the echo of a distant time

THE ECHO OF A DISTANT TIME: A MATHEMATICAL MODEL OF ACCOUNTANCY AND BOOKKEEPING USING MEASURE THEORY

© North Delta College 2015

Mathema'cs applied to Business Theory 1

INTRODUCTION

The no?on of Accoun?ng Circuit transcends the strictly Financial domain and belongs to the realm of Philosophy too.

In this presenta?on we will build a mathema?cal model of Accoun?ng and Bookkeeping using Measure theory. Please don't be too afraid as we will

remain very prac?cal and answer some fascina?ng ques?ons:

Where does Money come from? Is it possible that Money in some form existed before Man?

So, fasten your seat belts, and enjoy the ride...


SUMMARY PART 1: Survey of Accountancy and Bookkeeping A) What do Accountants and Bookkeepers do? B) Double-‐Entry Bookkeeping C) Accoun?ng Circuits D) 2 key principles of Accountancy PART 2: Survey of Measure Theory A) What is Measure Theory? B) 3 key proper?es PART 3: High end Model: an Echo from a distant ?me A) The spirit of the model PART 4: The le\er: Mathema?cal Details A) Chart of Accounts B) 2 new measures C) The Fundamental Principle D) What it means E) Illustra?on on an Example: Revenue Recogni?on PART 5: FINAL STATEMENT: Where does Money come from?


PART 1: SURVEY OF ACCOUNTANCY AND BOOKKEEPING

Accountancy is a well known profession visible to many in most major organisa?ons. Accountants' primary role remains to keep track of the money going into the company, the money going out, and also to a lesser degree the inner transac?ons. In modern ?mes, Accountants translate their analysis into financial statements: the most famous being the income statement, the balance sheet and the cash-‐flow. Accountants use Bookkeepers to do their daily job. Whereas the 3 aforemen?oned financial statements reveal a high-‐end synthesis of the financial situa?on, the day to day monitoring of every single accoun?ng transac?on is made inside the Accoun?ng books by the Bookkeepers.


What do Accountants and Bookkeepers do?

Double-‐Entry Bookkeeping

The reason why accoun?ng works is due to a magical insight: namely Double-‐Entry Bookkeeping. To keep track of an individual transac?on, Bookkeepers use 2 accounts at the same ?me. Accounts are basically the pigeon holes into which accountants store the monetary value of all the assets and liabili?es of the company. Double-‐Entry Bookkeeping consists for every transac?on to Credit a certain account and Debit another account with the same value and vice versa.



Figure 1: Example of an Account Figure 2: Double-‐Entry

Cash

Debit Credit

100 50

Cash Receivables

125 125

For anyone new with handling money, it is absolutely cri?cal to understand the mathema?cal structure of the accoun?ng flow. Money follows circuits. From one en?ty, the sender to the receiver. These circuits are air-‐?ght in between them. Furthermore, some financial en??es can have several circuits penetra?ng them. A set of en??es interlinked in between them with accoun?ng circuits forms a network.


Accoun?ng Circuits


Direc?on of money Figure 3: A reduced circuit

Figure 4: An en?ty with several circuits

Figure 5: An accoun?ng network

Accoun?ng Systems follow 2 key rules: 1) First, they have to be poten?ally closed. The reason behind this is to keep track of all in-‐going and all out-‐going transac?ons. 2) Second, there are always 2 flows accompanying any monetary transac?on. The proper money exchange and a reverse flow going back to the money sender manifested by a physical asset (goods, receipt, legal document .... )


2 key principles of Accountancy


Figure 6: Principle 1

Figure 7: Principle 2

What is Measure Theory?

Measure theory in modern Mathema?cs is the founda?on of Integra?on and Probability Theory. The key idea is to measure a wide array of Mathema?cal objects the same as we measure areas of figures in the plane or volumes in space. The way this is achieved is by associa?ng a number to every sub part of the Universe and by asking that this rela?onship verifies 3 basic axioms. Therefore Measure Theory is about expanding a concept we use in everyday life and to give it its full poten?al.


