A Essay On The Relationalism Of Space-time

What is the evidence for relationalism about space, and how should we critically evaluate it?

In this essay I will critically evaluate the evidence for 'relationalism about space' as expounded by three important figures, Gottfried Wilhelm Leibniz, Ernst Mach and Albert Einstein. I will do this by assessing some prominent arguments and objections, and subsequently discussing one key thought experiment in depth. I will argue that while there is very persuasive evidence to show that space is not a substance (in any definition applicable to matter) and not absolute, it is nevertheless more likely to be some sort of entity (independent of matter) in light of modern science. What is meant by 'relationalism', in general, is the belief that space is not an absolute, tangible entity, but rather the relations that happen to appear between objects or events. In particular, the specific relationalism I will consider puts immense importance upon mechanical descri ptions of motion, which are derived from assumptions about spatial relations (Kennedy, 2003). By 'spatial relations' is meant the distances and durations between events involving objects of a certain length, breadth and height.

I will begin by evaluating the evidence found in Leibniz's theories from the 17th century, in which space is conceived as a mind-dependent construct rather than a real 'thing'. Leibniz argues for a space-time model in which space is the collection of possible 'places' an object could inhabit, where the place is defined by its relations to other objects (assumed to be stationary). In our minds, a place is idealised in order that we perceive it as constant through time, though in reality Leibniz sees these places as being instantaneous, and thus unable to be ascribed to more than one object. It is my understanding that Leibniz did not deny the existence of external reality as such, merely the tangibility of our concept of space. These theories hinge on the notion that movement is produced by a primitive force, and most importantly, Leibniz showed that his findings were consistent with the Newtonian mechanics of the day (Huggard and Hoefer, 2009). The evidence for this theory seems to be the macroscopic consideration of Newtonian mechanics in which earthly objects can be observed as being at rest in comparison to another object. At the time, there was little scientific research which would render the theories inapplicable to 'outer' space so it was assumed that this extrapolation was reasonable (Huggard and Hoefer, 2009) (I will discuss this reasoning in conjunction with the later philosophies). Leibniz's conception of substance was mainly a type of 'atomism' using the mathematical phenomenon of one-dimensional points called 'monads', and these were imbued with the aforementioned primitive force so as to create matter (Ross, 1984). I find this concept persuasive due to its consistency with mechanics, and the definition of substance which is clearly inapplicable to space though retains an element of scientific realism; although it may prove difficult to incorporate the scientific beliefs of today. The force discussed here was quantified as modern-day kinetic energy, i.e. mass times speed squared, which can be scaled by one half to match up exactly, however scientists have recently found that matter is held together by many smaller particles which interact using at least two different forces (Nave, 2012). Though Leibniz's monads may be argued to constitute the smallest known particles, his theory of force would have to be modified considerably. One way is to say that force could be broken down into the atomic forces, though even then I would argue that these forces are not necessarily intrinsic properties of substance as Leibniz proposed, but rather properties of the particles which are affected by them. Another approach could be to suggest that Leibniz's force is that fundamental force, which can take on different properties in order to interact with different particles, yet this seems superfluous to scientific theories and would have to be confirmed by further investigation to become truly plausible. I feel it is unlikely that the detailed ideas about the primitive force can be rescued without becoming ad-hoc. What I think is right about Leibniz's theory, on the other hand, is the idea that 'places' are not consistent through time. This agrees with the generally accepted theory that the universe is expanding and accelerating, causing every 'place' or point to move away from each other through time (Nave, 2012). This expansion means that even the bare relations between objects and events are changing and there is essentially no temporal constancy. This aspect of the theory really pins down the necessity of space not being absolute, which I believe is a valid conclusion to draw. Einstein's research offers an objection to this way of thinking, as it implies space to be a solid "thing" existing independently of our perception (Ross, 1984). General Theory of Relativity (hereafter GTR) gives us a picture of space as being "a continuous manifold" (Zyga, 2009) implying that it has intrinsic properties such as curvature, which make it near impossible for any scientific realist to argue that it does not have an independent existence.

