Abstract
The question of the dimensionality of space has informed the development of physics since the beginning of the twentieth century in the quest for a unified picture of quantum processes and gravitation. Scientists have worked within various approaches to explain why the universe appears to have a certain number of spatial dimensions. The question of why space has three dimensions has a genuinely philosophical nature that can be shaped as a problem of justifying a contingent necessity of the world. In contrast to explanations of three-dimensionality based on anthropic arguments, we support the search for a theory that provides a justification for the dimensionality of space based on a combination of deductive and inductive reasoning applied to science. In doing so, we argue that Kant correctly approached the question in “Thoughts on the true estimation of living forces” (1747) by connecting space dimensionality and the inverse square law. In expounding the strategy of Kant’s argument, we describe the main features of a general Kantian explanation of the dimensionality of space and discuss them with respect to current accounts of explanation in the philosophy of science, such as inference to the best explanation and the deductive-nomological model.
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Notes
They are: (i) the direct or indirect observation of a Kaluza-Klein tower of states, or (ii) the observation of deviations in the inverse square law of gravity in short-range experiments. In our paper, we mostly review arguments provided in favour of the existence of extra-dimensions that appeal to this second kind of experimental search.
Manson (2003) offers a clear overview of teleological arguments in modern science.
In string theory, which is the most ambitious “theory of everything” proposed to date, the issue of spatial dimensions is central. It has been found that in most constructions it is required to have nine space dimensions and one time dimension for mathematical consistency of the theory. The experimental fact that we perceive three dimensions is accommodated by proposing various compactification alternatives to the remaining six dimensions.
A recent example of this approach is the following: “One of the most fundamental questions concerning our universe is why we live in a \((3+1)\)-dimensional space-time, and why the universe is expanding. The aim of this Letter is to provide some evidence that these facts can be derived from a nonperturbative formulation of superstring theory in \((9+1)\) dimensions based on matrix models” (Kim et al. 2012).
In referring to axioms of geometry, Poincaré pointed out that they are conventions and that “our choice among all possible conventions is guided by experimental facts; but it remains free and is limited only by the necessity of avoiding all contradiction” (Poincaré 1902, p. 65).
For an historical account of the inverse square law, see Henry (2011).
Other than three dimensional space as derived from the ISL are ruled out from Kant’s argument by virtue of the fact that we could not perceive or imagine other than 3D spaces. Euclidean space might be seen as a convention adopted by Kant, but, as we shall see, his argument aims at finding the physical necessary ground for the fact that the physical world is experienced to be three-dimensional.
Solutions to the field equations of Einstein’s theory of General Relativity reduce to Newton’s theory of gravitation and hence the ISL, \(F\propto 1/r^2\). Since 3D is a fixed input to General Relativity, just as it is for Newtonian gravity, the ISL is derived when 3D and other inputs needed to describe the theory are applied. If space had \(n\) dimensions, and the principle of general covariance were maintained, the force law would reduce to \(F\propto 1/r^{n-1}\). Therefore, one could just as legitimately start with GCTn (General Covariant Theory in \(n\) spatial dimensions) and call ISL an input, and 3D (and General Relativity) would be derived as a consequence.
Kant’s argument is concerned with a notion of space that is very different from the one adopted later on in the Critical period, both in the Dissertation (1770) and in the Critique of pure Reason (1781/1787). In Thoughts on the True Estimation of living forces, Kant is discussing properties of physical space, whereas in the Critique of pure Reason, specifically in the Transcendental Aesthetics (A19-49/B33-73, pp. 172–192) he considers space insofar as it is an a priori intuition of outer sensibility. As such, this pure intuition necessarily determines spatiality as the a priori form of outer appearances. For the account of empirical space as treated in physics, see Kant (1786, AA 4:480–481, pp. 15–16).
Kant discusses a notion of space that is similar to the one presented in the Physical Monadology (1756). In the latter, Kant distinguished the notion of extension from physical space: substances or physical monads do possess an extension, because they fill space not by virtue of their mere existence, but by virtue of their sphere of activity determined by attractive and repulsive forces. In supporting the distinction between extension and space, Kant combined Cartesian concepts of force, such as De Mairan’s (Massimi and De Bianchi 2013), with Leibniz’s metaphysics (see Schönfeld 2000; Watkins 2001), in order to modify Newton’s account of space and gravity. In particular, the notion of active force, as presented in 1747, is explicitly derived from Leibniz’s Specimen Dynamicum (1695).
