# Mathematics & Social Algebra

Basic Olicognograph: Complexification Emergence

## Pure Mathematics and/or many Tools for flexible use?

Logic(s) as a formal science(s) has(ve) reach a strange status. All people and even some animals have some abstract management, may be things start with the mirror's test: recognize that it is your picture there and when (human specialization ?) also noticing mirror's inversion: what is on the left is on picture's right. Any cultured human, whatever the culture, makes extraordinary imaginative constructions and many of these set systems of explainations and ontologies more or less'right'. But, even the most previously called primitive ones can have locally superior ontologies that is superior to the modern western global ones. Ethnographic investigations showed many to be be very functionnal, say like of taxonomies of animals in one's own environment. There are many complex systems of traditional agriculture that are much more sustainable than the modern promoted ones. Some traditional systems of numbering, reveal uses of concepts previously imagined only possible in the pure modern mathematics. Modern mathematics would be superior if commonly really well used at popular level. Use of formal sciences by common "modern humans" is quite questionning. No doubt that on one side all intent to practice logically but also culturally there are various relative attitudes toward formal sciences. Trainings vary a lot. In maths some students very talented with formal science may be not be the most efficient in cultural practices. Main reasons have to do diversity of intelligence and multiplicity of factors. Making that an self-efficient management of formal sciences is correlated to differents cognitive registers like: memory - motivation - interest - education trajectories - social and professional ambition and so on.

Within mathematics there is diversity of types of skills: geometric, analytic, combinatory. Recalling some L. Scharwtz comments, if commonly all mathematician reach some level of mastership better than any non trained curious in the most of variety of registers of mathematics; the best calculators are not necessarilly the best mathematicians. Specialists of some branch of mathematics, may have trouble in other registers. Any person and mathematicians also practice by affinity; but in modern practices it is to notice that people good at mathematics and culturally open minded (not focused on the pure beauty of mathematics and nothing more) there are well polyvalent and often also good at non mathematics sciences they approach more easilly than others. This has to do of course with the modern development of mathematics and logics as formal science supporting other scientific aprroaches. Such methods, mathematics especially, since the start of modern era, previous even to industrial revolution and rigorous symbolic logics only more recently, are essential aids to scientific research. The question is why much less or why so arbitrarilly used on social issues? Abused more by the common mathematics professionals 'self-justificationists' against democratic practices (on typically social issues). Engineers of superstructures of social containment rather than fair advisors of local democracies.

Almost a century ago, it was still possible to imagine that formal sciences could be comprehensive and successfully imagined close to achievment. The fundamental structures of arithmetic where almost all there: some forms of "pure" but still empirically minded management of mathematics were seemingly giving the feeling that the power of differential calculus could provide a perfect toolbox but all other non structurally solid engineering would have to be called irrational. Perfection has not been reached. Progress in logics and mathematics themselves contributed to the 'deconstruction'. Deconstruction but, more the one of the utopia of cartesianist mathematics as applied by pro-progress minded authoritarians. Many renowned mathematicians where so, after more than a century and half of industrial revolution, may be when achieved Universities' Revolution ('sturm-und-drang'). Universities were was educated the future elites of Nations (another emerging concept of that time). Universities' revolution started in german's ones: if Newton was one century before teaching at Cambridge most engineers of industrial revolution where more (unwise) experimentalist technique craftsmen. German universities achieved the combination of natural philosophy, laboratories investigation, mathematical formalism, logics and teaching to future elite whom dispersed thereafter in the country.

But mathematics faced a series of fundamental internal renewal questionning like: - the transition from euclidean to non euclidean geometry with Lobachevsky (reunified by Klein); - the set theory, which made Cantor win after death over the 'reasonable' Kronecker. Innovation is often followed by synthesis. Capabilities accumulate to treat rigorously more and more phenomena of mathematics as well as complicated physics. Diversity grows with either non simple formulations either strange patterns which to their turn, reveal naturally 'core'. Comprehension for one expert turns impossible. Previous methods reach their limits, produce insatisfaction, force paradigmes changes sometimes with much pain..

