THOUGHT AND ACTION FROM AXIOMATIC CALCULUS TO FRACTAL
by Siegfried Maser (translated by Meike Asbach)
As early as Aristotle (384-322) it is said that we humans are beings who think as well as act, and that these two activities are characteristic of human life. Nature and the cosmos are subject to a development=evolution which we observe in individual organisms (=ontogenesis) as well as in the whole of the organic world (= phylogenesis). As a principle for this development in nature, Aristotle postulates entelechie, meaning that all matter contains the seed of its own development and completion! According to Aristotle, each kind of matter and organism (the latter being a compound of matter) possesses within itself a force, which intractably leads it to self-development and self-completion. The inner aim is, so Aristotle, especially in humans, to develop into creative beings with the help of thought and action. The latter notion today still gains a lot of sympathy and conviction, regardless of whether we take the material / structural, the forces / functional or both as its causes.
In any case, 'thinking' and 'acting' are human activities, and we all know from our own experience, that there are different qualities involved in thinking and acting, yes, that we even make mistakes, for to err is human, after all! There is right and wrong thinking, thorough and superficial thinking, simple and complex thinking, there is linear and roaming thinking, aporetic and systematic thinking, provable/ universally applicable and subjective/individual thinking, analogous and digital thinking, converging and diverging thinking, and so on and so forth. There is good and bad acting, skilful and clumsy acting, careful and careless acting, social and egoistic acting, conscious / purposive acting, planned and spontaneous acting and so forth.
It is known that in everyday life we learn and use all these modes of thinking and acting to different degrees of quality, and each situation in its concrete particularity then shows us whether we have thought right or wrong, whether we have done right or wrong, whether we have found the right words or accomplished a necessary deed, although this is usually revealed afterwards, when we can see the consequences of our actions.
In everyday usage, we most commonly differentiate between 'thinking' and 'acting' and separate them: like theory from practice, theorising from practising, saying from doing, verbally from manually, head from hand, talking about a problem from tackling it and theorists from practitioners. Here too it would of course be more correct to replace the 'either/or' with an 'as well as', since we tend to do both things together, sometimes simultaneously, sometimes one after the other, sometimes correspondingly, at other times contradictorily: as a rule, we think before we act; we reflect on what we have done. As a rule, we think about why and how we do something beforehand; and once we have done something, we think about whether we have achieved our objectives with it, whether it could have been done better, more easily, cheaply, quickly. This is how experience comes into being, which especially the practitioners claim as their own. 'Learning by doing' does after all also mean learning from mistakes: making mistakes is not supposed to be bad, but to repeat them would be stupid! And it is precisely because it is easier and cheaper to correct mistakes which have merely been thought and not yet carried out, that we learn the folowing rule of thumb from experience: first think, then act! This could also mean that the structure of our thinking plays a major role in determining the structure of our actions.
Even Aristotle had noticed, that next to 'pure thought' and 'pure action', there was a third important aspect of human activity, namely a field where thought and action continuously overlap and therefore cannot be separated from one another. He called this aspect poiesis, = an activity which is productive. To create and to design would surely fall under this heading.
There are thus three typical, human activities: (pure) thinking, creating and ( pure) action. Since 'creating' can be understood as an interaction between thinking and acting, it is necessary to understand 'thought' and 'action' as a preliminary.
Let us start by reflecting on 'thought', since this incidentally has a longer tradition than reflection on 'action'. In Aristotle's work, we find the following thesis, for example: thought attains its highest quality in science, and science works according to the laws of logic (= organon/ hand tool of thought).
Especially with the advent of rationalism, the ideal of science was expressed in an axiomatic calculus: René Descartes (1596 1650), Blaise Pascal (1623 1662) and Gottfried Wilhelm Leibniz (1646 1716). For every rationalist, 'ratio' or reason is the supreme source of cognition. Something is true if it is free of contradictions, consistent. Thought is conceptualised as follows:
The elements of thought are first of all terms, the meaning of which is fixed by definition. General terms have 'general' meaning, i.e. they have a large area of application, but little content/ information (e.g. 'thing'); specific terms have a 'specific' meaning, i.e. they have a small area of application but a large content / information (e.g. proper names). " General" and "specific" are gradual concepts, i.e. there is a sense of degree, a more or a less, inherent in both. The following stands a rule for definition: the meaning of a term is determined by the meaning of the closest general term plus its own specific properties (=genus proximum et differentiam specificam). Some examples: parallelogram=quadrilateral whose opposite sides are parallel and equal in length. Rectangle= a parallelogram having four right angles. Square= a rectangle having four equal sides. Through such defining processes the terms ideally form a pyramid of terms, at the apex of which the general terms can be found, and along the base of which the specific terms can be found. The resulting hierarchy is determined through the degree of generality.
