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Ancient double-entry bookkeeping: Lucas Pacioli's treatise
Ancient double-entry bookkeeping Lucas Pacioli's treatise
192 pages, Kindle Edition
Published April 5, 2014
2 people are currently reading
30 people want to read
About the author
Luca Pacioli
42 books17 followersItalian mathematician that was born between 1446 and 1448 and died in 1517.
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January 8, 2025
Why Must Debits = Credits?
I would like at first to state that the importance of this book cannot be overstated. The development of double-entry bookkeeping was a key casual driver in the commercial revolution the spurred the industrial revolution. Along with secure property and contract rights guaranteed by the rule of law, the modern state, political accountability and the scientific method, double-entry bookkeeping must rank as an important aspect of social development and advancement along the path to the modern world.
The accounting profession is often more concerned with the “how” rather than the “why” as mentioned by Page Lawrence in the introduction. Students of accounting are taught that Debits must equal Credits from their first accounting course but, most often, they are not told why this is the case; why this must be the case. Instead, too often students are told that it is just a basic concept of double entry bookkeeping. While this is true, it is by no means self-evident. Even more fundamental, students are taught that Assets = Liabilities + Equity and that this is nothing less than the fundamental accounting equation. Why is this the case? Students are often told that this just is the case as well or that it is just a feature of financing the Assets. What is not explained is the basic philosophical proposition behind this accounting equation. It is not an a priori assertion made on the part of accountants to get the system rolling. The assumption proper is really about the basic nature of material reality as we encounter it. These types of issues are not often discussed in most modern accounting courses. Principles such as Asset = Liabilities + Equity and Debits = Credits are often accepted on ‘faith’ so to speak.
As written in the introduction by Page Lawrence “It is a significant fact that the rules and principles elucidated by Pacioli are contained in a book given to mathematics. One cannot help but believe that the derivation of double-entry bookkeeping is an explanation of the algebraic equation used with such skill by the ancient Greek mathematicians, applied practically to the scientific recoding of business transactions, for just as in algebra, the equation once established cannot be changed but by addition of positive and negative quantities.” It is by the algebraic logic contained in this book that we can prove why Assets = Liabilities + Equity and why its follows logically that Debits = Credits.
Proof of these assertions can be especially helpful to the students new to the study of accounting because it aids new students in understanding the basic algebraic logic of the accounting discipline. The fact that Assets = Liabilities + Equity and that from this it follows that Debits = Credit is not an article of faith, but the result of these principles being based on fundamental features of material reality and algebraic logic. The answers are to be found in this book and go back to fundamental philosophical principles upon which accounting is based.
Assets = Liabilities + Equity
Why must this be the case?
The process of proof here has been called ‘conjecture and refutation’ by Karl Popper. We develop a falsifiable notion about material reality using our human cognitive powers. The conjecture is then tested against the observable material reality. By being falsifiable, it is meant that if the conjecture is indeed false, it can and will be found to be false, not that it is necessarily false. If it is not found to be false, we proceed as if it is true until it can be shown to be false. Our conjecture is very simply that the sum of the parts must equal the whole. Thus far, this conjecture about the nature of material reality has not been found to be false but if were indeed false, it would be easy to demonstrate it as such. The idea that “The whole is greater than the sum of its parts.” is a principle found in Aristotle’s Metaphysics but this is a principle of synergy with the most obvious example being teamwork. Mathematically, and this is how the principle is used in the material world of objects, things and numbers, the whole is equal to the sum of its parts, neither more nor less. This is an example of a linear system with independent components parts. This is the world of accounting. With this, we have a fundamental principle welded to material reality itself and the only axiom needed to provide the basis upon which the entire edifice of double-entry accounting is built.
Assets are the whole upon which claims are made. The claims made upon the Assets are either by made by creditors (Liabilities) or owners (Equity). If the Assets are more than or less than the sum of Liabilities and Equity, then there are more Assets claimed than actually exist or there are Assets left unclaimed. Obviously, neither can be the case without violating a basic feature of material reality that the sum of the parts must be equal to the whole. From here, we know that changes made to the parts must equal changes made to the whole so as not to violate this same basic material principle of nature. Debits and Credits are used to make changes to the whole (Assets) and to the parts (Liabilities and Equity), therefore Debits must equal Credits. If the Debits do not equal the Credits, then the sum of the parts will not equal the whole and a basic principle of material reality is violated. We can be confident in the logic of accounting and the fundamental principles upon which it is based such as Assets = Liabilities + Equity and Debits = Credits with the knowledge that such principles are tied to a basic feature of material reality. We can proceed to higher and more complex issues of accounting knowing that edifice is built upon a solid foundation, a principle welded to reality that we can prove to be the case. Interestingly, this method of constructing a structure of knowledge based on a singular fundamental truth is evocative of Rene Descartes’ Meditations on First Philosophy.
