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Is Physical Geometry Conventional?

When Euclid outlined his geometry, it was probably the best and most complete 'science' to date. His axioms, from which practically the whole of modern geometry is derived were held in such high regard, and seemingly so fruitful for describing the world around us, that for nearly two millennia they remained unchallenged, and were held to be simple mathematical or scientific truths. It came as something of a shock, therefore, to those mathematicians who has left the axioms uncontested, that it is possible to define an equivalent set of axioms that appear to be significantly different in that they describe a world with a radically different shape, which are not only just as consistent as Euclid's axioms, but also just as fruitful, able to be used successfully to answer the same geometric questions. An important discussion, therefore, in modern philosophy of physics, asks whether a Euclidean geometry or a non-Euclidean geometry is the correct description of the world, and, furthermore, if there is nothing to choose between them in merely experimental terms, what it means to say that one is 'true'.

One of Euclid's axioms states that, given a line going in any direction and any point, there is exactly one parallel line that goes through the point (taking any line to be parallel to itself)- this seems to be totally in keeping with how we see the world, and it seems difficult to imagine two different lines parallel, in the sense that they never cross, but which each went through a particular point. The reply often comes that it is difficult to imagine parallel lines never crossing at all- since this requires us to imagine the lines flowing all the way to infinity, a task which we do not have the faculties to perform. This sounds, at the very least, an interesting objection, but we will not trouble ourselves with it here, and shall take as given that such a conception of parallel lines is at least cogent, which seems all we need for the debate at hand. The problem occurs in that it is possible to construct a consistent non-Euclidean geometry in which more than one parallel line may pass through a single point, and whilst this may jar with our intuitions and (though some deny this) the way we experience the world, such a geometry proves a more than adequate replacement for Euclidean geometry, in that there are some situations in which its use greatly simplifies expressions and proofs, and makes easy work of some geometry which is very complex in its Euclidean exposition.

With the advent of relativity, many of our intuitions about how the universe is put together were challenged, such as the separation of space and time, and there being simple truths about events being simultaneous with one another. One particular problem with which physicists have struggled is that of how we might maintain our best definition of a 'straight line', namely the path taken by light travelling through a vacuum, in view of evidence that the path of light is affected by local gravitational fields through which the light passes. One possibility was to give up and simply to abandon our conception of light travelling in a straight line through empty space, and to posit a 'universal force', which warps the path of light around massive objects. Another, and relatively attractive1 possibility, however, is that light does travel in straight lines through vacuums, and that it is simply that space 'curves' due to gravity.

The question of what exactly we mean to say that space is curved is an interesting one. Intuitively, when told that space is curved, we might imagine a flat surface with lumps in it, yet this is, perhaps, the opposite of what we mean. In saying that space is curved we suggest that the lumps are 'built in' to space's fabric, so that what looks like a flat (hyper-)surface is, in fact, riddled with areas warped out of place by gravity. When attempting to get our heads round this, it is good to remember that we are not the outside observer, looking at the bumpy surface from above, but the two-dimension bug crawling across the surface. What might look from outside like a lumpy surface, to us, on the surface, looks just like a flat plane- we just find that weird things happen when we come to one of these 'bumps': it suddenly takes longer to move just as far, and objects that used to be stationary in front of us are no longer in the centre of our view, and appear to move as we cross the bump (just as we think the stars 'move' as we walk in 'straight lines towards them' across the Earth's surface). The fascinating thing about curved space is that things move as if along a curve when moving inertially in one direction only. Since the lumps are so inextricably woven into the fabric of space, rather the the objects contained within it, we cannot detect them visually, occurring as they do at a pre-experiential level.

We have a choice, therefore, between light continuing to move in straight lines through non-Euclidean space, and the preservation of Euclidean space, where light no longer moves in straight lines, instead acted upon by a new universal force. The interesting thing about these two theories, is that they make exactly the same empirical predictions. The preservation of 'straightness of light-paths' in the theory with non-Euclidean geometry plays off exactly against the reinterpreted meaning of space, which is no longer a homogenous, boring, flat hyper-surface, but is now riddled with 'interesting' areas of texture.

