1

1

 

1+1=3

1

 

1+1=3

1+1=3

Everybody knows that 1 + 1 = 2. However, in the 21st century, expressions such as 1 + 1 = 3 occurred to reflect important characteristics of economic and business processes. It seems that this contradicts core mathematical axioms and is incorrect from a mathematical point of view.

We find similarities in the history of mathematics―what had been considered strange, ungrounded and inconsistent with the existing mathematics, was incorporated later in the main body of mathematical knowledge. Here are some examples:

In China and India, mathematicians used negative numbers for centuries before these numbers came to Europe. However, when the European mathematicians encountered negative numbers, critics dismissed their sensibility. Some of the notable European mathematicians, such as d’Alembert or Frend, did not want to accept negative numbers until the 18th century and referred to them as “absurd” or “meaningless” (Kline, 1980 [5] , Mattessich, 1998 [6] ). Even in the 19th century, it was common practice to ignore any negative results derived from equations, on the assumption that they were meaningless (Martinez, 2006 [7] ). For instance, Lazare Carnot (1753-1823) affirmed that the idea of something being less than nothing is absurd (Mattessich, 1998 [6] ). Outstanding mathematicians such as William Hamilton (1805-1865) and August De Morgan (1806-1871) had akin opinions. Similarly, irrational numbers and later imaginary numbers were firstly rejected. Today these concepts are accepted and applied in numerous scientific and practical fields, such as physics, chemistry, biology and finance.

The interaction between physics and mathematics gives us one more example. In the 20th century, physicists started using functions that took infinite values at some points. At first, mathematicians objected by saying that there are no such functions in mathematics (cf., for example, von Neumann, 1955 [8] ). However, later they grounded utilization of these functions developing the theory of distributions and finding numerous new applications for this theory (Schwartz, 1950/1951 [4] ).

2. Problems with the Conventional Arithmetic

Human beings have used conventional arithmetic for millennia before the most inquisitive thinkers started questioning its validity in certain settings. It dates back to ancient Greece, where mathematicians and philosophers already started to doubt the convenience of conventional arithmetic. The Sophists, who lived from the second half of the fifth century B.C to the first half of the fourth century B.C. asserted the relativity of human knowledge and suggested many paradoxes, explicating complexity and diversity in the real world. One of them, the famous philosopher Zeno of Elea (490 - 430 B.C.), who was said to be a self- taught boy from the country side, invented very notable paradoxes, in which he questioned the popular knowledge and intuition related to fundamental essences such as time, space, and numbers.

An example of this type of reasoning is the paradox of the heap (or the Sorites paradox where σωρος is the Greek word for “heap”). It is possible to formulate this paradox in the following way.

1) One million grains of sand make a heap.

2) If one grain of sand is added to this heap, the heap stays the same.

3) However, when we add 1 to any natural number, we always get a new number.

There are analogies of this paradox in our time. For instance, if you have ten million dollars (a “heap” of money) and somebody will give you a dollar, will you say that you have ten million and one dollar when asked about your assets? No, you will most likely say that you have the same ten million dollars (the same “heap”).

As we know, Greek sages posed questions, but in many cases, including arithmetic, suggested no answers. As a result, for more than two thousand years, these problems were forgotten and everybody was satisfied with the conventional arithmetic. In spite of all problems and paradoxes, this arithmetic has remained very useful.

In recent times, scientists and mathematicians have returned to problems of arithmetic. The famous German researcher Herman Ludwig Ferdinand von Helmholtz (1821-1894) [1] was one the first scientists who questioned adequacy of the conventional arithmetic. In his “Counting and Measuring” (1887) [1] , Helmholtz considered an important problem of the applicability of arithmetic to physical phenomena. This was a natural approach of a scientist, who judged mathematics by the main criterion of science―observation and experiment.

The scientist’s first observation was that as the concept of a number is derived from some practice, usual arithmetic has to be applicable in all practical settings. However, it is easy to find many situations when this is not true. To mention but a few described by Helmholtz, one raindrop added to another raindrop does not make two raindrops. It is possible to describe this situation by the formula 1 + 1 = 1.

In a similar way, when one mixes two equal volumes of water, one at 40˚ Fahrenheit and the other at 50˚, one does not get two volumes at 90˚. Alike, the conventional arithmetic fails to describe correctly the result of combining gases or liquids by volumes. For example (Kline, 1980), one quart of alcohol and one quart of water yield about 1.8 quarts of vodka.

Later the famous French mathematician Henri Lebesgue facetiously pointed out (cf. Kline, 1980 [5] ) that if one puts a lion and a rabbit in a cage, one will not find two animals in the cage later on. In this case, we will also have 1 + 1 = 1.

However, since very few paid attention to the work of Helmholtz on arithmetic, and as still no alternative to the conventional arithmetic has been suggested, these problems were mostly forgotten. Only years later, in the second part of the 20th century, mathematicians began to doubt once more the absolute character of the ordinary arithmetic, where 2 + 2 = 4 and 2 × 2 = 4. Scientists and mathematicians again started to draw attention of the scientific community to the foundational problems of natural numbers and the conventional arithmetic. The most extreme assertion that there is only a finite quantity of natural numbers was suggested by Yesenin-Volpin (1960) [2] , who developed a mathematical direction called ultraintuitionism and took this assertion as one of the central postulates of ultraintuitionism. Other authors also considered arithmetics with a finite number of numbers, claiming that these arithmetics are inconsistent (cf., for example Van Bendegem, 1994 [3] and Rosinger, 2008 [4] ).

Van Danzig had similar ideas but expressed them in a different way. In his article (1956), he argued that only some of natural numbers may be considered finite. Consequently, all other mathematical entities that are called traditionally natural numbers are only some expressions but not numbers. These arguments are supported and extended by Blehman, et al. (1983) [9] .

Other authors are more moderate in their criticism of the conventional arithmetic. They write that not all natural numbers are similar in contrast to the presupposition of the conventional arithmetic that the set of natural numbers is uniform (Kolmogorov, 1961 [10] ; Littlewood, 1953 [11] ; Birkhoff and Bartee, 1967 [12] ; Dummett 1975 [13] ; Knuth, 1976 [14] ). Different types of natural numbers have been introduced, but without changing the conventional arithmetic. For example, Kolmogorov (1961) [10] suggested that in solving practical problems it is worth to separate small, medium, large, and super-large numbers.

Regarding geometry, it was discovered that there was not one but a variety of arithmetics, which were different in many ways from the conventional arithmetic. It is natural to call the conventional arithmetic by the name Diophantine arithmetic because the Greek mathematician Diophantus, who lived between 150 C.E. and 350 C.E., and who extensively contributed to the development of conventional arithmetic. Consequently, new arithmetics acquired the name non-Diophantine arithmetics.

Burgin built first Non-Diophantine arithmetics of whole and natural numbers (Burgin, 1977 [12] ; 1997 [15] ; 2007 [16] ; 2010 [17] [18] ) and Czachor extended this construction developing Non-Diophantine arithmetics of the real and complex numbers (Czachor, 2015 [19] ).

In the following section, we will show that non-Diophantine arithmetics occur in economics, starting with mergers and acquisitions.

3. Examples of Non-Diophantine Arithmetic

In the following section, we will show that non-Diophantine arithmetics occur in economics, business and social settings, starting with mergers and acquisitions.

Mergers and acquisitions (M & A) are processes in which the operating units of two companies are combined. Whereas in a merger, two approximately equally sized companies consolidate into one entity; in an acquisition, a larger company takes over the smaller one.

There are different motivations for M & As, the most typical being the following:

1) Synergies or economies of scale are achieved when the larger company may be able to lower the per unit purchasing cost due to higher bulk orders, or lower fixed cost by removing or combing duplicate departments such as research, accounting, or compliance.

2) Increased market share usually leads to the greater than before market power.

3) Technology driven M & As are aimed at gaining access to new technologies.

4) Tax reduction in M & A is the situation when a company may acquire a loss generating, yet presumably long term profitable company to reduce taxes.

5) In the case of CEO activism, a CEO may want to amplify his standing by increasing company size and seemingly showing leadership in M & A.

The success of M & As varies strongly. With respect to the cost saving due to synergy and economies of scale, many M & As do achieve their objective, as we can see in Figure 1.

However, with respect to expected revenue, only few companies achieve the desired objective, as we can see in Figure 2.

1

 

1+1=3

1+1=3

Everybody knows that 1 + 1 = 2. However, in the 21st century, expressions such as 1 + 1 = 3 occurred to reflect important characteristics of economic and business processes. It seems that this contradicts core mathematical axioms and is incorrect from a mathematical point of view.

