August 12, 2015

From Hilbert’s hotel to Hilbert’s method

Comment on ‘Infinite Debt, Future Generations, and Hilbert’s Hotel’


There are many ways to abuse economics, the three most popular are politics, entertainment, and kindergarten. In the economics kindergarten, they love to play with tiny toy models. And, you know, kids are a bit ad hoc; they start with beer, add the medium of exchange and then IOUs, introduce interest out of the blue, and finalize with the Erlang number. However, what looks like utter confusion is the well-established economic method: “... twentieth-century neoclassical theory resembles nothing so much as the child's game of Mr. Potatohead — the fun comes in mixing and matching components with little or no concern for the coherence of the final profile.” (Mirowski, 1995, p. 294)

Coherence and consistency were never the strong points of the representative economist (2013). But infinite gallimaufry is the privilege of the philosophical economist.

It should be, first of all, clear that there is no such thing as a ‘real’ economy. Because the economy comes into being as the interaction of real and nominal variables, all real models are garbage — even beer models. Second, in economics, infinity should not be used to push a problem out of sight. It is silly to argue that debt is not a problem if it never has to be redeemed.

Hilbert’s hotel is a fine example of all that is wrong with economics. The irony is that Hilbert was the most famous proponent of the correct scientific method, which came to be known as the axiomatic-deductive method. It works as follows.

“When we assemble the facts of a definite, more-or-less comprehensive field of knowledge, we soon notice that these facts are capable of being ordered. This ordering always comes about with the help of a certain framework of concepts... The framework of concepts is nothing other than the theory of the field of knowledge. ... If we consider a particular theory more closely, we always see that a few distinguished propositions of the field of knowledge underlie the construction of the framework of concepts, and these propositions then suffice by themselves for the construction, in accordance with logical principles, of the entire framework. ... The procedure of the axiomatic method, as it is expressed here, amounts to a deepening of the foundations of the individual domains of knowledge — a deepening that is necessary for every edifice that one wishes to expand and to build higher while preserving its stability.” (Hilbert, 2005, pp. 1107-1109)

Needless to say that Hilbert’s method has never been applied properly by either orthodox or heterodox economists. Both camps are still stuck with the ‘child's game of Mr. Potatohead.’ This explains why economics is a failed science.

The fundamental mistake in the actual discussion of debt is that the pivotal relationship between increase/decrease of debt and profit/loss is entirely missing. For the correct theory see (2014).

Egmont Kakarot-Handtke

Hilbert, D. (2005). Axiomatic Thought. In W. Ewald (Ed.), From Kant to Hilbert. A Source Book in the Foundations of Mathematics, Vol. II, 1107–1115. Oxford, New York: Oxford University Press.
Kakarot-Handtke, E. (2013). Confused Confusers: How to Stop Thinking Like an Economist and Start Thinking Like a Scientist. SSRN Working Paper Series, 2207598: 1–16. URL
Kakarot-Handtke, E. (2014). Mathematical Proof of the Breakdown of Capitalism. SSRN Working Paper Series, 2375578: 1–21. URL
Mirowski, P. (1995). More Heat than Light. Cambridge: Cambridge University Press.

For more about Hilbert see AXECquery.