Complexity also increased and turned unmanageable. Meanwhile essential units, especially the humane ones, seems to loose more en more sense of and right to wholeness, despite the wide availability of tools for abstract management. Meanwhile bureaucracies prospered and still do over these artificial and virtual complications. Choice is harder, technologies may seem to resolve part of those troubles, but simultaneously, increasing unfitness. Rigidities are produced in abundance; varying in scale across cultures, may grow even more by the globalization of the paradigms of management and the systemic efforts to substitute natural ways to solve complexity. Natural mechanisms that have developped there, but not for the lone unanchored speculative benefit of human species, and now deeply disturbed. The rigidites that have on the material side of forms of constructions and artificial, implicit and explicit, structures to maintain imbalance (like transformed ecosystems, from natural climax). This too for registers of subjective distorsions, which produce behaviorally objective effects. All are producing many potential sources of catastrophic effects; more of less commeasured, incommensurate or unimaginable. So regression of civilizations much higher negative catastrophic regressions, compared to positive ones; end of humans' civilizations' may accelerate.

Applied mathematics, if well distributed between engineers are oddly used and strongly theoretically debating in economics and political issues, weird employments and mostly dreadful with social issues. Part of essential questions at the respect would be:

- Are we using the right mathematics, that is phenomenologically close to what we are and as helping tools, not for making fake dialectics, rethoretic and demonstrations for imposing specific non common interests?
- Do we use models for selfish academic positions and vain gestures or do we use them as substantial social aids?
- Are we trying to ignore complexities for pretending to be right, rather than helpful and honest?
- Which kind of other reasons there are with so many people taking tools and scientific lessons for unfair?
- What would be the good ways, in pedagogic frameworks, to support decisions so as to apply better in social and humane supports to democratic local processes?

For example same adjective of “pure” mathematics can be seen as ambiguous. It seems to address only rigorous formalism out of reality. But most problems explored by mathematicians are about the proper conditions of use of methods and in practical problems like modeling in physical sciences or computerized algorithms. So what is called pure mathematics or pure physics formulation may be closer to right formalism of reality. They care a lot about conditions and constraints of application of formalism in a way that could mean a lot to proportionate better formal methods used in soft sciences.

- One important question is: if mathematics are theoretically able to give enough comprehensive views of complex world ? The theoretical answer have been demonstrated no in absolute, but there are plenty of things to do with wholeness at level of complex scaled units, finite basic networks, diversity of algebra (like Lie's ones),
- Logics have shown able to scatter into second order and higher order logic, treating about special problems, with their limits and means of formal methods, another actually pending question is with the sort of formal use they could have. At least to make our mind more correct about our world, another with the sort of aid they could received thanks to personnel computer capabilities (when they have shown to expand so much the possibilities of engineering),
- How to use formal methods into softwared tools to capture, cover and care the essential of humane sustainable transformations supportive of democratic processes ? (rather than as in powerpoint copycat trickeries)

## More realistic Mathematics: Freer but Closer and Shorter ?

Mathematics as applied in Economics came to create a pure economics that its own brilliant authors insisted not to abuse by overextending the interpretations of what people should do in real economics, as would like 'pure markets ideologists'. In our opinion the main interesting things in those exercises would be in helping understand limits of present practice of policies systems; out the 'terrorism of people' that considered common people too simple and too irrational and so that the most brutal ways to promote efficient monetarism was enough. Even when they themselves quite more contained in their own universities the means of reduction of global planetary systems gave them a too caricature-like rent position and 'pensée-unique' amplified echos. Considering globalization these systems 'harmonizing economic policies' were more trying to turn their specialized register into a comprehensive simple logic, having answers to any social objections. They have received an extraordinary but well deficient and distorting social roles as well as exhausted the over-ambition of their original promoters; meanwhile stayed improper means for humane development. But the Leviathans of their snowballing effects still roll on and we are turning more anxious to were it goes. Just take the fifty years ahead and half dozen of most critical raw material and, think about how things will turn if just maintaining the current trends.

