8. Geometries and space
1. Introduction
Most people are unaware that around a century and a half ago a revolution took place in the field of geometry that was
as scientifically profound as the Copernican revolution in astronomy and, in its impact, as philosophically important
as the Darwinian theory of evolution. (Greenberg 1974, p. ix)
It is a general conviction that geometry, with all its truths, is valid with unconditioned generality for all men, all times,
all peoples, and not merely for all historically factual ones but all conceivable ones. (Husserl 1989, p. 179)The aim of this chapter is to consider one of the classic ‘stories’ in the history of mathematics:
the origin of non-Euclidean geometry. Although in some ways a part of the arrival of modern,
abstract mathematics (which is generally thought to be about itself ) as a replacement for tra-
ditional mathematics (which is, again in our usual version, about things and the world), the
story has traditionally been told as one of a particular process of discovery; the problem of Euc-
lid’s parallel postulate, and the invention of the non-Euclidean geometries by Lobachevsky and
Bolyai in the 1820s. It has the virtues of good stories: a connected thread, even a hero/heroes,
gropings for a solution followed by an unexpected twist. Its defects, as historians are sometimes
anxious to point out, are that history is more complex, and to construct such a story some
important details must be left out or simplified. The ‘classical’ history (Bonola 1955) which is
full, careful, and scholarly, is nearly 100 years old, and not surprisingly for some time there have
appeared criticisms, attempts to tell the story differently, or to tell a different story altogether.
The questions raised are typical ones in the history of scientific revolution, which were already
discussed by Bachelard in the 1930s: when mathematicians discover a completely different way
of doing mathematics (in this case, geometry), are they adding to the old mathematics, repla-
cing it, or giving us a new perspective from which the old (Euclidean) is a special case of the
new? To what extent is the previous pursuit of Euclidean geometry made invalid or irrelevant?
And so on.
Before we start, we face various problems; one particular one concerns geometry itself. Partly
(but not only) as a result of the ‘revolution’ to which the quote refers, the study of geometry has
gradually become something of a second-class subject, at least in universities. True, Pythagoras’s
theorem and the criteria for congruent triangles are still part of ‘general culture’, but their
epistemological status tends to be hazy. So the reader might take a moment to reflect on two
questions.
- What is geometry about—what is its subject-matter?
- How do we know that its results are true?
The answers to these will of course be influenced by your education as well as by personal opinion,
but to have thought about them may help. In previous chapters, a too definite knowledge of