Complexity
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Comments on Qualitative Complexity

Below, find a few remarks made on this paper by the reviewers of Complexity International, followed by my replies.

Reviewer 1:

In the section dedicated to "Structure", before the first formulae, there is the idea that the mathematical description only allows for a static view of the objects described. I would relativize that issue, since mathematical descriptions can well include within their own formulation a possible evolution of the model (see logics of action, temporal logics, or some categorical treatments in theoretical computer science).

Reply:

To indicate my acquaintance with such ideas, I should have made a reference to a book containing an overview of these branches of mathematics — e.g. D.A.Pospelov, Situation-driven control: Theory and practice (Moscow: Nauka, 1986). I will not relativize that issue, since all the attempts of incorporating time (movement, functioning, evolution) in mathematics are completely structural, as far as I know. Today, mathematical reasoning is not suited to speak of dynamics or development — and the present tendency towards still more abstract formality indicates that functional and developmental ideas will hardly get any expression in the mathematics of the nearest future. More considerations on the subject can be found in: P.B.Ivanov, Computability in developing systems (1996).

There is one more aspect: common prejudice about the structure of scientific work. Traditionally, a scientist is supposed to treat a very narrow and special subject, and any generalizations are to be presented as if they had been made on the basis of many such special investigations. However, the actual logic of scientific research is quite opposite: special research is always directed by some general considerations which determine the choice of subject, the methods of research and the ways of interpreting the results. There is no science without such conceptual background, and the attempts to hide that under the primitively inductive style are the relics of a centuries-old philosophy (F.Bacon etc.), renewed by the logical positivism of the XX century. I suppose that it's high time to abandon that primitive line and make scientific papers more logical, showing the place of particular statements in a wider picture. Thus, the way of reporting the results might reflect the natural direction of activity, from general ideas to their implementation.

Reviewer 1:

The following use of mathematical notation is rather awkward, and a much more elegant presentation could be done using category theory for instance (however I doubt it would really be necessary, when one looks at what is achieved using this pseudomathematical model: mere introduction of the notion of structural complexity as an entropy-like function).

Reply:

Complexity International started as a methodological journal, trying to synthesize the variety of applied research into an integral view on complexity in general. My paper was an attempt to revive this orientation, which has become completely dissolved in the rush of highly technical papers of the last volumes. Mathematics serves as a mere illustration of general ideas in this paper, and I didn't intend to obtain any formal results, which is stressed by the very title of the paper. Simple notions are better suited for illustration, so that I don't have to waste two thirds of the paper explaining the notation. Yes, category theory is a nice toy — but it develops in the same conceptual frame as the rest of mathematics, and brings nothing new into the discussion of fundamental problems.

I should stress that structural complexity is not a number, since any numerical estimates of complexity belong to another level, that of multiplicity. Structural complexity can be represented by a hierarchy of numerical measures of complexity, but it can never be reduced to any one of them. Rather, different structures might be used as the units of structural complexity, just like numbers measure multiplicity.

Reviewer 1:

In the section on "System", there is a reference to Goedel, which could be skipped. The idea that functional complexity is only revealed dynamically and that a mathematical description is a priori static, is not very closely related to the incompleteness theorems.

Reply:

I agree, that the relation between the incompleteness theorem and the insufficiency of the structural approach of mathematics should be clarified. In this paper, I express my opinion without any substantiation, which is not acceptable for the present style of "scientific" reports. According to the current norms, I should first publish a paper on Goedel theorem, and then refer to it in context of this discussion. Unfortunately, there is a loop: such paper on imcompleteness would involve the ideas expressed in this paper and could hardly be published before it. The sentence might be skipped to please the reviewer — still, I may leave it as it stands, since the paper will not be published anyway.

Reviewer 1:

I enjoyed the multiple foldings involved in the section on "Hierarchy", it is an interesting idea of the whole paper.

Reply:

Thanks. This is a very important feature of hierarchies, with numerous examples in the literature — but no awareness yet.

Reviewer 1:

The conclusion is very speculative, and personnally I do not share such dream visions. I would greatly recommend a rewriting of that conclusion.

Reply:

Sorry, but I hate the style of concluding sections that just list the "main results" of the paper, duplicating the abstract or/and the introduction. For some papers this may be alright, but only for those in the traditional line, pretending to establishing the final truths. The work that is not (and cannot ever be) finished should better indicate the ways of further development (or suggest more associations to think over) in the end of the text.

Reviewer 2:

English: a lot of mistakes (about 16 on the front page only)

Reply:

Since English has become the language of international communication in science, there is a tendency to a kind of language chauvinism in the academic journals: English natives are more likely to pass the barrier, the references to non-English sources are discouraged, and the reviewers have an easy way to reject the paper without being too particular in their critique. Everybody knows that efficient communication does not require perfect phrasing and correct spelling — all the misunderstandings can be fixed in a private talk. Those who are interested in the contents of the paper would not count language errors in it.