As we shall see, Kant arrived at the solution by excluding Leibniz’s geometrical argument and proofs based on the properties of numbers, and (1) by following a metaphysical principle, and (2) by assuming that the cause of gravity can be known. The latter assumption was compatible with Euler’s position on Newtonian physics. According to Euler, indeed, Newtonian mechanics should have been enriched by Cartesian elements, and physics must have been able to explain phenomena via causal principles (see Henry 2011).
In Sect. 351 of the Theodicy, Leibniz claims: “With the dimensions of matter it is not thus: the ternary number is determined for it not by the reason of the best, but by a geometrical necessity, because geometricians have been able to prove that there are only three straight lines perpendicular to one another which can intersect at one and the same point” (Leibniz [1710] 1985, p. 335).
The argument is the following: “Because everything found among the properties of a thing must be derivable from what contains within itself the complete ground of the thing itself, the properties of extension, and hence also its three-dimensionality, must also be based on the properties of the force substances possess in respect of the things with which they are connected” (Kant 1747, AA 1:24).
According to Lessing, “Kant undertook the difficult business of educating the world. He estimated the living forces, without first estimating his own”. The statement appeared in Das Neueste aus dem Reiche des Witzes, Lessing 1751, p. 32
In order to be aware of the hostile environment in which Kant’s work was received, it is worthwhile looking at the review that appeared in Nova Acta Eruditorum (Anonymous 1752, pp. 177–179), where an anonymous reviewer focused on two main points of Kant’s meditations. The first one concerns the notion of action of substances. According to the anonymous reviewer, Kant’s notion of a force as being determined to act outwards is striking (“What namely is that force of the same body A, determined to act outwardly?” [Quid enim est vis illa ipsius A, ad extra se agendum determinata? ], p. 178). The sarcastic attitude of the reviewer is then directed towards Kant’s conception of three-dimensional space (Anonymous 1752, pp. 178–179). The reviewer stated “It seems that the author, who in one way or another heard of action in inverse square of the distances and of three dimensions, unites them as in the same way as we bring together dreamed-up ideas.” [Videas hic hominem, qui aliqua de actione in reciproca duplicata distantiarum, et de trina dimensione, sando inaudivit, haec ita conjungere, ut somniando ideas conjungimus] (Anonymous 1752, p. 179). The main criticism on this point is that the necessary reciprocal connection among substances is not perspicuous and the same holds for the mechanics underlying higher-dimensional spaces. Therefore, this argument is pure speculation and overall, the reviewer claims, Kant’s works “deserve no more of his paper and time”.
See Hall & Hall (1962, pp. 133–134).
Even if it has been claimed that Kant abandoned the spirit of his 1747 argument and then treated space in a very different way (Friedman 1992; Watkins 2001; Schönfeld 2000), there are still traces of this argument in his 1755 cosmology. In Universal Natural History and Theory of the Heavens, Kant claims: “Attraction is without doubt a quality of matter that is just as pervasive as the coexistence that makes space in that it combines substances by reciprocal dependences, or, to put it more accurately, attraction is precisely that universal relationship that unites the parts of nature in one space: it therefore extends to the entire expanse of space into all the reaches of its infinity” (Kant 1755, AA 1:308). The order of space depends on attraction, which in turn regulates reciprocal fundamental interactions. The latter are the constraints that unify space into one expanse of space. The counterbalance of attraction is to be found in repulsive forces. Again in 1756, in the Physical Monadology, Kant has the same problem of balancing the idea of attraction at a micro-level and solves it via the concept of repulsive force, which responds to the inverse cube of the distance. This ratio is also used by Kant in 1786 and later on in the Opus postumum.
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Acknowledgments
This work has made been possible also thanks to the grant offered by CERN TH-Division to Dr De Bianchi in 2012 and 2013. We are grateful to two anonymous referees whose stimulating comments improved the final version of our paper. We would like to thank Brigitte Falkenburg and Daniele Cozzoli for fruitful discussions.
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De Bianchi, S., Wells, J.D. Explanation and the dimensionality of space. Synthese 192, 287–303 (2015). https://doi.org/10.1007/s11229-014-0568-1
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DOI: https://doi.org/10.1007/s11229-014-0568-1