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 probably have made a reference to some book containing an overview of the related branches of mathematics — e.g. D.A.Pospelov, Situation-driven control: Theory and practice (Moscow: Nauka, 1986). I am not inclined to relativizing the issue, since, up to now, any attempts of incorporating time (movement, functioning, evolution) in mathematics were entirely structural, as far as I can judge. Today, mathematical reasoning is poorly suited to discuss dynamics, or development; the present trend towards ever more abstract formality indicates that functional and developmental ideas can hardly receive an adequate expression in the mathematics of the nearest future. Additional considerations on the subject could be found in: P.B.Ivanov, Computability in developing systems (1996).

There is yet another aspect, the common prejudice about the essence of science. 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 the bulk of such special investigations. But the true logic of scientific research is exactly the opposite: any special research is always regulated 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 a conceptual background, and the attempts to disguise this important circumstance by a primitive inductive style are mere relics of a centuries-old philosophy (F.Bacon etc.), slightly renewed by the logical positivism of the XX century. I suppose that it's high time to abandon that stylish primitivism in order to make scientific papers more logical, and indicating the place of any special choice in a wider picture. In particular, the arrangement of the results might better 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 of complexity in general. My paper was an attempt to revive this orientation, which has eventually become dissolved in the rush of highly technical papers of the last volumes. Here, mathematics serves as a mere illustration of general ideas, 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 text explaining the notation. Yes, the category theory is an amusing toy; but it develops in the same conceptual frame as the rest of mathematics, and brings no principal novelty in the discussion of fundamental problems.

I would stress once again that structural complexity is not a number, since any numerical estimates of complexity belong to a different level, namely, that of multiplicity. Structural complexity can be represented by a hierarchy of numerical measures of complexity, but it can never be reduced to any specific measure. 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 Gödel, 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 additionally clarified. In this paper, I express my opinion without a due substantiation, which is is far from the current style of "scientific" reports. According to the present norms, I should first have publish a paper on the Gödel theorem, and then quote it in context of this discussion. Unfortunately, there is logical circularity: such a paper on incompleteness would necessarily involve the ideas expressed in the present paper, which should hence have been presented beforehand... A phrase might well be skipped, to please the reviewer; still, still, I'd rather 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, albeit without a clear awareness.

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, that could do; but only for those in the traditional line, pretending to establishing final truths. A piece of work that has not yet been (and, in fact, can never be) finished should better conclude by indicating the ways of further development, or suggesting additional conceptual links to think over.

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 flavor of language chauvinism in academic journals: native English-speakers are more likely to pass the barrier, and the references to non-English sources are discouraged; in any way, reviewers now have a quick excuse for rejecting a paper without going into any particulars. Everybody knows that efficient communication does not require perfect phrasing, or correct spelling: all the misunderstandings can be fixed in the flow of communication. Those interested in the ideas would not waste time counting language lapses.

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 Gödel's theorem"). The math parts are a bit basic.

Reply:

Well, I have to admit it. The discussion is mainly methodological, with minimum mathematics, just for illustration. It would be nice to learn which particular ideas have attracted the reviewer's attention. Still, I do not find any inaccurateness in the quoted sentence on the essence of structure; in fact, it is much more accurate than the usual mathematical definitions, reducing structure as such to arbitrary special 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 well acquainted with the Prigogine's line (and I reference 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 of the possible kinds of development, and that it is insufficient in other cases (especially, in social sciences). This is a topic for a special discussion.

With all that, why should I discuss somebody else's views in my text instead of those of my own? And why should my views merely expand somebody's ideas, rather than follow my own way of 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 my paper was to clarify the ideas related to complexity, and for that purpose, I suggest a new way of definition, relating any notions to their hierarchical context. The reviewer has missed that point, because of the common prejudice that the only precision possible is that of syllogistic deduction. However, neither deduction is the only source of rigor, nor can it be rigorous enough, as my previous paper, Computability in developing systems, clearly indicates. The standard form of discourse imposed by "scientific" journals is nothing but a 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 numerous 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. Certainly, the text should be as clear as possible. However, following the above formal requirements would rather obfuscate any problem.

  1. An overview of the previous work in the same field adds nothing to the contents of the paper, while increasing its size and obscuring the author's intentions. Such historical issues should rather be treated in a special appendix, or even in a separate paper devoted specifically to the history of science.

  2. Problems that have not yet been studied to any considerable extent cannot have a clear preliminary formulation. A concise exposition of the principal line of research would restrict the problem to a particular approach, which may prove utterly inadequate in the end. A general indication of the scope is quite enough for an introduction, since it is the whole body of the paper that is to exactly specify the issues to consider. Scientific research differs from engineering in that the latter starts with a (more or less comprehensive) list of features to implement, while science is bound to ramble in the dark to discover phenomena yet unknown.

  3. "Precise" definitions are only possible within a very narrow portion of research, where nothing really new is expected to come, beyond a rearrangement of the already available. Activities like that may be of some use for certain pragmatic purposes, but they have nothing to do with science. Likewise, the methods to employ will normally crystallize in the course of their employment, and not before real research. To demonstrate a clearly worked and economical argument, one needs to entirely solve the problem, which is impossible in any serious study. Such "sharpened" descriptions appear much later, as simplified accounts, in the educational context.

  4. A summary of the author's contribution in the conclusion would mean that a final solution is already at hand, which is almost never the case. The only honest kind of summary is to remind what the author did (but never have done) in the body of text; still, isn’t it what an abstract is for? The author can never enumerate the remaining open questions, just because there is an infinity of such questions, including those considered in the text. Of course, the author is free to speak of his personal interests and priorities; the reader may however be interested in anything else.

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 did not pay attention to the specific type of definition employed in this work: the categories mutually define each other through being used in the same context, in different positions. This method of definition is no less precise than the traditional reduction to something previously introduced (and which always needs to be somehow defined in its turn). An illustration of some particular viewpoint (say, a set theoretical model of structure) may happen to be much less useful to illustrate a different idea; one does not need to stick to a single picture throughout the whole. "Metaphysical speculations" (methodological research) are absolutely necessary to avoid blind technicality, purposeless manipulation with empty symbols and arbitrary terminology.

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 real creativity.


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