# Contemporary reactions to the rise of axiomatization in the 19th/20th centuries

Starting somewhere in the 19th century, mathematics turned from the study of concrete objects to the study of objects satisfying enough properties to lead to interesting theorems. For example:

1. From permutation groups and groups of geometrical transfomations we got the group axioms.

2. From ad hoc methods of approximating functions in analysis we got metric spaces.

3. From metric spaces we got topological spaces.

Are there documented examples of mathematicians from this period explicitly noticing and commenting on this trend (either in mathematics in general, or in a particular field)? In particular, are there examples of resistance to the idea?

• You can try with : Leo Corry, Modern Algebra and the Rise of Mathematical Structures (2004) and Jeremy Gray, Plato's Ghost: The Modernist Transformation of Mathematics (2008). – Mauro ALLEGRANZA Feb 22 '15 at 21:03
• It's a bit unclear if you are interested in the turn from constructive to non-constructive/axiomatic methods, that met a lot of resistance in the early 20th century, or in the turn from concrete to abstract structures that happened a bit later, 1920s-30s, with much less resistance. In particular, would you consider Cantor's sets "concrete" or "abstract"? – Conifold Feb 23 '15 at 1:14
• @Conifold I would probably consider a set to be concrete, when studied for its own sake. I don't know what you mean by "non-constructive/axiomatic methods". Surely the turn towards abstract structures happened earlier, no? I thought the group axioms were from the late 19th century. – Jack M Feb 23 '15 at 9:19
• What people objected to the most was the use of excluded middle, infinities, axiom of choice, etc. ("this is not mathematics, this is theology", the rise of intuitionism and finitism). This was inextricably linked to abstract structures because they were defined axiomatically, and most required non-constructive methods to build meaningful theories. But this was not resistance to abstract structures as such. Galois already wrote "group axioms", but very few thought of them as defining "abstract structure" before van der Waerden's text of 1930. en.wikipedia.org/wiki/Moderne_Algebra – Conifold Feb 23 '15 at 21:05