OKSIMO and BOURBAKI. A Metamathematical Perspective on Oksimo. Part 1

eJournal: uffmm.org
ISSN 2567-6458, 22.Sept – 24.Sept  2021
Email: info@uffmm.org
Author: Gerd Doeben-Henisch
Email: gerd@doeben-henisch.de

(Some minor corrections: 23.Sept 2021)

(A substantial extension: 24.Sept.2021)


This text is part of a philosophy of science  analysis of the case of the oksimo software (oksimo.com). A specification of the oksimo software from an engineering point of view can be found in four consecutive  posts dedicated to the HMI-Analysis for  this software.[*]


Covered under the pseudonym of N.Bourbaki [1] appeared 1970 the French edition of a book which 1968 already had been translated into English  (reprinted 1970) called  Theory of Sets.[2] This book is the first book of a series about ELEMENTS OF MATHEMATICS.

To classify this book about set theory as a book of Metamathematics and as such as a book in the perspective of Philosophy of Science will become clear if one starts reading the book.[3]


It is the basic conviction of the Bourbaki book, that “… it is known to be possible … to derive practically the whole of known mathematics from a single source the Theory of Sets.” (p.9) And from this Bourbaki concludes, that it will be sufficient “… to describe the principles of a single formalized language, to indicate how the Thory of Sets could be written in this language, and then to show how the various branches of mathematics  … fit into this framework.”(p.9)

Thus, the content of mathematics — whatever it is — can according to Bourbaki be described in one single language [Lm] and the content will be called Theory of Sets [T] .


Because the one single language Lm used to describe the Theory of Sets shall be a language with certain properties one has to define these properties with some other language, which is talking about Lm. As language for this job Bourbaki is using the ordinary language [Lo].(p.9) But the reasoning within which one is using this ordinary language is called metamathematics (cf. P.10f). Within the metamathematical point of view the language Lm under investigation is seen as a set of previously given objetcs without any kind of meaning, where only the assigned order is of importance.(cf. p.10): “… metamathematical ‘arguments’ usually assert that when a succession of operations has been performed on a text of a given type, then the final text will be of another given type.”(p.10)

What looks here at first glance  as the complete formalization of mathematics it is not. Bourbaki states clearly that “formalized mathematics cannot in practice be written down in full“(p.11) There has to be assumed as ‘last resort’ the assumption of a common sense of the mathematician and the intuition of the reader. (cf. p.11)


This conflict between at one hand of  the idea of a formalization of  Mathematics by a formalized language Lm  and on the other hand by the well known proof of Gödel [4] of the incompleteness of the axioms for classical arithmetic  (cf. p.12) is not a real conflict as long as one takes into account — as Bourbaki points out — that the ‘content of mathematics’ is only given in different layers of languages (Lm, Lo, …) which again are embedded in a presupposed ‘common sense’ which is nothing else as the cognitive machinery of human persons including an embedded meaning function relating different kinds of knowledge into different kinds of — internal as well as external — expressions of some language L. Thus any kind of a  ‘reduction of meaning’ seems never to be a ‘complete reduction’ but only a ‘technical reduction’ to introduce some ‘artificial (abstract) objetcs’ which can only work because of their embedding in some richer context.

This new perspective can be called the cognitive-semiotic turn which became possible by new insights of modern brain sciences in connection with pysychology and semiotics.

From this new point of view one can derive the idea of embedding metamathemics in a more advanced actor theory providing all the ingredients to make metamathematics more ‘rational’.

Actor theory first outline
Figure 1: Actor theory first outline

The details of the Actor Theory [AT] can become quite complex. Here a first outline of the basic ideas and what this can mean for a metamathematical point of view of mathematics.

World is not World

The main idea is founded in the new insights of Biology and Neuro-Psychology of the handling of body-world interactions as exercised by humans. One of the main insights is rooted back to von Uexküll [5] more than 100 years ago, when he described how every biological organism perceives and handles some world outside of the body  with the inner neuronal structures given! Thus different life forms in the same outside world  W will peceive and act neuronally in different worlds! Brain X acts in world X which is somehow related to the outside world W as well as Brain Y acts in world Y which also is  somehow related to the outside world W.

These basic insights relate as well to more developed life forms as such as  humans are. We as humans do not perceive and think the world W outside of our bodies ‘as it is’ but only as our brain inside our body can process all the body states related to the outside world in the mode of the inside brain. Thus if the different human individuals would have different brains they would live in different worlds and their would be no chance of a simple communication. But as we know from physiological and behavioral  studies humans can to some extend communicate successfully. Thus there exists inside of every human individual a human-processed world h(W) which is different from other life-forms like a rat, a worm, an octopus, etc.