We find similarities in the history of mathematics―what had been considered strange, ungrounded and inconsistent with the existing mathematics, was incorporated later in the main body of mathematical knowledge. Here are some examples:

In China and India, mathematicians used negative numbers for centuries before these numbers came to Europe. However, when the European mathematicians encountered negative numbers, critics dismissed their sensibility. Some of the notable European mathematicians, such as d’Alembert or Frend, did not want to accept negative numbers until the 18th century and referred to them as “absurd” or “meaningless” (Kline, 1980 [5] , Mattessich, 1998 [6] ). Even in the 19th century, it was common practice to ignore any negative results derived from equations, on the assumption that they were meaningless (Martinez, 2006 [7] ). For instance, Lazare Carnot (1753-1823) affirmed that the idea of something being less than nothing is absurd (Mattessich, 1998 [6] ). Outstanding mathematicians such as William Hamilton (1805-1865) and August De Morgan (1806-1871) had akin opinions. Similarly, irrational numbers and later imaginary numbers were firstly rejected. Today these concepts are accepted and applied in numerous scientific and practical fields, such as physics, chemistry, biology and finance.

The interaction between physics and mathematics gives us one more example. In the 20th century, physicists started using functions that took infinite values at some points. At first, mathematicians objected by saying that there are no such functions in mathematics (cf., for example, von Neumann, 1955 [8] ). However, later they grounded utilization of these functions developing the theory of distributions and finding numerous new applications for this theory (Schwartz, 1950/1951 [4] ).

2. Problems with the Conventional Arithmetic

Human beings have used conventional arithmetic for millennia before the most inquisitive thinkers started questioning its validity in certain settings. It dates back to ancient Greece, where mathematicians and philosophers already started to doubt the convenience of conventional arithmetic. The Sophists, who lived from the second half of the fifth century B.C to the first half of the fourth century B.C. asserted the relativity of human knowledge and suggested many paradoxes, explicating complexity and diversity in the real world. One of them, the famous philosopher Zeno of Elea (490 - 430 B.C.), who was said to be a self- taught boy from the country side, invented very notable paradoxes, in which he questioned the popular knowledge and intuition related to fundamental essences such as time, space, and numbers.

An example of this type of reasoning is the paradox of the heap (or the Sorites paradox where σωρος is the Greek word for “heap”). It is possible to formulate this paradox in the following way.

1) One million grains of sand make a heap.

2) If one grain of sand is added to this heap, the heap stays the same.

3) However, when we add 1 to any natural number, we always get a new number.

There are analogies of this paradox in our time. For instance, if you have ten million dollars (a “heap” of money) and somebody will give you a dollar, will you say that you have ten million and one dollar when asked about your assets? No, you will most likely say that you have the same ten million dollars (the same “heap”).

As we know, Greek sages posed questions, but in many cases, including arithmetic, suggested no answers. As a result, for more than two thousand years, these problems were forgotten and everybody was satisfied with the conventional arithmetic. In spite of all problems and paradoxes, this arithmetic has remained very useful.

In recent times, scientists and mathematicians have returned to problems of arithmetic. The famous German researcher Herman Ludwig Ferdinand von Helmholtz (1821-1894) [1] was one the first scientists who questioned adequacy of the conventional arithmetic. In his “Counting and Measuring” (1887) [1] , Helmholtz considered an important problem of the applicability of arithmetic to physical phenomena. This was a natural approach of a scientist, who judged mathematics by the main criterion of science―observation and experiment.

The scientist’s first observation was that as the concept of a number is derived from some practice, usual arithmetic has to be applicable in all practical settings. However, it is easy to find many situations when this is not true. To mention but a few described by Helmholtz, one raindrop added to another raindrop does not make two raindrops. It is possible to describe this situation by the formula 1 + 1 = 1.

In a similar way, when one mixes two equal volumes of water, one at 40˚ Fahrenheit and the other at 50˚, one does not get two volumes at 90˚. Alike, the conventional arithmetic fails to describe correctly the result of combining gases or liquids by volumes. For example (Kline, 1980), one quart of alcohol and one quart of water yield about 1.8 quarts of vodka.

Later the famous French mathematician Henri Lebesgue facetiously pointed out (cf. Kline, 1980 [5] ) that if one puts a lion and a rabbit in a cage, one will not find two animals in the cage later on. In this case, we will also have 1 + 1 = 1.

However, since very few paid attention to the work of Helmholtz on arithmetic, and as still no alternative to the conventional arithmetic has been suggested, these problems were mostly forgotten. Only years later, in the second part of the 20th century, mathematicians began to doubt once more the absolute character of the ordinary arithmetic, where 2 + 2 = 4 and 2 × 2 = 4. Scientists and mathematicians again started to draw attention of the scientific community to the foundational problems of natural numbers and the conventional arithmetic. The most extreme assertion that there is only a finite quantity of natural numbers was suggested by Yesenin-Volpin (1960) [2] , who developed a mathematical direction called ultraintuitionism and took this assertion as one of the central postulates of ultraintuitionism. Other authors also considered arithmetics with a finite number of numbers, claiming that these arithmetics are inconsistent (cf., for example Van Bendegem, 1994 [3] and Rosinger, 2008 [4] ).

Van Danzig had similar ideas but expressed them in a different way. In his article (1956), he argued that only some of natural numbers may be considered finite. Consequently, all other mathematical entities that are called traditionally natural numbers are only some expressions but not numbers. These arguments are supported and extended by Blehman, et al. (1983) [9] .

Other authors are more moderate in their criticism of the conventional arithmetic. They write that not all natural numbers are similar in contrast to the presupposition of the conventional arithmetic that the set of natural numbers is uniform (Kolmogorov, 1961 [10] ; Littlewood, 1953 [11] ; Birkhoff and Bartee, 1967 [12] ; Dummett 1975 [13] ; Knuth, 1976 [14] ). Different types of natural numbers have been introduced, but without changing the conventional arithmetic. For example, Kolmogorov (1961) [10] suggested that in solving practical problems it is worth to separate small, medium, large, and super-large numbers.

Regarding geometry, it was discovered that there was not one but a variety of arithmetics, which were different in many ways from the conventional arithmetic. It is natural to call the conventional arithmetic by the name Diophantine arithmetic because the Greek mathematician Diophantus, who lived between 150 C.E. and 350 C.E., and who extensively contributed to the development of conventional arithmetic. Consequently, new arithmetics acquired the name non-Diophantine arithmetics.

Burgin built first Non-Diophantine arithmetics of whole and natural numbers (Burgin, 1977 [12] ; 1997 [15] ; 2007 [16] ; 2010 [17] [18] ) and Czachor extended this construction developing Non-Diophantine arithmetics of the real and complex numbers (Czachor, 2015 [19] ).

In the following section, we will show that non-Diophantine arithmetics occur in economics, starting with mergers and acquisitions.

3. Examples of Non-Diophantine Arithmetic

In the following section, we will show that non-Diophantine arithmetics occur in economics, business and social settings, starting with mergers and acquisitions.

Mergers and acquisitions (M & A) are processes in which the operating units of two companies are combined. Whereas in a merger, two approximately equally sized companies consolidate into one entity; in an acquisition, a larger company takes over the smaller one.

There are different motivations for M & As, the most typical being the following:

1) Synergies or economies of scale are achieved when the larger company may be able to lower the per unit purchasing cost due to higher bulk orders, or lower fixed cost by removing or combing duplicate departments such as research, accounting, or compliance.

2) Increased market share usually leads to the greater than before market power.

3) Technology driven M & As are aimed at gaining access to new technologies.

4) Tax reduction in M & A is the situation when a company may acquire a loss generating, yet presumably long term profitable company to reduce taxes.

5) In the case of CEO activism, a CEO may want to amplify his standing by increasing company size and seemingly showing leadership in M & A.

The success of M & As varies strongly. With respect to the cost saving due to synergy and economies of scale, many M & As do achieve their objective, as we can see in Figure 1.

 

However, with respect to expected revenue, only few companies achieve the desired objective, as we can see in Figure 2.

