Introduction to the New Series

Introduction to the New Series
On Logic, Mind and Complex Systems

1. The new principles of editing "Mathesis Universalis"

The New Series starts from issue 9, the numbering being continued after the Old Series. The new editorial principles are as follows.

(1) Each issue is concerned with one chosen subject; the subject is somehow related to the ideas of Mathesis Universalis project, originated with Leibniz; a modern version of that project is meant here as possibly widest use of computational modelling and simulation to cope with complex systems.

(2) Unlike the old series, the new series does not pretend to invite original contributions on the subject in question. Instead, as a rule, references are offered to the most useful WWW documents concerning the problem involved. Such a hint consists of links to respective sites, and information about the main contents of the document referred to. Thus, to some extent, this site will be of the kind called "reader's digest", thus being an Internet guide.

(3) Original contributions are welcome, provided that they offer something new that is not found in WWW sites. Especially, readers are encouraged to comment and discuss items being referred to through links. Thus "Mathesis Universalis" is to play an additional role of dialog forum.

(4) Each issue is introduced by a study of its main problem. This is to include a statement of the problem, discussing its theoretical and practical import, reporting on solutions found in the literature, hinting at relations to the Mathesis Universalis project, giving reasons of the choice of references in the issue so introduced.

2. What should mean the conjunction "logic, mind, complex systems"

          Motto. How can we ever exclude the possibility of our being presented someday by (perhaps by some extraterrestrial visitor) with a (perhaps extremely complex) device or "oracle" that "computes" a noncomputable function?
          Martin Davis, Computability and Unsolvability, 1958.

Let X and Y be entities produced by an algorithm, eg, mathematical demonstrations done by a prover; or translations of a text, etc. Then Y is said to be more complex than X if the shortest algorithm to produce Y is longer than the shortest algorithm to produce X.

This simple definition of complexity, due to Kolmogorov and (independently) the IBM worker Chaitin, provides us with a nexus between complexity and logic. For, any algorithmic procedure requires a decidable theory, say, a part of arithmetic, while a theory is said to be decidable if for any formula of it either this formula or its negation is logically derivable from the axioms of this theory. The steps whose number is to measure the size of an algorithm are those of logical derivation, and thus algoritmic behaviour keeps in step with what is directed by logic.

Are there any things in the earth or the heaven not being produced by algorithmic processes? This question, born in the first half of the 20th century, caused a new great divide between philolosophers, a divide of which the older philosophers did not even dream. If there are any such things, let us call them non-algorithmic. Now we face a question which is fundamental for the philosophy of mind: is the human mind a non-algorithnic thing? If the answer is given in the negative, then nothing speaks against matching humans by digital machines, provided a due technological progress. Thus the strong AI position would prove triumphant.

If one likes expressing the issue in terms of complexity, the mentioned definition seems conceptually "too short" since it does not deal with non-algoithmic processes. However, we can extend it with saying that if there is a non-algorithmic process, it is more complex than any algorithmic one, in this sense that the shortest procedure to produce it would require infinitely many steps, thus being longer than any algorithmic procedure. In such a way, the mind can be defined in terms of complexity as a thing more complex than any digital (or Turing) machine.

Mathesis Universalis in its new shape should help in approaching answers to such questions as posed above and those which follow from them. Are there any non-algoritmic things in the empirical world, as quantities which can be rendered by uncompytable numbers alone? And relations to be rendered by uncomputable functions? If there is a class of such things does the humam mind belong to that class? Are there, apart from non-computability, any other kinds of complexity that made some things intractable for our methods of research? Especially for social research which seems to deal with most complex systems?

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