Positioning a Firm
A very natural question arises: “Why should strategic factors be considered along and, in fact, on par with profit in positioning a firm?” It seems that profit encompasses all aspects of the firm’s performance and its interaction with the environment. Optimal market share, for example, is then a function of the maximum level of profit. In fact, putting a firm’s market share on par with profit gives rise to some nasty problems of commensurability between these two sometimes-conflicting indicators because the growth of market share may require a sacrifice in profit.
A reader even slightly familiar with the workings of an economic mechanism will immediately respond with general and, for the most part, sensible arguments in favor of evaluating market share independently and on par with profit. His argument will be based on the obvious statement: “Profit is the general criterion to measure a firm’s performance but is often impossible to link in a complete and consistent way (even in probabilistic terms) to future profit with the present situation.” Meanwhile, a reader would be stumped if asked to explain the proper amount of a sacrifice in profit in favor of increasing a firm’s market share.
      The prevailing approach to firm performance, expounded in the literature on economic theory and mathematical economics, in particular, operates under the assumption that profit is the criterion of firm performance along the whole trajectory of its dynamics. For the most part, this category of economic literature eschews a conceptual explanation of the very critical question of why it is necessary to introduce criteria other than profit when evaluating a firm’s performance (for instance, market share). I want to point out in this connection that there is a gap between economic literature and business literature, which does touch on this topic. However, the latter also lacks a conceptual framework within which to study this problem. It distances the business literature from the real-life work of successful business leaders who solve these perplexing problems intuitively. In fact, on those occasions when this far-from-perfect literature on business does make its way into business practices, individuals who are not sufficiently critical of it can find themselves in trouble.
      Let me clarify the last statements. I propose that the explanation may hinge on one and the same poorly elaborated two-fold problem: introduction of the notion of indeterminism and its degree. But here I avoid a philosophical discussion on this subject. I will try to clarify the problem of implementation of strategic factors via the analogy with chess; in its turn chess is becoming more and more an established source for solving business problems. The reader has to forgive me for the following lengthy elaboration of the chess model. I hope he/she will be rewarded for their kindness by a concept and a language that will save them a lot of time for understanding a firm’s position.

                                            Chess as a Disruptive System’s Model
Chess is simple because it has fixed boundary conditions – a fixed goal and fixed starting position – and also fixed rules of possible pieces interactions. (The word piece is used in a broad sense, including also squares.) Meanwhile chess is a disruptive system, i.e., time and space in this system, using Hamlet’s words, is “out of joint.” We are unable to cre-ate an operationally viable procedure that could completely and consistently link the beginning of the game with its end in such a system. It means that in spite of the fact that chess is a finite game we are unable to check all of the possible permutations (even the fastest computers operating at the speed of light might spend many years to check ~ 10120 permutations) or to find an algorithm that allows us to elaborate optimal local solutions that are simultaneously globally optimal.
      As a result, the players have to proceed from start to finish without knowing the consequences of a given move. This indeterminacy makes the formulation and the solution of an intermediate local problem the crucial aspect of the process of move selection. Many ingenious devices have been invented to make a more satisfying move. In other words, the chess players avoided the temptation to find one exhaustive method for playing the whole game and created different methods relevant to different structures that appear along the game.
      In this way the game of chess has been divided into several stages--the opening, the middle game, and the end game. The opening is associated mainly with the reflective style that is based on the repetition of the past. Case studies in business remind us of this style. Extrapolations are just a modification of the reflective style. But such a style cannot be used on other stages of the game because they are disposed to many fast changes in the structure of the situations.
      Those who play chess know that two styles prevail after an opening: combinational and positional.
Setting a concrete narrow goal (as usual material), and elaborating a detailed program to achieve this goal distinguishes the combinational style. This style reminds the solution of local business problems concerning some investment projects, marketing, etc. especially if they can be solved by optimization methods.
But what can be done if a combination could be paralyzed or a vulnerable position has been created afterwards? This is as relevant to chess as it is to business when an impossibility occurs to create such intermediate goals that could link in a complete and consistent way the future with the starting conditions.
      The positional style in chess, which was originated at the end of the nineteenth century by Paul Morphy and William Steinitz (unfortunately, both of them ended their days in a mad house), was an answer to the last question. This style assumes that the development of the game is going via step-by-step improvements of a position by disposing the future development in unknown situations. This does not preclude the usage of other styles; the positional style only postpones the usage of the combination until the position is mature enough to shift to the combinational style.
      Thus, the function of a position is to induce the development of the game in a certain direction (offensive or defensive), absorb unexpected outcomes in a desirable way, and minimize the negative influence of these outcomes.
The choice of a better position among many possible ones requires its evaluation. The obvious solution to the problem is to find probabilities (frequencies) of successful continuations on the basis of statistics of previous games. But such a solution is not always possible because new positions appear, and any attempts to make the last position similar to the previous one are failing because the criteria for similarity are not clear.
      This situation becomes very dramatic when taking into account that a position has to keep its wholeness. A widely spread opinion is that for holistic situations it is not permissible to dissect a position because the holistic effect can be lost.
      The originality of the positional style is first of all expressed via a non-trivial procedure that allows dissecting a position and reviving it as a whole in such a way as to preserve its holistic effect.
      Two major features distinguish a structure of a position in this procedure. First is the recognition of material parameters (the pieces) along with the relations between them as independent (controlled) variables.
When one deals with a combination it is sufficient to know only the coordinate and sometimes the number of moves (analog to time) that each piece can make; the linkage between them will be done via a procedure that is complete and consistent. Now let me come back to the evaluation of the pieces in a combination. These values are fully conditional, i.e., they are reflections of the complete and consistent way in which to link the given goal, starting conditions, and the rules of pieces interaction. As a matter of fact, prices of equilibrium in economics, or the so-called shadow prices, have the same meaning.
      The following example should help an uninitiated reader to under-stand the meaning of fully conditional values of the pieces. It concerns a chess problem usually given to the beginners. Suppose we have a position on the board (Figure1) and it is white’s turn to move.
                                                                 Figure 1. Initial state

