Perspective
Luca Pacioli is known for his works on accounting and on the mathematical theory of perspective.
These two subjects have in common to be applications of mathematics in the real world. Luca Pacioli was a mathematician, who had an encyclopedic knowledge of mathematics. His seminal work on the double entry bookkeeping method is part of an authoritative book on mathematics, where he summarized the state of the art at his time. The bookkeeping method describes the method used by the merchants in Venice, and its work on perspective is inspired by the works of artists who were both versed into mathematics and painting.
Luca Pacioli was close to Leonardo da Vinci, who once wrote that "the eye is a window of the soul". Perspective is a specific image produced by observing a scene from a certain vintage point. It implies that there is a viewing process at work. Similarly, the notion of "debit" and "credit" as described in his work on accounting depends on the position we look at things. From the point of view of business owners, their credit is what they owe to others, their debit is what others owe to them. From the opposite point of view, for example from a bankers' perspective, their credit is what the they lend to business owners, and their debit is what business owners have lent to them. A bank account is a debit account, from the bank's point of view, because it corresponds to what the banks are liable to their customers. A credit account is a loan to their customers.
Hence, conceptually speaking, there is a strong connection between how perspective is used in painting and how it is used in accounting. The position in which we are defines how we see things.
The importance that the notion of perspective played in Luca Pacioli's and Leonardo da Vinci's world view is the fact that they were engaged in a debate to raise perspective as a full-fledged science per se, in a discussion known as the "Controversy of Perspective".
In that sense, the conceptual foundations of the theory misnamed "theory of relativity" as developed first by Galileo and then by Einstein insists on developing a description of reality which is expressed differently depending on the position of the observer. The theory of relativity is misnamed, because it describes what is invariant. Philosophical relativism, which consists in asserting that there is no objective reality, because everybody can define his or her own, is antithetic to the scientific theory known as relativity.
In accounting, whether a business experiences a loss or a profit is not subject to interpretation. It is based on a thorough, systematic, organized recording of the financial transactions that have occurred in a period of time. No accountability is imaginable if no facts can be recorded and agreed upon.
Accountability of information follows the same thread. Perspectives on information assets depend on the relative positioning of the viewer. If information is considered to be transactional, and is recorded into accounts, then the perspectives are defined by the various ways in which it is possible to look at a piece of information. It doesn't negate the fact that the information itself exists.
Miscellaneous notes.
Links between perspective in mathematics and accounting.
The notion of perspective is invisible but present in the accounting theory of Luca Pacioli.
The mathematical disciplines are applied in multiple areas, including "Perspective" and "Commerce". The Great Art is the algebra and the Minor Art is the Practica Negotiara (Practical Commercial). Perspective is subaltern to the Geometry and Arithmetic.
Predominance of mathematics. What counts is the "eye" and perspective is the way things are interpreted. This goes into the heart of mathematics. In accounting, there is an ethical motivation (get 'good credit'), which is the way a merchant is seen by the others, which implies a perspective. See Fabio, p. 519.
Leonardo Da Vinci: "The eye is the window of the soul".
Double entry bookkeeping is about who sees what from which point of view.
Accounting is not only just adding number, it's about presenting a good picture of yourself and the business. It's like a good painting. A painting with no or bad perspective would be as bad as a ledger where there is a difference between credit and debit.
In Fabio Maia Bertato et al., perspective "initiated a phase in which the reality became understood in mathematical terms".
Luca Pacioli: Particularis de computis et scripturis:
The third and last thing is to arrange all the transactions in such a systematic way that one may understand each one of them at a glance, i.e., by the debit (debito--owed to) and credit (credito*-- owed to) method. This is very essential to merchants, because, without making the entries systematically it would be impossible to conduct their business, for they would have no rest and their minds would be troubled. For this purpose I have written this treatise, in which, step by step, the method is given of making all sorts of entries.1
This is a proposal for a visual clue on how to arrange the reporting of the monetary transactions in such a way that make visible the results of their business activities. It is arranged in such a way that it offers a perspective on a complex reality. The notions of "credit" and "debit" are views that can be captured at a glance.
First, we must assume that every action is determined by the end in view, and in order to pursue this end properly, we must use every effort. The purpose of every merchant is to make a lawful and reasonable profit so as to keep up his business.2
Also it said that the head of the merchant has a hundred eyes, and still they are not sufficient for all he has to say or to do.3
The basic principle of the difference between debit and credit is a point of view. From the point of view of a business owner, what he owes is his credit, -- he owes it to his creditors -- and what is owed to him is his debit. From the other side point of view, a creditor considers that the business owner has a debt to him, and a debitor considers that the business owner has made a credit to him. The different points of view is key to the double entry accounting. Therefore it is possible to say: "from the perspective of the business owner", the credit is money that he owes, and from the perspective of the creditor, the credit is money that the business owner owes him. In the last sentence, a perspective is understood as a point of view. In mathematics, the perspective is a view defined from the specific position of the observer.