 

眾所周知，1 + 1 = 2。然而，在21世紀，諸如1 + 1 = 3之類的表達式卻被用來反映經濟和商業流程的重要特徵。這似乎與核心數學公理相矛盾，從數學角度來看是不正確的。 我們在數學史上發現了相似之處——那些曾經被認為是奇怪、毫無根據且與現有數學不符的東西，後來卻被納入了數學知識的主體。以下是一些例子： 在中國和印度，數學家在負數傳入歐洲之前的幾個世紀裡一直使用負數。然而，當歐洲數學家遇到負數時，批評者卻對其意義不屑一顧。一些著名的歐洲數學家，如達朗貝爾或弗倫德，直到18世紀才開始接受負數，並稱其“荒謬”或“毫無意義”（Kline，1980 [5] ，Mattessich，1998 [6] ）。即使在19世紀，人們普遍認為忽略任何由方程式得出的負結果毫無意義（Martinez，2006 [7]）。例如，拉扎爾·卡諾（Lazare Carnot，1753-1823）斷言「有小於無」的想法是荒謬的（Mattessich，1998 [6]）。威廉·漢密爾頓（William Hamilton，1805-1865）和奧古斯特·德·摩根（August De Morgan，1806-1871）等傑出數學家也持有類似的觀點。同樣，無理數以及後來的虛數最初也被否定。如今，這些概念已被接受並應用於許多科學和實踐領域，例如物理、化學、生物和金融。 物理學和數學之間的相互作用為我們提供了另一個例子。在20世紀，物理學家開始使用在某些點上取無窮值的函數。起初，數學家反對這種說法，認為數學中不存在這樣的函數（例如，馮諾依曼，1955 [8]）。然而，後來他們利用這些函數發展了分佈理論，並發現了該理論的許多新應用（施瓦茨，1950/1951 [4]）。 2. 傳統算術的問題 在最富於好奇心的思想家開始質疑其在某些情境下的有效性之前，人類已經使用傳統算術數千年。這種算術可以追溯到古希臘，當時的數學家和哲學家已經開始質疑傳統算術的便利性。生活在公元前五世紀下半葉至公元前四世紀上半葉的智者們主張人類知識的相對性，並提出了許多悖論，闡明了現實世界的複雜性和多樣性。其中一位是著名哲學家埃利亞的芝諾（西元前 490-430 年）。據說他是一位來自鄉下的自學成才的男孩，他提出了一些非常著名的悖論，質疑與時間、空間和數字等基本本質相關的普遍知識和直覺。 這種推理的一個例子是堆悖論（或稱 Sorites 悖論，其中 σωρος 是希臘語中「堆」的意思）。我們可以用以下方式來表達這個悖論。 1) 一百萬粒沙子可以堆成一堆。 2) 如果在這個堆裡再加一粒沙子，這個堆仍然保持不變。 3) 然而，當我們將 1 加到任何自然數上時，我們總是會得到一個新的數。 在我們這個時代，也存在著類似這個悖論的情況。例如，如果你有1000萬美元（一堆錢），有人給你1美元，當被問及你的資產時，你會說你擁有1000萬美元零1美元嗎？不會，你很可能會說你擁有同樣的1000萬美元（同樣的「一堆」錢）。 眾所周知，古希臘聖賢提出問題，但在許多情況下，包括算術，卻沒有給出答案。結果，兩千多年來，這些問題被遺忘，每個人都滿足於傳統的算術。儘管有種種問題和悖論，算術仍然非常有用。 近年來，科學家和數學家們又重新回到了算術問題。著名的德國學者赫爾曼·路德維希·費迪南德·馮·亥姆霍茲（1821-1894）[1]是最早質疑傳統算術有效性的科學家之一。亥姆霍茲在其著作《計數與測量》（1887）[1]中，思考了一個重要的問題：算術在物理現像中的適用性。對於以科學的主要標準——觀察和實驗——來評判數學的科學家來說，這是一種自然的方法。 科學家的第一個觀察結果是，由於數字的概念源自於某種實踐，因此通常的算術必須適用於所有實際場景。然而，我們很容易發現許多情況並非如此。僅舉亥姆霍茲描述的幾個例子，一滴雨滴加到另一滴雨滴上並不會變成兩滴雨滴。這種情況可以用公式 1 + 1 = 1 來描述。 類似地，當兩個等量的…混合時…

1

 

1+1=3

1+1=3

Everybody knows that 1 + 1 = 2. However, in the 21st century, expressions such as 1 + 1 = 3 occurred to reflect important characteristics of economic and business processes. It seems that this contradicts core mathematical axioms and is incorrect from a mathematical point of view.

We find similarities in the history of mathematics―what had been considered strange, ungrounded and inconsistent with the existing mathematics, was incorporated later in the main body of mathematical knowledge. Here are some examples:

In China and India, mathematicians used negative numbers for centuries before these numbers came to Europe. However, when the European mathematicians encountered negative numbers, critics dismissed their sensibility. Some of the notable European mathematicians, such as d’Alembert or Frend, did not want to accept negative numbers until the 18th century and referred to them as “absurd” or “meaningless” (Kline, 1980 [5] , Mattessich, 1998 [6] ). Even in the 19th century, it was common practice to ignore any negative results derived from equations, on the assumption that they were meaningless (Martinez, 2006 [7] ). For instance, Lazare Carnot (1753-1823) affirmed that the idea of something being less than nothing is absurd (Mattessich, 1998 [6] ). Outstanding mathematicians such as William Hamilton (1805-1865) and August De Morgan (1806-1871) had akin opinions. Similarly, irrational numbers and later imaginary numbers were firstly rejected. Today these concepts are accepted and applied in numerous scientific and practical fields, such as physics, chemistry, biology and finance.

The interaction between physics and mathematics gives us one more example. In the 20th century, physicists started using functions that took infinite values at some points. At first, mathematicians objected by saying that there are no such functions in mathematics (cf., for example, von Neumann, 1955 [8] ). However, later they grounded utilization of these functions developing the theory of distributions and finding numerous new applications for this theory (Schwartz, 1950/1951 [4] ).

2. Problems with the Conventional Arithmetic

Human beings have used conventional arithmetic for millennia before the most inquisitive thinkers started questioning its validity in certain settings. It dates back to ancient Greece, where mathematicians and philosophers already started to doubt the convenience of conventional arithmetic. The Sophists, who lived from the second half of the fifth century B.C to the first half of the fourth century B.C. asserted the relativity of human knowledge and suggested many paradoxes, explicating complexity and diversity in the real world. One of them, the famous philosopher Zeno of Elea (490 - 430 B.C.), who was said to be a self- taught boy from the country side, invented very notable paradoxes, in which he questioned the popular knowledge and intuition related to fundamental essences such as time, space, and numbers.

An example of this type of reasoning is the paradox of the heap (or the Sorites paradox where σωρος is the Greek word for “heap”). It is possible to formulate this paradox in the following way.

1) One million grains of sand make a heap.

2) If one grain of sand is added to this heap, the heap stays the same.

3) However, when we add 1 to any natural number, we always get a new number.

There are analogies of this paradox in our time. For instance, if you have ten million dollars (a “heap” of money) and somebody will give you a dollar, will you say that you have ten million and one dollar when asked about your assets? No, you will most likely say that you have the same ten million dollars (the same “heap”).

As we know, Greek sages posed questions, but in many cases, including arithmetic, suggested no answers. As a result, for more than two thousand years, these problems were forgotten and everybody was satisfied with the conventional arithmetic. In spite of all problems and paradoxes, this arithmetic has remained very useful.

In recent times, scientists and mathematicians have returned to problems of arithmetic. The famous German researcher Herman Ludwig Ferdinand von Helmholtz (1821-1894) [1] was one the first scientists who questioned adequacy of the conventional arithmetic. In his “Counting and Measuring” (1887) [1] , Helmholtz considered an important problem of the applicability of arithmetic to physical phenomena. This was a natural approach of a scientist, who judged mathematics by the main criterion of science―observation and experiment.

The scientist’s first observation was that as the concept of a number is derived from some practice, usual arithmetic has to be applicable in all practical settings. However, it is easy to find many situations when this is not true. To mention but a few described by Helmholtz, one raindrop added to another raindrop does not make two raindrops. It is possible to describe this situation by the formula 1 + 1 = 1.

In a similar way, when one mixes two equal volumes of water, one at 40˚ Fahrenheit and the other at 50˚, one does not get two volumes at 90˚. Alike, the conventional arithmetic fails to describe correctly the result of combining gases or liquids by volumes. For example (Kline, 1980), one quart of alcohol and one quart of water yield about 1.8 quarts of vodka.

Later the famous French mathematician Henri Lebesgue facetiously pointed out (cf. Kline, 1980 [5] ) that if one puts a lion and a rabbit in a cage, one will not find two animals in the cage later on. In this case, we will also have 1 + 1 = 1.

However, since very few paid attention to the work of Helmholtz on arithmetic, and as still no alternative to the conventional arithmetic has been suggested, these problems were mostly forgotten. Only years later, in the second part of the 20th century, mathematicians began to doubt once more the absolute character of the ordinary arithmetic, where 2 + 2 = 4 and 2 × 2 = 4. Scientists and mathematicians again started to draw attention of the scientific community to the foundational problems of natural numbers and the conventional arithmetic. The most extreme assertion that there is only a finite quantity of natural numbers was suggested by Yesenin-Volpin (1960) [2] , who developed a mathematical direction called ultraintuitionism and took this assertion as one of the central postulates of ultraintuitionism. Other authors also considered arithmetics with a finite number of numbers, claiming that these arithmetics are inconsistent (cf., for example Van Bendegem, 1994 [3] and Rosinger, 2008 [4] ).

Van Danzig had similar ideas but expressed them in a different way. In his article (1956), he argued that only some of natural numbers may be considered finite. Consequently, all other mathematical entities that are called traditionally natural numbers are only some expressions but not numbers. These arguments are supported and extended by Blehman, et al. (1983) [9] .

