Notes on Geometry and Epistemology, Phil 2234:
1. The Tradition: From the time of Euclid, geometry was regarded as the best example of just
how human knowledge of the world should work. Beginning with the axioms (and
definitions), Euclidean geometers set forth to prove all the theorems of geometry. Since the
truth of the axioms was generally regarded as self-evident, and since the theorems followed
from those axioms (as we could see simply by following through each proof), geometry
seemed to show that we could know substantial facts about the world (about the results of
certain measurements, for example) by means of “pure reason”. Thus geometry seemed to
support the idea of a priori knowledge, that is, the idea that we can know things about the
world without depending on experience to tell us about them. The notion that we have such
knowledge (and its employment in the construction of systems of metaphysics) is one of the
central themes of rationalism, one of the two influential forms of philosophy to emerge in
early modern times. (Descartes, in particular, held that we had innate ideas of the basic
natures or properties that things could possess, and that we could clearly and distinctly grasp
how these natures had to be related to each other without any need for experience to show us.)
2. The sticking point: But from very early on, geometers were concerned about Euclid’s fifth
axiom. This axiom says that given any line and a point not on that line, one and only one line
can be drawn through the point, parallel to the line. Unlike the other four (which say simple
and straightforward things like “Given any two points, one and only one (straight) line can be
drawn connecting them.” And “A circle of any radius can be drawn with any given point as its
centre.”, this “parallels” axiom is complex enough and hard enough to be intuitively certain
about that many thought it really ought to be proven as a theorem, rather than left as an
axiom.
3. Efforts to prove the parallel postulate: Efforts to prove the axiom go way back, including
many of the great Arab geometers, and continue up to Saccheri, a Jesuit priest who actually
thought he had succeeded in “cleansing Euclid of all blemish”. Saccheri took the approach of
trying to prove the axiom by reductio. That is, he assumed that it was false and tried to derive
a contradiction. There are (of course) two ways that it could be false. There could be no
parallel lines at all, through the given point, or there could be more than one. From both of
these, Saccheri tried to show that some contradiction would follow. And in fact, from the first
(the “no parallels” assumption) he was able to do it. The other axioms do require that there be
at least one parallel (though they can easily be adjusted to remove this impediment). But from
the second, “many parallels” assumption, though he did derive some odd theorems (one of
which he mistook for a contradiction), in fact he derived no contradiction at all. In fact, as has
been shown since, there are systems of geometric axioms that include the “no parallels”
axiom, and others that include the “many parallels” axiom, which are provably consistent IF
Euclid’s axioms are consistent.
4. Non-Euclidean Geometry: The upshot of all this was the recognition, early in the nineteenth
century, that there were non-Euclidean systems of geometry. The geometrical truth, it
seemed, could no longer be decided a priori, by pure reason. There were now three different
“classes” of geometry: Bolyai-Lobachevskii, Riemannian, and Euclidean. Riemann, in
particular, systematized the whole range of geometries for “curved” space. These geometries
are Euclidean in the limit, that is, by confining ourselves to a small enough area, we can make
the geometry as close to Euclidean as we want. But as the area gets larger, the geometry
differs more and more from Euclid’s.
Geometry Bolyai-Lobachevskii
Euclid
Rieman
n
Parallels
Many
One
None
C/D for a
circle
>
=
<
Sum of
interior
angles for
a triangle
< 180o
= 1800
> 180o
5.
6.
7.
8.
Physical Space and Visualization: A further objection was also raised to the idea that threedimensional physical space could really be non-Euclidean. We can easily picture nonEuclidean surfaces—the surface of a sphere obeys the principles of Riemannian geometry:
lines are great circles (i.e. circles that cut the sphere in half—these give shortest distance paths
along the surface), there are no parallels (all great circles intersect), triangles have angle sums
greater than 180o, etc. And the saddle surface (or the pseudo-sphere) gives us BolyaiLobaschevski geometry, with many parallels, etc. But these surfaces are embedded (when we
imagine them) in a Euclidean 3-space. How can we imagine the embedding space itself being
non-Euclidean? Some (Kant, for example) thought this was just impossible to do. But
Helmholtz suggested instead that what was required was not to imagine a three dimensional
space “from the outside”, in the fourth dimension (in parallel to what we do with these
surfaces embedded in 3-space), but instead to ask what measurement results we would get in a
non-Euclidean space. If we imagine our physical measurements giving us non-Euclidean
results, that just is imagining 3-space to be non-Euclidean.
Gauss’ Test: Gauss proposed to test which geometry applies (at least on the scale of the
earth) by measuring the angles contained in a large triangle formed by shining lights on the
peaks of three mountains. (This is entirely in the Helmholtz spirit, though it predates
Helmholtz’ work on the issue.) The results (unsurprisingly) came out to 180o, within
experimental error. But this result’s interpretation needs some real care. After all, the method
used seems to assume that light rays travel in straight lines. If they don’t, then the figure
enclosed by the light rays would not really be a triangle, and the sum of the angles contained
in it could differ from the sum of the angles in a real triangle. Of course we could test
whether the light was travelling in straight lines by finding some other way to measure
whether the lines the light followed were straight. For example, we could try to measure the
length of the paths the light followed. If these paths were the shortest paths from peak to
peak, then (since a line is the shortest distance between any two points) we would have to say
the lines were straight after all. But when we make this measurement, we are assuming that
nothing is distorting our measuring rods, making one path seem shorter when in fact it’s really
longer. And if the distortion is caused by a universal force, i.e., a force which distorts the
results of all measuring devices to exactly the same extent, then there is no way we can use
our observations to rule out the possibility that such a force has affected our measurements.
Poincarre’s Conventionalism: Poincarré’s response to this problem was to accept a form of
conventionalism. According to conventionalism, there is no objectively right geometry for
physical space. Instead, we can choose (conventionally) any geometry we want. Of course,
to make that geometry fit with actual measurement results, we may have to invoke universal
forces. These forces can be chosen in such a way as to “adjust” whatever measurements
(corrected for any differential forces affecting their outcomes) we make to fit our choice of
geometry. Given that we can make this choice, and fix any problems using universal forces,
Poincarre suggested that we ought to use Euclidean geometry because it is so simple and
convenient. But that is not to say that physical space is really Euclidean. According to
Poincarre, space is not really one way or another; it’s our choice of geometry that decides how
we will go about measuring the world.
Einstein vs. Lorentz: Einstein’s interpretation of the Lorentz equations is an interesting
switch. Lorentz interpreted his transformations as the effect of a universal force, caused by
motion through the ether, which would prevent us from detecting that motion in any way. But
Einstein derives them from the principle of relativity (for Galilean frames) and the constancy
of the velocity of light for all observers. So for Lorentz a universal force has distorted the
results of our measurements, but for Einstein, our measurements are just as they should be,
and the transformations simply express how our measurements will compare with the
measurements of someone moving with respect to us (at some constant velocity). Einstein
has dispensed with the ether, which Lorentz claimed was there but couldn’t be measured. So
in a clear sense, Einstein has a more economical theory.