Abstraction is a powerful mathematical technique that has influenced everything from quantum theories of gravity to nuclear deterrence theory. But has it gone too far?
Axiomatics: Mathematical Thought and High ModernismThe 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 300Yet mathematics has undergone tremendous changes, especially during the twentieth century, when it pushed ever deeper into the realm of abstraction.
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.
Often, this meant not applying existing mathematical concepts, but inventing new ones. An influential example was the 1944 book, which established modern game theory and included very non-playful implications for strategies of nuclear defence.
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