Statements come into existence through the association of elementary terms, through eg. and/conjunction, or/disjunction, if-then implication, and others. Rose is red, chair is comfortable; today is not tomorrow, etc. It is the subject of conceptual logic which concerns itself with the laws of such associations of terms; when it only observes the areas of application (extensionally), it functions as 'class logic'/ set theory; when it also considers the content (intensionally), it functions as 'predicate logic'.
The association of statements produces conclusions: if all men are mortal, and if Plato is a man, then Plato is mortal. The laws of such associations of statements are governed by the logic of statements; it takes the form of "formal logic" when it only differentiates between true and false statements; it is known as " modal logic", when modalities ( possible, real, necessary, impossible) are taken into account; as "probability logic", when it considers information as function of its unlikeliness (cf. algebra of events and information theory).
Finally, through the association of conclusions, we get proofs/ chains of proof. The laws of such proofs are described by 'proof logic/epistemology. (e.g. deduction, induction, reduction.)
If human thought then attains its highest quality in science, and in attaining it only has recourse to the rules and laws of logic (cf. Aristotle), we can now say more precisely: In any science, all terms/ elements have to be defined and all statements have to be proved!
But to define terms means to trace them back to other terms via the rule of definition. These terms then have to be defined/traced back in turnad infinitum! At some point, in reality, this endless process has to be broken off, and there will remain what are called basic terms, which cannot be defined themselves, but starting from which all other terms can be defined: The "tip" of the pyramid of terms thus remains empty, the problems arising out of this are put 'outside the brackets'. Basic terms are supposedly obvious, directly comprehensible.
In analogy proving statements means to trace them back to other statements via rules of proof. These statements then have to be defined/traced back in turnad infinitum! Again, this endless process has to be broken off, and there will remain what are called principles (=axioms), which cannot be proved themselves, but starting from which all other terms can be proved. All chains of proof must therefore start with the unprovable, the problems arising out of this are put 'outside the brackets' once more. Axioms are supposedly obvious, directly comprehensible.
Leibniz finally defines the ideal of science as an axiomatic calculus, an axiomatic system, on the basis of these findings:
1. should wherever possible define all its terms using the rule of definition. Those basic terms which are not defined (these should be kept to a minimum), must be independent of each other and obvious.
2. should prove wherever possible all its statements using explicit rules of proof. Those principles / axioms which are not proven, should be kept to a minimum, and must be independent of each other and obvious.
3. phrases its rules of definition and proof in such a way, that the system of statements as a whole (=theory=science) is free of contradictions, consistent (=true) and complete.
Examples of sciences which would fit this ideal of science most of all, i.e. those which attain the highest possible quality of human thought, would first of all be logic itself, to which Aristotle gave this shape in his "Syllogistics; then there would be the geometry to which Euclid gave this shape in his "Elements"; the mechanics of Isaac Newton, and more recently a whole range of axiomatic calculus in mathematics, information technology and the natural sciences. The problems of consistency and completeness, thoses of simplicity, of independence and obviousness were to become major themes in 20th century meta-mathematics.
This way of thinking in systems can also be applied to the domain of human action: we think of human actions as a system, i.e. there are elementary actions (movements) which we sum up into simple and complex actions. Working the opposite way, 'Taylorism', for example dissects complex chains of action into their elementary steps and thus organises the productive areas (with all the advantages and disadvantages this entails.)
Naturally, there were and are critics of this ideal of science as a whole: Once very early on, from the domain of the humanities, but then also from the domain of natural sciences, e.g. biology. Those contents which rationalism takes for granted as basic premisses for science do not apply at all in some areas: next to differentiation there is integrality, next to isolation there is inter-connectivity, next to simplicity complexity, next to unity diversity, next to sameness individuality, next to stability instability, next to durability there is passage, next to necessity there is chance , next to causality there is history, next to reversible irreversible, next to repeatability there is uniqueness, next to mechanics self-organisation, etc. New terms move into the foregroud of interest: evolution, interactivity, self-organisation, chaos and catastrophes (big oaks from small acorns grow!). All that which had been put outside the brackets so far, suddenly arouses special interest. Fractals, which were first defined in mathematics, then observed in nature and have long since become symbols, serve as paradigms for this.
