- An a priori truth is such that it’s impossible to conceive of anything contrary to it. For example, it’s a priori true that there’s one dimension of time, for it would be impossible to imagine more than one dimension of time—at least I can’t figure out how to do so. An a posteriori truth, on the other hand, is such that it’s possible to conceive of something (or more than one thing) contrary to it. Here gravity works as an example: It’s a posteriori true that gravity exists and does what it does, for it’s possible to imagine a world without gravity or a world with a different kind of gravity.
- “Mathematics” starts out with its a priori axioms, and then in becoming “physics” it adds its physics-relevant a posteriori postulates. “Economics,” by contrast, stays “economics” through that same transition. (Note: “Axioms,” under my definition, are by definition a priori, which means that the phrasing “a priori axiom” is redundant. In the same way, “postulates” are by definition a posteriori, which means that the phrasing “a posteriori postulate” is redundant.)
- However, analogous to the distinction between “mathematics” and physics” can be made the distinction between “praxeology” and “economics.”
- Each axiom is either true or false—actually, it may be better to say that each axiom is either coherent or incoherent—in that each axiom is either (a) “fundamental” or not and (b) a priori true or not—i.e. coherent or not. The postulates, on the other hand, are chosen relatively freely, for a system of postulates can describe either reality or a hypothetical; a system of postulates, as long as each postulate is “fundamental,” can be a coherently counterfactual system made out of the minimal number of assumptions. Axioms, being minimal a priori assumptions, have no conceivable alternatives, whereas postulates, being minimal a posteriori assumptions, do have conceivable alternatives.
- From the axioms and postulates come the theorems. From something “minimal” comes something “maximal.”
- Mathematics is the pure logic of space and time, number and shape. In other words, mathematics is the a priori foundation of any a posteriori field involving space, time, number, or shape. What’s praxeology, then? I’d say that praxeology is the pure logic of action, i.e. the a priori foundation of any a posteriori field involving action. That is, mathematics is to physics (and some other fields) as praxeology is to economics (i.e., the study of money), linguistics (i.e., the study of words), etc.
- Note, though, that with good enough notation people may start to think of praxeology as part of mathematics.
- To use pure reason is to reason purely from the a priori. Mathematics and praxeology, then, are exercises in pure reason.
- An impressive feat of redundancy: “The pure theory of action, i.e. praxeology, partakes of apodictic certainty, gets its truth a priori, and makes use of axioms only (with no postulates).”
- Mises talks about the importance of using “imaginary constructions.” Interestingly, an “imaginary construction” is a counterfactual postulate.
- Consider the distinctions between (a) “reason” and “experience” and (b) “rationalism” and “empiricism.”
- An a priori truth comes “prior” to experience, and an a posteriori truth comes “posterior” to experience.
Praxeology and mathematics
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