Abstract
According to a famous Leibnizian dictum, space is nothing but “an order of situations, or an order according to which situations are disposed”. This so called “spatial relationism” is sometimes understood as a form of relativism (space is relative to the possible dispositions of bodies), which seems to entail the possibility of a spatial pluralism (possible worlds with non-equivalent dispositions of bodies might have non-equivalent spatial structures). An overly exclusive focus on the exchange between Leibniz and Clarke, considered to be the locus classicus for understanding Leibniz’s views on space, did much to lend credence to such a conception. In addition, in several passages, Leibniz talked of other possible worlds as lacking this or that spatial property. In the exchange with De Volder, for example, he explained that God could have been pleased with a phenomenal world with gaps in it. In this description, as in other passages, it seems that we can imagine without contradiction a world with a geometrical structure different from ours. This stance was supported, as we will see, by influential scholars who concluded, without much trouble, that a plurality of spaces was conceivable for Leibniz. Yet this idea runs counter to two other no less famous Leibnizian dicta: first, that geometrical truths are absolutely necessary (one cannot deny them without contradiction); second, that geometry should be considered as the science of space and that we can specify various properties of geometrical space (such as tridimensionality, homogeneity, isotropy or continuity). Combined together, these last two claims tend to prevent any form of pluralism: space is the object of a science which describes its essential properties and these properties are truths which are absolutely necessary. These eternal truths apply to all possible worlds. Our goal in this paper is to confront this tension in Leibniz’s description of space.
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Notes
- 1.
“Leibniz’s Fifth Paper”, §104 (GP VII, 415). Translated by Leroy E. Loemker, in Loemker (1989, 714).
- 2.
To De Volder, 24 March/3 April 1699 (A II 3, 545).
- 3.
See De Risi (2007).
- 4.
Loemker (1989, 639), GP VI, 603. Emphases in bold are ours, those in italic are Leibniz’s. We have modified Loemker’s translation. The French word “terrain” has been rendered by Loemker as “situation”. This is confusing for the reader, since “situation” is usually used to render “situs”. We have reestablished the word terrain, which is used by Leibniz in other texts.
- 5.
Loemker (1989, 487), GP VII, 303.
- 6.
On the idea that space and time act as a “framework”, see Réponse aux reflexions contenues dans la seconde Edition du Dictionnaire Critique de M. Bayle, article Rorarius (1702), GP IV, 568: “But space and time taken together constitute the order of possibilities of the one entire universe, so that these orders – space and time, that is – frame not only what actually is but also anything that could be put in its place, just as numbers are indifferent to anything that can be res numerata.” (Loemker 1989, 583). We modify Loemker’s translation, in particular we render the verb “quadrer” by its literal meaning, “to frame”, instead of the overly vague “to relate”.
- 7.
Loemker (1989, 700), GP VII, 395.
- 8.
Loemker (1989, 714), GP VII, 415.
- 9.
Loemker(1989, 708), GP VII, 407.
- 10.
Note, however, that these texts do not state that space “with things” in it is “real”, and that this reality has to be related to material bodies. They rather affirm that there is no proper actuality or reality of space in and of itself (against Newton’s reality of absolute space), and that space can only be said to be “actual” when it is considered as that in which actual things are. In this sense, talking about the actuality of space does not amount to claiming either that space is actual in the sense of being a material or physical entity or that space depends totally upon a certain disposition of actual bodies and has to change in some way with every change in the said disposition.
- 11.
- 12.
- 13.
Explicitly employing Rescher’s arguments, E. Vailati claims that the plurality of spaces is obvious in Leibniz’s thought, in Vailati (1997, 116).
- 14.
This argument seems nowadays particularly weak: a better knowledge of the texts on the Characteristica Geometrica has shown that Leibniz’s project of analysis situs was much more an attempt at generalizing Euclidean geometry than it was the invention of an alternative geometry. Interestingly enough, the position that space is only one amongst several other orderings of possible worlds has been defended nonetheless by one of the main editors of Leibniz’ texts on analysis situs: Echeverría (1999, 430, and note 8, where he discusses Rescher and Belaval). We will come back to this question in the last section.
