Spherical Accounting: Using geometry to embody developmental integrity

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It is universally accepted that the most basic form of accounting lies in the single answer to the question "what does it cost?" A more elaborate approach, leading to that figure, is to itemize in a list the costs which total to that single budget figure. The great historical innovation in ensuring control over financial accounts was the shift to double entry bookkeeping, subsequently taking the form of the spreadsheets that are basic to the project and program management of any modern institution. Spreadsheets facilities have of course been embodied into spreadsheet software.

The following text is concerned with how the third dimension is reflected in an actual geometrical representation of accounting -- moving beyond one-dimensional "budgets", and two-dimensional budget lines and spreadsheets. It is correct that standard spreadsheets offer 3-D graphics in addition to 2-D displays -- as a means of obtaining an overview of sets of datapoints. The focus here however is on the potential significance of the actual geometry in integrating disparate (or potentially incommensurable) preoccupations of an organization in such a manner as to heighten the coherence and integrity of the operation.

The only resources relating tangentially to this possibility seem to be materials referenced by Robert Grace (The Over-all Picture, 2000). The exercise here is intended to be suggestive of possibilities -- as a stimulus to imagination and creativity -- rather than implying closure on a well-defined new method.

Chris Lucas (Multidimensional Economics, 1999) outlines a complexity-based economics in the light of the following assessment.

Current economic theory reduces all things to one dimension, that of monetary value, in fact that seems to be the whole basis of the science. Yet value is not in itself a one dimensional concept, we value many things that are impossible to classify in such linear monetary terms, air or sunlight for example. Complex notions of value require a type of economics that is itself complex and multi-dimensional, a value system that goes beyond the trade based concept of material exchange and takes into account the wider needs of people and planet.

Vahe Poladian, et al (Time is Not Money: The case for multi-dimensional accounting in value-based software engineering, 2003) argue that:

Indeed, in theoretical economics all costs can, in principle, be expressed in dollars. Software engineering problems, however, often present situations in which converting all costs to a common currency is problematical. In this paper we pinpoint some of these situations and the underlying causes of the problems, and we argue that it is often better to treat costs as a multidimensional value, with dimensions corresponding to distinct types of resources. We go on to highlight the differences among cost dimensions that need to be considered when developing cost-benefit analyses, and we suggest mechanisms for mediating among heterogeneous cost dimensions.

Richard B. Dull and David P. Tegarden (A Comparison of Three Visual Representations of Complx Multidimensional Accounting Information, 1998) have investigated the relationship between three visual representations (two-dimensional, three-dimensional fixed, and three-dimensional rotatable) of multidimensional data, and the subjects' ability to make predictions based on the data (namely "cognitive fit"). Output of a momentum accounting system was simulated and graphics were rendered based on that information. They concluded:

The results of this study indicate that the form of the representation of data affects the accuracy of the predictions novices make based on that data. Additionally, one can conclude that multidimensional visual representation of complex multidimensional data results in greater decision making accuracy because it facilitates the direct examination of the complex relationships in the data. This conclusion has implications within the realm of accounting with respect to decision-making when multiple variables are involved. As variables increase in complexity (defined as dimensionality), there should be representations that show the interaction among the variables to help enhance decision-making accuracy. Furthermore, with the trend towards supporting decision-making with multidimensional data using on-line analytical processing (OLAP), knowledge discovery, and data mining tools, this conclusion implies that in the future, designers of advanced accounting information systems should be cognizant of the need to "fit" the dimensionality of the data to the dimensionality of the visual representation. Otherwise, the effectiveness of the decision maker may be compromised.

David Ellerman (Double-Entry Bookkeeping: The Mathematical Formulation and Generalization, 1986) explores the lack of any mathematical exploration of standard double entry bookkeeping which might enable its generalization:

With [one] exception, the author has not been able to find a single mathematics book, elementary or advanced, popular or esoteric, which notes that the ordered pairs of the group of differences construction are the T-accounts used in the business world for about five centuries. And this mathematical basis for DEB is totally unknown in the "parallel universe" of accounting. This almost complete lack of cross-fertilization between mathematics and accounting is a topic of some interest for intellectual history and the sociology of knowledge. The story is probably rather simple from the mathematics side. Double-entry bookkeeping is apparently too mundane to hold the sustained attention of mathematicians. The real question lies on the accounting side. Over the last century, how could professional accountants and accounting professors have failed to find the mathematical basis for DEB even though it was part of undergraduate algebra? [see also below]

In arguing for "a more flexible approach to logic chains", Réal Lavergne (Results-Based Management and Accountability for Enhanced Aid Effectiveness, 2002) states:

One of the dangers of the textbook logical framework analysis (LFA) is that it seems to imply a degree of orderliness and certainty about managing for development that often belies reality. At least superficially, the logic model assumes that there is consensus about the goals and objectives of the project and about the choice of strategies to achieve those goals and objectives. It also implies a linear chain of causality from inputs and activities to outputs and successive levels of outcomes, of the sort that one can observe in the HIV/AIDS example cited above. It suggests a degree of predictability that makes it possible to plan in advance and a capacity to measure outcomes that may be unrealistic in many cases.

Xavier Bry and Jean-François Casta (Synergy Modelling and Financial Valuation: Contribution of Fuzzy Integrals, 2003) point out that financial assessments are characterized by: the importance of the role assigned to human judgement in decision making, the use of qualitative information and the dominant role of subjective evaluation. They examine the specific problems raised by the modelling of synergy between the assets of a firm:

As a process which aggregates information and subjective opinions, the financial evaluation of the company raises very many problems relating to issues such as measurement, imprecision and uncertainty. The methods used in the process of financial evaluation are classically based on additivity. By construction, these methods abandon the idea of expressing phenomena of synergy (or redundancy, nay mutual inhibition) linked to over-additivity (or under-additivity) that may be observed between the elements of an organised set such as a firm's assets. This synergy (respectively redundancy) effect may lead to a value of the set of assets greater (resp. lower) than the sum of the values of all assets. This is particularly the case in the presence of intangible assets as good will.

"Triple-entry bookkeeping": Henning Kirkegaard (The Logic of Double-Entry Bookkeeping) proposes a new understanding of the inherent uncertainties of double-entry bookkeeping system in the light of the study on triple-entry bookkeeping by Yuji Ijiri (Momentum Accounting and Triple-Entry Bookkeeping: Exploring the Dynamic Structure of Accounting Measurements. American Accounting Association, 1989). He stresses that for five centuries the questions about the causes of claims and agreements have been left unanswered in accounting since double-entry bookkeeping has only two logical measurement points.

As a result, double-entry bookkeeping can o

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