At the same time, at local levels, systems of quantifications are too incipient, weak or wrong respect to whole effects and complex lessons. Same comments on, above mentioned, pure mathematics applied to economics but it is not that way that they are used in policies. To observe that smartest mathematicians have been put to support financial market systems to produce more instability, at the opposite of what they were trained and 'formated'. Some of the problems of the ways that mathematics and logics are used in economics may be:

- The Frame of real numbers (for subjective contrary balance), infinite arithmetic (for mostly finite ones, complex valued and finite complex users), asymptotically ruling functions (when so interesting 'emergencies', margins 'not of the same type as their ground', critical phenomena and transformations occuring at bordering interfaces are ignored). This frame provided a coherence for perfect formal system that is: too fixed and too rigid and when reality is so easilly switching, thresholding, confusing determinisms and underteminisms, having probabilistic behaviors of said capricious actors acting with their own brain.
- Simple overall perfect first logic-arithmetic-reductionist frames helped to establish logical interpretations, but shown at the same time their limits in their conditions, so very improbable with real fluidities. May be even more important for having some use with imitations, tautological short private discrete or partial concepts: to use with convictions and will, if valued important, anchored but and for a short while, within the best intervals provided by physical phenomena allowances. In practice they have been shown inconsistent and revealed not compatible with most phenomenological observations of how actors can behave,
- Probably we have most of the knowledge and the formal tools needed to cope with our problems; out of part of methods and formal ways economics as traditionnaly applied and politically used, but there is also the big problems of normative policies which, despite wrong, prevent better democratic algebraic trails.
- Also we have not enough of the methods of wholeness, like aids to overview and choose between common understanding of what and how could be used formal methods. This question of course multidisciplinary teams and their consensus out of democracy,
- Axiomatic demonstrations procedures may stay the right pathways to formal truth and remain some way to discriminate between formal fallacies and clear quantitative comprehension it is to see that properly applied to naturally complex level, these demonstrations will be very complex in the computer program algorithmic sense,
- Neither deductive demonstrations of theories with nothing more than overreduced models based on unbelievable asumptions rather than as partial use for cautious care of fair practices are really handful. They could be indirectly usefull if helping to lower down absurd primary positivist determinists. They are practically weak. Deductive social sciences are too prepared, too short, too marked by one narrowminded culture for being taken for true. They may be against the usefulness but also against more sincere efforts,

I have dispersed in other webpages of introduction to this complex democratic package some intuive proposals about what could be, in my opinion, a formalism consistent to reality. Yet, it is not consistently formulated nor socially perceived as phenomenology inspiring. Now to give a physical ground to our social exercices of democratic transformation you may seek to use either complicated thermodynamics (do not think that we can prejudgingly fully abstract our will or utilitarism or preferences from physical reality), probably in the sense of Prigogine either biothermodynamics with some logic from Girard and sense of life's logic like from Kauffmann, Longo, Bailly or enaction. For more special tools there are plenty as started by Mandelbrot, Ruelle, Wolfram, Bejan.

It will be hard to be comprehensive but with a clear thinking wholesome views incorporated on sort of proper measures of information and communication of scales or gauges of these estimations can be quite usefull if provided to legitimate human actors. That is, potentially close to practical issues, wholesome and disposed at reachable complex places. There they could be of better use rather as united nations agencies burocratic gentleness. Calling these gauges or scales of estimations like tables of energetics'costs' (exergy-values, energetic returns, neguentropy gains, estimates of entropic release prevented by actions or preventions) of transformations. This may not prevent any transformations but help shared choice and expression of serious responsibility according potential damages. Best saving solutions would stay in minds, and not the maximisation of energetic waste one wich is often implicit in non sustainable maximisation of profits speculations .

Formal basic approach to physical world and biological world we are transforming, could be completed looking at the sort of formal concepts as given by Penrose in his 'Road to Reality'. Also some (not free from profane's mistakes) starters are with following olicognographs:

- The method of physics is a come and go between mathematical model and experiment called “Galilean Heuristic",
- “Energy of Changing Physical Systems:Thermodynamics,” showing a frame of basic formula,
- “Physical systems and formalism in reflection” interprets relationships between disciplines tocomprehend complex world and the limits of formalism.

This could be first level or anchor of modelling over which to put upon a mesolevel of decisions like designed as "games" from the theory of games. But different kind of mathematical behaviors may have to apply, mixtures of anticipations, stochasticity or determinism, decisions managed in the way of fuzzy logic.