                                                                     
      The player with the white pieces knows the rule that allows a pawn to be promoted to any piece upon reaching the eighth rank. The beginner's first impulse is to promote his/her pawn to a queen, for he/she knows the queen is generally the strongest piece.
      But if he falls victim to his natural impulse, black will mate white in one move (Figure 2). However, if white was to promote his pawn to a knight, black king would be checked and mated immediately (Figure 3.)

                                                                 Figure 2. The Tempting Move

                                        

      The above example vividly shows that if all conditions for a problem are specified, and it is typical for a combination, the value of a piece is fully determined by these conditions, i.e. the value of the piece has to reflect its direct contribution to the goal and must be subjected to the given constraints (initial state and rules of interaction).
Another situation is in disruptive systems. Along with the coordinates and time have been elaborated many other dimensions. The last ones characterize relations between the pieces – so called positional parameters. If, for example, we have on the chessboard on the side of each player a king and two pawns we cannot say that their position is equal. The value of a position in a combinational style, as I have shown above, is distinguished by its direct contribution to the goal. It is not so for the positional style because it is impossible to link in a complete and consistent way the given position with the end game. In this case along with the pieces – material parameters - have to be elaborated positional parameters.
      To explain the last statement I will continue the consideration of the previous example. One can say that if one player has two pawns that are located on the same row and close to each other and the second player’s pawns are sporadically dispersed, generally speaking, the first player has a different position: his pawns could help each other.
In chess, there are hundreds of positional parameters like closeness, doubling pawns, backward pawns, open file, etc. As a matter of fact, a deductive concept for elaboration of positional parameters does not yet exist. Professional chess players are setting positional parameters on the basis of their knowledge and intuition.
Now we are facing the next problem: the evaluation of the material and positional parameters in a way that preserves the holistic effect. The chess players developed a new system of evaluations of these parameters that we don’t see in other fields. In other words, the chess players developed two systems of evaluations; respectively, for the combinational and the positional styles. The first system, so-called, fully conditional values, has been mentioned above during the discussion concerning the combinational style.
      A different situation appears along the positional style. Here we cannot use fully conditional prices because we cannot set a proper inter-mediate goal and connect it with starting conditions of a given situation. Meanwhile, we have to measure the influence of a given position on the end game. The chess players for this purpose dissected the full conditional value and elaborated a two-part system of unconditional values for material and positional parameter. Different methods of setting values are used for each part.
      The first part is concerning material parameters and is based on un-conditional values for single pieces. Of enormous interest from the general methodological perspective is the fact that chess is, apparently, the only area where there has been any success in revealing in a rigorous fashion the necessity of unconditional values for material parameters and how to construct them analytically.
      To be more precise, these values should be called semi conditional. Such a clarification is arising from the analytical procedure to find these values offered by Henry Taylor. This procedure is based only on one condition - the rules of pieces’ interaction; the boundary conditions (intermediate goals and starting conditions) are not taken into account.
      These semi conditional values are well known to the chess players – they are normed relative to pawn=1 and evaluate a queen by 9 pawns, rook - 5 pawns, bishop and knight - 3 (3.5) pawns.
The second part of the set of unconditional values concerns values of positional parameters. These values are formed in the same way as positional parameters: professional chess players are setting these values on the basis of their knowledge and intuition.
      Generally speaking, a high degree of subjectivity is intrinsic to the positional style. The role of the executor, his/her strengths and weaknesses, become vital because the actual realization of the position is not known be-forehand, and future moves have to be based on the contingent situation at hand. The role of subjectivity is clearly seen in a positional sacrifice, the sacrifice of material on the name to improve the positional parameters and in general the value of a position. To the reader not familiar enough with chess, please recall that positional sacrifice is differentiated from combinational in that, at the moment of sacrifice, it is not clear how the loss will be concretely compensated, that is, how the positional advantage gained at the cost of the sacrifice will transfer into a possible effective combination in the unknown future. That is why, the chess textbooks recommend a positional sacrifice only for highly skilled chess players because ordinary chess players will be unable to realize a positional sacrifice.
      Here we can clarify the differences between subjectivity and objectivity. It is very important for further business considerations, and enlightening the title of this paper. Some novelty in these definitions is related to their operational aspect. Objectivity means that the evaluation of the result will not depend on the subjective values of the observers, it is impartial to them; in other words, the evaluation of an object could be separated from the person who will further elaborate it. Subjectivity means the impossibility to separate the evaluation of an object from the person who will further elaborate it.
      Thus, everything was done to create a weight function for an evaluation of a position. Claude Shannon elaborated a formal expression of this function.