Reviewer 2:

A paradigm is discussed which defines complexity in terms of "integrity", "structure" and "system". Part of the discussion is mathematical, the rest being merely philosophical. Despite very interesting ideas, its content is often inaccurate (for instance: "Structure is more than just elements and links, it is a kind of wholeness, a level in the hierarchy of integrity") and conclusions are hazy ("Functional complexity leads to the numerous forms of Godel's theorem"). The math parts are a bit basic.

Reply:

Well, I have to admit it. The discussion is mainly methodological, with minimum mathematics for illustration. It would be nice to find out which ideas seemed interesting to the reviewer. Still, I do not see any inacurateness in the contents of the quoted sentence about structure — actually, it is much more accurate than the usual mathematical definitions, reducing structures to their formal models.

Reviewer 2:

My advise would be to read "Chaos and Information Theory: an heuristic outline", Nicolis and Prigogine, World Scientific, 1990.

Reply:

I am acquainted with the Prigogine's line (and there is a reference to one of Prigogine's books in the paper). My approach is quite different — though, of course, I appreciate the value of Prigogine's ideas for the comprehension of the necessity of incorporating development in science. However, I suppose that his theory deals with only one kind of development and is insufficient in other cases (especially in social sciences). This is a topic for a special discussion.

By the way, why should I discuss somebody else's views instead of my own? And why my views should necessarily be based on somebody's ideas and not on my own thought?

Reviewer 3:

While the subject of making explicit various aspects of the notion of complexity is an appealing one, I think that the paper lacks precision in its use of terms, and fails to present a logical and rigorous argument from well defined contentions to conclusions.

Reply:

The principal goal of the paper is to make the terms related to complexity more precise, and a new way of definition is suggested, relating any notions to their hierarchical context. The reviewer did not understand that, since he is convinced that the only precision possible is that of syllogistic deduction. However, neither deduction is the only rigorous argument, nor it is rigorous indeed, as indicated in my previous paper, Computability in developing systems. The standard form of discourse demanded by "scientific" journals is mere tribute to an obsolete tradition.

Reviewer 3:

In a paper of this kind one would expect to find:

1. In the introduction, a clear statement of the research question being tackled including how the research builds on the existing body of research. I would expect to find reference to specific papers (rather than whole volume citations) defining the point of departure of the research, what has gone before and what new methods are to be employed. There is a substantial literature concerning the definition of complexity.

2. In the body, clear definitions of terms, clear descriptions of analytical models and methods to be used to make the argument, and an economy of description demonstrating a clearly worked out argument.

3. A clear conclusion summarising the contribution of the research and remaining open questions.

Reply:

This is a good summary of bad style. The articles like that pretend that they really contribute into development of science, while timidly hiding any valuable thought in the haze of references to the predecessors. Well, the text should be as clear as possible. However, following the above formal requirements would rather make the problem more obfuscated.

  1. An overview of previous work on the subject adds nothing to the contents of the paper, while increasing its size and hiding the author's intentions. Such historical issues should better be discussed in a special appendix, or even in a separate paper devoted specifically to the history of science.

  2. For the problems that have not been solved yet, there cannot be any preliminary formulation. A brief statement of the research question would restrict the problem to a particular form, which may prove utterly inadequate in the end. An general indication of the scope is quite enough for the introduction, and it is the whole body of the paper that is to exactly specify the problem considered. Scientific research differs from engineering in that the latter has a list of properties to achieve in an invention, while the former has to ramble in the dark to discover the properties yet unknown.

  3. Clear definitions are only possible in a very narrow region of research, where nothing new is to be found, except for more combinations of the elements already known. Such activity may be of use for some immediately pragmatic purposes, but it has nothing to do with science. Likewise, the methods used may only crystallize in the course of their employment, and not before the research. To demonstrate a clearly worked and economical argument, one has to solve the problem completely — which is impossible for any serious problem. Such clear descriptions appear much later, as simplified accounts for educational needs.

  4. A summary of the author's contribution in the conclusion assumes that some problems have been sovled completely, which is almost never the case. The only honest summary would be a brief description of what the author did (not have done) in the body of text — but this is what the abstract serves for. The author can never list the remaining open questions, since there is an infinity of such questions, including those considered in the text. All that can be done is to indicate which questions are of primary interest for the author, while the reader may be interested in something quite different.

Reviewer 3:

Instead we find many undefined terms which have precise meaning only for the author, a set theoretical model that is introduced and then largely abandoned, and a rambling development which includes a number of irrelevant metaphysical speculations.

Reply:

The reviewer has not noticed the new type of definition employed in this work: the categories mutually define each other being used in the same context, in different positions. This method of definition is not less precise than the traditional reductions to the previously introduced notions (which will always have to be somehow defined as well). An illustration of some points (set theoretical model of structure) would not be as useful to illustrate a different idea, and there was no need of sticking to it throughout the paper. "Metaphysical speculations" (methodological research) are necessary to avoid blind technicality, aimless manipulation with empty symbols and terms.

Reviewer 3:

The paper demonstrates a lack of understanding of the incremental nature of scientific investigations and the degree of precision required for publication in a scientific journal.

Reply:

The reviewer demonstrates a lack of understanding of the impossibility of incremental evolution in science, and the inevitability of scientific revolutions. He confuses science with engineering, or mere craftsmanship, denying any actual creativity.


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