Questions for the methodologists out there

 I am seeking some clarification on the nature of true a priori synthetic propostions, in particular the axiom of action, and I was hoping that some of you who are more methodologically inclined can help me out.

I understand the following:  The axiom of action is a true a priori synthetic proposition.  That is, observations are not necessary to know that it's true, and yet, it tells us something valuable about the real world.  We know that it is true because any attempt to disprove the axiom of action necessarily involves a performative contradiction.  In other words, the attempt to disprove action involves acting. True a priori synthetic propositions might appear to pose a problem, however.    Since propositions like these rest on knowledge based on reflective understanding that comes from the mind, and yet tell us something concrete about the real world, it would seem that they must be idealistic instead of realistic.  That is, the real world must somehow conform to our minds, instead of the other way round.  However,  this is not so. According to Hoppe, true synthetic a priori propositions are not idealistic. (This is where my understanding starts to get a little hazy).  From  Economic Science and the Austrian Method:

"We must recognize that such necessary truths are not simply categories of our mind, but that our mind is one of acting persons. Our mental categories have to be understood as ultimately grounded in categories of action. And as soon as this is recognized, all idealistic suggestions immediately disappear. Instead, an epistemology claiming the existence of true synthetic a priori propositions becomes a realistic epistemology."

"Acting is a cognitively guided adjustment of a physical body in physical reality. And thus, there can be no doubt that a priori knowledge, conceived of as an insight into the structural constraints imposed on knowledge qua knowledge of actors, must indeed correspond to the nature of things. The realistic character of such knowledge would manifest itself not only in the fact that one could not think it to be otherwise, but in the fact that one could not undo its truth."

This is where I'm not sure.  I think Hoppe means the following, but please correct me if I'm wrong (and I probably am):  Many ideas and concepts spring from the mind, but only a very small amount of a priori knowledge has relevence in reality.  There's a limitless amount of a priori knowledge that results in any number of analytic propostions, but in order for an a priori axiom to qualify as a synthetic proposition, something else is required.  And that something else is the performative contradiction, "performative" being an important word. The attempt to disprove the axiom of action involves doing something as opposed to just thinking something. The performative contradiction, not only confirms the truth of the proposition, but acts as a filter, sifting out any a priori knowledge that doesn't conform to reality and allowing only that which does to make the grade. Since this process is grounded in reality, anything which does make the grade must be grounded in reality also.    Is my understanding correct? 

My seond question is this.  There are examples of a priori synthetic propositions in mathematics and geometry.  Can anyone give me examples of performative contradictions concerning axioms in these fields?

Just to note something: synthetic a priori truths are not true because they cannot be contradicted without presupposing their very validity; it is rather the opposite, that they cannot be contradicted because they are true. Without the notion of action and of causality observational experience is an impossibility. This evidences their essential nature. Russel considered universals (e.g. relations such as "in", "and" etc.) to be synthetic a priori truths. I think your understanding is more or less correct though; whenever one must presuppose a truth to deny it, it is an indicator of its nature as an a priori truth. Mises argued something along these lines in The Ultimate Foundation of the Economic Science, as have Hollis and Nell in Rational Economic Man.

Inquisitor:

Just to note something: synthetic a priori truths are not true because they cannot be contradicted without presupposing their very validity; it is rather the opposite, that they cannot be contradicted because they are true.

Don't you have that kind of backwards, or am I missing something? Here's an excerpt from Hoppe from the same paper.

"On the contrary, like Kant before him, Mises very much stresses the fact that it is usually much more painstaking to discover such axioms than it is to discover some observational truth such as that the leaves of trees are green or that I am 6 foot 2 inches. Rather, what makes them self-evident material axioms is the fact that no one can deny their validity without self-contradiction, because in attempting to deny them one already presupposes their validity."  (my emphasis)

He's saying: If you attempt to disprove it, you have to presuppose the very thing that you're trying to disprove. In other words, you engage in a performative contradiction. Therefore, it must be true. If I'm wrong please correct me. 

Anyway, my primary question had to do with explaining why a priori synthetic propositions are not idealistic.  Any thoughts on that?

 

He's referring to how we may discover such truths; my point was that one mustn't confuse how they may be discovered with why their denial entails performative contradictions (i.e. because they are essential truths.) Note how Hoppe mentions that one cannot undo their truth. Something is a contradiction not because we could not visualize otherwise, but rather because it really does not hold in the world.

Again, I think you're correct in your interpretation of Hoppe; that is how I understood him as well.