                       f(P)= 200(K – K') + 9(Q – Q') + 5(R – R') + 3(B – B' + N – N')_
                              – .5(D – D' + S – S' + I – I') + .1(M – M') + . . . ,

in which K, Q, R, B, N, P is the number of White kings, queens, bishops, knights, and pawns on the board; D, S, I are doubled, backward, and isolated White pawns; M- white’s mobility (measured, say, as the number of legal moves available to White). Prime letters are the similar quantities for Black. The coefficients 9, 5, and 3 are the widely accepted valuations of relevant chess pieces; the coefficient 200 for the king allows recognizing the end of the game; the coefficients .5 and .1 are merely Shannon’s rough estimates of valuations of corresponding positive and negative positional parameters.
       Chess computer programming based on the idea of choosing the best (satisfying) position via a weight function combined with other devices (e.g., the library of openings) had experimentally proven that such a con-cept works and even have bitten the world chess champion Gary Kasparov. But we have to keep in mind that a chess computer program is to a great extent a formalized version of certain subjective views of an expert who includes in the weight function the set of the positional parameters and their values.

                                Application of the chess model to the behavior of a firm
First of all, the chess model clarified that solutions to problems in disruptive systems are found by a diversity of styles. On each level of the hierarchy of a firm all three styles are used. Meanwhile different layers (a group of levels) in a hierarchy are distinguished by different prevailing styles. On the lower layer the reflective style is prevailing; on the middle layer – the combinational style; on the top layer (presidents, vice presidents, etc.) – the positional style. If the combinational style is mainly oriented to elaborate an efficient project, the positional style is mainly oriented to setting the strategic position of a firm.
      Such a simple statement concerning different styles that are used in a firm could have some practical applications.
Since different styles require different types of people, it is necessary to be careful with the promotion policy. A manager who is very good in the combinational style (usually left hemisphere oriented) could be bad in the positional style (usually right hemisphere oriented), and vice versa.
      Moreover, the understanding of the relation between different styles could help to avoid some corporate conflicts. It is well known, that new comers with a MBA degree have in their disposal modern sophisticated tools to solve business problems. The middle age and old people that mainly occupy the top layer as usual don’t know the modern techniques. That is why the new comers, especially from first class business schools, often treat the top layer managers as “dead wood.” But if one keeps in mind that the new comers tools are mainly oriented to the solutions of problems that require the combinational style and these people don’t have enough experience and intuition that is required for the positional style, it is becoming clear that the conflicts between generations are to a great ex-tend artificial and could be relaxed.
      The understanding of the differences in style could help to avoid misunderstanding in communication between different layers. Sometimes top layer managers could face difficulties explaining a certain task to the mid-dle layer manager, and it may be difficult for a middle layer manager to understand the ideas of the top layer manager. These kind of misunderstandings especially concern the role of investments in human relations that are so vital for top layer managers. The conflict between Lee Iacocca, as Ford’s company President, and Henry Ford, as its chairman of the board, to a great extent was stipulated by Iacocca’s inertial middle layer kind of thinking. The last one misses the importance of human relations to which Henry Ford paid enormous attention.
      Below, I mainly want to concentrate on the usage of the chess positional style for setting an evaluation of a corporate strategic position. The corporate weight function also includes along with inputs-outputs (material parameters) strategic factors (positional parameters). The value of this function requires a special term. Let us call it predis (shortening the term predisposition.) Setting a price in the process of buying a business could justify the relevance of the introduction of the term predis: the price for a business includes not only its material components but also strategic factors. Meanwhile, the elaboration of these prices was not conceptualized.
But let us further develop the structure of this function. The set of material parameters is given. The set of strategic factors was defined in an empirical way, mainly by acceptance of the well-established indicators. The ingenuity of a businessman is stipulating the further development of this set.
It is more complicated to find the proper values for the material and positional parameters.
      The prices for the material parameters have to be semi conditional. The economists did not yet develop the concept of such prices. Meanwhile some elaborations that have been done in economics for different purposes could be used for setting the proper prices for the material parameters of a weight function. The development of the concept of dynamic equilibrium by John von Neumann exactly satisfies the requirement for such semi conditional prices. The trajectory of dynamic equilibrium and their dual characteristics – prices - are revealed in an assumption about their limitations only by technologies (equivalent to the rules of interaction), regard-less of the boundary conditions - initial resources and goals.
