My favorite Bertrand Russell book is Introduction to Mathematical Philosophy, not least for the perspective shift he pulls off just in the title. He elaborates on it like this (I’ll paraphrase): This isn’t philosophy of mathematics, where we’ll sit around and ask deep, open-ended, metaphysical questions about whether or not numbers really exist, or if they’re just in our heads. It’s mathematical philosophy, where we’re going to try to think about philosophy (including the philosophy of logic and mathematics) like mathematicians would, using mathematicians’ tools.
Here’s an example of how this works. There’s a famous proof of the existence of God by St Anselm, called the Ontological Argument. Let’s say God is just our idea of the most perfect thing possible. Everything that could be good, God is: he’s all-knowing, all-powerful, all-good. Well, then this most perfect thing possible would have to exist, because something that exists is better than something that doesn’t — so an idea of a God who doesn’t exist wouldn’t really be completely perfect, would it?
Kant had already said that this proof was baloney, because “existence” wasn’t a predicate like goodness or knowledge. But Russell and analytic philosophy took it one step further. In math and formal logic, existence isn’t a predicate — it’s a quantifier. Like in the sentence, “For every natural number, there is a larger natural number.” We’re not making deep existence claims here, just singling out an element in a system.
So if we can come up with a model that’s foundationally and structurally sound, and works, let’s use it. What looked like an impossible problem wasn’t a problem after all; we’d just gotten twisted up in the way we talked about it.
So philosophy of mathematics => mathematical philosophy. Change of grammar => change of perspective.
You can imagine all kinds of variations on this. For instance:
- science of politics
- politics of science
- scientific politics
This gets you three totally different approaches.
Sometimes, we get lots of ambiguity because we can’t pull this reversal off. For instance, “digital history” means both the history of digital technology (usually done using recognizably traditional historical methods) AND using digital tools to do historical research.
What else could we switch around so we could see things differently?