Below, find a few remarks made on this paper by the reviewers
of Complexity International, followed by my replies.
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
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.
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
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.
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.
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.
I enjoyed the multiple foldings involved in the section on
"Hierarchy", it is an interesting idea of the whole paper.
Thanks. This is a very important feature of hierarchies,
with numerous examples in the literature — but no
The conclusion is very speculative, and personnally I do not share
such dream visions. I would greatly recommend a rewriting of that
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.
English: a lot of mistakes (about 16 on the front page only)
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.
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.
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
My advise would be to read "Chaos and Information Theory: an heuristic
outline", Nicolis and Prigogine, World Scientific, 1990.
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?
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.
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.
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
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.