      Certainly the application of such prices could not be done in the pre-sent time because it requires the development of a huge statistical base and its elaboration. That is why this part of the weight function does not have a practical application at this time. But it has a practical application in the long term.
The ideas relevant to the other part of the weight function, i.e., the positional parameters, could be used today. They could help us clarify the controversy that has been mentioned in the beginning of the paper concerning the usage of strategic factors as a part of the criterion for a firm’s evaluation. It seems to me that failure of many firms to exploit the sacrifice of profit on the name of improving the value of strategic factors is rooted in their attempt to objectify this process rather than to subjectively it.
      Certainly, everything said above concerning subjectivity is not easy to apply to the real business world. Because managers usually do not own the firms they work for, they either have to come up with some objective (statistical) justification of their subjective decisions or simply reject the efficient subjective decisions in favor of the more standard, but scientifically corroborate strategies. Sometimes, the managers simply make up the data to support their decisions, which eventually ruin the business ethics of the firm.
      Thus, there are many attractive scientific methods for objectifying evaluations of a firm's positional variables that have been undertaken by some leading consulting firms, e.g., Mackenzie International Consulting, Boston Consulting Group, etc. In this context, the model PIMS (Profit in Market Strategies) merits a close examination.
Sidney Schoeffler , the founding director of The Strategic Planning Institute (SPI), established in 1975, is one of the people who initiated the development and the promotion of PIMS. The PIMS model is based on enormous amount of information from its firms’ clients — hundreds of leading firms from North America and Europe, represent¬ing several thou-sands of businesses (by business I mean division, product line, etc.), each characterized by more than 100 indicators. At first, the SPI focused on the analysis of the effect of separated positional parameters, such as the market share, the degree of vertical integration, etc. on the firm's ROI (Return on Investment). Later on, the research shifted to assessing the linkages between, on the one hand, interrelated positional parameters, and on the other, profitability.
      Ambitious as it may sound, PIMS model was designed to provide, as in natural sciences, a scientific approach to business management based on the objective market laws. Meanwhile, the concept of such laws as leading rules governing a system is more relevant to natural sciences, and, particularly, to physics, because the physical system is relatively stable. It seems to me, that this concept is not relevant to economic systems because the last are highly dynamic.
      There is generally nothing wrong with the idea of collecting and analyzing data about different firms performances in order to determine what appears to be in average a good strategic decision. On the contrary, many firm’s leaders can benefit from having access to such a data bank if they are using it as an auxiliary means (but not as a substitution) to their subjective decisions; in other words, if they compare the value of their subjective decisions with statistical ones and use the differences between these values only as warnings for their subjective decisions.
The conversation between Sidney Shoeffler and chess grandmaster Sergey Kudrin can help to clarify the statements made above concerning the subjectivity of strategic decisions. The conversation took place at the symposium Calculus of Predisposition that I have organized in 1992 at the University of Pennsylvania.
      Shoeffler asked Kudrin the following question: “Suppose that you have a data bank that contains information about every chess game ever played. Also, suppose that you can instantaneously retrieve any information on either material or positional parameters that you might be interested in. Do you think that such information would be sufficient for you to make an accurate evaluation of the position?” Kudrin's answer was as follows: “If one compares the evaluations of the same common positions given by the two grandmasters their verdict regarding a given position will coincide. In that respect, this data bank might be quite useful to me. However, the information contained in this data bank concerns mostly standard and well-known positions. A grandmaster usually wins because he/she can introduce original positions or come up with an unexpected series of moves in a well-known position. Then, if we get two grandmasters to analyze novel positions, their subjective evaluations will most likely be different, reflecting each player's individual style as well as his/her intuition and his/her ability to develop the new position.”
      Finally the understanding of the role of subjectivity in the process of evaluating the firm’s behavior could help to reorganize the accounting sys-tem. The balance sheets under the name of the firm’s top leader also have to contain the set of the subjectively chosen positional parameters and their values. It could be very productive to put on paper what the leaders have in their mind.

