# 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 attractive^{1}
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...