Other authors are more moderate in their criticism of the conventional arithmetic. They write that not all natural numbers are similar in contrast to the presupposition of the conventional arithmetic that the set of natural numbers is uniform (Kolmogorov, 1961 [10] ; Littlewood, 1953 [11] ; Birkhoff and Bartee, 1967 [12] ; Dummett 1975 [13] ; Knuth, 1976 [14] ). Different types of natural numbers have been introduced, but without changing the conventional arithmetic. For example, Kolmogorov (1961) [10] suggested that in solving practical problems it is worth to separate small, medium, large, and super-large numbers.

Regarding geometry, it was discovered that there was not one but a variety of arithmetics, which were different in many ways from the conventional arithmetic. It is natural to call the conventional arithmetic by the name Diophantine arithmetic because the Greek mathematician Diophantus, who lived between 150 C.E. and 350 C.E., and who extensively contributed to the development of conventional arithmetic. Consequently, new arithmetics acquired the name non-Diophantine arithmetics.

Burgin built first Non-Diophantine arithmetics of whole and natural numbers (Burgin, 1977 [12] ; 1997 [15] ; 2007 [16] ; 2010 [17] [18] ) and Czachor extended this construction developing Non-Diophantine arithmetics of the real and complex numbers (Czachor, 2015 [19] ).

In the following section, we will show that non-Diophantine arithmetics occur in economics, starting with mergers and acquisitions.

3. Examples of Non-Diophantine Arithmetic

In the following section, we will show that non-Diophantine arithmetics occur in economics, business and social settings, starting with mergers and acquisitions.

Mergers and acquisitions (M & A) are processes in which the operating units of two companies are combined. Whereas in a merger, two approximately equally sized companies consolidate into one entity; in an acquisition, a larger company takes over the smaller one.

There are different motivations for M & As, the most typical being the following:

1) Synergies or economies of scale are achieved when the larger company may be able to lower the per unit purchasing cost due to higher bulk orders, or lower fixed cost by removing or combing duplicate departments such as research, accounting, or compliance.

2) Increased market share usually leads to the greater than before market power.

3) Technology driven M & As are aimed at gaining access to new technologies.

4) Tax reduction in M & A is the situation when a company may acquire a loss generating, yet presumably long term profitable company to reduce taxes.

5) In the case of CEO activism, a CEO may want to amplify his standing by increasing company size and seemingly showing leadership in M & A.

The success of M & As varies strongly. With respect to the cost saving due to synergy and economies of scale, many M & As do achieve their objective, as we can see in Figure 1.

 

However, with respect to expected revenue, only few companies achieve the desired objective, as we can see in Figure 2.

 

眾所周知，1 + 1 = 2。然而，在21世紀，諸如1 + 1 = 3之類的表達式卻被用來反映經濟和商業流程的重要特徵。這似乎與核心數學公理相矛盾，從數學角度來看是不正確的。 我們在數學史上發現了相似之處——那些曾經被認為是奇怪、毫無根據且與現有數學不符的東西，後來卻被納入了數學知識的主體。以下是一些例子： 在中國和印度，數學家在負數傳入歐洲之前的幾個世紀裡一直使用負數。然而，當歐洲數學家遇到負數時，批評者卻對其意義不屑一顧。一些著名的歐洲數學家，如達朗貝爾或弗倫德，直到18世紀才開始接受負數，並稱其“荒謬”或“毫無意義”（Kline，1980 [5] ，Mattessich，1998 [6] ）。即使在19世紀，人們普遍認為忽略任何由方程式得出的負結果毫無意義（Martinez，2006 [7]）。例如，拉扎爾·卡諾（Lazare Carnot，1753-1823）斷言「有小於無」的想法是荒謬的（Mattessich，1998 [6]）。威廉·漢密爾頓（William Hamilton，1805-1865）和奧古斯特·德·摩根（August De Morgan，1806-1871）等傑出數學家也持有類似的觀點。同樣，無理數以及後來的虛數最初也被否定。如今，這些概念已被接受並應用於許多科學和實踐領域，例如物理、化學、生物和金融。 物理學和數學之間的相互作用為我們提供了另一個例子。在20世紀，物理學家開始使用在某些點上取無窮值的函數。起初，數學家反對這種說法，認為數學中不存在這樣的函數（例如，馮諾依曼，1955 [8]）。然而，後來他們利用這些函數發展了分佈理論，並發現了該理論的許多新應用（施瓦茨，1950/1951 [4]）。 2. 傳統算術的問題 在最富於好奇心的思想家開始質疑其在某些情境下的有效性之前，人類已經使用傳統算術數千年。這種算術可以追溯到古希臘，當時的數學家和哲學家已經開始質疑傳統算術的便利性。生活在公元前五世紀下半葉至公元前四世紀上半葉的智者們主張人類知識的相對性，並提出了許多悖論，闡明了現實世界的複雜性和多樣性。其中一位是著名哲學家埃利亞的芝諾（西元前 490-430 年）。據說他是一位來自鄉下的自學成才的男孩，他提出了一些非常著名的悖論，質疑與時間、空間和數字等基本本質相關的普遍知識和直覺。 這種推理的一個例子是堆悖論（或稱 Sorites 悖論，其中 σωρος 是希臘語中「堆」的意思）。我們可以用以下方式來表達這個悖論。 1) 一百萬粒沙子可以堆成一堆。 2) 如果在這個堆裡再加一粒沙子，這個堆仍然保持不變。 3) 然而，當我們將 1 加到任何自然數上時，我們總是會得到一個新的數。 在我們這個時代，也存在著類似這個悖論的情況。例如，如果你有1000萬美元（一堆錢），有人給你1美元，當被問及你的資產時，你會說你擁有1000萬美元零1美元嗎？不會，你很可能會說你擁有同樣的1000萬美元（同樣的「一堆」錢）。 眾所周知，古希臘聖賢提出問題，但在許多情況下，包括算術，卻沒有給出答案。結果，兩千多年來，這些問題被遺忘，每個人都滿足於傳統的算術。儘管有種種問題和悖論，算術仍然非常有用。 近年來，科學家和數學家們又重新回到了算術問題。著名的德國學者赫爾曼·路德維希·費迪南德·馮·亥姆霍茲（1821-1894）[1]是最早質疑傳統算術有效性的科學家之一。亥姆霍茲在其著作《計數與測量》（1887）[1]中，思考了一個重要的問題：算術在物理現像中的適用性。對於以科學的主要標準——觀察和實驗——來評判數學的科學家來說，這是一種自然的方法。 科學家的第一個觀察結果是，由於數字的概念源自於某種實踐，因此通常的算術必須適用於所有實際場景。然而，我們很容易發現許多情況並非如此。僅舉亥姆霍茲描述的幾個例子，一滴雨滴加到另一滴雨滴上並不會變成兩滴雨滴。這種情況可以用公式 1 + 1 = 1 來描述。 類似地，當兩個等量的…混合時…

 

1+1=1

5+7=1

12+12=1

1

 

1+1=3

1+1=3

Everybody knows that 1 + 1 = 2. However, in the 21st century, expressions such as 1 + 1 = 3 occurred to reflect important characteristics of economic and business processes. It seems that this contradicts core mathematical axioms and is incorrect from a mathematical point of view.

We find similarities in the history of mathematics―what had been considered strange, ungrounded and inconsistent with the existing mathematics, was incorporated later in the main body of mathematical knowledge. Here are some examples:

In China and India, mathematicians used negative numbers for centuries before these numbers came to Europe. However, when the European mathematicians encountered negative numbers, critics dismissed their sensibility. Some of the notable European mathematicians, such as d’Alembert or Frend, did not want to accept negative numbers until the 18th century and referred to them as “absurd” or “meaningless” (Kline, 1980 [5] , Mattessich, 1998 [6] ). Even in the 19th century, it was common practice to ignore any negative results derived from equations, on the assumption that they were meaningless (Martinez, 2006 [7] ). For instance, Lazare Carnot (1753-1823) affirmed that the idea of something being less than nothing is absurd (Mattessich, 1998 [6] ). Outstanding mathematicians such as William Hamilton (1805-1865) and August De Morgan (1806-1871) had akin opinions. Similarly, irrational numbers and later imaginary numbers were firstly rejected. Today these concepts are accepted and applied in numerous scientific and practical fields, such as physics, chemistry, biology and finance.

The interaction between physics and mathematics gives us one more example. In the 20th century, physicists started using functions that took infinite values at some points. At first, mathematicians objected by saying that there are no such functions in mathematics (cf., for example, von Neumann, 1955 [8] ). However, later they grounded utilization of these functions developing the theory of distributions and finding numerous new applications for this theory (Schwartz, 1950/1951 [4] ).

2. Problems with the Conventional Arithmetic

Human beings have used conventional arithmetic for millennia before the most inquisitive thinkers started questioning its validity in certain settings. It dates back to ancient Greece, where mathematicians and philosophers already started to doubt the convenience of conventional arithmetic. The Sophists, who lived from the second half of the fifth century B.C to the first half of the fourth century B.C. asserted the relativity of human knowledge and suggested many paradoxes, explicating complexity and diversity in the real world. One of them, the famous philosopher Zeno of Elea (490 - 430 B.C.), who was said to be a self- taught boy from the country side, invented very notable paradoxes, in which he questioned the popular knowledge and intuition related to fundamental essences such as time, space, and numbers.

