[umutisik]

I agree that it's hard to imagine an alternate history when things happened through a mixture of pure and application-motivated work. In each example, I can see how people arrive at these notions through an application-driven mind-set (transformation groups, GCD through simplifying fractions during calculations, solving polynomial equations that come up in physics calculations). Computability and complexity, in the flavor of Turing's and subsequent work, I already see as application-driven work, as they were building computing machines at the time and wanted to understand what the machines could do.

Related to this topic. I highly recommend this speech / article by Von Neumann: https://www.zhangzk.net/docs/quotation/TheMathematician.pdf