In the domain of mathematics, for example, it is common practice to analyse regular and continuous functions, " decent" functions, i.e. those which can be calculated, differentiated, integrated etc. Stairs, jumps, peaks and similar are "indecent" problems, calculations become complex and can turn out differently in each individual case, tangents are indeterminable, etc. Such problems are therefore excluded for being chaotic. (Jumps are apparently unknown in nature!) It was precisely these "indecent" cases, which captured the interest of Benoit Mandelbrot: Starting with a straight line, he divided it into three parts and made a peak out of the middle one. Now having four sub-parts of line, he divided each one of these into three in turn, and again changed the middle one into a peak. Having now got to a total of 16 sub-parts he treated each of these as above and so on. A computer which calculated this simple algorithm for the duration of one night then produced a curve which had peaks everywhere, which could no longer be differentiated or integrated, a curve which was thoroughly 'indecent', irregular, discontinuous. Mandelbrot named this highly aesthetic result a fractal, from the latin 'fractus' meaning 'broken-up, fragmented'. Today we often speak of "Mandelbrot-manikin."
In the analysis of such fractal structures (fractal geometry), the following terms take on central importance:
Self-similarity: In fractal structures, the structure of the whole is mirrored in its details (in regular, continuous structures, on the other hand, the closer the detail, the simpler it gets!); each detail contains the structure as a whole, but they are only similar, not identical!
Self-organisation: complex, dynamic systems organise themselves from within through continuous, dynamic construction (networking) of structures of order and their disintegration. The parameters for this order can be obtained through analysis.
Dynamics: the dynamics and the vitality of the whole are always maintained through manyfold processes of feedback and interaction within the self-organisation.
In his book " The fractal factory The revolution of business culture" (Springer Verlag, Berlin, 1992), Hans Jürgen Warnecke has transferred such fractal structures, albeit in the form of models, to his search for new forms of organisation in business. It would then be quite obvious to transfer these models (first of all verbally) into the productive domain of knowledge and thereby apply them to thought and action. This I have attempted very briefly below in accordance with the theses of Warnecke (ibid, p.142ff):
Definition: A fractal is an independently active scientific domain, whose aims and achievements are clearly definable.
Fractals are self-similar, each one provides a distinct service.
Fractals engage in self-organisation:
Operatively: The procedures are organised to an optimum with the help of adapted methods. (Self-organisation)
Tactically and strategically: fractals recognise and formulate their aims as well as their internal and external relationships in a dynamic process. Fractals reshape, regenerate and disintegrate themselves (Self-optimisation).
The system of goals, which describes the goals of the fractals, must serve to reach the goals of the science itself: it is about interest in recognition and interest in use. (Goal-orientation).
Fractals are connected through a highly efficient information and communications network. They determine themselves how data is accessed and how much is accessible. The performance of the fractal is continuously monitored and evaluated. (Dynamics).
Result (after H.-J. Warnecke):
Just like the science itself, the scientist has to be all-encompassing in the way in which he goes about his task.
Self-organisation necessitates autonomy: i.e. self-government and self- determination.
The process of structuring can only be partly objectified: there is no ideal in science, if anything, there is a range of optima
From the primitive via the complex to the simple!
So far the " venture "science (e.g. university, especially design / theory and practice), has not been thus understood: But maybe then, from this material we can actually draw up real approaches for reform -at least a discussion would make sense- especially in the field of design.
The author about himself
(Biographical notes in the German text.)
I was originally interested in mathematics, for I was fascinated by its thoroughness and exactness. In search of the nature and the causes behind this exactness, the centre of my interest shifted from close scrutiny of abstract mathematical calculation in the beginning, via general axiomatics, to the study of the principles of philosophy. I reached the following fundamental insights:
1. In the pure, theoretical sciences, the progress of scientific development is identified with the increasing precision of its findings. Historically, this led to the diversification into many separate, autonomous disciplines, in each of which the problems are defined with an ever increasing degree of complication and complexity, with inevitable consequences for the findings, especially as fara as proving their truth and their inter-human communication are concerned. Common ground now only exists in the goals shaped by scientific theory: increasing exactness in findings, or 'science for the sake of science'. (cf. in art: l'art pour l'art.)
2. In the applied, practical sciences, the progress of scientific development is identified with an increasing degree of perfection in technology, civilisation and culture. The historical development from manual to industrial to automised production of food therefore followed logically, i.e. science is a means to an end, it serves to improve the quality of human life.
3. In reality, a deep rift exists between these two idealised fundamental philosophies, it is the rift between theoreticians and practitioners, idealists and pragmatists.
I wanted to bridge this gap. For me the creative domain was exemplary for this: town planning, architecture, design, art. My approach is to work in an interdisciplinary team of experts. My aids are systems theory, planning theory, communications theory, cybernetics and ecology. My aim is to improve the quality of life, especially where people have to rely on the help of others. This led to my commitment to the "so-called" developing countries including my involvement in university politics, particularly with regard to university reform.