With the internal algebraic logic built into the double entry bookkeeping system, we can prove that Debits must equal Credits in algebraic fashion. This is one of those rare cases where we can make a statement about the material world by proving a mathematical / logical theorem.
Debits = Credits > The Proof (Simplified Form):
Proof Steps >>>>>>>>>>>>>>>>>>>>>>> Reasoning
1. Assets = Liabilities + Equity >>>>>>>>>>Premise: Sum of the parts equals the whole
2. Δ Assets = Δ Liabilities + Δ Equity >>>>> A change in the parts is equal to the change in the whole
3. Δ Assets = Asset(i) – Asset(d) >>>>>>>> A change is an increase (i) less a decrease (d) in an amount
Δ Liabilities = Liability(i) – Liability(d)
Δ Equity = Equity(i) – Equity(d)
4. Asset(i) – Asset(d) = Liability(i) – Liability(d) +Equity(i) – Equity(d) >>>>>By substituting the result of step 3 into the result of step 2
5. Asset(i) + Liability(d) + Equity(d) = Asset(d) + Liability(i) + Equity (i) >>>>> Add A(d), L(d) and E(d) to both sides of the equation
6. Debits = Credits >>>>>>>>> Asset(i), Liability(d), Equity(d) are all debits and Asset(d), Liability(i), Equity(i) are all credits
Q.E.D.
The detail of moving from Step 4 Step 5:
Step 4: Asset(i) – Asset(d) = Liability (i) – Liability(d) + Equity(i) – Equity(d)
+ Asset(d) + Asset(d)
+ Liability(d) + Liability(d)
+ Equity(d) + Equity(d)
Step 5: Asset(i) + Liability(d) + Equity(d) = Asset(d) + Liability(i) + Equity(i)
Step 6: Debits = Credits
I would like at first to state that the importance of this book cannot be overstated. The development of double-entry bookkeeping was a key casual driver in the commercial revolution the spurred the industrial revolution. Along with secure property and contract rights guaranteed by the rule of law, the modern state, political accountability and the scientific method, double-entry bookkeeping must rank as an important aspect of social development and advancement along the path to the modern world.
The accounting profession is often more concerned with the “how” rather than the “why” as mentioned by Page Lawrence in the introduction. Students of accounting are taught that Debits must equal Credits from their first accounting course but, most often, they are not told why this is the case; why this must be the case. Instead, too often students are told that it is just a basic concept of double entry bookkeeping. While this is true, it is by no means self-evident. Even more fundamental, students are taught that Assets = Liabilities + Equity and that this is nothing less than the fundamental accounting equation. Why is this the case? Students are often told that this just is the case as well or that it is just a feature of financing the Assets. What is not explained is the basic philosophical proposition behind this accounting equation. It is not an a priori assertion made on the part of accountants to get the system rolling. The assumption proper is really about the basic nature of material reality as we encounter it. These types of issues are not often discussed in most modern accounting courses. Principles such as Asset = Liabilities + Equity and Debits = Credits are often accepted on ‘faith’ so to speak.
As written in the introduction by Page Lawrence “It is a significant fact that the rules and principles elucidated by Pacioli are contained in a book given to mathematics. One cannot help but believe that the derivation of double-entry bookkeeping is an explanation of the algebraic equation used with such skill by the ancient Greek mathematicians, applied practically to the scientific recoding of business transactions, for just as in algebra, the equation once established cannot be changed but by addition of positive and negative quantities.” It is by the algebraic logic contained in this book that we can prove why Assets = Liabilities + Equity and why its follows logically that Debits = Credits.
