Insofar as logic consists in information-preserving operations, the non-physically-applicable parts of math still correspond to the real world, in that they preserve the information about the real world which was put into formulating/locating the starting formal system in the first place.
This is what makes mathematics so wondrously powerful: formality = determinism, and determinism = likelihood functions of 0 or 1. So when doing mathematics, you get whole formal systems where the theorems are always at least as true as the axioms. As long as any part of the system corresponds to the real world (and many parts of it do) and the whole system remains deterministic, then the whole system compresses information about the real world.
Insofar as logic consists in information-preserving operations, the non-physically-applicable parts of math still correspond to the real world, in that they preserve the information about the real world which was put into formulating/locating the starting formal system in the first place.
Whereas the physically inapplicable parts don't retain real-world correspondence. Correspondence isn'ta n intrinsic, essential part of maths.
Among my friends interested in rationality, effective altruism, and existential risk reduction, I often hear: "If you want to have a real positive impact on the world, grad school is a waste of time. It's better to use deliberate practice to learn whatever you need instead of working within the confines of an institution."
While I'd agree that grad school will not make you do good for the world, if you're a self-driven person who can spend time in a PhD program deliberately acquiring skills and connections for making a positive difference, I think you can make grad school a highly productive path, perhaps more so than many alternatives. In this post, I want to share some advice that I've been repeating a lot lately for how to do this:
That's all I have for now. The main sentiment behind most of this, I think, is that you have to be deliberate to get the most out of a PhD program, rather than passively expecting it to make you into anything in particular. Grad school still isn't for everyone, and far from it. But if you were seriously considering it at some point, and "do something more useful" felt like a compelling reason not to go, be sure to first consider the most useful version of grad that you could reliably make for yourself... and then decide whether or not to do it.
Please email me (lastname@thisdomain.com) if you have more ideas for getting the most out of grad school!