An example of this type of reasoning is the paradox of the heap (or the Sorites paradox where σωρος is the Greek word for “heap”). It is possible to formulate this paradox in the following way.

1) One million grains of sand make a heap.

2) If one grain of sand is added to this heap, the heap stays the same.

3) However, when we add 1 to any natural number, we always get a new number.

There are analogies of this paradox in our time. For instance, if you have ten million dollars (a “heap” of money) and somebody will give you a dollar, will you say that you have ten million and one dollar when asked about your assets? No, you will most likely say that you have the same ten million dollars (the same “heap”).

As we know, Greek sages posed questions, but in many cases, including arithmetic, suggested no answers. As a result, for more than two thousand years, these problems were forgotten and everybody was satisfied with the conventional arithmetic. In spite of all problems and paradoxes, this arithmetic has remained very useful.

In recent times, scientists and mathematicians have returned to problems of arithmetic. The famous German researcher Herman Ludwig Ferdinand von Helmholtz (1821-1894) [1] was one the first scientists who questioned adequacy of the conventional arithmetic. In his “Counting and Measuring” (1887) [1] , Helmholtz considered an important problem of the applicability of arithmetic to physical phenomena. This was a natural approach of a scientist, who judged mathematics by the main criterion of science―observation and experiment.

The scientist’s first observation was that as the concept of a number is derived from some practice, usual arithmetic has to be applicable in all practical settings. However, it is easy to find many situations when this is not true. To mention but a few described by Helmholtz, one raindrop added to another raindrop does not make two raindrops. It is possible to describe this situation by the formula 1 + 1 = 1.

In a similar way, when one mixes two equal volumes of water, one at 40˚ Fahrenheit and the other at 50˚, one does not get two volumes at 90˚. Alike, the conventional arithmetic fails to describe correctly the result of combining gases or liquids by volumes. For example (Kline, 1980), one quart of alcohol and one quart of water yield about 1.8 quarts of vodka.

Later the famous French mathematician Henri Lebesgue facetiously pointed out (cf. Kline, 1980 [5] ) that if one puts a lion and a rabbit in a cage, one will not find two animals in the cage later on. In this case, we will also have 1 + 1 = 1.

However, since very few paid attention to the work of Helmholtz on arithmetic, and as still no alternative to the conventional arithmetic has been suggested, these problems were mostly forgotten. Only years later, in the second part of the 20th century, mathematicians began to doubt once more the absolute character of the ordinary arithmetic, where 2 + 2 = 4 and 2 × 2 = 4. Scientists and mathematicians again started to draw attention of the scientific community to the foundational problems of natural numbers and the conventional arithmetic. The most extreme assertion that there is only a finite quantity of natural numbers was suggested by Yesenin-Volpin (1960) [2] , who developed a mathematical direction called ultraintuitionism and took this assertion as one of the central postulates of ultraintuitionism. Other authors also considered arithmetics with a finite number of numbers, claiming that these arithmetics are inconsistent (cf., for example Van Bendegem, 1994 [3] and Rosinger, 2008 [4] ).

Van Danzig had similar ideas but expressed them in a different way. In his article (1956), he argued that only some of natural numbers may be considered finite. Consequently, all other mathematical entities that are called traditionally natural numbers are only some expressions but not numbers. These arguments are supported and extended by Blehman, et al. (1983) [9] .

Other authors are more moderate in their criticism of the conventional arithmetic. They write that not all natural numbers are similar in contrast to the presupposition of the conventional arithmetic that the set of natural numbers is uniform (Kolmogorov, 1961 [10] ; Littlewood, 1953 [11] ; Birkhoff and Bartee, 1967 [12] ; Dummett 1975 [13] ; Knuth, 1976 [14] ). Different types of natural numbers have been introduced, but without changing the conventional arithmetic. For example, Kolmogorov (1961) [10] suggested that in solving practical problems it is worth to separate small, medium, large, and super-large numbers.

Regarding geometry, it was discovered that there was not one but a variety of arithmetics, which were different in many ways from the conventional arithmetic. It is natural to call the conventional arithmetic by the name Diophantine arithmetic because the Greek mathematician Diophantus, who lived between 150 C.E. and 350 C.E., and who extensively contributed to the development of conventional arithmetic. Consequently, new arithmetics acquired the name non-Diophantine arithmetics.

Burgin built first Non-Diophantine arithmetics of whole and natural numbers (Burgin, 1977 [12] ; 1997 [15] ; 2007 [16] ; 2010 [17] [18] ) and Czachor extended this construction developing Non-Diophantine arithmetics of the real and complex numbers (Czachor, 2015 [19] ).

In the following section, we will show that non-Diophantine arithmetics occur in economics, starting with mergers and acquisitions.

3. Examples of Non-Diophantine Arithmetic

In the following section, we will show that non-Diophantine arithmetics occur in economics, business and social settings, starting with mergers and acquisitions.

Mergers and acquisitions (M & A) are processes in which the operating units of two companies are combined. Whereas in a merger, two approximately equally sized companies consolidate into one entity; in an acquisition, a larger company takes over the smaller one.

There are different motivations for M & As, the most typical being the following:

1) Synergies or economies of scale are achieved when the larger company may be able to lower the per unit purchasing cost due to higher bulk orders, or lower fixed cost by removing or combing duplicate departments such as research, accounting, or compliance.

2) Increased market share usually leads to the greater than before market power.

3) Technology driven M & As are aimed at gaining access to new technologies.

4) Tax reduction in M & A is the situation when a company may acquire a loss generating, yet presumably long term profitable company to reduce taxes.

5) In the case of CEO activism, a CEO may want to amplify his standing by increasing company size and seemingly showing leadership in M & A.

The success of M & As varies strongly. With respect to the cost saving due to synergy and economies of scale, many M & As do achieve their objective, as we can see in Figure 1.

 

However, with respect to expected revenue, only few companies achieve the desired objective, as we can see in Figure 2.

 

眾所周知，1 + 1 = 2。然而，在21世紀，諸如1 + 1 = 3之類的表達式卻被用來反映經濟和商業流程的重要特徵。這似乎與核心數學公理相矛盾，從數學角度來看是不正確的。 我們在數學史上發現了相似之處——那些曾經被認為是奇怪、毫無根據且與現有數學不符的東西，後來卻被納入了數學知識的主體。以下是一些例子： 在中國和印度，數學家在負數傳入歐洲之前的幾個世紀裡一直使用負數。然而，當歐洲數學家遇到負數時，批評者卻對其意義不屑一顧。一些著名的歐洲數學家，如達朗貝爾或弗倫德，直到18世紀才開始接受負數，並稱其“荒謬”或“毫無意義”（Kline，1980 [5] ，Mattessich，1998 [6] ）。即使在19世紀，人們普遍認為忽略任何由方程式得出的負結果毫無意義（Martinez，2006 [7]）。例如，拉扎爾·卡諾（Lazare Carnot，1753-1823）斷言「有小於無」的想法是荒謬的（Mattessich，1998 [6]）。威廉·漢密爾頓（William Hamilton，1805-1865）和奧古斯特·德·摩根（August De Morgan，1806-1871）等傑出數學家也持有類似的觀點。同樣，無理數以及後來的虛數最初也被否定。如今，這些概念已被接受並應用於許多科學和實踐領域，例如物理、化學、生物和金融。 物理學和數學之間的相互作用為我們提供了另一個例子。在20世紀，物理學家開始使用在某些點上取無窮值的函數。起初，數學家反對這種說法，認為數學中不存在這樣的函數（例如，馮諾依曼，1955 [8]）。然而，後來他們利用這些函數發展了分佈理論，並發現了該理論的許多新應用（施瓦茨，1950/1951 [4]）。 2. 傳統算術的問題 在最富於好奇心的思想家開始質疑其在某些情境下的有效性之前，人類已經使用傳統算術數千年。這種算術可以追溯到古希臘，當時的數學家和哲學家已經開始質疑傳統算術的便利性。生活在公元前五世紀下半葉至公元前四世紀上半葉的智者們主張人類知識的相對性，並提出了許多悖論，闡明了現實世界的複雜性和多樣性。其中一位是著名哲學家埃利亞的芝諾（西元前 490-430 年）。據說他是一位來自鄉下的自學成才的男孩，他提出了一些非常著名的悖論，質疑與時間、空間和數字等基本本質相關的普遍知識和直覺。 這種推理的一個例子是堆悖論（或稱 Sorites 悖論，其中 σωρος 是希臘語中「堆」的意思）。我們可以用以下方式來表達這個悖論。 1) 一百萬粒沙子可以堆成一堆。 2) 如果在這個堆裡再加一粒沙子，這個堆仍然保持不變。 3) 然而，當我們將 1 加到任何自然數上時，我們總是會得到一個新的數。 在我們這個時代，也存在著類似這個悖論的情況。例如，如果你有1000萬美元（一堆錢），有人給你1美元，當被問及你的資產時，你會說你擁有1000萬美元零1美元嗎？不會，你很可能會說你擁有同樣的1000萬美元（同樣的「一堆」錢）。 眾所周知，古希臘聖賢提出問題，但在許多情況下，包括算術，卻沒有給出答案。結果，兩千多年來，這些問題被遺忘，每個人都滿足於傳統的算術。儘管有種種問題和悖論，算術仍然非常有用。 近年來，科學家和數學家們又重新回到了算術問題。著名的德國學者赫爾曼·路德維希·費迪南德·馮·亥姆霍茲（1821-1894）[1]是最早質疑傳統算術有效性的科學家之一。亥姆霍茲在其著作《計數與測量》（1887）[1]中，思考了一個重要的問題：算術在物理現像中的適用性。對於以科學的主要標準——觀察和實驗——來評判數學的科學家來說，這是一種自然的方法。 科學家的第一個觀察結果是，由於數字的概念源自於某種實踐，因此通常的算術必須適用於所有實際場景。然而，我們很容易發現許多情況並非如此。僅舉亥姆霍茲描述的幾個例子，一滴雨滴加到另一滴雨滴上並不會變成兩滴雨滴。這種情況可以用公式 1 + 1 = 1 來描述。 類似地，當兩個等量的…混合時…