Therefore, accounting uses perspectives at its core. In any transaction there are two points of view. The relation Manhattan is a borough of New York City can be reversed as New York City contains the borough of Manhattan. What's essential here is not so much the order (it's equivalent whatever way you see it), but the fact that there is one relation (monetary transaction in the first case, containment in the second one), and that the interpretation of the semantics depends on the relative position of the participant.
History of perspective
Italian painters of the Renaissance were also interested in science, and especially in mathematics. The theory of perspective was considered part of geometry, or an addition to geometry, and they applied it to their painting.
Brunelleschi
Filippo Brunelleschi (1377-1446), an architect who lived and worked in Florence, Italy, invented perspective, a way to represent three dimensional objects in two dimensions. Parallel lines converge to a common vanishing point on the horizon line.4 Leonardo da Vinci was interested by perspective, but he didn't accept Brunelleschi's one-point linear perspective, which applies when an observer is directly facing an object, because it didn't work for objects held very close to the eye. Leonardo da Vinci wrote: "Practice must always be built upon strong theory, of which perspective is the signpost and the gateway, and without perspective nothing can be done well in the matter of painting."5 Leonardo's was introduced to mathematics by Luca Pacioli (1445-1527) , who later became his close friend. Pacioli explained in a book the practice of "double entry bookkeeping", i.e., the notion that every transaction should be recorded twice, as a debit and as a credit, and that the balance should always be null. For that reason, Luca Pacioli is considered the "father of accounting". The new accountability gained by financial transactions played a major role in the flourishing of commerce from that period on.
Donatello
Masaccio
Leon Battista Alberti
Piero della Francesca
Piero della Francesca (~1415-1492) was an early Renaissance painter who was trained in mathematics, most likely for mercantilism6 and developed a strong interest in the theoretical study of perspective.
Luca Pacioli
Luca Pacioli was a mathematician who was deeply influenced by Piero della Francesca, and reproduced his work on perspective. A controversy followed because Luca Pacioli was accused to have plagiarized della Francesca about his writings on perspective.
The Controversy of Perspective
In Fabio...:
Domenico da Chivasso (c. 1350) proposes its inclusion among the arts of the Quadrivium too. 33 Michele Savonarola (c. 1385 - 1468), Marsilio Ficino (1433 - 1499), Girolamo Savonarola (1455 - 1498), Luca Pacioli and Leonardo Da Vinci (1452 - 1519) also defended this position. We denominate the debate about the inclusion of the Perspective among the arts of the Quadrivium as “Controversy of the Perspective”.
References and Reading Notes
Sabrina Sforza Galitzia, The secret perspective of Leonardo and Pacioli
Pacioli and Perspective, Part I, Vico Montebelli, Lett Mat Int (2015) 3:135–141 DOI 10.1007/s40329-015-0090-4
Pacioli and Perspective, Part II, Vico Montebelli, Lett Mat Int (2015) 3:189–195 DOI 10.1007/s40329-015-0098-9.
"LUCA PACIOLI AND THE “CONTROVERSY OF THE PERSPECTIVE”: THE CLASSIFICATION OF THE MATHEMATICS FROM THE CLASSICAL ANTIQUITY TO THE END OF THE QUATTROCENTO, by Fábio Maia Bertato, Campinas State University – Brazil and Itala Maria Loffredo D'Ottaviano, Campinas State University – Brazil, Revista Brasileira de História da Matemática Especial no 1 – Festschrift Ubiratan D‟Ambrosio – (dezembro/2007), Publicação Oficial da Sociedade Brasileira de História da Matemática ISSN 1519-955X
Biography of Luca Pacioli: http://www-history.mcs.st-and.ac.uk/Biographies/Pacioli.html
THE ENIGMA OF LUCA PACIOLI'S PORTRAIT
NPR, Episode 407: A Mathematician, The Last Supper And The Birth Of Accounting, October 2, 2012
POLYHEDRA & PLAGIARISM in the RENAISSANCE
Dr. Tomás García-Salgado, Faculty of Architecture/ UNAM/ SNI-III, THE PERSPECTIVE of LEONARDO’S LAST SUPPER.
Pacioli, Luca, Encyclopedia.com. Complete Dictionary of Scientific Biography, ©2008, Charles Scribner's Sons.
William L. Weis and David E. Tinius, "Luca Pacioli: Accounting's Renaissance Man," Management Accounting, July 1991, pp. 54-56. Copyright by Institute of Management Accountants, Montvale, N.J.
Gerald L. Alexanderson, Santa Clara University, About the cover: Luca da Pacioli and Leonardo’s drawings of polyhedra, 2010, DOI: 10.1090/S0273-0979-10-01308-X
JJ O'Connor and E.F. Robertson, Mathematics and art - perspective, January 2003, MacTutor History of Mathematics.
Saad Imran, An accountant’s perspective on The Blockchain.
Christine M. Pense, Pacioli's Bookkeeping, Dissertation, Chapter 3
The Accounting Franciscan, catholicism.org
Lee Miller, A View of Contemplative Geometry in Freemasonry, Part III, Fra Luca Pacioli and Others, 2002
New World Encyclopedia: Luca Pacioli