Today, I was working on a piece I’m writing about 19th century developments in mathematics and I saw something interesting. In the piece, I draw particular attention to a few things. One of these is the precision Weierstrass brought to the concept of a limit, removing all references to motion or geometry, and giving it a very focused arithmetic definition. At the same time, Riemann found the foundation of geometry in large general notions like manifold or manifoldness. His general notions supported the non-Euclidean geometry that had already developed from investigations of Euclid’s parallel postulate. Finally I wrote about Cantor’s insights into infinities – his observation that, for example, the set of rational numbers is equivalent to the set of natural numbers (the whole numbers we count with) while the set of real numbers is not countable and therefore a larger infinite collection. The set theoretic ideas that Cantor finally shaped provide the ground for nearly all of modern mathematics.
These are very sketchy references to just a few of the prolific accomplishments of this period, a time during which a more conceptual approach to problems became favored. But each of them had a significant impact on the future of the discipline. What I found interesting today is that they each wrestle with some of the fundamental features that define mathematics – arithmetic, generalization and order. And they each also point back to (and solve) difficulties created by the notion of infinity.
The arithmetic Weierstrass brought to calculus relieved any lingering discomfort there may have been with hazy geometric definitions and infinitesimal quantities. While Riemann had no interest in the axiomatic issues in geometry, his conceptual manifold gave new meaning to the non-Euclidean geometries that were born of attempts to investigate the parallel postulate (the one that requires that parallel lines will not intersect no matter how far they are extended). There is no way to actually look at this infinite extension and there is no way to prove the statement using other axioms and definitions. (About 30 years before Riemann’s famous lecture, Janos Bolyai and Lobachevsky independently confirmed that entirely self-consistent “non-Euclidean geometries” could be created in which the parallel postulate did not hold). And Cantor could see that there was some order (and arithmetic) to be found in otherwise unmanageable infinite collections.
Riemann acknowledged the influence of the philosopher Herbart who argued that space is not a container that holds the world around us, but more a kind of organizing principle of cognition. Riemann’s manifold, as a collection of objects, also points to the idea of sets, important to the future of the discipline.
Infinite lengths, or infinite collections, or an infinite number of times, cannot actually be seen, but they can be precisely imagined. Perhaps this is because they grow out of experiences (like repetition) that keep getting worked on by overlapping and complex cognitive mechanisms. Like the energy of wind and water, that seems to just be there, without direction or intent (and particularly unresponsive to our needs), an imagined infinity, after careful observation, can be captured, harnesses and directed.
In his history of set theory (Labyrinth of Thought), José Ferreirós Dominguez writes about Riemann’s most general ideas:
By establishing the theory of magnitudes upon the foundation of manifolds, Riemann transgressed the limits of the traditional conception of mathematics, turning it into a discipline of unlimited extent and applicability, since it embraced all possible objects.
These ideas would also lay the groundwork for topology, where a new kind of geometric intuition, together with set-theoretic ideas, again recasts the idea of continuity.
A nice essay on the significance of Riemann’s conceptual mathematics can also be found here.
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