 

5+7=1

12+12=1

 

因為

5月+ 7月= 1年

12小時 + 12小時

1

 

1+1=3

1+1=3

Everybody knows that 1 + 1 = 2. However, in the 21st century, expressions such as 1 + 1 = 3 occurred to reflect important characteristics of economic and business processes. It seems that this contradicts core mathematical axioms and is incorrect from a mathematical point of view.

We find similarities in the history of mathematics―what had been considered strange, ungrounded and inconsistent with the existing mathematics, was incorporated later in the main body of mathematical knowledge. Here are some examples:

In China and India, mathematicians used negative numbers for centuries before these numbers came to Europe. However, when the European mathematicians encountered negative numbers, critics dismissed their sensibility. Some of the notable European mathematicians, such as d’Alembert or Frend, did not want to accept negative numbers until the 18th century and referred to them as “absurd” or “meaningless” (Kline, 1980 [5] , Mattessich, 1998 [6] ). Even in the 19th century, it was common practice to ignore any negative results derived from equations, on the assumption that they were meaningless (Martinez, 2006 [7] ). For instance, Lazare Carnot (1753-1823) affirmed that the idea of something being less than nothing is absurd (Mattessich, 1998 [6] ). Outstanding mathematicians such as William Hamilton (1805-1865) and August De Morgan (1806-1871) had akin opinions. Similarly, irrational numbers and later imaginary numbers were firstly rejected. Today these concepts are accepted and applied in numerous scientific and practical fields, such as physics, chemistry, biology and finance.

The interaction between physics and mathematics gives us one more example. In the 20th century, physicists started using functions that took infinite values at some points. At first, mathematicians objected by saying that there are no such functions in mathematics (cf., for example, von Neumann, 1955 [8] ). However, later they grounded utilization of these functions developing the theory of distributions and finding numerous new applications for this theory (Schwartz, 1950/1951 [4] ).

2. Problems with the Conventional Arithmetic

Human beings have used conventional arithmetic for millennia before the most inquisitive thinkers started questioning its validity in certain settings. It dates back to ancient Greece, where mathematicians and philosophers already started to doubt the convenience of conventional arithmetic. The Sophists, who lived from the second half of the fifth century B.C to the first half of the fourth century B.C. asserted the relativity of human knowledge and suggested many paradoxes, explicating complexity and diversity in the real world. One of them, the famous philosopher Zeno of Elea (490 - 430 B.C.), who was said to be a self- taught boy from the country side, invented very notable paradoxes, in which he questioned the popular knowledge and intuition related to fundamental essences such as time, space, and numbers.

An example of this type of reasoning is the paradox of the heap (or the Sorites paradox where σωρος is the Greek word for “heap”). It is possible to formulate this paradox in the following way.

1) One million grains of sand make a heap.

2) If one grain of sand is added to this heap, the heap stays the same.

3) However, when we add 1 to any natural number, we always get a new number.

There are analogies of this paradox in our time. For instance, if you have ten million dollars (a “heap” of money) and somebody will give you a dollar, will you say that you have ten million and one dollar when asked about your assets? No, you will most likely say that you have the same ten million dollars (the same “heap”).

As we know, Greek sages posed questions, but in many cases, including arithmetic, suggested no answers. As a result, for more than two thousand years, these problems were forgotten and everybody was satisfied with the conventional arithmetic. In spite of all problems and paradoxes, this arithmetic has remained very useful.

In recent times, scientists and mathematicians have returned to problems of arithmetic. The famous German researcher Herman Ludwig Ferdinand von Helmholtz (1821-1894) [1] was one the first scientists who questioned adequacy of the conventional arithmetic. In his “Counting and Measuring” (1887) [1] , Helmholtz considered an important problem of the applicability of arithmetic to physical phenomena. This was a natural approach of a scientist, who judged mathematics by the main criterion of science―observation and experiment.

The scientist’s first observation was that as the concept of a number is derived from some practice, usual arithmetic has to be applicable in all practical settings. However, it is easy to find many situations when this is not true. To mention but a few described by Helmholtz, one raindrop added to another raindrop does not make two raindrops. It is possible to describe this situation by the formula 1 + 1 = 1.

In a similar way, when one mixes two equal volumes of water, one at 40˚ Fahrenheit and the other at 50˚, one does not get two volumes at 90˚. Alike, the conventional arithmetic fails to describe correctly the result of combining gases or liquids by volumes. For example (Kline, 1980), one quart of alcohol and one quart of water yield about 1.8 quarts of vodka.

Later the famous French mathematician Henri Lebesgue facetiously pointed out (cf. Kline, 1980 [5] ) that if one puts a lion and a rabbit in a cage, one will not find two animals in the cage later on. In this case, we will also have 1 + 1 = 1.

However, since very few paid attention to the work of Helmholtz on arithmetic, and as still no alternative to the conventional arithmetic has been suggested, these problems were mostly forgotten. Only years later, in the second part of the 20th century, mathematicians began to doubt once more the absolute character of the ordinary arithmetic, where 2 + 2 = 4 and 2 × 2 = 4. Scientists and mathematicians again started to draw attention of the scientific community to the foundational problems of natural numbers and the conventional arithmetic. The most extreme assertion that there is only a finite quantity of natural numbers was suggested by Yesenin-Volpin (1960) [2] , who developed a mathematical direction called ultraintuitionism and took this assertion as one of the central postulates of ultraintuitionism. Other authors also considered arithmetics with a finite number of numbers, claiming that these arithmetics are inconsistent (cf., for example Van Bendegem, 1994 [3] and Rosinger, 2008 [4] ).

Van Danzig had similar ideas but expressed them in a different way. In his article (1956), he argued that only some of natural numbers may be considered finite. Consequently, all other mathematical entities that are called traditionally natural numbers are only some expressions but not numbers. These arguments are supported and extended by Blehman, et al. (1983) [9] .

Other authors are more moderate in their criticism of the conventional arithmetic. They write that not all natural numbers are similar in contrast to the presupposition of the conventional arithmetic that the set of natural numbers is uniform (Kolmogorov, 1961 [10] ; Littlewood, 1953 [11] ; Birkhoff and Bartee, 1967 [12] ; Dummett 1975 [13] ; Knuth, 1976 [14] ). Different types of natural numbers have been introduced, but without changing the conventional arithmetic. For example, Kolmogorov (1961) [10] suggested that in solving practical problems it is worth to separate small, medium, large, and super-large numbers.

Regarding geometry, it was discovered that there was not one but a variety of arithmetics, which were different in many ways from the conventional arithmetic. It is natural to call the conventional arithmetic by the name Diophantine arithmetic because the Greek mathematician Diophantus, who lived between 150 C.E. and 350 C.E., and who extensively contributed to the development of conventional arithmetic. Consequently, new arithmetics acquired the name non-Diophantine arithmetics.

Burgin built first Non-Diophantine arithmetics of whole and natural numbers (Burgin, 1977 [12] ; 1997 [15] ; 2007 [16] ; 2010 [17] [18] ) and Czachor extended this construction developing Non-Diophantine arithmetics of the real and complex numbers (Czachor, 2015 [19] ).

In the following section, we will show that non-Diophantine arithmetics occur in economics, starting with mergers and acquisitions.

3. Examples of Non-Diophantine Arithmetic

In the following section, we will show that non-Diophantine arithmetics occur in economics, business and social settings, starting with mergers and acquisitions.

Mergers and acquisitions (M & A) are processes in which the operating units of two companies are combined. Whereas in a merger, two approximately equally sized companies consolidate into one entity; in an acquisition, a larger company takes over the smaller one.

