### Double entry bookkeeping again

I have had some interesting correspondence with Professor David Ellerman about our arxiv paper on double entry bookkeeping (a version of which was also included in the published paper

also available here).

Professor Ellerman has written extensively (prior to our work) on the mathematical treatment of double-entry bookkeeping. For his views one should consult his papers and blog, but his main slogan seems to me to be that double-entry bookkeeping is an elementary application of the group of differences construction (that is, the construction of the additive group of integers from the abelian monoid of natural numbers).

Our view was that there is another context in which to study the subject, namely the algebra of concurrent distributed processes. There accounting amounts to measuring such processes, not simply in an abelian group but in a kind of distributed abelian group.

However, from the discussions with Professor Ellerman I see that in the section where we give an example of traditional accounting we have omitted an important aspect.

It was difficult to write the section because an accounting system is a complicated thing with a lot of structure and there is debate about just what is the minimum to constitute a double-entry accounting system. This debate has bedevilled the historical research into the origins of partita doppia.

Before describing the main matter there is a minor thing I might like to change, namely that we used negative and positive numbers instead of credits and debits. That unfortunately suggests to a mathematician that we are using the ring of integers rather than the abelian group. The distributive law destroys the duality between positive and negative, between debits and credits. Professor Ellerman calls the group of debits and credits the Pacioli group in honour of Luca Pacioli who wrote at a time (1494) when there was no ring of integers.

We believe that the principal aspects of partita doppia is that a system of accounts is closed, and that in any transaction a pair (or more) of accounts is involved, a debit being made to one account whereas a credit is made to another, and further that the total is invariant. These are the key properties of a system of accounts in our algebra. To me these are the properties which characterize double-entry accounting, and the origin of the name.

However there is another crucial feature of partita doppia in practice, namely that the state of an account is held as a pair of numbers, a debit and a credit total, and not just as the (debit or credit) balance. (In the history of transactions in an account the debits are written to the left and credits to the right.) Further the accounts in a conventional system are divided into accounts which are normally credit accounts and those which are normally debit accounts, which helps, via the "golden rules of accounting", to keep track of which account should be credited and which debited.

When I get time I will add some new material in the example in the arxiv paper in which the state of a conventional T-account will be a pair of natural numbers, debit and credit, rather than an integer.

*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*,also available here).

Professor Ellerman has written extensively (prior to our work) on the mathematical treatment of double-entry bookkeeping. For his views one should consult his papers and blog, but his main slogan seems to me to be that double-entry bookkeeping is an elementary application of the group of differences construction (that is, the construction of the additive group of integers from the abelian monoid of natural numbers).

Our view was that there is another context in which to study the subject, namely the algebra of concurrent distributed processes. There accounting amounts to measuring such processes, not simply in an abelian group but in a kind of distributed abelian group.

However, from the discussions with Professor Ellerman I see that in the section where we give an example of traditional accounting we have omitted an important aspect.

It was difficult to write the section because an accounting system is a complicated thing with a lot of structure and there is debate about just what is the minimum to constitute a double-entry accounting system. This debate has bedevilled the historical research into the origins of partita doppia.

Before describing the main matter there is a minor thing I might like to change, namely that we used negative and positive numbers instead of credits and debits. That unfortunately suggests to a mathematician that we are using the ring of integers rather than the abelian group. The distributive law destroys the duality between positive and negative, between debits and credits. Professor Ellerman calls the group of debits and credits the Pacioli group in honour of Luca Pacioli who wrote at a time (1494) when there was no ring of integers.

We believe that the principal aspects of partita doppia is that a system of accounts is closed, and that in any transaction a pair (or more) of accounts is involved, a debit being made to one account whereas a credit is made to another, and further that the total is invariant. These are the key properties of a system of accounts in our algebra. To me these are the properties which characterize double-entry accounting, and the origin of the name.

However there is another crucial feature of partita doppia in practice, namely that the state of an account is held as a pair of numbers, a debit and a credit total, and not just as the (debit or credit) balance. (In the history of transactions in an account the debits are written to the left and credits to the right.) Further the accounts in a conventional system are divided into accounts which are normally credit accounts and those which are normally debit accounts, which helps, via the "golden rules of accounting", to keep track of which account should be credited and which debited.

When I get time I will add some new material in the example in the arxiv paper in which the state of a conventional T-account will be a pair of natural numbers, debit and credit, rather than an integer.

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