As to the broader questions of formal methods in the social sciences, well, that is a huge topic with a vast literature. BTW I agree that the kind of dynamics you’re talking about can’t be modeled verbally. We may disagree about the value added of such modeling, however.

As for explaining convergence to equilibrium, I’m not familiar with Menger’s theory. If it looks anything like the neo-Austrians’, though, it is pretty rough and doesn’t lead to specific predictions or give much better intuition than the non-mathematical “just-so” stories in the textbooks about gluts and shortages.

Convergence to equilibrium is a pretty hard and complex problem if you actually want to say something about relative speed of convergence in different markets, price dispersion along the way to equilibrium, impact of different information conditions, etc. The idea that we could make much headway on these questions without some kind of formal modeling (at least an agent-based computer model, if nothing else) seems far-fetched to me. I would be delighted to be proved wrong about this, and people are of course welcome to try, but it seems quixotic at best..

And yes, you’re right, simply invoking the word “cause” doesn’t make a causal argument. But Menger Sr. does, in fact, offer a causal explanation for price determination. It’s quite distinct from, e.g., Walras’s. And he and his followers do have a theory of price dynamics. It’s only since the post-WWII formalization of the discipline that we’ve lost these notions, IMHO.

Reading Israel Kirzner’s work I was struck by how much his verbal arguments rely on the reader already knowing the pictures for supply and demand, indiference curves, etc. That’s smart on his part.

As for causality and math versus causality and words: Come on. Saying the word “cause” doesn’t capture anything about the mechanism by which prices are set. And “causal” processes can be described in formal terms with dynamic models, at least if you’re not a strict Humean skeptic.

It is a fair critique of the simple supply-demand math that it doesn’t include the stories about how shortages and gluts lead to price dynamics, but that isn’t a problem with math. That’s a problem with the theory. (You can add price-adjustment mechanisms, but I think it is safe to say that individual optimizing models of price dynamics are not fully satisfactory.)

As for the more general issue under discussion, well, there is a huge literature on this. For now, if you want an economic proposition that many find true and useful, but that cannot be expressed mathematically, how about:

“Prices are determined by supply and demand.”

The concept of (causal) determination is not easily expressed mathematically. The standard supply-and-demand models, from the Econ 101 version to Arrow-Debreu and beyond, say simply that if the price is equal to some specified price, then the quantity supplied equals the quantity demand, and not otherwise. We can define an equilibrium price mathematically, but such models do not explain what “caused” it.

Of course, the usual response from mathematical social scientists is that causality is not an operationally meaningful concept. It is a non-scientific (or pre-scientific), “philosophical,” or even worse, “literary” concept. Again, that begs the question of what constitutes a scientific explanation.

Along these lines, readers might find the Cohn article cited in this post of interest.

Are there any important recent example of how “fallible” mathematics is? Even mistakes humans make, don’t matter much to the discussion. This is why linguists separate performance and competence.

“Thus, there could easily be a book written on the “Rhetoric of Mathematics.””

There is one: Imre Lakatos, Proofs and Refutations

A book worth a look is by Philip Davis and Reuben Hersh titled: THE MATHEMATICAL EXPERIENCE. Davis and Hersh write that: “the line between complete and incomplete proof is always somewhat fuzzy, and often controversial” (1981: 34; cf. p. 40).

Real proofs “are established by ‘consensus of the qualified’” and are “not checkable … by any mathematician not privy to the gestalt, the mode of thought in the particular field … It may take generations to detect an error” (1981: 354).

Mathematical knowledge is “fallible, corrigible, tentative, and evolving, as is every other kind of human knowledge” (1981: 406)

The main point is that Economists and Strategic Management Scholars (like you, Nicolai and myself) need not be overly concerned that we have no bedrock of ultimate Truth. Standards of mathematical proof also change.

Another book by mathematician Morris Kline, titled: MATHEMATICS: THE LOSS OF CERTAINTY says much the same: “There is no rigorous definition of rigor. A proof is accepted if it obtains the endorsement of the leading specialists of the time and employs the principles that are fashionable at the moment. But no standard is universally acceptable today” (1980: pp. 6 and 315)

These quotes can be found in D. McCloskey, THE RHETORIC OF ECONOMICS. Thus, there could easily be a book written on the “Rhetoric of Mathematics.”

(p > p’) => f(p) =Man, Economy, and State, he tries to show that everything follows from one single assumption. When translating his arguments into math, it becomes obvious that he is constantly introducing new assumptions. You can’t do that with formal economics.

I have never seen a single example of a economic theory that is both consistent and cannot be translated to mathematics.
I have however seen many examples of verbal “theorems” that are not theorems at all (the Coase theorem, the rotten kid theorem…)