Math History – Understanding Hermann Weyl’s Statements

Question

"The introduction of numbers as coordinates is an act of violence." – Hermann Weyl.

A lot of people like this quote, apparently. They also seem to associate it to the manifold context in the obvious way: they interpret the quote as saying that focusing on coordinate charts is insightless, or at least dry. Although I aggree with this, I feel like the way the quote is phrased is not inclined to that interpretation. If not by the fact that Weyl is connected to geometry (and that this quote is in Bredon), I would interpret this phrase as saying that "Representing numbers by a given basis is insightless and can lead to mistakes, like thinking that divisibility is dependent of the number basis you are using." I think the words numbers and coordinates are strangely placed.

Therefore, my question is: What was the precise context on which this quote arose and, if possible, can we pinpoint exactly what did Weyl mean?

Answer 1

From a Google search, it appears the quote is from Hermann Weyl's Philosophy of Mathematics and Natural Science. I found a copy online here; the relevant passage is on page 90 (search "act of violence"):

The introduction of numbers as coordinates by reference to the particular division scheme of the open one dimensional continuum is an act of violence whose only practical vindication is the special calculatory manageability of the ordinary number continuum with its four basic operations. The topological skeleton determines the connectivity of the manifold in the large.

His meaning appears to turn on the idea of a "division scheme." In a preceding paragraph, he wrote:

In general a coordinate assignment covers only part of a given continuous manifold. The 'coordinate' $(x_1, \ldots, x_n)$ is a symbol consisting of real numbers. The continuum of real numbers can be thought of as created by iterated bipartition. In order to account for the nature of a manifold as a whole, topology had to develop combinatorial schemes of a more general nature.

To paraphrase, I believe he's saying that the topological structure of both the real number line and manifolds more generally is determined by how it breaks into smaller pieces, and how those pieces fit together (in modern terminology, we might look at a space's locale of open sets). To add a coordinate system rooted in arithmetic operations, either to the real line or to a manifold, in his view, is to add too much structure (although it does help with calculations).