- 15.
Earman (1979).
- 16.
Vailati (1997).
- 17.
Khamara (2006, 40-41). The author surprisingly introduces the notion of “point-particles” to clarify Leibniz’s paragraph 47 in the fifth letter to Clarke about the explanation of what it is for one thing to be at the “same place as” another thing. Such an introduction is necessary because the author systematically replaces “things” in space by material “bodies”. But it becomes superfluous as soon as “things” are conceived of in the most general sense, including material bodies, abstract figures, and dimensionless points.
- 18.
Echeverría, Parmentier (1995).
- 19.
De Risi (2007).
- 20.
See E. Slowik: “On the whole, De Risi and Arthur have contributed greatly to the cause of disassociating Leibniz from the traditional, reductive, and external relationism that most modern philosophers of space and time have tended to read into his philosophy” (Slowik 2019, 111, note 16).
- 21.
Hartz, Cover (1988).
- 22.
Futch (2008) considered the issue of the plurality of spaces. As his analyses did not refer to Leibniz’s geometry, they are not relevant for our survey. Besides, he accorded the primary place to time, not to space. For this reason, the arguments he used to explain the unity of space are not consistent with an approach focused on space itself.
- 23.
Arthur (2013).
- 24.
Arthur (2013, 500-501): “Such a space is therefore a three-dimensional partition, or system of boundaries or figures, corresponding to any one of the actually infinite divisions or foldings of matter at any given instant. Insofar as space is regarded as perduring, on the other hand, it is a phenomenon, in that it is an accidental whole that is continuously changing and becoming something different. It can be represented mathematically by supposing some set of existents hypothetically (and counterfactually) to remain in a fixed mutual relation of situation, and gauging all subsequent situations in terms of transformations with respect to this initial set.”
- 25.
Ibid., p. 514: “Phenomenal space is universal space, which consists in a different partition of places from one instant to another. […] viewed through time it is continuously changing and becoming something different, so that it is never the same thing from one instant to the next.”
- 26.
Belkind (2013).
- 27.
Belkind (2013, 474): “A body occupies a place when it has a particular set of distance relations to other bodies that are not moving (in Leibniz’s language bodies that are fixed existents). One problem that immediately comes to mind is whether or not this definition of place is circular given that motion is ordinarily understood as change of place. If distance relations are determined in relation to bodies in which there is no motion, and if place is defined using distance relations, then it is not clear whether place or motion is the more fundamental concept.”
- 28.
Belkind (2013, 464).
- 29.
E. Slowik: “On a straightforward interpretation of reductive relationism, the extension within bodies and the relative configuration of the bodies can remain invariant, whereas the actual distance relations among bodies (i.e. the geometry) can vary significantly. Likewise, one could employ the observations of various rigid body motions as a means of determining the geometrical structure of space as a whole. Yet, in contrast, there are a number of discussions in Leibniz’s late corpus that single out (infinite) Euclidean geometry as the only possible spatial structure” (Slowik 2012, 121; see also note 70 attached to this paragraph).
- 30.
This is related to the logical conception of necessity as the impossibility of being something else. But, as we will see in Sect. 5.4, the conception of necessity requires some more subtle analyses than this.
- 31.
E. Slowik: “Interestingly, it would seem to follow that a limited material world with, say, a spherical shape would have a non-Euclidean metric on that surface, given his notion of relative distance” (Slowik 2012, 121).
- 32.
Garber (2015, 243).
- 33.
Loemker (1989, 690), GP VII, 376.
- 34.
Loemker (1989, 703), GP VII, 400.
- 35.
Ibid.
- 36.
3rd letter to Clarke §5, Loemker (1989, 682), GP VII, 364.
- 37.
See Arthur (1994, 237): “Thus the hypothesis of fixed existents allows us to define place in terms of an equivalence: it is the equivalence class of all things that bear the same situation to our (fictitious) fixed existents. And when we take all possible situations relative to these fixed existents, we have a manifold of places, or abstract space.” On the characterization of situational order by means of the concept of the equivalence of mutual situations, see also Winterbourne (1982, 203).