Proof of these assertions can be especially helpful to the students new to the study of accounting because it aids new students in understanding the basic algebraic logic of the accounting discipline. The fact that Assets = Liabilities + Equity and that from this it follows that Debits = Credit is not an article of faith, but the result of these principles being based on fundamental features of material reality and algebraic logic. The answers are to be found in this book and go back to fundamental philosophical principles upon which accounting is based.
Assets = Liabilities + Equity
Why must this be the case?
The process of proof here has been called ‘conjecture and refutation’ by Karl Popper. We develop a falsifiable notion about material reality using our human cognitive powers. The conjecture is then tested against the observable material reality. By being falsifiable, it is meant that if the conjecture is indeed false, it can and will be found to be false, not that it is necessarily false. If it is not found to be false, we proceed as if it is true until it can be shown to be false. Our conjecture is very simply that the sum of the parts must equal the whole. Thus far, this conjecture about the nature of material reality has not been found to be false but if were indeed false, it would be easy to demonstrate it as such. The idea that “The whole is greater than the sum of its parts.” is a principle found in Aristotle’s Metaphysics but this is a principle of synergy with the most obvious example being teamwork. Mathematically, and this is how the principle is used in the material world of objects, things and numbers, the whole is equal to the sum of its parts, neither more nor less. This is an example of a linear system with independent components parts. This is the world of accounting. With this, we have a fundamental principle welded to material reality itself and the only axiom needed to provide the basis upon which the entire edifice of double-entry accounting is built.
Assets are the whole upon which claims are made. The claims made upon the Assets are either by made by creditors (Liabilities) or owners (Equity). If the Assets are more than or less than the sum of Liabilities and Equity, then there are more Assets claimed than actually exist or there are Assets left unclaimed. Obviously, neither can be the case without violating a basic feature of material reality that the sum of the parts must be equal to the whole. From here, we know that changes made to the parts must equal changes made to the whole so as not to violate this same basic material principle of nature. Debits and Credits are used to make changes to the whole (Assets) and to the parts (Liabilities and Equity), therefore Debits must equal Credits. If the Debits do not equal the Credits, then the sum of the parts will not equal the whole and a basic principle of material reality is violated. We can be confident in the logic of accounting and the fundamental principles upon which it is based such as Assets = Liabilities + Equity and Debits = Credits with the knowledge that such principles are tied to a basic feature of material reality. We can proceed to higher and more complex issues of accounting knowing that edifice is built upon a solid foundation, a principle welded to reality that we can prove to be the case. Interestingly, this method of constructing a structure of knowledge based on a singular fundamental truth is evocative of Rene Descartes’ Meditations on First Philosophy.
With the internal algebraic logic built into the double entry bookkeeping system, we can prove that Debits must equal Credits in algebraic fashion. This is one of those rare cases where we can make a statement about the material world by proving a mathematical / logical theorem.
Debits = Credits > The Proof (Simplified Form):
Proof Steps >>>>>>>>>>>>>>>>>>>>>>> Reasoning
1. Assets = Liabilities + Equity >>>>>>>>>>Premise: Sum of the parts equals the whole
2. Δ Assets = Δ Liabilities + Δ Equity >>>>> A change in the parts is equal to the change in the whole
3. Δ Assets = Asset(i) – Asset(d) >>>>>>>> A change is an increase (i) less a decrease (d) in an amount
Δ Liabilities = Liability(i) – Liability(d)
Δ Equity = Equity(i) – Equity(d)
4. Asset(i) – Asset(d) = Liability(i) – Liability(d) +Equity(i) – Equity(d) >>>>>By substituting the result of step 3 into the result of step 2
5. Asset(i) + Liability(d) + Equity(d) = Asset(d) + Liability(i) + Equity (i) >>>>> Add A(d), L(d) and E(d) to both sides of the equation
6. Debits = Credits >>>>>>>>> Asset(i), Liability(d), Equity(d) are all debits and Asset(d), Liability(i), Equity(i) are all credits
Q.E.D.
The detail of moving from Step 4 Step 5:
Step 4: Asset(i) – Asset(d) = Liability (i) – Liability(d) + Equity(i) – Equity(d)
+ Asset(d) + Asset(d)
+ Liability(d) + Liability(d)
+ Equity(d) + Equity(d)
Step 5: Asset(i) + Liability(d) + Equity(d) = Asset(d) + Liability(i) + Equity(i)
Step 6: Debits = Credits
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