## Social Algebra ?

We would call “social algebra” the kind of social rules of calculus put in evidence in social mechanisms. An agreement on their use has to come partly, at least, from democratic consensus. A ground of scientific contains should be neutral to social manipulations. Not to say that this “agreed social algebra” will change much of today’s calculus of present models or that these last ones have to be very different from simple rules. Just that previously, we do not know if they are better than a social consultation, including the sort of algebra citizens can accept. Social algebra should not be fixed. Once corrected, the mistake is to think that negociating formulas could be all what people need to do and if so that there would be more social potential in what practical technicians are doing; rather than in specific social projects designed by foreigners. If technicians could socially implement feasible technical solutions together, with communities' understanding and support, this can feedback communities, skill and choices, to design their own solutions and care the maintenance of civil works turned their own.

Plenty of formalization or quantifications produce more irrational beliefs and expect methods managed by experts as miracles of science. Any number obtained from most simple algebra is suggested to be taken for primary truth and the discussion about them turned to a debate of non relevant arguments. In real life, you face vague problems; the design of rules of quantification and of calculus are important and more delicate. You have to consider many semi-qualitative dimensions, such as precision, appropriate reductions and simplifications, calculable, incompleteness, complexity, non-solvability (by formal methods), diversity, heterogeneity, and so on. All this is also mixed with many assumptions, limiting the application of rigorous methods.

- Intends to evade determinism since the figured level, when realist actions must fix,
- Tries to avoid detailed abstract definitions (it is the right of communities or groups involved in the problem to objective their definitions),
- Provides reasoned structures but insists their management shall neither finish nor be closed at an abstract level; it is up to those close to the situation to commit themselves and make sense of any projects,
- Suggests resources of diversification of ideas in applications, adaptations, redesigns of pictures, and apply qualified frames,
- Intends to provide schemes covering enough variety of practical perspectives on methods.

Accountancy in modern social organization of democracies and other kinds of political systems have become a major issue. Not only because of its importance, but also because many serious problems of information have emerged with the globalization of determinism and virtual systematizations (called rationalization) of illusions. One major mistake has been to think that it was possible to reach a comprehensive view of essential situations from some computerized office, just helping the conduction of general comparison and driving from there actors onto the right path. This produced two major problems of perspective. First, it promoted systems of accounting information that have missed that socio-technical systems and acting communities are complex and locally all different or specific, even when using quite similar tools. Geopolitical national administrations, transnational firms, emerging markets and international financial organizations have pushed globalization, with many transfers of technology and investments, to build these sorts of transnational systems of information. But transfers of means of communication have revealed that expectations and local needs of information have been quite misunderstood.

"The information paradox arises when we want to assess how many resources to dedicate to information search. Whereas we can calculate before and how many resources to use in order to obtain a desired output from a firm or factory, given that we have all relevant information on how the farm or factory functions, such a calculus of optimization cannot be made for the production of knowledge. We simply do not have the necessary information prior to incurring the expense of acquiring it. We can normally only go by past experience and seek limited information before making a decision' And once the knowledge has been acquired, the costs of information search are sunk costs: they have no effect on the further use of the information".

With the intent to be as simple as possible, academic ways to teach lacks of pragmatism, are defective on intuitive management of quantitative methods, shared symbolic manipulations and robustness. The point is how to assume a rigorous formalism ? - without trying to convert people into blind believers. Nor it is to make them confused with falselly sophisticated, when operations are too simple and and not requiring concensus (as free algebra) or when at other times special algebra, not satisfying properties asked by logics. Here the problem is that we have to know both universal and special algebra. Balance also are consistently managed in models of game theory and since years they are developing view on rigid bargaining, economic sense of cooperation, assymmetries of information. Now, if showing the many situations may happen and have economics sense if is also to put these models and in wise, virtuous and positive practices, so people may choose, like how they want to make things, with good 'scientific speculations' of the natural transformations they manage.

"Much of formal game theory utilizes one of 3 forms of description, as the primitive concept accepted from which the formal economic model is developed. The 3 forms are: The extensive form of the game; the strategic form and the coalitional form. Each can be utilized without reference to the other, but the level of abstraction clearly flows in one direction.