PART 2: SURVEY OF MEASURE THEORY

Intui?ve Understanding Enlarged Understanding

Figure 8: Extension of a concept

Figure 9: Measures

Universe A number = 2.74

A number = 5.23

Example of Measures

The usual measure of length, area and volume we use in daily life are the most common examples of measures. If one scales these measures by a real number, one again gets a measure. We can see how we have very naturally extended a concrete concept into the mathema?cal realm.



Figure 10: Areas and Lengths Length = 2 cm Area = 6.6 cm2

Figure 11: Scaling by 10

Usual Measure = Area = 6.6 cm2 New Measure = Area mul?plied by 10 = 66 cm2

Atypical Measures

Let us now give 2 non-‐trivial examples:



Probability measure. When we say the probability of tossing a coin on head is 0.5, we are using a new measure: the probability measure. The mathema?cal objects we are measuring are taken from the universe of all possible future events and their eventuality to happen.

Dirac mass. The Dirac mass is used in Quantum physics. It is a very strange measure whereby the measure is equal to 1 if the measured object contains a certain given point, and equal to 0 otherwise

Figure 12: Tossing a coin.

Figure 13: Dirac mass of point A

Measure = 1 Measure = 0

A AX X

Discrete Measures

We will now consider 3 proper?es that we would like the measures of our model to abide by. The first property we are interested in is discreteness. Discrete measures take only integer values. The most famous example is the Dirac mass. However there are plenty other such measures.



Figure 14 : Discrete measures

Universe

Measure = 2 Measure = 6

Measure = 8

Peripheral Measures

The second very compelling property we want to put forward is the peripheral nature of some measures. Peripherality manifests itself in the fact that regardless of the object measured, the peripheral measures always puts more weight on the boundary of the object rather than its interior. What we mean by such a statement is the fact that for peripheral measures: The epicentre of ac?on is always on the border of the measured object.



Picture 15: More weight on the borders

Border Interior

Topological Measures

Third and last property we are requiring is some form of topological symmetry. In par?cular we want such measures to remain the same on objects of different shapes provided these objects are synchronic. In our accoun?ng example, this will be the case when 2 different accounts gets hit by the same transac?on.



Figure 16: Topological Invariance

Different topological shape but synchronic because a\ached

The spirit of the model

Let us now give a high end understanding of the model. The key ideas. Our mathema?cal model will take the Universe of all possible transac?ons and assign measures on them. As we have already seen all accoun?ng transac?ons are cons?tuted of 2 movements: the first movement underlines where the money is going; the second movement indicates a tangible asset (either the goods or a receipt et cetera) with monetary value going reverse towards the source. Theorem : What we will show is that the money goes from the intangible realm to the material one.


PART 3: HIGH END MODEL: AN ECHO FROM A DISTANT TIME

Figure 17: The model -‐ high end

Object B

Transac?on 2

Object A

Transac?on 1

Source Figure 18: An accoun?ng transac?on

The spirit of the model

There are 3 stakeholders in any transac?on 1) The ins?gator ( or the client receiving goods) 2) The funder ( en?ty holding the ini?al capital) 3) The working force ( those performing the task related to the transac?on) The ins?gator is ini?ally in the material realm but starts driming towards the intangible one. The funder is in the intangible realm and sinks more towards it. The working force is always and ever in the material world Mathema'cs applied to Business Theory 15

PART 3: HIGH END MODEL: AN ECHO FROM A DISTANT TIME

Figure 19: The 3 stakeholders

Working Force Funder

Ins?gator

Chart of Accounts

The Chart of Accounts lists the accounts in the accoun?ng system. These are basically the pigeon holes in which accountants store the monetary value of all the assets and liabili?es of the company. For Example there is an account called Cash represen?ng the amount of cash available at a given ?me. There is an account called receivables storing the total value of the receivables of the company. And so on... All the elements of the balance sheet are segmented into these pigeon holes.


PART 4: THE LETTER: MATHEMATICAL DETAILS

Figure 20: Example of Chart of Accounts.

