## Wednesday, February 16, 2011

### Mathematical economics - double-entry bookkeeping

I want to talk a little about a mathematical description of accounting rather than economics in general. When I was making the move from Australia to Italy, which I finally did in 1998, I had some particular trouble understanding my financial resources. I thought about the method of keeping accounts introduced in Italy, and first described by Luca Pacioli in 1494, called double-entry bookkeeping or partita doppia.

It seemed very closely related to the work we had been doing on compositional concurrent systems using the compact closed bicategory of spans of graphs, so in 1997 I spoke at New Trends in Semantics in Bologna, and in 1998 we wrote a paper about partita doppia which was accepted by TAC subject to revisions which we never carried out. Much later (2008) I did make this paper available on arxiv, but in the mean time we had written about accounts in the published paper P. Katis, N. Sabadini, R.F.C. Walters, On the algebra of systems with feedback & boundary, Rendiconti del Circolo Matematico di Palermo Serie II, Suppl. 63 (2000), 123-156. We had also published a paper giving some details of the mathematical structure involved, namely P. Katis, R.F.C. Walters, The compact closed bicategory of left adjoints, Math. Proc. Camb. Phil. Soc., 130, 77-87, 2001.
This all came back to mind since I happened by chance on a blog by John Baez on Mathematical Economics dated January 8, 2011. He mentions that a student of his, Miguel Carrión Álvarez, had moved to economics after his PhD in physics, and he quotes Álvarez as follows:
"I believe the proper way to look at macroeconomics is as a flow network and as such ideas from category theory may be useful at least to organize one’s thinking. The economy is a network whose nodes are “agents” and links are “economic relations”. Economic relations are flows, of goods and services in one direction and of money in the other direction (opposite categories via arrow reversal?).
Each node also has a balance sheet: assets and liabilities, andit’s the liabilities that are the key here, because they are mostly monetary." You can find the rest here.
In the comments that followed this post our paper in arxiv was mentioned rather critically - the fact that the title was "On partita doppia" was clearly viewed as an affectation. Perhaps so. I was trying to emphasize the historical origins. Our introduction begins:
In 1494 Fra Luca Pacioli published in Venice one of the first printed mathematical books. One section, Computis e Scripturis, is the first published description of partita doppia or double-entry bookkeeping, the foundation of accounting. Double-entry bookkeeping had been developed over a period of  years by Italian merchants and bankers. The aim of accounting is the measurement of a distributed concurrent system, and it is our contention that it is one of the earliest and most successful mathematical theories of concurrency.
The other criticism by Baez was the lack of some higher categorical definitions. His criticism is partially justified - the abstract formal notions and their applications were developed slowly by a handful of mathematicians - but in any case he seems to be unaware of Cambridge Philosophical Society paper mentioned above.
I finish this post by recommending to Álvarez these old papers. I think they make some progress in the direction he also is considering. (I should not omit mentioning that in concurrency theory the use of (compact closed) monoidal categories is well-known beginning with David Benson in the 70's, then Montanari and Meseguer 1990, and later Samson Abramsky among others, and of course our work.)

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Todd Trimble said...

Bob, I plan to leave a link to this post over at Azimuth; I think it might be helpful.

9:47 AM
Miguel said...

I just found this - thanks, I'll make sure to read the paper on Paccioli.

4:45 PM
Unknown said...

This indeed is a very great piece of work and tactfully written. It is not just going to help accountants but common man as well

8:45 PM