- The extensive form is clearly process oriented and essentially institutional. All moves and information conditions are spelled out. The rules of the game are given in detail. It is a matter of a modeling decision as to whether to include items such as verbal statements as part of the formal moves of the game. Usually they are excluded because it is extremely difficult to reduce them to formal mathematics, but in a limited way it can be done.
- The strategic form can be derived from the extensive form by utilizing the construction of a set of strategies derived from the extensive form. The game theoretic concept of a strategy is a complete book of instructions that could be given by a player to a proxy player to play on his behalf. The instructions must cover every contingency feasible in the game systematically.
- A solution in the general sense describes outcomes of the bargaining process. This may involve varying vantage points. A solution may represent an evaluation of the bargaining power of players deduced from the game, it may respect fairness considerations or principles of equity, expected gains in some (vaguely defined) stochastic environment, or results of a specified procedure involving arguments, counter arguments, objections and counter objections.
- Solutions may also be defined as the result of a noncooperative game which is based on the data of the cooperative game and represents a bargaining process. A Nash equilibrium of such a game may result in a solution of the cooperative game. The interpretation of this noncooperative Nash equilibrium may furnish a justification of the cooperative bargaining solution resulting. Agreements may be registered with respect to certain types of mechanisms. The first formal version of a mechanism was that of an adjustment process.
- Yet, the availability of a concept of a mechanism led to several interesting questions starting from the basic theorems of welfare economies. These theorems state that perfectly competitive prices induce a Pareto efficient allocation and that any Pareto efficient allocation may be induced by suitable competitive prices in neoclassical economic environments. The sets of strategies for the n agents (now players) are sets of preference relations on allocations including their true preferences.
- This specific kind of game called direct revelation game or just direct game allows to interpret a player's message as his lying or telling the truth. Whenever truth telling is a Nash equilibrium in a direct game it is even an equilibrium in dominant strategies. This means that truth telling, if it is consistent with the Nash equilibrium, it is even optimal for each player independent of whether the others tell the truth or lie.
- Repeated games with incomplete information denotes a type of game in which the players are facing an information structure which is established by chance at the beginning of the game. For the zero sum case an essential result is presented by the vex-cav Theorem: Consider the value of the expected game presented by the mixture of the states of nature. Now, when the distribution varies, the value of the expected game appears as a function on the probability simplex. We may define the lower convex envelope of this function to be the largest convex function dominated by the value function. The vex-cav Theorem states that the value of the repeated game exists if the successive formation of lower convex and upper concave envelope of the value function as described above, commutes.
- One important strength of noncooperative game models lies in the fact that, although equilibrium behavior of players is defined and of central concern in the analysis, also non-equilibrium behavior is possible. The very definition of a Nash equilibrium does by no means imply that it is realized in an actual play of the game nor that persons playing that game have to act as rational players. It is exactly this feature which allows it to analyze non-equilibrium behavior and, in a dynamic context, convergence or divergence properties of chosen strategy profiles. Because of the preeminence of scientific facts, it may can give more sense toward a deduced “administration,” that is, after basic properties and if and only if there are no oversimplifications or overgeneralizations of scientific facts.

Finally out existing models and other methods we will present in next page it is important to include in these formalism the sense of complexities, which may have dual (complex) perspective. First with the weight of theoretical lessons respect to confusion of realities. This implying to have exercices of interpretations, definitions, designs and simulations mostly in a democratic way, at best with scientific concepts, knowledge, patterns (including visual) pratical situations, kits, maps and so on. The pedagogic part which may be not so primary than thought. Another kind of involvement of complexity has to do on how to include better complex issues and kinds of such simulations that do not just look like the interesting things already explored by formal sciences as catastrophes, bifurcations, instabilities, phase diagrams (of regimes), strange attractors, stochastic models, stochastic probilized scales, multiple levels, structural models, heterogenous (un)logics trees, fractals, percolations, evolutionnary algorithms, cellular automata, artificial neural networks, complex networks, blurred complex social networks, complex social matrices, and so on, but how to managed them sustantial to the situations, not just manipulated for another determinist purpose even if hidden.