The Sophists, who lived from the second half of the fifth century B. C to the first half of the fourth century B. One of them, the famous philosopher Zeno of Elea - B. It is possible to formulate this paradox in the following way. There are analogies of this paradox in our time. 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 [1] was one the first scientists who questioned adequacy of the conventional arithmetic.
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. Alike, the conventional arithmetic fails to describe correctly the result of combining gases or liquids by volumes. For example Kline, , one quart of alcohol and one quart of water yield about 1. Later the famous French mathematician Henri Lebesgue facetiously pointed out cf. Kline, [5] that if one puts a lion and a rabbit in a cage, one will not find two animals in the cage later on.
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. 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 [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. Van Danzig had similar ideas but expressed them in a different way. In his article , 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. 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, [10] ; Littlewood, [11] ; Birkhoff and Bartee, [12] ; Dummett [13] ; Knuth, [14]. Different types of natural numbers have been introduced, but without changing the conventional arithmetic.
For example, Kolmogorov [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 C. Consequently, new arithmetics acquired the name non-Diophantine arithmetics. Burgin built first Non-Diophantine arithmetics of whole and natural numbers Burgin, [12] ; [15] ; [16] ; [17] [18] and Czachor extended this construction developing Non-Diophantine arithmetics of the real and complex numbers Czachor, [19].
In the following section, we will show that non-Diophantine arithmetics occur in economics, starting with mergers and acquisitions. In the following section, we will show that non-Diophantine arithmetics occur in economics, business and social settings, starting with mergers and acquisitions.
Whereas in a merger, two approximately equally sized companies consolidate into one entity; in an acquisition, a larger company takes over the smaller one. However, with respect to expected revenue, only few companies achieve the desired objective, as we can see in Figure 2.
Figure 1. Mergers and Acquisitions achieving the expected cost savings Source: McKinsey [20]. Figure 2. Mergers and Acquisitions meeting the percentage of expected revenue Source: McKinsey [20]. Therefore, synergy emerges when the cooperation of two systems gives a result greater than the sum of their individual components. This state of affairs reflects negative synergy or system friction.
Michael Angier [26] defines synergy as the phenomenon of two or more people getting along and benefiting one another, i. For example in an expression of two words, e. Its absence changes the meaning, e. Adding two words symbols , i. Furthermore, there are examples of non-Diophantine arithmetic in everyday life. It actually means that you can buy two items for the price of one. Such advertisement may refer to almost any product: bread, milk, juice etc. This is incorrect in the conventional arithmetic but is true for some non-Diophantine arithmetic, as we will show below.
Another example: when a cup of milk is added to a cup of popcorn then only one cup of mixture will result because the cup of popcorn will very nearly absorb a whole cup of milk without spillage. This is impossible to replicate with conventional arithmetic but it is true for some non-Diophantine arithmetics.
Coming to the dealership, you find that the price is five dollars more. Do you think that the new price is different from the initial one or you consider it practically one and the same price? It is natural to suppose that any sound person has the second opinion. Critics may object that we use Non-Diophantine arithmetics to explain these phenomena.
They may say that we use the conventional arithmetic but only transform its operations according to some formulas. This objection is similar to the 18 th century Europe claim that people do not use negative numbers but only employ positive numbers with additional symbols. Now let us look whether the laws of non-Diophantine arithmetic can reflect the economic and psychological phenomena considered above.
Here we consider only arithmetics in the set W of whole numbers. We start with a more general concept of a prearithmetic Burgin, [23]. If X is a subset of the set R of all real numbers, then the arithmetical completion of X consists of all sums and products of elements from X. Functions f and g allow defining two new operations in the set W:.
Let us take the set A which is the domain of f, i. Naturally the conventional arithmetic W is a whole-number prearithmetics. Another example of whole-number prearithmetics is a modular arithmetic, which is studied in mathematics and used in physics and computing.
For instance, when the modulus is equal to 10, the modular arithmetic Z 10 contains only ten numbers 0, 1, 2, 3, 4, 5, 6, 7, 8, 9 and when the result of the operation the conventional arithmetic is larger than 10, then it is reduced to these numbers in the modular arithmetic.
Readers can find information about modular arithmetics in many books and on the Internet. Here we only give some examples for the modular arithmetic Z 10 :. When a whole-number prearithmetic satisfies the following conditions, it becomes a general Non-Diophantine arithmetic.
Condition A3. For instance, condition A2 is true when g is also a total function, e. An important class of Non-Diophantine arithmetics is formed by projective arithmetics.
For instance, taking the number 2. This prearithmetic is a projective whole-number arithmetic if the following conditions are satisfied:. It is possible to find a theory of this and other Non-Diophantine arithmetics in Burgin, ; ; ; and Here we consider only simple examples and some properties of Non-Diophantine arithmetics because the goal of this work is a demonstration of a possibility of the rigorous mathematics to correctly and consistently interpret seemingly paradoxical statements, which describe situations in various spheres of real life.
Example 1. Example 2. Example 3. Example 4. One more unusual property of Non-Diophantine arithmetics is related to physics. However, these relations do not have an exact mathematical meaning and are used informally. In contrast, Non-Diophantine arithmetics provide rigorous interpretation and formalization for these relations.
Namely Burgin, [28] ,. One of this type of arithmetic is considered in Example 4. Sign up to join this community. The best answers are voted up and rise to the top. Stack Overflow for Teams — Collaborate and share knowledge with a private group.
Create a free Team What is Teams? Learn more. Asked 6 years, 11 months ago. Active 4 years, 1 month ago. Viewed k times. Improve this question. Should be easy from there Show 2 more comments. Active Oldest Votes. Improve this answer. David Richerby David Richerby 6 6 silver badges 11 11 bronze badges. Add a comment. Xenocacia 7, 2 2 gold badges 20 20 silver badges 53 53 bronze badges. Show 4 more comments. This is very easy
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