What constitutes space? : The development of Leibniz's theory of constituting space

Transcription

1 What constitutes space? : The development of Leibniz's theory of constituting space Hiroyuki INAOKA (Kobe, Japan) X.Internationalen Leibniz Kongress: Leibniz Universität Hannover, July 2016

2 Outline 1:Introduction 2:The Leibniz-Clarke correspondence and Analysis Situs 3:The theory of space before the latest period 4:The three points argument in New System 5:The three kind of the point: which is an element of space?

3 1:Introduction

4 Leibniz s theory of space and time in the correspondence with Clarke. Relational view: Space and time as relationship among things Research question: How and why does Leibniz find his final view of space?

5 1:Introduction 2:The Leibniz-Clarke correspondence and Analysis Situs 3:The theory of space in the middle period 4:The development of the concept of space and substance 5:The three kind of the point: which is an element of space?

6 The Leibniz-Clarke correspondence and Analysis Situs The final theory of space = the relational view + analysis situs (De Risi [2007])

7 The letter to Des Bosses, 21 July 1707 That is, a simple substance, even though it does not have extension in itself, nonetheless has position, which is the foundation of extension, since extension is the simultaneous continuous repetition of position, just as we say that a line comes to be from the flux of a point, since in the trace left by a point its different positions are connected. (GP II, 339)

8 The 4 th letter to Clarke Clarke contends that space doesn t depend on the situation of bodies. I reply that it s true that space doesn t depend on this or that particular spatial lay-out of bodies, but it is the order that makes it possible for bodies to be situated, and by which they have a lay- out among themselves when they exist together, just as time is that order with respect to their successive position. (GP VII 376)

9 The 5 th letter to Clarke This is how men come to form for themselves the notion of space: They consider a number of things existing at once, and find in them a certain order of coexistence, according to which things relate to one another more or less simply. This order is their situation or distance [ ].

10 When one of those coexisting things x changes its relation to a number of others that don t change their relations among themselves, and another thing y comes to have the same relation to the others that x previously had, then we say that y has come into the place of x [ ] And the totality of all those places is called space. (GP VII 400)

11 X A C B

12 Y A C B

13 According to De Risi, this theory of constituting space could be embedded in his theory of substance. Each monad expresses all other monads. The situational relationship among monads is represented by each monad as a phenomenon. Each monad is individuated by the set of such representations. Thus, the network of the situation among monads is constituted and then it makes itself the framework of space.

14 Analysis Situs contributes to such constitution of space by monads. Leibniz argues that space could be defined as the set of all situations in a manuscript on Analysis Situs in the latest period. In fact, the definition of space as the largest place is found in a manuscript written in Such a definition of space as attempted in Analysis Situs is necessarily relates to the discussion between Leibniz and Clarke. Leibniz s theory of constituting space would be reformulated as a two-step theory:the definition of space in Analysis Situs and its realization made by the representation of monad.

15 1:Introduction 2:The Leibniz-Clarke correspondence and Analysis Situs 3:The theory of space before the latest period 4:The development of the concept of space and substance 5:The three kind of the point: which is an element of space?

16 Richard T. W. Arthur attempts to show that Leibniz, before the latest period, has already the almost same theory of space as in the latest period.(arthur[2013]) Arthur focuses texts written in about 1679 when Leibniz reintroduced the idea of substantial form in his metaphysics and sets up to develop Analysis Situs seriously.

17 An extensum is what has magnitude and situation. (A VI 4, 1987) Situation is defined by the concept of congruence. Congruence is defined by co-perceiving of things. Thus, the definition of extensum could be derived as a whole with co-existing parts that have mutual situation (Arthur [2013, p.518])

18 Arthur s interpretation is controversial because De Risi seems to take a skeptical view of whether there can be some connection between Analysis Situs and the theory of space before the latest period. To explicate the development of the theory of space in Leibniz in detail, we must consider that which is right.

19 The narrower and closely related problem: what does Leibniz think about fundamental elements of space? Both De Risi and Arthur think that space is an abstract order of actual entities which represents all possible relations of situations. This idea implies that a space is not the entity which exists in itself but constituted from representations of entities, because a space as an order could be thought as an order of something which plays a role as elements of a space. So, we need to identify such elements.

20 What is this? X A monad! C B

21 Our claim: Leibniz could not hold the idea of constituting space until he could present what is the alternative to Cartesian concept of extension in 1700 s.

22 The concept of space The early period: pure extension, set of points, absolute space De Veritates, de Mente, de Deo, de Universe, 1676 (cf. Mercer[2001, p.439]) the idea of space is recognized by this: namely, that it is that by which we separate the place and, as it were, the world of dreams from our own. As this is so, it does not follow that there exists anything but sensation, and the cause of this sensation and of its consistency. (A VI, 3 511)

23 However, as Parkinson claims, we can read a sign of the theory of space of the Leibniz- Clarke correspondence in the manuscripts known as De Summa Rerum. (Parkinson[1992, pp.xi-liii])

24 The middle period: space as maximum, point as minimum Specimen Geometriae Luciferae 1695 (GM VII 283), Analysis Geometrica Propria 1698 (GM V 174)

25 The early period(1679):pure extensa, set of points The middle period(1695): space as maximum The latest period(1715-6): network as representation of monads

26 The concept of substance In the tradition of Leibniz study, the concept of substance has a long-discussed history. According to Samuel Levey, in the early period, Leibniz emphasizes unity of substance (substantial form). But from 1700, simplicity of substance (monad) is emphasized. (Levey[2007][2008][2011][2012])

27 Arthur claims that Leibniz holds the concept of monad from very young days to the end.(arthur[2014]) Which is a right picture of Leibniz? A philosopher of monad or a philosopher of a substantial form-and-then-monad?