                                                    Aesthetics and a Firm’s Position
The analogy between the behavior of a firm and the progress of a chess game seemed obvious: in each case both participants are facing competition from one another and the achievements of their remote goals are exploiting a process where “the time is out of joint”. Later, when I be-came familiar with George Birkhoff's works on aesthetics, I was amazed by the similarity of the formal structures of chess positions, the positions of a firm, and aesthetic measures by Birkhoff. His formula for the aesthetic measure M is a function of two variables, C – complexity (relevant to material parameters) and O –order (relevant to positional parameters). To illustrate his ideas Birkhoff uses, for example, a convex polygonal tile. The measure of its complexity is determined by the number of its sides - (equivalent to material parameters) and the measure of order — by such parameters as repetition, similarity, contrast, equality, symmetry, balance, and sequence - (equivalent to positional parameters.)
       Needless to say, the issue is not the simple assertion of the formal similarity of these structures. A deeper consideration of these structures shows that they have a surprisingly similar meaning.
      From the history of various sciences, we know that the homogeneity of mathematical formulae can be used as a pretext for pronouncing a hypothesis on the homogeneity of the phenomena concealed behind them. Just that sort of phenomenon took place in the nineteenth century when James Maxwell, on the basis of the discovered homogeneity of the differential equations describing wave forms of electric, magnetic, and light waves demonstrated their common nature.
      Starting with an understanding of Birkhoff’s mathematics, I began my search for isomorphisms. I found that in all three cases we deal with predispositions—the structures, which are not completely and consistently linked to pragmatic goals but, at the same time, are somehow organized. Thus, I assumed that in the light of the contemporary view of a chess game and a firm's behavior, it might be beneficial to search for an analogy between the criterion used to evaluate a position in chess, a firm's strategic position and an aesthetic measure.
I supposed that the generic term for the value of a predisposition is relevant to beauty.
Intuitively many professionals and amateurs, when it concerns beauty, agree that there are some obvious features of beauty: beauty can be applied to any object; beauty is not pragmatic (please, don't confuse it with applied art); meanwhile beauty has some order; beauty is unique; some features (elegance, symmetry, asymmetry, etc) are accepted as features that belong to beauty; beauty is holistic (synergetic): beauty cannot be dissected in a simple mechanical way; the holism of beauty can be recognized via its influence on the whole; beauty is perceived by intuition along with rationality; beauty is subjective; beauty has a sign (positive-negative) and a degree.
But until today there is not even an intuitive clear reply to the major question that concerns the essence of beauty. The question "Where is the ‘meat’ (‘seed’) of beauty?” does not yet have an answer.
      My answer is: “Beauty is a predisposition, i.e. a value of a position via evaluation of its parameters in unconditional (semi conditional) values.” I tried to find the core of beauty, i.e., its rational part, as an invariant of any disrupted system. A reader could grasp the meaning of this core from the above descriptions of the positional style in different systems. This core could be later covered by artistic means and human beings perceptions. In the latest case beauty is represented in a most complete way and is becoming the leading category in the domain of art.
Such an approach to beauty could help to elaborate some hints concerning the linkage between aesthetic and business that is very badly developed. Russell Ackoff recognized beauty/fun along with the other three major aspects of development – truth, plenty, the good. Meanwhile, Ackoff mentioned
      “The Aesthetics is the least understood aspect of development. The terms “management science," "management tech-nology," "management education," "managerial economics," and "managerial ethics” convey at least some meaning to most managers. On the other hand, the “aesthetic of management” conveys little if any meaning to anyone, let alone managers.” (p.277)

