2019 BJUP-41-Spring-2011.46-52.pdf

1/7

British Journal of Undergraduate Philosophy - Issue 4(1) Spring 2011

The Substantivalists Razor: Cutting Geometry from Space

Douglas J. M. Battersby

University of Leeds

Contemporary metaphysics seems to have reached an impasse between rela-tionism and substantivalism about space. The basic conceptions these termsrepresent are so fundamentally opposed, and their defences so complex, thatit seems unlikely that proponents of either view will seek a position of com-promise, let alone be receptive to defection. Relationism claims for itself aless cumbersome ontology than substantivalism, using the principle of Oc-

cams razor to cut space from existence. For that reason, this investigationis structured as an interrogation of a staunch relationist position, and pur-sues one particular argument (concerning geometric relations) to its logicalconclusions. By the papers close, I hope to have demonstrated that the re-lationists position necessarily entails logical fallacies, which in turn make itsontology the more cumbersome of the two. I will open with a brief (andrather generalised) synopsis of the two opposing philosophies, before detail-ing the argument I am advancing. I will endeavour to be as concise as possible

in explicating that which is likely to be familiar to the reader, and correspond-ingly expansive in examining the more esoteric areas of the discussion whichI feel have been significantly overlooked.

Substantivalism tells us that the existence of space is non-contingent uponthe existence of objects interior to it. For the substantivalist, objects havean absolute location, and the spatial relation between two objects can be ex-pressed in such terms. The relationist, however, believes that space is simplythe spatial relations that hold between objects - the existence of space is solely

entailed by the existence of those objects. According to the relationist, spaceexists only in terms of relations, and has no ontological status. The distancebetween two objects is the irreducible fact of the matter, and no explanatorymediation to a third entity (space) is necessary. These simple opposing propo-sitions produce some surprisingly complex consequences. One difficulty forrelationism is that the very terms of language presuppose the substantivalistsposition: we think and talk about points of space, moving through it, increas-ing it, as if it were a substance. Indeed since Aristotles abhorred vacuum,orthodox metaphysics has largely served to reinforce such presuppositions, ashave many prominent figures in the field of psychics, most notably Newtonand Einstein. This should not, of course, unduly prejudice us towards the

39

8/14/2019 BJUP-41-Spring-2011.46-52.pdf

2/7


substantivalist, but it is indicative of the kind of difficulties the relationistneeds to work through, to convince us that these ways of thinking (and talk-ing) really are just features of thought (and language), and hold no bearing onontological commitment.

Let us take a simple sentence by way of example: put that object (o) in thatspace (x). For the substantivalist, xrepresents a genuine point in absolutespace, regardless of whether it is occupied by any object, or even if thereare no objects in existence at all. The substantivalist says to the relationistthat the very terms of such simple direction implicitly admit the existenceofx, and therefore space-as-object. The relationist responds that xactuallyrepresents a set of possible spatial relations which an object couldbear to

other existent objects. This set of relations is defined, and each unoccupiedlocation (point) is differentiated from other points, by reference to geometriclaws. The premise implicitly relies on geometric mathematical truths beingnecessarytruths - they would be true regardless of the way the world obtains.The relationist can therefore talk about spatial points without admitting oftheir existence, and without being contingent upon the way the world ac-tually obtains - put that object (o) in those geometric relations with otherobjects (x).

However, the emergence of papers which successfully advance the logicalconsistency of non-Euclidian geometries makes the premise that geometricmathematical truths are necessary doubtful: if space is curved, Euclids lawswill fail to identify midpoints, angular relations, parallels, etc., in the defini-tional function that the relationist requires in order to talk about points. In-deed many contemporary cosmological theories hold that certain observablephenomenon indicate that space is curved. Now this is not immediately dis-astrous for the relationist, who can claim that the peculiarities non-Euclidian

geometry seems to introduce are somehow intrinsic to the relation betweenthe two objects.1 Poincars doubling universe experiment, which has tradi-tionally been used as areductio ad absurdumproof by relationists, can in factbe used by the substantivalist in conjunction with notions of non-Euclidianspace to promote his position.