I will introduce Kant's philosophy briefly; in my opinion it is not a strong contender in providing a valid case for the nature of space, so I will set out some criteria which relationalism must overcome in order to prove its worth. Kant's transcendental idealism is comparable to Leibniz's relationalism in its idealist quality. This can be found in the idea that the external world is perceived and then made sense of through ideal constructions of our minds. The major difference in the theories is that Kant sees all objects as merely appearances, with no necessity attached to their being as we perceive them. While Leibniz takes the realist scientific view that knowledge of the world is possible, Kant prefers to be epistemologically cautious, especially with regards to space. With our apparent innate knowledge of geometry, Kant believes space must have its conceptual source in our minds, thus the spatial quality of objects cannot be anything but the product of our a priori knowledge (Nerlich, 1994). Indeed the evidence for Kant's idealism must be the metaphysical instinct to doubt our senses, with logic being the only recompense. However, the claim Kant makes that objects are idealized is dubious if we apparently cannot know the nature of their existence. I feel Kant's main tenet of scepticism leads only to the conclusion that one cannot make any claims about the nature of space i.e. being sceptical precludes the ability to be aware of the true nature of space, and so only scientific realists have the capacity to produce valid arguments on the subject. To assess Leibniz on this criteria then, leads me to think that his idealism must stem from a worryingly similar innate knowledge of geometry. Though he never explicitly states anything of the sort, it seems necessary to the idealisation of space that we have prior experience of geometry, and it seems only our minds are prior to experience of the world. Leibniz does try to distance himself from the phenomenalist through his postulation of monads as being "the perceivers on which material bodies ultimately depended" (Ross, 1984). While it seems phenomenalistic to suggest that matter perceives itself, this certainly must be a realist claim as the monads are nevertheless contrived to be independent of human perception.

Another famous account of relationalism, this from the 19th century, was heralded by Mach. Huggard and Hoefer (2009) argue that there is a preliminary reading of Mach's views which they term 'Mach-lite'. In this version, Mach criticises "Newton's bucket experiment" (Huggard and Hoefer, 2009) in order to show that absolute space is an unnecessary concept in mechanics, and he does this through pointing out that one might just as well have any motion happening relative to the total mass of the universe - which is undoubtedly tangible - as to an absolute space. Mach only needs to remove the assumption of absolute space from Newton's theories, leaving the majority of the mechanics intact; the reference frame then becomes "the fixed stars" (Huggard and Hoefer, 2009). In this case the evidence for relationalism seems to lie only in the relation of motion to other objects, and the relationalist position seems obsolete. Despite this being a legitimate standpoint, it seems there is evidence in the same work of Mach's which points more strongly towards an extreme version. This interpretation boils down to the belief that everything is only "relatively constant" (Becher, 1905), by which is meant that no object or relation can be absolute or independent of others. This is made possible through the belief that objects are in fact ever-changing due to their environments, for example light may change the colour we perceive. Mach's theories of substance are such that there are only our sensations and our interpretations of them. This varies from Leibniz's relationalism in the sense that even substance is given ideal qualities, such as permanence, by the mind (Becher, 1905). While Mach has taken pains that these conceptions fit with the science of his day, it is unavoidable that there is a measure of 'brute fact' about them, in particular Becher (1905) says in Mach's opinion "it is not the business of science to explain the existence of the elements . but to recognize these as ultimately 'given' in experience", elements being things such as emotions, sensations, temperature and colour (Becher, 1905). Modern-day biology and chemistry also seem to posit a problem in the form of their definitions of substance which assume constancy independent of perception. In a way, Mach has treated substance as being the same as space - by saying that neither is self-contained - and this seems to conflict with the majority of science and tend away from being scientifically realistic. Thus I find that Mach's definitions of substance are more contrived than those of Leibniz, though again they are successful in that space cannot be seen as substantial. The main influence of the latter interpretation was creating the idea that in order to create a purely relativistic account of motion, all absolute quantities, such as velocity, must be removed from mechanics and replaced with relative quantities, this idea was later called 'Mach's Principle' by Einstein (Huggard and Hoefer, 2009). In this case, instead of objects having distinct lengths, breadths and heights, the mechanics should be able to work with lengths relative to other objects or some arbitrary measure of length.

Einstein's scientific research was guided by the Machian principle, as Sklar (1977) says he wanted "to find a set of laws that could accurately describe the evolution of systems from the viewpoint of any reference system whatever". The research was successful in following the principle for the most part, in that the Special Theory of Relativity (STR) mirrored the suitability of Newtonian and Galilean mechanics (Huggard and Hoefer, 2009), however GTR comes with a few weighty problems for the relationalist. The first is mentioned by Sklar (1977), that in GTR "a new invariant notion of absolute acceleration is well-defined" which undoubtedly is incompatible with strict relationalism. Moreover, the spacetime utilised in GTR is assumed to be substantial in order to describe its curvature and nature mathematically, and the most serious result is the bending of spacetime by gravity and gravity's causal effects on celestial bodies (Sklar, 1977). For Leibniz's relationalism it seems possible that the properties of space could perhaps be treated relatively just like the properties of objects and events, but it still seems dubious whether this would become purely ad-hoc and require dramatic recasting of Leibniz's theories. Mach's relationalism is even further removed from this possibility and could not inaugurate the slightest measure of space having any properties. Since these results seem insurmountable for the realist relationalist, the only way I see to preserve one's relationalism is therefore to take a step back from epistemological certainty and assume a more phenomenalistic position. With this in mind I shall return to Mach and Leibniz's theories in a very useful thought experiment.

