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The first examples that spring to mind are groups of numbers. But groups crop up in every area.
Work on the notion of infinity led to paradoxes. This forced mathematicians to formalize set theory. Progressive axiomatization led to the current ZFC system, which nonetheless remains subject to various shortcomings following the work of Kurt Gödel and Paul Cohen.
When we first start working with groups, we patiently draw up the tables for those with only a few elements. A one-element group consists solely of the identity element and is therefore unique. Similarly, groups with two or three elements are unambiguously determined. The surprises begin with four elements...
The era of so-called "modern mathematics" brings back many memories, some pleasant, others decidedly less so. But does that mean we should throw the baby out with the bathwater?