Poincar asks us to consider the following: if everything in the universe (in-cluding distances) doubled in size over night, would we notice a change?Relationists have typically used the experiment to argue that no discernible

1This emerged from a conversation with Professor Le Poidevin.

40

8/14/2019 BJUP-41-Spring-2011.46-52.pdf

3/7


change would occur, which demonstrates the non-existence of space. Nerlichinsightfully critiques this argument, saying that it implicitly relies on whathe calls the inconsequentiality principle: IP: inconsequential propertiesare not properties.2 A universal doubling in a non-Euclidian space would

result in consequential, perceptibly evident changes, most obviously in theshape of objects. Not even an appeal to something intrinsic in the nature ofa relation between two objects can defend the relationist from this objection,as the change would be consequential, and therefore indicate mediation withabsolute space.

Robin Le Poidevin raises further issues of geometry which the relationistfaces in terms of overdetermined truthmakers. Le Poidevins argument de-pends on what he calls the explanatory principle:

a natural extension of the truthmaker principle [is] the explanatoryprinciple: Where p entails q, there is a corresponding connectionbetween the truthmakers ofpand qthat explains the entailment.3

He considers the following geometric situation:

1. A and C are two objects on a flat surface which are at a distance of 10"from each other.

2. The relationship of the proposition AC = 10 is represented by R1, andexists independently of any other object.

3. On the straight line AC there is an object, B, which is 4" from A and 6"from C.

4. There are two further relationships: R2(AB) and R3(BC).

5. AB = 4" and BC = 6" entail AC = 10".

6. According to the explanatory principle, there is a corresponding connec-tion between the truthmakers of AB = 4" and BC = 6", and its entail-

ment, AC = 10", that explains the relationship that is entailed, R4.7. R4expresses the proposition AC = 10. This relationship is wholly su-

pervenient upon R2and R3.4

The problem for the relationist is that the simple proposition AC = 10 is ap-parently entailed by two independent truthmakers, R1and R4. The relation-ist obviously wants to admit only of the existence of R1, but the explanatory

2

Nerlich [11] p. 171.3Le Poidevin [7] p. 191.4Ibid. [7] p. 193.

41

8/14/2019 BJUP-41-Spring-2011.46-52.pdf

4/7

8/14/2019 BJUP-41-Spring-2011.46-52.pdf

5/7


ceases to exist, and is replaced by a new relationship, R

1.9 The existential

dependence of R1and R4dissolves, and the relationist is in a position to denythere is supervenience, and in turn that the truthmakers of AC are overde-termined. She uses the Leibnizian analogy of a childless couple to illustrateher conception of persistence.10 The weakness of the analogy reveals theconceptual weakness which underlies Conroys position. I do not think thatchildless couple effectively reflects spatial relations as the relationist wouldconceive them - it is a relation whose existence is dependent on the non-existence of a third object, and thus it is implicitly supervenient. RegardingLe Poidevins assumption that the relationist would accept the only x and yprinciple, I fail to see why Conroy believes the relationist would reject sucha principle.

In fact, to deny the principle would violate the relationists essential notionthat spatial relations exist without mediation. Conroys argument, in its ea-gerness to utilise theories of persistence, seems to have lost relevance to thediscussion at hand.

Gibb criticises Le Poidevins explanatory principle as ontologically extrava-gant11, and suggests that a more modest modal strict implication conceptionof entailment (as formulated by C. I. Lewis) is sufficient:

P Q iff (P Q)

She argues primarily that Le Poidevin misconstrues a mathematical entail-ment for an ontological one. Using the above construction, she asserts thatif AB + BC, then AC represents a mathematically necessary truth. Gibbsays that the demand of Le Poidevins explanatory principle to account fortruth-making principles is simply unnecessary, and it is on this point that Ifind fault with her argument.

As I will demonstrate, Gibb misinterprets the function of formal languages,and by doing so confuses ontological entailment for its modal equivalent.Barcan Marcus offers a valuable assessment of the function we should expectfrom modal languages:

An ideally adequate interpretation of a formal language would be[...] that truth as well as meanings are preserved [...] informally

9

Mortensen [9].10Conroy [2] p. 370.11Gibb [3] p. 175.

