If you study the history of modern mathematics one of the recurring themes is the collapse of the foundations. A realisation that the assumptions underlying a topic were not as strong as might be hoped. There are three classics, which (with a broad brush) might be described as:
- The collapse of Analysis
The problems with infinity and infinitesimals had been known to the Greeks who discussed their paradoxes. In the seventeenth century, however, arguments and proofs involving infinitesimals became more acceptable. A powerful system started to emerge developed most clearly by Newton and Liebniz: the Calculus. Some still had issues, Berkeley famously mocked the “ghosts of departed quantities”, but most persisted and were greatly rewarded. The problems did eventually arise in mathematical settings, for example in Fourier’s study of heat, and it required most of the nineteenth century for Cauchy and Weierstrass to give a rigorous version based on limits. - The collapse of Geometry
With issues arising in the basic assumptions of Analysis, mathematicians looked for firmer ground. Many felt that Geometry and, in particular the great work of Euclid’s Elements might provide this. Unfortunately there was a wrinkle in the Elements, often known as the Parallel Postulate, a statement that seemed far too complicated to be an assumption. Many had tried to show that the parallel postulate could be proved from the other axioms, and failed. Some had actually glimpsed something further, notably Khayyam and Saccheri. In fact, as Bolyai and Lobachevsky would eventually show, there are perfectly good geometries that obey all the axioms apart from the Parallel postulate. With the zoo of examples that started to be considered, could geometry did not look like such a good foundation. - The collapse of Arithmetic
Instead of geometry, therefore other systems were considered, ideas from logic, and Cantor’s set theory were brought into play. Hilbert hoped that a system could be created that was consistent, complete and decidable. Many took on the challenge, Russell and Whitehead created the magnificent Principia Mathematica that famously does not prove that 1+1=2 until page 362. As the ideas started to become clearer, however, the way was left clear for a deeper issue. The results of Kurt Gödel showed that, if we want to include arithmetic, we cannot hope for a system for mathematics that is both complete and consistent. Furthermore, we cannot even prove the consistency of our systems without resorting to a more powerful one (who’s own consistency cannot be proved without a more powerful system still). Some even wonder whether arithmetic is consistent!
As these collapses hit the ideas that fields rest on, one would expect there to be some consequences. Some areas that turn out to be fallacies. Yet this does not seem to be the case. The fundamental ideas of calculus remained the same, although one had to be careful about the exact functions you were talking about. The discovery of non-Euclidean geometries simply revealed additional worlds, all the old results held but some now needed to note an additional assumption. Even the work on undecidability leads, most obviously through Alan Turing, to the theoretical underpinnings of computers. In fact studying the deeper issues seems to open up new areas but not harm those that have been established.
I therefore have a question:
Have we ever lost any mathematics?
Are there mathematical areas that have simple collapsed, having been accepted widely as true, even rigorous? I would like to rule out the case where an area has been rendered unimportant by the development of different techniques. In that case the results still hold, but are no longer as interesting.
Edit: 10/5/12
Some great discussion on math overflow, including one serious candidate, Italian algebraic geometry.
