- Logic as a field of study is concerned with the question of what implies what—more precisely, the question of what kinds of propositions imply what kinds of propositions—and logic as a tool (e.g., the notation) makes it easier to figure out and keep track of what implies what.
- To be a logical person is to be good at taking logical implication into account. You don’t let mutually contradictory beliefs take up residence in your head, for example. But in using the tools of logic in order to be logical, there must be something that you’re being logical about. It’s misleading to talk about reducing mathematics to logic because mathematics isn’t just about what implies what or making tools to help with that. Mathematics is about what implies what when thinking about number, shape, space, and time. That is, logicism is misleading because logic is only about how to think; it doesn’t say what to think about. Mathematics does use that kind of thinking, i.e. the logical kind of thinking, but for the purpose of certain kinds of inquiry. In short: Mathematics isn’t just logic. Mathematics is logical thinking about certain things; viz., it’s the pure logic of number and shape, space and time. It’s everything that can be figured out about those concepts. It’s the (precisely formulated) a priori substratum of any a posteriori field making (precise) use of those concepts, physics being the most obvious example.
- According to Mises, logic, mathematics, and praxeology are the a priori fields. To that I’d add: If mathematics is the pure logic of number, shape, etc., then praxeology is the pure logic of action.
Logic, mathematics, and praxeology
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