43

8/14/2019 BJUP-41-Spring-2011.46-52.pdf

6/7


valid arguments would go into formally valid arguments [and] mean-ing would be preserved in translating out of the artificial languagebackinto those segments of discourse for which it serves as a for-malized equivalent.12

It is on the latter point that I believe Gibb does not appreciate the essenceof Le Poidevins argument. The proposition p represents the following geo-metric truths AB = 4" and BC = 6" (and ABC is a straight line). Necessar-ily, for the truth-functional connective to hold, the fundamental truthscontained in p are sufficient to make q (AC = 10) true also. The propo-sition q and the truths it represents are true by virtue of the complex set oftruths (andmeanings) which the proposition prepresents. I am unsure how

Gibb would satisfactorily de-modalize the entailment back in to ordinary lan-guage, withoutinformalallusion to the constituents of p. It seems that shepermits the proposition p to represent some state of affairs, but restrictsthe implication of the entailment q to a merely formal function, a sort ofvacant signifier. Gibbs formal constructions are insufficient to disprove thenecessity of truth-making principles.

In his response to Gibb, Le Poidevin admits the shortcomings in his formula-

tion of the explanatory principle, and replaces it with a more refined version:

The correspondence principle: Where p1. . . pn, entails q, and p1. . . pn,are true, the truthmakers of p1. . . pn constitute a truthmaker forq.13

The reformulation strengthens the truthmaker principle by cementing thetie between geometrical inferences and the modalization which Gibb found

fault with. I would also add that I consider her objections to be aimed atissues in the philosophy of logic, not metaphysics.

I hope to have illustrated that the apparently simple geometric premises whichrelationism depends upon necessarily produce regressively more convolutedstances. The relationists boast of a simpler ontology is outweighed by theunwieldy excesses which his philosophy demands of him.

12Barcan Marcus [1] p. 240.13Le Poidevin [8] p. 186.

44

8/14/2019 BJUP-41-Spring-2011.46-52.pdf

7/7


References

[1] Barcan Marcus, R. (1972) Quantification and Ontology, in Nos 6(3):

240-250.[2] Conroy, C. (2008) No Lacuna and No Vicious Regress: A Reply to Le

Poidevin,Acta Analytica23(4): 367-72.

[3] Gibb, S. C. (2006) Space, Supervenience and Entailment, inPhilosophi-cal Papers35(2) 171-184.

[4] Huggett, N. & Hoefer, C. Absolute and Relational Theories of Spaceand Motion, inThe Stanford Encyclopedia of Philosophy (Fall 2009 Edi-tion), [http://plato.stanford.edu/archives/fall2009/entries/spacetime-theories].

[5] Kingsley, A. C. (2004) The Only X And Y Principle, inInquiry47(4):338-359.

[6] Le Poidevin, R. (2003)Travels in Four Dimensions. United States: OxfordUniversity Press.

[7] Le Poidevin, R. (2004) Space, Supervenience and Substantivalism, inAnalysis64(3): 191-198.

[8] Le Poidevin, R. (2006) Truth-makers and Geometrical Inference: Replyto Gibb, inPhilosophical Papers35(2): 185-192.

[9] Mortensen, C. Change, in The Stanford En-cyclopedia of Philosophy (Fall 2008 Edition),[http://plato.stanford.edu/archives/fall2008/entries/change].

[10] Nerlich, G. (1979) What Can Geometry Explain?, inThe British Journalfor the Philosophy of Science30(1): 69-83.

[11] Nerlich, G. (1991) How Euclidean Geometry Has Misled Metaphysics,inThe Journal of Philosophy88(4): 169-189.

[12] Spade, p. V. William of Ockham, in The Stan-ford Encyclopedia of Philosophy (Winter 2009 Edition),[http://plato.stanford.edu/archives/win2009/entries/ockham].

45

BJUP-41-Spring-2011.46-52.pdf

Documents

Transcript of BJUP-41-Spring-2011.46-52.pdf