There are different motivations for M & As, the most typical being the following:

1) Synergies or economies of scale are achieved when the larger company may be able to lower the per unit purchasing cost due to higher bulk orders, or lower fixed cost by removing or combing duplicate departments such as research, accounting, or compliance.

2) Increased market share usually leads to the greater than before market power.

3) Technology driven M & As are aimed at gaining access to new technologies.

4) Tax reduction in M & A is the situation when a company may acquire a loss generating, yet presumably long term profitable company to reduce taxes.

5) In the case of CEO activism, a CEO may want to amplify his standing by increasing company size and seemingly showing leadership in M & A.

The success of M & As varies strongly. With respect to the cost saving due to synergy and economies of scale, many M & As do achieve their objective, as we can see in Figure 1.

 

However, with respect to expected revenue, only few companies achieve the desired objective, as we can see in Figure 2.

 

眾所周知，1 + 1 = 2。然而，在21世紀，諸如1 + 1 = 3之類的表達式卻被用來反映經濟和商業流程的重要特徵。這似乎與核心數學公理相矛盾，從數學角度來看是不正確的。 我們在數學史上發現了相似之處——那些曾經被認為是奇怪、毫無根據且與現有數學不符的東西，後來卻被納入了數學知識的主體。以下是一些例子： 在中國和印度，數學家在負數傳入歐洲之前的幾個世紀裡一直使用負數。然而，當歐洲數學家遇到負數時，批評者卻對其意義不屑一顧。一些著名的歐洲數學家，如達朗貝爾或弗倫德，直到18世紀才開始接受負數，並稱其“荒謬”或“毫無意義”（Kline，1980 [5] ，Mattessich，1998 [6] ）。即使在19世紀，人們普遍認為忽略任何由方程式得出的負結果毫無意義（Martinez，2006 [7]）。例如，拉扎爾·卡諾（Lazare Carnot，1753-1823）斷言「有小於無」的想法是荒謬的（Mattessich，1998 [6]）。威廉·漢密爾頓（William Hamilton，1805-1865）和奧古斯特·德·摩根（August De Morgan，1806-1871）等傑出數學家也持有類似的觀點。同樣，無理數以及後來的虛數最初也被否定。如今，這些概念已被接受並應用於許多科學和實踐領域，例如物理、化學、生物和金融。 物理學和數學之間的相互作用為我們提供了另一個例子。在20世紀，物理學家開始使用在某些點上取無窮值的函數。起初，數學家反對這種說法，認為數學中不存在這樣的函數（例如，馮諾依曼，1955 [8]）。然而，後來他們利用這些函數發展了分佈理論，並發現了該理論的許多新應用（施瓦茨，1950/1951 [4]）。 2. 傳統算術的問題 在最富於好奇心的思想家開始質疑其在某些情境下的有效性之前，人類已經使用傳統算術數千年。這種算術可以追溯到古希臘，當時的數學家和哲學家已經開始質疑傳統算術的便利性。生活在公元前五世紀下半葉至公元前四世紀上半葉的智者們主張人類知識的相對性，並提出了許多悖論，闡明了現實世界的複雜性和多樣性。其中一位是著名哲學家埃利亞的芝諾（西元前 490-430 年）。據說他是一位來自鄉下的自學成才的男孩，他提出了一些非常著名的悖論，質疑與時間、空間和數字等基本本質相關的普遍知識和直覺。 這種推理的一個例子是堆悖論（或稱 Sorites 悖論，其中 σωρος 是希臘語中「堆」的意思）。我們可以用以下方式來表達這個悖論。 1) 一百萬粒沙子可以堆成一堆。 2) 如果在這個堆裡再加一粒沙子，這個堆仍然保持不變。 3) 然而，當我們將 1 加到任何自然數上時，我們總是會得到一個新的數。 在我們這個時代，也存在著類似這個悖論的情況。例如，如果你有1000萬美元（一堆錢），有人給你1美元，當被問及你的資產時，你會說你擁有1000萬美元零1美元嗎？不會，你很可能會說你擁有同樣的1000萬美元（同樣的「一堆」錢）。 眾所周知，古希臘聖賢提出問題，但在許多情況下，包括算術，卻沒有給出答案。結果，兩千多年來，這些問題被遺忘，每個人都滿足於傳統的算術。儘管有種種問題和悖論，算術仍然非常有用。 近年來，科學家和數學家們又重新回到了算術問題。著名的德國學者赫爾曼·路德維希·費迪南德·馮·亥姆霍茲（1821-1894）[1]是最早質疑傳統算術有效性的科學家之一。亥姆霍茲在其著作《計數與測量》（1887）[1]中，思考了一個重要的問題：算術在物理現像中的適用性。對於以科學的主要標準——觀察和實驗——來評判數學的科學家來說，這是一種自然的方法。 科學家的第一個觀察結果是，由於數字的概念源自於某種實踐，因此通常的算術必須適用於所有實際場景。然而，我們很容易發現許多情況並非如此。僅舉亥姆霍茲描述的幾個例子，一滴雨滴加到另一滴雨滴上並不會變成兩滴雨滴。這種情況可以用公式 1 + 1 = 1 來描述。 類似地，當兩個等量的…混合時…

 

5+7=1

12+12=1

 

因為

5月+ 7月= 1年

12小時 + 12小時

 

12小時+12小時 = 1天

1

 

1+1=3

1+1=3

Everybody knows that 1 + 1 = 2. However, in the 21st century, expressions such as 1 + 1 = 3 occurred to reflect important characteristics of economic and business processes. It seems that this contradicts core mathematical axioms and is incorrect from a mathematical point of view.

We find similarities in the history of mathematics―what had been considered strange, ungrounded and inconsistent with the existing mathematics, was incorporated later in the main body of mathematical knowledge. Here are some examples:

In China and India, mathematicians used negative numbers for centuries before these numbers came to Europe. However, when the European mathematicians encountered negative numbers, critics dismissed their sensibility. Some of the notable European mathematicians, such as d’Alembert or Frend, did not want to accept negative numbers until the 18th century and referred to them as “absurd” or “meaningless” (Kline, 1980 [5] , Mattessich, 1998 [6] ). Even in the 19th century, it was common practice to ignore any negative results derived from equations, on the assumption that they were meaningless (Martinez, 2006 [7] ). For instance, Lazare Carnot (1753-1823) affirmed that the idea of something being less than nothing is absurd (Mattessich, 1998 [6] ). Outstanding mathematicians such as William Hamilton (1805-1865) and August De Morgan (1806-1871) had akin opinions. Similarly, irrational numbers and later imaginary numbers were firstly rejected. Today these concepts are accepted and applied in numerous scientific and practical fields, such as physics, chemistry, biology and finance.

The interaction between physics and mathematics gives us one more example. In the 20th century, physicists started using functions that took infinite values at some points. At first, mathematicians objected by saying that there are no such functions in mathematics (cf., for example, von Neumann, 1955 [8] ). However, later they grounded utilization of these functions developing the theory of distributions and finding numerous new applications for this theory (Schwartz, 1950/1951 [4] ).

2. Problems with the Conventional Arithmetic

Human beings have used conventional arithmetic for millennia before the most inquisitive thinkers started questioning its validity in certain settings. It dates back to ancient Greece, where mathematicians and philosophers already started to doubt the convenience of conventional arithmetic. The Sophists, who lived from the second half of the fifth century B.C to the first half of the fourth century B.C. asserted the relativity of human knowledge and suggested many paradoxes, explicating complexity and diversity in the real world. One of them, the famous philosopher Zeno of Elea (490 - 430 B.C.), who was said to be a self- taught boy from the country side, invented very notable paradoxes, in which he questioned the popular knowledge and intuition related to fundamental essences such as time, space, and numbers.

An example of this type of reasoning is the paradox of the heap (or the Sorites paradox where σωρος is the Greek word for “heap”). It is possible to formulate this paradox in the following way.

1) One million grains of sand make a heap.

2) If one grain of sand is added to this heap, the heap stays the same.

3) However, when we add 1 to any natural number, we always get a new number.

There are analogies of this paradox in our time. For instance, if you have ten million dollars (a “heap” of money) and somebody will give you a dollar, will you say that you have ten million and one dollar when asked about your assets? No, you will most likely say that you have the same ten million dollars (the same “heap”).

As we know, Greek sages posed questions, but in many cases, including arithmetic, suggested no answers. As a result, for more than two thousand years, these problems were forgotten and everybody was satisfied with the conventional arithmetic. In spite of all problems and paradoxes, this arithmetic has remained very useful.

In recent times, scientists and mathematicians have returned to problems of arithmetic. The famous German researcher Herman Ludwig Ferdinand von Helmholtz (1821-1894) [1] was one the first scientists who questioned adequacy of the conventional arithmetic. In his “Counting and Measuring” (1887) [1] , Helmholtz considered an important problem of the applicability of arithmetic to physical phenomena. This was a natural approach of a scientist, who judged mathematics by the main criterion of science―observation and experiment.