Chart of Accounts -‐ From Intangible to Material


A key aspect of our model is that these accounts can be ordered. The order rela?on is: bigger value for more material or tangible and lower values for more intangible. For Example we consider the most tangible account to be cash because it is money that can be physically seen. In a similar vein, Receivables are more intangible than Sales because as an asset the financial counterpart is less palpable.

PART 4: MATHEMATICAL DETAILS

Figure 21: The order rela?on : From Material to Intangible

More Material More Intangible

Assets

Liabili?es

Cash

Interest on a Bond COGS

Receivables

2 new measures

There are 2 measures that we associate to the Chart of Accounts: 1) The Gate measure which measures how much and in which direc?on the money is flowing 2) The Median measure which measures if the transac?on is credited or debited The Universe they can be applied upon is the universe of all possible accoun?ng transac?ons in a given accoun?ng system. Proper?es : Gate is a peripheral measure which becomes non-‐null when actual money is flowing into the accounts. Median is a discrete measure which gives the same value on 2 different accounts provided these accounts are part of the same accoun?ng circuit.




The Fundamental Principle

The main theorem says the following: Let A and B be two accounts. In any case we either have Gate(A) = Gate(B) or Median(A) = Median(B) If for any reason it is the former case which is true, i.e. Gate(A) = Gate(B) we are in a situa?on where there is actually money flowing in the account, and we indeed have a double entry in the books. Furthermore, in this precise situa?on, there is a fundamental structural inversion during transfer of ownership of the physical monetary counterpart. Gate(border of A) = Median(interior of B) Gate(border of B) = Median(interior of A)


Figure 22: Structural Inversion

Account A Account B

interior

border

Gate Gate

Median Median

What it means

Understanding properly this principle is key. As we have said there are 2 flows accompanying any monetary transac?on. The proper money exchange and a reverse flow going back to the money sender manifested by a physical asset (goods, receipt, legal document .... ) The principle says that during the transac?on, money goes up the rela?on order while the physical counterpart goes down.



Figure 23: Structural Inversion

Time

+ -‐ Order Rela?on

= Transac?on

Money Physical Trace

Money Physical Trace

Illustra?on on an Example: Revenue Recogni?on

Let us illustrate this structural theorem on an example. We suppose we look at the chart of accounts of a company PPP for a given project with revenue recogni?on. Let us describe this project:



Revenue Recogni?on -‐ Chart of Accounts

If we look at the Chart of Accounts, we find 6 main accounts to consider: Cash, Receivables, COGS (Cost Of Goods Sold), Billing, Sales Plus a Non-‐standard account that we just labelled Revenue Account. Using Double-‐Entry Bookkeeping the project unfolds in this chart of accounts the following way:



Revenue Recogni?on -‐ The measures

On the Maths side, The Universe to consider are all possible double entry on these 6 accounts star?ng from March 15 un?l project comple?on. We can thus consider the 2 measures Gate and Median.



Median

Gate Circuit

X = Transac?on

X fixes over A if there is Double-‐Entry in the Books

Evolu?on of same account A over ?me

X

Figure 24: Revenue Recogni?on Measures

Time

Money

Revenue Recogni?on -‐ Illustra?on of the Principle

The key saying of our theorem claims that: There are 2 sources of money flow inside the organisa?on: 1) one coming from the pool of cash available (ini?al fund) and used to pay the cost of the project. 2) an other one coming from sales (customers) and turning into profit for the company In both cases, the money moves towards the more material realm.



Figure 25: Financial flows inside the organisa?on

Where does Money come from?

Thus we have proven money moves from the intangible realm to the material one. We can now re-‐ask the ques?ons posed in the introduc?on. Namely: Where does money come from? Did Money exist before Man? We see the origin point, we will call it the Aleph point, is situated at the junc?on between the material realm and the intangible realm. That Money existed before Man in some form or an other is a fact we are deeply convinced of. Mathema'cs applied to Business Theory 25

PART 5: FINAL STATEMENT

Figure 26: The theorem

Figure 27: The Aleph point

Intangible Realm

Material Realm

Intersec?on Area

Aleph Point

the echo of a distant time

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