An important piece of evidence the classical relationalists might use to their advantage is the 'incongruent counterparts' thought experiment. Kant originally used this to try and prove substantivalism victorious and later transcendental idealism (Lowe, 2002), but I will show that, in its original form, it is better suited to relationalism. In Kant's example of a solitary hand consisting a universe, Lowe (2002) argues that the relationalist must determine 'handedness' (whether it is right or left handed) by the relations of the parts of the hand, which obviously mirror those of the hand of opposite 'handedness'. This leads only to the conclusion that relationalists cannot account for independently existing incongruent counterparts through their relations. However, the relationalist might point out that, just as any dichotomy of concepts e.g. good and evil, north and south... etc., one extreme is only defined in terms of not being the other, i.e. good can be defined as the opposite of evil (Sheldon, 1922). In fact such relationships between concepts are the only way that they gain any meaning, and I believe that this is paralleled in the definition of "left hand" and "right hand", or indeed any incongruent counterpart. The consequence of this, is that 'handedness', or incongruence, are both relational properties since their truth-makers are their counterparts. A solitary hand, therefore, has no 'handedness' property. What Lowe (2002) would say to this is that I have merely described the arbitrary nature of naming 'left' and 'right' without giving a reason for the difference in their nature or appearance, however what I'm trying to say goes deeper than this. There are many objects which are asymmetrical and yet do not have an existing incongruent counterpart, a shell for example, and yet we do not say that the shell has an intrinsic property of "handedness". Only once an identical mirrored shell came into existence could we define what it meant to be of the original type or the other. Essentially, I believe it is possible for asymmetry to exist without the human need to classify it in terms of what it is not, and ascribing "handedness" is merely a projection until concrete distinction or relation is needed. If one is going to insist, however, on the innateness of handedness, then the relationalist can refer to mathematics. It is assumed that such logic is contained in any world based on our own, and if this is the case then it is also the case that incongruent counterparts can often be mathematically described. If one takes the group of all real numbers R which can symbolise the set of all real objects, and the arbitrary subgroup S, one can create a "left coset" in which a number, x, in R that's not in S is left-multiplied by S, giving xS= {xy_1,xy_2,.,xy_n?n=|S|} (Weisstein, 2014). While the naming of each left and right cosets may be arbitrary, the nature of each is unique and very difficult to classify. Thus left-handed objects can be defined as those resembling the asymmetry in the left coset and likewise for the right, moreover I believe that since mathematicians have left their definition of what it means to be incongruent to 'brute fact', so can the realist philosopher. Indeed the idea of incongruence as a phenomenon was really founded in the geometry of congruent and incongruent shapes, thus it should be left to mathematics to explain such instances. Though an admittedly weak position to conclude on, I believe the opposition has just as weak a position in that they will be forced to say that it is 'brute fact' in their eyes that incongruent counterparts contain an intrinsic property of handedness. Be that as it may, an objection to any relationalist interpretation of this experiment remains. In light of recent research e.g. GTR, space is often seen as a 3-dimensional manifold, and as such might have a topology with a 2-dimensional, unbounded but closed surface. If this is the case, Lowe (2002) argues that an object can be transported about the surface and when it gets to the same set of co-ordinates it will be the mirror image of itself. Therefore incongruent counterparts could always be made congruent by "transporting it around the universe" (Lowe, 2002) and would no longer appear to be fundamentally different. Though the argument follows the same lines as I have used previously, the key assumption is that space has a shape, implying a substantial quality. This seems to be indubitable evidence for the opposition, and, as I have discussed alongside Einstein's philosophy, more scientifically plausible.

To conclude, I believe the evidence for the types of relationalism mentioned is abundant in both selected parts of science (Newtonian mechanics) and in thought experiments, much of which it has not been possible to address. However, it seems necessary for the relationalist to deny the modern research concerning the nature of space and become somewhat of a phenomenalist. My own opinion is that relationalism cannot stand up against modern science as I am foremost a realist, and therefore other contenders for theories about space must be taken more seriously. As Nerlich (1994) says, Leibniz and Kant are two 'purist' reductionists in their idealism. He claims that this comes at the cost of plausibility, and that the pure reduction theories cannot be reconciled with impure, plausible ones. I agree with this wholeheartedly, and believe the same can also be said of Mach.