Proposed rule: People writing about the history of mathematics, should learn something about the history of mathematics.
Mathematicians didn't just randomly decide to go to abstraction and the foundations of mathematics. They were forced there by a series of crises where the mathematics that they knew fell apart. For example Joseph Fourier came up with a way to add up a bunch of well-behaved functions - sin and cos - and came up to something that wasn't considered a function - a square wave.
The focus on abstraction and axiomatization came after decades of trying to repair mathematics over and over again. Trying to retell the story in terms of the resulting mathematical flow of the ideas, completely mangles the actual flow of events.
I have to disagree with this. Modern (pure) mathematics is abstract and very often completely detached from practical applications because of culture and artistic inspiration. There is no "objectivity" driving modern pure mathematics. It exists mostly because people like thinking about it. Any connection to the real world is often a coincidence or someone outside the field noticing that something (really just a tiny-tiny amount) in pure maths could be useful.
> forced there by a series of crises where the mathematics that they knew fell apart
This can be said to be true of those working in foundations, but the vast majority of mathematicians are completely uninterested in that! In fact, most mathematicians today probably can't cite you the set-theoretic (or any other foundation) axioms that they use every day, if you ask them point-blank.
Mathematicians didn't just randomly decide to go to abstraction and the foundations of mathematics. They were forced there by a series of crises where the mathematics that they knew fell apart. For example Joseph Fourier came up with a way to add up a bunch of well-behaved functions - sin and cos - and came up to something that wasn't considered a function - a square wave.
The focus on abstraction and axiomatization came after decades of trying to repair mathematics over and over again. Trying to retell the story in terms of the resulting mathematical flow of the ideas, completely mangles the actual flow of events.