      The inability to elaborate the category of aesthetics in business has deep roots. It seems to me that this elaboration requires a sharp turn in the way of treating beauty. I guess that my approach to beauty via the concept of a predisposition allows finding a linkage between business and aesthetics.
I understand that there are no short ways of elaborating these linkages. One way – from the final end - pure art, e.g., classical music. (As matter of fact, Birkhoff in his previously quoted book, has analyzed the aesthetic measure of music by the same terms as ceramic and architecture.) The other way – from the beginning, i.e., very specific human business activities. To avoid vulgarization several intermediate stages should be put between these two ends.
At the present time a lot is being done to such a linkage between both ends, i.e., in a "tunnel manner." On the one end we see John Cimino's and his colleagues activities. The company, Creative Leaps International, that Mr. Cimino is heading, is oriented to stimulate a starting predisposition to creativity via primarily musical performance accompanied with some mimes performance. As Mr. Cimino declared during his recent visit to the Wharton School, the major idea of his performance is to ignite the creativity (the imaginary power) of managers before they are becoming involved in a session of solutions for certain business problems.
      The Center for Creative Leadership is developing the next stage in the linkage of art and business in particular by involvement of the business participants in performances.
      I highly appreciate all the above-mentioned artistic activities. But let me repeat that there is a large gap between the ignition of imagination and a solution for certain business problems. It may be that some managers could intuitively jump via the gap between these two ends and successfully solve a certain business problem. But I guess for the majority of managers it is impossible.
       It seems to me, that my work together with Dr. Vera Zubarev creates an opportunity to attack the problem of increasing imagination from the other end. We have taught together a course Aesthetics in Decision Making for Benjamin Franklin undergraduates students at the University of Pennsylvania; I myself taught a course of strategic planning, limited to the rational part of beauty,. for MBA students at Wharton school and for executives at the Organizational Dynamics Center at the University of Pennsylvania.
      In our course for undergraduate students we are trying to build a bridge between art and business starting from the core of beauty (on the basis of chess), developing it to the formation of a strategic position of a firm (as it was described above) and mapping these ideas to decision making by major protagonists in literature pieces. On the other hand, we are trying to use the great ideas and artistic means from art to enrich the core of beauty as it is represented in chess and business. Since the ideas and explanations of this concept reach to great depth we concentrate only on one piece of literature, Romeo and Juliet by Shakespeare. Along with the text of the play we analyze two different versions of the movie Romeo and Juliet by Zeffirelly and by Luhrman.
      I think that the above mentioned understanding of the linkage between art and business will stimulate business people to deserve more of their very limited time to art. The business people often recognize art as a luxury. The long obscure journey from art to the solution of certain business problems seems to me is the major reason why the business people resist spending their time on art even if they understand that art broadens their vision. It is known that such prestigious schools as Sloan and Dartmouth tried to link business and art. But mainly they failed. May be the absence of a more pragmatic linkage between art and solving business problems was the major reason of these failures.
      I hope that further elaboration of the linkage between art and business in a “tunnel way” could make art more pragmatic for business people and bring provocative means to increase theirs imagination.

                                                    Instead of a Conclusion

    The existing mathematical methods are unable to solve the general problems in disruptive systems to which business belongs. The solutions to problems in such systems are done sporadically and specifically for each system and even for each problem. Often it was done under the name heuristic programming.
     As it is well known, classical mathematics assumes that a mathematical result has to be complete and consistent. The famous mathematician Israel Gelfand called attention to the necessity to create a new mathematics that could conceptualize the ways of solving problems for systems that I called disruptive ones. As Gelfand said, perhaps, it makes sense “to resist the dangerous and careless use of exact mathematical and logical systems outside the domain of their applicability.” He contends that new ideas must come from mathematics. He mentions two: “One is a radical shift in the concept of space, quantum gravitation, and the like. The other emanates from the ancient field of combinatorics, which is still lurking in the backwaters of mathematical sciences” (p. 16).
Gelfand also spoke about the need to elaborate a universal language—a language that will help reconcile the two mathematics and bridge the void fraught with so much pain and suffering for mankind.
      To my best knowledge, independently only Ron Atkin under the principle of connectivity had done an essentional attempt to find a new mathematical approach to the solutions of problems for disruptive systems. Unfortunately, his work is far from being finished.