The development of geometry and its philosophical implications
Geometry has its historical origins in empirical findings about the world, but was
subsequently developed as a theory of theoretical and mathematical interest. The
success of Euclid’s project to axiomatise geometry and thereby provide rigorous
deductive proofs of its various theorems suggested that laws of geometry, unlike the
empirical sciences, possess a universality akin to those of other mathematical fields.
Given that the axiomatic bases that can be provided for Euclidean geometry were
accepted as obviously true of ordinary space, the necessary truth of the theorems
followed, so the study seemed to be expressing facts about the world that were both
synthetic and a priori. In fact, Euclidean geometry and the further non-Euclidean
geometries that have been discovered are consistent branches of pure mathematics
and as such generate analytic truths which only on a given interpretation can be seen
to express factual truths about the world.
The first expressions of geometrical laws were undoubtedly arrived at through
observation and inductive reasoning from measurements and calculations about
physical objects. Euclid sought to formalise the study and express all its truths in
universal form, concerned with what holds true of geometrical concepts in principle
rather than in practise. Clearly, in order to arrive at conclusions through strictly
logical deductive reasoning, we must start with some premises, so a number of
unproved statements must be held without proof. Euclid selected as these a small
number of postulates and axioms, the former being statements directly relating to
geometrical concepts, the latter more general truths, but all considered to be so clearly
true that no reasonable person could doubt them. He also saw it as necessary to begin
with definitions of all terms employed, and while he did not explicitly recognise the
need to accept without further explanation some ‘primitive’ terms by which the rest
can be defined, refinements of his system by others makes this clear. The set of
primitive terms and set of axioms (there is conventionally no longer a distinction
drawn between axioms and postulates as in ‘The Elements’) adopted are not uniquely
suitable, rather there are many equally fruitful choices. Since the truth of all theorems
derived in the system depends simply on the truth of the axioms, the axioms need to
be simple and evident as possible. That Euclid’s axioms are a priori truths was a
widely accepted belief, as unlike empirically founded propositions they seem to have
a necessary, universal quality. For example, that the sum of interior angles of a
triangle is equal to 2 right angles is a fact that we would like to say could never be
disconfirmed by experiential evidence, we think it is an undeniable truth.
One of the postulates of Euclidean geometry, and its various logical equivalents,
stands out as anomalous. Without it the system is incomplete, however it is of a more
complex and less self-evident nature than the others adopted, requiring slightly more
reflection on the concepts involved to accept it as true. The axioms states, roughly,
that two non-parallel straight lines eventually meet at one unique point. Attempts to
show it can be removed from the list of postulates, instead perhaps derived as a
theorem, while unsuccessful in that specific aim, led to the shocking discovery of
other mathematically interesting systems of geometry adopting postulates different to
and incompatible with Euclid’s. Given the certainty with which we adopt Euclidean
theorems as true, it seems natural to suppose these conflicting geometries necessarily
false, hence inconsistent. In fact, closer inspection reveals some of Euclid’s original
proofs are not as logically rigorous as intended, but rather appeal to diagrams and our
intuition of geometrical forms. Conversely, our intuition regarding the falsity of nonEuclidean theorems may lead us to overlook their logical deducibility and
consistency. In order to prove the consistency of any geometrical system, it must be
reformulated in as abstract a form as possible, so that primitive terms become logical
variables and all attention is on the logical interrelations between sentences rather
than the meanings of the sentences themselves. This schematic form of geometry can
be referred to as pure or uninterpreted geometry, whereas our ordinary interpretation
is applied or interpreted geometry. Any postulates are clearly not ‘self-evident’ truths,
but devoid of any assertion about specific objects, and all deduced propositions are
analytic ones. An absolute proof of consistency may not be possible as it requires a
model to be found under which all postulates of the system are true, and it is likely
such a model must be an infinite one of which we can never have perfect knowledge.
However, a relative proof can be given, which shows remarkably that non-Euclidean
geometries are consistent if Euclidean geometry is. The consistency of the latter can
be shown to be dependent on the consistency of the mathematical theory of real
numbers.
Now we are faced with several incompatible systems of geometry each as consistent
as the others, the question as to which accurately represents the physical world seems
to be a straightforwardly empirical matter. Of course, on a physical interpretation the
postulates assert universal laws, so could never be proved beyond all doubt, merely
corroborated by evidence. The first step in interpreting geometry is to assign
meanings to the primitive terms. Taking as an example the term straight line, it can be
seen that there are various ways of explaining its meaning. We could interpret it as the
shortest distance between two points, or as the path of a ray of light through a medium
of uniform refractive index, or as the path along which a stretched cord lies as its
tension increases without limit. When all terms are described in a similar way the
geometrical theorems become theorems about the universe in just the same way as
those of empirical science, capable of being refuted by experimental results. The
Newtonian laws of physics imply that it is Euclidean geometry that is true of the
physical world; however today Einstein’s physics lead us to conclude that Euclidean
geometry is false, Riemannian true. It would seem that the matter is settled, the
geometry of space empirically decided.
But is it just an empirical question whether or not the universe is Euclidean in nature?
As previously mentioned, it does not seem possible that we could come across a
triangle whose interior angles did not add up to equal two right angles, and if we did
seem to do so we would assume some error in our measuring rather than take it as
disconfirmation of the proposition. The previously illustrated way of describing terms
does render geometrical theorems empirical in nature, but this is not the only way of
interpreting them. Instead, we might simply say that it is an essential part of the
meaning of ‘straight line’ that a triangle composed of straight sides has interior angles
adding up to two right angles. Then evidence to the contrary would not lead us to
conclude that this is a false statement, but that the figure we have observed is not a
triangle in this sense, that the sides aren’t really straight lines.
In testing any interpreted geometry, we are actually presupposing a physical theory. A
result that conflicts with the geometrical hypotheses must lead us to correct our
theory, but this correction could be made within the geometry or the underlying
physics. If we wished, we could retain Euclidean geometry and adjust our theories
regarding light rays for example. Under this interpretation, the truths of the geometry
would not be empirical ones but analytic ones, true by virtue of the language
employed. This approach may seem appealing when considering the relative
simplicity of Euclidean geometry as compared to Riemannian, however it is the
overall simplicity of our theories about the world that should be taken into account.
Adjusting physics to accommodate Euclidean geometry may not result in a more
simple theory of space, and is not preferred by practising physicists. Given that we
can interpret geometrical terms in the first empirical sense detailed above, or the
second analytic sense, the important question is which reflects a deeper tendency in
our ordinary discussion of space. Certainly physically interpreted geometry can not
imbue theorems with the necessity they were once considered to possess, but it is only
if reduced to analytic truths that they can do so.
Although geometry seems at first glance to be a branch of mathematics that is ‘about’
the physical world, it is now recognised that it can be studied in a pure or an
interpreted way. Geometrical axioms on the ordinary physical interpretation are not
known a priori, as is clear from the discovery of non-Euclidean systems.
Alternatively, the necessity of the axioms can be preserved, but then the facts they
express are no longer synthetic ones about the world. Without considering the truth or
falsity of the axioms, geometrical theories are of great mathematical interest and
significance. The various applications of different systems of geometry to the physical
universe show the remarkable way in which pure mathematics is indispensable to the
empirical sciences. That an abstract mathematical theory can be developed without
consideration of its possible interpretations and then prove to be a model for physical
reality, as in the case of Riemannian geometry, can be seen as support of a realist
conception of mathematics.