- 38.
Nouveaux essais sur l’entendement humain, II, ii, § 5 (1704), A VI 6, 127. Our translation.
- 39.
Loemker (1989, 487, modified), GP VII, 303-304.
- 40.
See Tentamen anagogicum (1695), section 2.1.
- 41.
Loemker (1989, 484), GP VII, 278-279.
- 42.
See Sect. 5.2.3.
- 43.
A VI 4, 1664. Our translation.
- 44.
Réponse aux reflexions contenues dans la seconde Edition du Dictionnaire Critique de M. Bayle, article Rorarius, sur le systeme de l’Harmonie preétablie (1702), Loemker (1989, 583, modified), GP IV, 568.
- 45.
- 46.
A II 3, 16-17. Our translation.
- 47.
Two questions then naturally follow: would such a world still be spatial? If not, what would be a non-spatial world? Both the idea of a world endowed with space but not with incommensurable magnitudes and the idea of a non-spatial world without any order of co-existence are intriguing ideas. But they do not relate to our topic, especially as Leibniz’s use of such a possibility is mainly analogical and not to be considered in itself.
- 48.
Loemker (1989, 700), GP VII, 396.
- 49.
Leibniz to De Volder, 24th March/3rd April 1699, transl. Lodge (2013, 311), A II 3, 545. See also “I don’t say that the vacuum, the atom, and other things of this sort are impossible, but only that they are not in agreement with divine wisdom” (Letter to J. Bernoulli, Jan. 1699, GM III, 565; transl. Ariew, Garber (1989, 170).
- 50.
Principia logico-metaphysica (ca. 1689), A VI 4, 1645. Our translation.
- 51.
This world would play the same role as what we now call a “Euclidean model” of a non-Euclidean geometry. It does not suffice to ground an alternative geometrical science, but it certainly shows that a geometry in which the parallel postulate is not satisfied cannot be contradictory without Euclidean geometry’s being so as well. See V. De Risi: “To push Leibniz up a road he had certainly not thought of taking, we may conclude by saying that, according to his logical and epistemological principles, non-Euclidean geometries would be viewed as coherent systems of axioms; for negating the parallel axiom does not in the least imply a contradiction” (De Risi 2016, 119).
- 52.
B. Riemann, Sur les hypothèses qui servent de fondement à la géométrie, in Riemann (1854, 200-299). Y. Belaval briefly referred to Riemann’s work.
- 53.
- 54.
Nouveaux essais sur l’entendement humain, II, xiii, § 3 (1704), A VI 6, 146-147. Our translation.
- 55.
In the same way, the notion of congruence can be defined on a sphere or a cylinder, see De Risi (2007, 181, n. 52), GM V, 189.
- 56.
De Risi (2016).
- 57.
De primis Geometriae elementis (1680), in Echeverría, Parmentier (1995, 281-283); Essais de Théodicée, § 351 (1710), GP VI, 307-308.
- 58.
Translated by De Risi (2016, 175-176, our emphasis).
- 59.
Loemker (1989, 666-667), GM VII, 18.
- 60.
Loemker (1989, 671), GM VII, 25.
- 61.
Kripke (1963).
- 62.
De natura veritatis, contingentiae et indifferentaeque atque de libertate et praedeterminatione, ca. 1685-1686; A VI 4, 1517. Our translation.
- 63.
See Generales inquisitiones de analysi notionum et veritatum, §56–61 (1686), A VI 4, 757–758.
- 64.
De natura veritatis, contingentiae et indifferentaeque atque de libertate et praedeterminatione (ca. 1685–1686), A VI 4, 1518. Our translation.
- 65.
See Lin (2016).
- 66.