The scientist’s first observation was that as the concept of a number is derived from some practice, usual arithmetic has to be applicable in all practical settings. However, it is easy to find many situations when this is not true. To mention but a few described by Helmholtz, one raindrop added to another raindrop does not make two raindrops. It is possible to describe this situation by the formula 1 + 1 = 1.

In a similar way, when one mixes two equal volumes of water, one at 40˚ Fahrenheit and the other at 50˚, one does not get two volumes at 90˚. Alike, the conventional arithmetic fails to describe correctly the result of combining gases or liquids by volumes. For example (Kline, 1980), one quart of alcohol and one quart of water yield about 1.8 quarts of vodka.

Later the famous French mathematician Henri Lebesgue facetiously pointed out (cf. Kline, 1980 [5] ) that if one puts a lion and a rabbit in a cage, one will not find two animals in the cage later on. In this case, we will also have 1 + 1 = 1.

However, since very few paid attention to the work of Helmholtz on arithmetic, and as still no alternative to the conventional arithmetic has been suggested, these problems were mostly forgotten. Only years later, in the second part of the 20th century, mathematicians began to doubt once more the absolute character of the ordinary arithmetic, where 2 + 2 = 4 and 2 × 2 = 4. Scientists and mathematicians again started to draw attention of the scientific community to the foundational problems of natural numbers and the conventional arithmetic. The most extreme assertion that there is only a finite quantity of natural numbers was suggested by Yesenin-Volpin (1960) [2] , who developed a mathematical direction called ultraintuitionism and took this assertion as one of the central postulates of ultraintuitionism. Other authors also considered arithmetics with a finite number of numbers, claiming that these arithmetics are inconsistent (cf., for example Van Bendegem, 1994 [3] and Rosinger, 2008 [4] ).

Van Danzig had similar ideas but expressed them in a different way. In his article (1956), he argued that only some of natural numbers may be considered finite. Consequently, all other mathematical entities that are called traditionally natural numbers are only some expressions but not numbers. These arguments are supported and extended by Blehman, et al. (1983) [9] .

Other authors are more moderate in their criticism of the conventional arithmetic. They write that not all natural numbers are similar in contrast to the presupposition of the conventional arithmetic that the set of natural numbers is uniform (Kolmogorov, 1961 [10] ; Littlewood, 1953 [11] ; Birkhoff and Bartee, 1967 [12] ; Dummett 1975 [13] ; Knuth, 1976 [14] ). Different types of natural numbers have been introduced, but without changing the conventional arithmetic. For example, Kolmogorov (1961) [10] suggested that in solving practical problems it is worth to separate small, medium, large, and super-large numbers.

Regarding geometry, it was discovered that there was not one but a variety of arithmetics, which were different in many ways from the conventional arithmetic. It is natural to call the conventional arithmetic by the name Diophantine arithmetic because the Greek mathematician Diophantus, who lived between 150 C.E. and 350 C.E., and who extensively contributed to the development of conventional arithmetic. Consequently, new arithmetics acquired the name non-Diophantine arithmetics.

Burgin built first Non-Diophantine arithmetics of whole and natural numbers (Burgin, 1977 [12] ; 1997 [15] ; 2007 [16] ; 2010 [17] [18] ) and Czachor extended this construction developing Non-Diophantine arithmetics of the real and complex numbers (Czachor, 2015 [19] ).

In the following section, we will show that non-Diophantine arithmetics occur in economics, starting with mergers and acquisitions.

3. Examples of Non-Diophantine Arithmetic

In the following section, we will show that non-Diophantine arithmetics occur in economics, business and social settings, starting with mergers and acquisitions.

Mergers and acquisitions (M & A) are processes in which the operating units of two companies are combined. Whereas in a merger, two approximately equally sized companies consolidate into one entity; in an acquisition, a larger company takes over the smaller one.

There are different motivations for M & As, the most typical being the following:

1) Synergies or economies of scale are achieved when the larger company may be able to lower the per unit purchasing cost due to higher bulk orders, or lower fixed cost by removing or combing duplicate departments such as research, accounting, or compliance.

2) Increased market share usually leads to the greater than before market power.

3) Technology driven M & As are aimed at gaining access to new technologies.

4) Tax reduction in M & A is the situation when a company may acquire a loss generating, yet presumably long term profitable company to reduce taxes.

5) In the case of CEO activism, a CEO may want to amplify his standing by increasing company size and seemingly showing leadership in M & A.

The success of M & As varies strongly. With respect to the cost saving due to synergy and economies of scale, many M & As do achieve their objective, as we can see in Figure 1.

 

However, with respect to expected revenue, only few companies achieve the desired objective, as we can see in Figure 2.

 

眾所周知，1 + 1 = 2。然而，在21世紀，諸如1 + 1 = 3之類的表達式卻被用來反映經濟和商業流程的重要特徵。這似乎與核心數學公理相矛盾，從數學角度來看是不正確的。 我們在數學史上發現了相似之處——那些曾經被認為是奇怪、毫無根據且與現有數學不符的東西，後來卻被納入了數學知識的主體。以下是一些例子： 在中國和印度，數學家在負數傳入歐洲之前的幾個世紀裡一直使用負數。然而，當歐洲數學家遇到負數時，批評者卻對其意義不屑一顧。一些著名的歐洲數學家，如達朗貝爾或弗倫德，直到18世紀才開始接受負數，並稱其“荒謬”或“毫無意義”（Kline，1980 [5] ，Mattessich，1998 [6] ）。即使在19世紀，人們普遍認為忽略任何由方程式得出的負結果毫無意義（Martinez，2006 [7]）。例如，拉扎爾·卡諾（Lazare Carnot，1753-1823）斷言「有小於無」的想法是荒謬的（Mattessich，1998 [6]）。威廉·漢密爾頓（William Hamilton，1805-1865）和奧古斯特·德·摩根（August De Morgan，1806-1871）等傑出數學家也持有類似的觀點。同樣，無理數以及後來的虛數最初也被否定。如今，這些概念已被接受並應用於許多科學和實踐領域，例如物理、化學、生物和金融。 物理學和數學之間的相互作用為我們提供了另一個例子。在20世紀，物理學家開始使用在某些點上取無窮值的函數。起初，數學家反對這種說法，認為數學中不存在這樣的函數（例如，馮諾依曼，1955 [8]）。然而，後來他們利用這些函數發展了分佈理論，並發現了該理論的許多新應用（施瓦茨，1950/1951 [4]）。 2. 傳統算術的問題 在最富於好奇心的思想家開始質疑其在某些情境下的有效性之前，人類已經使用傳統算術數千年。這種算術可以追溯到古希臘，當時的數學家和哲學家已經開始質疑傳統算術的便利性。生活在公元前五世紀下半葉至公元前四世紀上半葉的智者們主張人類知識的相對性，並提出了許多悖論，闡明了現實世界的複雜性和多樣性。其中一位是著名哲學家埃利亞的芝諾（西元前 490-430 年）。據說他是一位來自鄉下的自學成才的男孩，他提出了一些非常著名的悖論，質疑與時間、空間和數字等基本本質相關的普遍知識和直覺。 這種推理的一個例子是堆悖論（或稱 Sorites 悖論，其中 σωρος 是希臘語中「堆」的意思）。我們可以用以下方式來表達這個悖論。 1) 一百萬粒沙子可以堆成一堆。 2) 如果在這個堆裡再加一粒沙子，這個堆仍然保持不變。 3) 然而，當我們將 1 加到任何自然數上時，我們總是會得到一個新的數。 在我們這個時代，也存在著類似這個悖論的情況。例如，如果你有1000萬美元（一堆錢），有人給你1美元，當被問及你的資產時，你會說你擁有1000萬美元零1美元嗎？不會，你很可能會說你擁有同樣的1000萬美元（同樣的「一堆」錢）。 眾所周知，古希臘聖賢提出問題，但在許多情況下，包括算術，卻沒有給出答案。結果，兩千多年來，這些問題被遺忘，每個人都滿足於傳統的算術。儘管有種種問題和悖論，算術仍然非常有用。 近年來，科學家和數學家們又重新回到了算術問題。著名的德國學者赫爾曼·路德維希·費迪南德·馮·亥姆霍茲（1821-1894）[1]是最早質疑傳統算術有效性的科學家之一。亥姆霍茲在其著作《計數與測量》（1887）[1]中，思考了一個重要的問題：算術在物理現像中的適用性。對於以科學的主要標準——觀察和實驗——來評判數學的科學家來說，這是一種自然的方法。 科學家的第一個觀察結果是，由於數字的概念源自於某種實踐，因此通常的算術必須適用於所有實際場景。然而，我們很容易發現許多情況並非如此。僅舉亥姆霍茲描述的幾個例子，一滴雨滴加到另一滴雨滴上並不會變成兩滴雨滴。這種情況可以用公式 1 + 1 = 1 來描述。 類似地，當兩個等量的…混合時…

 

5+7=1

12+12=1

 

因為

5月+ 7月= 1年

12小時 + 12小時

 

12小時+12小時 = 1天

good 

E01