Conventionalists argue that, without any empirical grounds to choose between these two theories, it is simply a matter of convention which of them is correct. We simply choose, as arbitrarily as we like, which of the theories is the truth. There may seem to be grounds to choose one, such as neatness for a particular problem or overall simplicity in stating the theory, but we may well find (as might seem obvious) that such notions of neatness and simplicity are also relative to the theory, and just as much a matter of convention. What seems like a simple exposition of the theory in its own terms, will turn out to be very complicated using the language of the other theory, and we are likely, therefore, to remain unable to find convincing reasons to choose one above the other. Reichenbach tries to justify ways that he suggests give good grounds for choosing one empirically equivalent theory above another, but he seems to assume just what is being debated. He argues that in terms of inductive simplicity we are able to find standards by which to judge one theory as better than another, but, in my opinion, fails to address the possibility, which Friedman stresses, that whilst he may produce plausible reasons for taking a particular standard of inductive simplicity as justified, such reasons are themselves theory-laden (even if we're too hooked on a particular theory to doubt it) and there will be ways of justifying virtually any standard of inductive simplicity.

Let me outline two other points of view, and then discuss conventionalism's prospects in light of them. The first other position that one might (and many do) hold is scepticism. A sceptic asserts that there simply is a truth about what geometry the world has, and that the fact that we have no way of knowing which this is is beside the point. This realist position has various intuitions strongly in its favour- it is easily understood, and most (non-philosophers) are happy with the idea of truths that we just can't know. The sceptics, furthermore, claim that their position is the least arrogant, in that it freely admits that man's abilities are limited, and that there are some things that we just can't do, rather than limiting facts about the world to what we can know. I am unconvinced by this claim- after all, isn't it arrogant in itself to claim that we can talk meaningfully about things without any conception of whether they are true or not, and no way of finding out? It seems that the sceptic wants to allow us free rein to talk about things without any justification, which is just what various other theories do not allow.

The second other possible view goes by many names, but it would be confusing to attempt to justify my choice of one, since each name seems to come with baggage and nuances particular to that theory, so I shall (without justification or apology) refer to the theory as 'reductionism', as Sklar does. Reductionism asserts that when two theories make all the same empirical claims, when they are empirically equivalent, they aren't two theories at all, but simply two ways of expressing the same theory. If there really can be no way of choosing between two theories, then they are not making any opposing claims, and anything that seems to be a disagreement isn't in fact one. Reductionists deny that there is a debate about which geometry really applies to the world- instead they suggest that non-Euclidean space is the same as Euclidean space with a universal force. Effectively, reductionists deny that meaning can be approached piecemeal, instead taking any word to hold its place only within the context of a holistic theory.

I would like to suggest that there isn't, in fact, any space between these two possible epistemological outlooks. It seems to me that either there are facts that we can't know, or nothing that we cannot know amounts to substantial part of a theory. Given this, it is clear that we should pick between one of these views, and that we cannot have it both ways- hold that there are substantial parts of theories that we cannot know, but that these do not amount to facts that we cannot know, but instead are simply 'up for grabs', and we can decide by convention how to proceed. My personal sympathies tend towards the reductionist view, that two theories that make all the same empirical claims and predictions are, in fact, the same theory differently expressed. As Sklar puts it:

To speak of adjudicating between these theories [Euclidean geometry with a universal force and non-Euclidean geometry] is as misleading as if one were to try to decide whether Newtonian mechanics expressed in English or Newtonian mechanics expressed in German was the correct dynamical theory.

It seems obvious, however, that a sensible denial of the reductionist position is that there are facts which we cannot come to know experimentally, and not that one theory is correct, even if we can't know experimentally which, not based on any facts, but simply by choice. Furthermore, the conventionalists appear to be using, without any attempt to justify this use, the word 'true' in a way that is completely orthogonal to its actual usage. It isn't the first bit clear what exactly conventionalists mean when they say that a theory is 'true' by convention. To say that a theory way used by convention, or that by convention we talk as if a theory were true (such as pretending that Newtonian mechanics was correct for the purposes of GCSE science lessons) makes perfect sense, but nowhere else do we say that things are true by convention. Put simply, it is understandable what conventionalists may be saying if they are merely suggesting that there are different ways of talking about the world, and that these two theories involving different geometries are both perfectly adequate for mathematicians' and scientists' purposes – which seems rather uncontroversially to restate the reductionist position. Otherwise, conventionalists seem to be suggesting that when we read the world as having a Euclidean geometry it becomes the case that this is so, and such a suggestion will clearly require a good deal more argument than conventionalism currently has supporting it, and I cannot see what form such an argument would take. After all, there will be no way to tell that the convention has become the truth- that is just where the debate began!

1...please excuse the pun...