Nouveaux essais sur l’entendement humain, IV, xi, § 10 (1704), A VI 6, 446–447. (Our translation and our emphasis.) The following passage is also very interesting: “But one will still ask on what such a connection is based, since there is surely some reality in it which is not deceptive. The answer will be that it is based on the relation between ideas. But one will ask in reply where would these ideas be if no mind existed and what would then become of the real foundation for the certainty of eternal truths. This finally leads us to the ultimate foundation of truths, namely, to that Supreme and Universal Mind which cannot fail to exist and Whose understanding truly is the sphere of eternal truths, as St Augustine recognized and expressed in a very vivid way. And so that it may not be thought that there is no need to have recourse to this notion of a Supreme and Universal Mind, it must be remembered that these necessary truths contain the determining reason and the regulating principle of existences themselves, or, in a word, the very laws of the Universe. Thus, these necessary truths being prior to the Existences of contingent Beings, they must necessarily be grounded in the existence of a necessary substance.” (Our translation and our emphasis.)
- 67.
On “conditional” truth and the problem De constantia subjecti, see Rauzy (2001, Chapter II, §6), where one can find a survey and a discussion of the literature. See also: Mates (1986); Adams (1994). None of these authors deal in detail with the specific problems related to the application of this category of “conditional truth” to mathematical theorems.
- 68.
In his analysis of the question, Rauzy emphasizes that one should relate existence to the actuality of the thing and not to the concept alone (2001, 119-120). However, when commenting upon the passage quoted from the Nouveaux essais (2001, 125) and the fact that the subject of a conditional truth might be “non-existent”, he follows another interpretation in terms of non-contradiction alone. This is at odds with the numerous passages in which Leibniz talked of mathematical truths as concerning not only existent things, but also possibilities (see Nouveaux essais II, xiv, §26 or II, xiii, §17). We shall thus stick to the interpretation of existence in terms of actuality – in accordance with the discussion above on the “actualization” of mathematical truths.
- 69.
This idea evocates what is called ‘if-thenism’ by contemporary philosophers of science. According to them, mathematical truths ultimately can be formulated as if p, then q. This thesis historically stemmed from the difficulty to conciliate the plurality of geometries with the logical nature of mathematics. On that topic, see A. Musgrave on Russellian logicism: ”After he had adopted the logicist thesis, Russell sought a way to bring geometry into the sphere of logic. And he found it in what I shall call the If-thenist manoeuvre: the axioms of the various geometries do not follow from logical axioms (how could they, for they are mutually inconsistent?), nor do geometrical theorems; but the conditional statements linking axioms to theorems do follow from logical axioms. Hence geometry, viewed as a body of conditional statements, is derivable from logic after all.” (Musgrave 1977, 110).
- 70.
De Risi (2007, 260).
- 71.
Leibniz’s central argument in this text consists in explaining that our world is the best since it is the richest regarding phenomena, even if the incommensurability of some magnitudes or things can be regarded as a flaw, as an imperfection. This example is used by reason of its comparability to the existence of sin and evil in the world.
- 72.
“Satisfaction” is taken here in the sense of “actuality”, which we describe above as holding for mathematical entities.
- 73.
See “Remarques sur le Livre sur l’origine du mal, publié depuis peu en Angleterre” (§14): “I have elsewhere made this remark, which is one of the most important in philosophy,drawing attention to the fact that there are two Great Principles: namely, that of Identicals or of contradiction, which states that, where there are two contradictory enunciations, one is true and the other is false; and that of Sufficient Reason, which states that there is no true enunciation the reason for which cannot be seen by who possesses all the knowledge required to perfectly conceive of it. Both principles occur not only in necessary truths but also in contingent ones.” (Essais de Théodicée, 1710, GP IV, 413–414. Our translation.)
- 74.
Note however that this does not mean that there is no geometry true “in every world” since on this interpretation also there may be a set of geometrical truths acting as necessary conditions for the possibility of a spatio-temporal structure in and of itself. But the point is that this “absolute geometry”, to take up a modern vocabulary, would have to hold prior to the intervention of architectonic principles, and thus prior to what, according to Leibniz, makes space Euclidean.
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Debuiche, V., Rabouin, D. (2019). On the Plurality of Spaces in Leibniz. In: De Risi, V. (eds) Leibniz and the Structure of Sciences. Boston Studies in the Philosophy and History of Science, vol 337. Springer, Cham. https://doi.org/10.1007/978-3-030-25572-5_5
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