Witold Marciszewski

The paper is meant as a survey of issues in computational complexity from the standpoint of its relevance to social research. Moreover, the threads are hinted at that lead to computer science from mathematical logic and from philosophical questions about the limits and the power both of mathematics and the human mind. Especially, the paper addresses Turing's idea of oracle, considering its impact on computational (i.e., relying on simulations) economy, sociology etc. Oracle is meant as a device capable of finding the values of uncomputable functions. Such an idealized entity is exemplified by the human mind's procedure of recognizing the truth of the Gödelian sentence, of identifying uncomputable numbers through Turing's diagonal procedure, etc. Since such procedures are strictly defined and are as reliable as any calculations, they are worth to be called computation as well. From the computation in the strict sense, that defined as purely algorithmic (mechanical) process, one distinguishes them with the term "hipercomputation". Now the following questions arise. - Are there undecidable problems (ie. not decidable with appropriate algorithms) in social research as are (according to what is reported esp. By S. Wolfram) in natural sciences? The answer in the negative would impose limitations on computer simulations (as entirely relying on algorithms). - If there are, then we have the next question: can such problems be addressed with hipercomputational procedures? - How such hipercomputational procedures would be related to analog computation (coextensive, everlappiing, etc.)? Another set of issues is stated in terms of tractability of decidable problems, that is, the efficiency of algorithms needed for solutions. As inefficient are regarded those which require more resources (time, memory, etc.) than is available in a foreseeable future. In this context, one discusses methods of such an efficient organizing computational processes to overcome the scarcity of resources; thus parallel, distributive, interactive, etc. computing are used as remedies. The paper claims, hinting at F.Hayek's ideas, that in some social systems (e.g., stock exchange, and free market in general) such an efficient organization of their computational activities spontaneously evolves. And this is the main source of its advantages over the central economic planning (as defended by O. Lange). This noticing (in terms of complexity theory) of analogy between Hayek's point and the current discussion of efficiency of algorithms is what may count as an original contribution of the present paper

When discussing Kazimierz Ajdukiewicz's role in philosophy, it is worthwhile recalling his participation in scholarly controversies. It was characteristic of his open mind that his taking part in debates was motivated by a vivid interest in various ways of thinking. Ajdukiewicz's intellectual power consisted, so to speak, in his ability of not to understand. This ability has brought him success in some important debates, concerning i.a. the classical logical concept of contradiction and the debate on universals raised in modern Poland with the nominalistic program of Stanislaw Lesniewski and Tadeusz Kotarbiński. In this latter debate Ajdukiewicz shows that when one says that individuals exist, the word „exist" refers to something different that in the statement that universals exist. In other words, the functor „is" has a different category in the definition of an individual from that appearing in the definition of a universal; hence there must be two different senses of the word „exist"

We discuss the existence of universal spaces (either in the sense of embeddings or continuous images) for some classes of scattered Eberlein compacta. Given a cardinal κ, we consider the class Sκ of all scattered Eberlein compact spaces K of weight ≤ κ and such that the second Cantor-Bendixson derivative of K is a singleton. We prove that if κ is an uncountable cardinal such that κ = 2&ltκ, then there exists a space X in Sκ such that every member of Sκ is homeomorphic to a retract of X. We show that it is consistent that there does not exist a universal space (either by embeddings or by mappings onto) in Sω₁. Assuming that ∂ = ω₁, we prove that there exists a space X ∈ Sω₁, which is universal in the sense of embeddings. We also show that it is consistent that there exists a space X ∈ Sω₁, universal in the sense of embeddings, but Sω₁ does not contain an universal element in the sense of mappings onto

There seems to be something mysterious about applications of for- mal systems, including those of logic, to empirical reality 1 . If logic is to be applied to empirical situations, like those described in an ordinary lan- guage, then { it seems to some people { its statements cannot be necessary, or analytic, propositions. However, they are both applicable and necessary. This supposed puzzle constitutes a signicant part of the problem of philo- sophical foundations of logic 2 . To this mind of the present writer, there is no mystery of applications, since any empirical, or even ostensive, predicate can be involved in cer- tain meaning postulates, e.g. \No red is green", which are both empirical and necessary propositions; empirical as they involve ostensibly dened predicated; necessary, as it is enough to know their meaning to state, their validity. The case of logical theorems can be considered at the same footing, logical constants being treated as dened ostensively. Nevertheless, not an individual state of mind but the actual state of scholarly discussions makes something a problem; in the present state of foundational discussions on logic, the \mystery of applications" preserves its vitality, hence any new way of dealing with it proves welcome

Intelligent problem-solving depends on consciously applied methods of thinking as well as inborn or trained skills. The latter are like resident programs which control processes of the kind called (in Unix) daemons. Such a computational process is a fitting reaction to situations (defined in the program in question) which is executed without any command of a computer user (or without any intention of the conscious subject). The study of intelligence should involve methods of recognizing those beliefs whose existence is due to daemons. Once having been aware of so produced belief, one can assess it critically and, if possible and necessary, make it more rational. Eg, beliefs concerning properties of time are produced by a daemon-like intuition, likewise the Euclidean properties of space. The merit of getting aware of such daemon's activities, and so transforming implicit beliefs into explicit ones, lies mainly in the axiomatic characterization of the properties involved. That makes possible to improve a daemon-like conceptual equipment (producing beliefs) by suitable modifications of the axioms, or postulates. Such postulate sets can also define artificial daemons to either emulate or improve natural intelligence.