Maths and abstraction

There's a book review at Nature that is somewhat interesting - Axiomatics: Mathematical Thought and High Modernism, which apparently argues this:

The work of mathematicians from centuries or even millennia ago speaks to their living peers in ways that practitioners of other disciplines must find baffling. Euclid’s proof that the list of prime numbers never ends is just as elegant and clear now as it was in around 300 bc, when it appeared in his book Elements.

Yet mathematics has undergone tremendous changes, especially during the twentieth century, when it pushed ever deeper into the realm of abstraction. This upheaval even involved a redefinition of the definition itself, as Alma Steingart explains in Axiomatics.

A historian of science, Steingart sees this revolution as central to the modernist movements that dominated the mid-twentieth century in the arts and social sciences, particularly in the United States. Mathematicians’ push for abstraction was mirrored by — and often directly triggered — parallel trends in economics, sociology, psychology and political science. Steingart quotes some scientists who saw their liberation from merely explaining the natural world as analogous to how abstract expressionism freed painting from the shackles of reality.

Further down it notes this:

To the mathematical-theory builder, abstraction is not a destination, but a journey. As Steingart puts it, ‘abstract’ is not an adjective but a verb: ‘to abstract’. In the 1930s, owing largely to the influence of German mathematician Emmy Noether, mathematicians began to construct axiomatic systems that were increasingly abstract and general. This revealed familiar objects such as numbers, card shuffles and geometrical symmetries to be special cases of the same concept.

The trend towards abstraction and generalization is often associated with a school of mathematics that blossomed in France after the Second World War. But, as Steingart shows, it took root in the 1930s in the United States and came to define the country’s mid-century mathematical culture. Steingart exemplifies the trend with the story of Foundations of Algebraic Topology, a 1952 book by US mathematicians Samuel Eilenberg and Norman Steenrod. It dealt with various calculation techniques to distinguish between geometric shapes, but the authors introduced the subject backwards, claiming that students should first familiarize themselves with highly technical algebraic tools and only later learn their relevance to shapes, or why the tools existed in the first place.

This reminds me of the argument Paul Johnson made in Modern Times (his history of most of the 20th century) - that the relativism in Einstein's physics introduced (or helped spread) moral relativism to the masses.   (I know many dispute that, but I think technology and science probably does have subtle, not always recognised, effects on the psychology of the masses.)