28 1:Introduction 2:The Leibniz-Clarke correspondence and Analysis Situs 3:The theory of space in the middle period 4:The three points argument in New System 5:The three kind of the point: which is an element of space?

29 Introduction to monad in New System (1695) atoms of substance are what a substantial thing is made of (and without which it wouldn t be substantial), are what pull it together so that it is (though in a non-basic way) one thing, and are what makes it active, what has it doing things. They might be called metaphysical points: they are related to mathematical points, which are their points of view for expressing the universe, but they are not themselves mathematical points because they have something alive about them, and a kind of perception. (GP IV 482-3)

30 The three points argument First Step: Leibniz shows that there are atoms of substance in nature by reductio ad absurdum argument: If such atom does not exist, there is no unity in the world. Reductio argument could guarantee only the existence of atoms. Second Step: Leibniz shows that there are three candidates for atoms of substance, that are physical point, mathematical point and metaphysical point.

31 Mathematical points themselves cannot make substances because of continuum problem. Physical points themselves also cannot do because this is not primitive elements. Conclusion: only metaphysical points can be such atoms.

32 1:Introduction 2:The Leibniz-Clarke correspondence and Analysis Situs 3:The theory of space in the middle period 4:The development of the concept of space and substance 5:The three kinds of the point: which is an element of space?

33 Arthur presents his interpretation that Leibniz already established the constituting theory of space before the latest period. We think that this interpretation depends on the assumption that Leibniz in the early or middle period holds the same concept of substance as in the latest period.

34 Leibniz s space before the latest period could not avoid the continuum problem. Textual evidences: Characteristica Geometrica, The letter to Des Bosses in 21 July 1707.

35 If we want to justify the claim that Leibniz in the middle period held that the constituting theory of space as similar to the theory in the period of the correspondence with Clarke, we need to demonstrate that it is metaphysical points which are fundamental elements of matter that make space. His ongoing arguments that aim to establish the concept of substance begin at the early period. Firstly his arguments target on the unity of substance, and then shift to the simplicity.

36 This teaches us that before focusing on the simplicity of substance, it is unclear that what Leibniz thinks as a fundamental element of matters has the nature of simplicity. So, we claims that Leibniz could not find what fundamental element of matters is until the 1700s.

37 In fact, in the correspondence with Clarke, Leibniz thinks that metaphysical points (monads) could be fundamental elements of space. However, before the latest period, Leibniz presents no argument that explains how to make extension from non-extensional points. The failure of identifying what is an alternative to Cartesian concept of extension also justify to our claim. (cf. Correspondence with De Volder)

38 If Leibniz could answer to the question of how does metaphysical points constitute space, he also could answer to the question of what is prior to Cartesian extension.

39 What nature does space have? What is alternative to Cartesian extension? What is essence of substance? Early period (1679) Middle period (1695) Absolute space maximum Latest period (1715-6) Network of representations among monads ambiguous ambiguous Simple substance (monad) Unity Unity or Simplicity Simplicity

40 Can be atoms of substances? Can be fundamental elements of space? Physical points No? Mathematical points Metaphysical points No? Yes?

41 Can be atoms of substances? Can be fundamental elements of space? Physical points No No Mathematical points Metaphysical points No Yes No Yes How does metaphysical points constitute space?

42 Conclusion We tried to settle the problem of whether Leibniz holds the constituting theory of space before the latest period and gave a negative answer. Certainly Leibniz continues to search the best definition of geometrical objects, however, from this it could not follow that he has the idea that space is an entity which is constituted by something. So, we need further investigation to make clear how he got to his final view.

43 References Richard T. W. Arthur: "Leibniz's Theory of Space", Foundations of Science, vol.18, : Leibniz, Polity Press, Vincenzo De Risi: Geometry and Monadology: Leibniz's Analysis Situs and Philosophy of Space, Birkhauser, Christia Mercer: Leibniz's Metaphysics: Its Origins and Development, Cambridge University Press, 2001.

44 G.H.R. Parkinson: "Introduction", De Summa Rerum: Metaphysical Papers, , Yale University Press, 1992, pp.xi-liii. Samuel Levey: On Unity and Simple Substance in Leibniz, The Leibniz Review, Vol.17, 2007, pp : Why Simples?: A Reply to Donald Rutherford, The Leibniz Review, Vol.18,2008, pp : On Two Theories of Substance in Leibniz: Critical Notice of Daniel Garber, Leibniz: Body, Substance, Monad, Philosophical Review, Vol.120, No.2, 2011, pp : On Unity, Borrowed Reality and Multitude in Leibniz, The Leibniz Review,Vol.22, 2012, pp