While there certainly exists math for the sake of math, there is a trickle down effect that is quite real (there’s also a trickle up effect that is real but that’s unrelated). Someone does some math for the sake of math. Later on, someone who is slightly more applied sees a link between that math and a more applied problem they’re working on. If the idea is truly useful, it propagates down all the way to application-focused practitioners. Researchers exist on a spectrum, generally, between pure theory and pure application.

Math has no application until you find an application for it. Differential equations are just equations until you pair them with physics. Formal logic is just an abstract discussion of human reasoning until you build a circuit, etc.