From this basic insight it follows that if we speak about the world W we do indeed  not speak about the world  W directly but about the world W as it is processed in a human-specific manner, the  world h(W). This has many implications.

  1. Because we know already that the world h(W) is not a static but a dynamic world depending from our learning history it can happen — and it happens all the time — that different individuals have different learning histories.  This can result in quite strong differences of experience and knowledge attached to different individuals, which can prevent a simple understanding between such individuals: the learned world h1(W) can to some degree be different from the learned world  h2(W) such that a simple and direct understanding will not be possible.
  2. This difference between the outside world W and the processed inside world h(W) relates to the communication too! The spoken or written expressions E of some language L are belonging to the outside world. They have a counterpart in the inner world as inner expressions E*, which can be associated with all kinds of processed inner states of the inner world h(W) = W*. These possible — and learned — associations between inner expressions and inner states belonging to h(W) is assumed here to be that what commonly is called meaning. Thus one has to assume an internal meaning function μ which maps the internal expressions E* of some internal language L*  into parts of the internally processed world h(W)=W* and vice versa. Thus we have μ: E* <—> W*. Thus μ(e*) would point to some part w* of the internally processed world W* as the ‘meaning’ of the internal expression e*.
  3. This semiotic architecture of human beings enables a nearly infinite space of expressions as well as associated meanings definable during learning processes. This is powerful, but it is also very demanding for the speaker-hearer: to enable a succesful communication between different speaker-hearer these have to train their language usage under sufficient similar conditions thereby constructing individual meaning functions which work — hopefully — sufficiently similar. If not then communication can slow down, can produce lots of misunderstandings or can even break down completely. [6]
  4. In the case of mathematics it is a long debated question whether mathematics can be reduced to the expressions Em of some mathematical language Lm or if mathematics has some mathematical objects on its own which are different from the expressions. If one would assume that mathematics has no objects on its own but only some expressions Em, then it would become difficult to argue whether exactly these expressions Em should be used and not some other expressions Ex. Moreover to classify expressions as ‘axioms’ or ‘theorems’ would be completely arbitrary.   The only ‘anchor’ of non-arbitrariness would consist in some formal criteria of a formal consistency which would disable the formal generation of pairs of expressions {a,a*} where one is excluding the other. But even such a formal consistency presupposes some criteria which are beyond the expressions as such! Thus mathematics would need some criteria outside mathematics. This can be understood as an argument for metamathematics.  But according to Bourbaki  metamathematics is defined as a set of operations on given expressions without a specific meaning.  This is not enough to establish formal consistency! Thus even metamathematics is pointing to something outside of given mathematical expressions.  What can this be?

To be continued …


[*] More recent versions of the specification of the oksimo oftware can be found in the bolg oksimo.org. Unfortunately are the texts in that blog  — at the time if this writing — still only in German. Hopefully this will change in the future.

[1] Bourbaki group in Wikipedia [EN]: https://en.wikipedia.org/wiki/Nicolas_Bourbaki

[2] N.Bourbaki (1970), Theory of Sets, Series: ELEMENTS OF MATHEMATICS, Springer, Berlin — Heidelberg — New York (Engl. Translation from the French edition 1970)

[3] The first time when the author of this text has encountered the book was some time between 1984 – 1987 while being a PhD-student at the Ludwig-Maximilians Univesty [LMU] in Munich. It was in a seminar with Prof. Peter Hinst about structural approaches to Philosophy of Science. The point of view at that time was completely different to the point of view applied in this text.

[4] Kurt Goedel. Über formal unentscheidbare Sätze der Principia
Mathematica und verwandter Systeme, i. Monatshefte fuer
Mathematik und Physik, 38:173–98, 1931.

[5] Jakob von Uexküll, 1909, Umwelt und Innenwelt der Tiere. Berlin: J.Springer.

[6] Probably everybody has made the experience in his life of being part of a situation where nobody speaks a language, which one is used to speak …




eJournal: uffmm.org, ISSN 2567-6458, 1.December  2020
Email: info@uffmm.org
Author: Gerd Doeben-Henisch
Email: gerd@doeben-henisch.de


The starting point of view in this blog has been and still is the point of engineering, especially the perspective of man-machine interface [MMI], later as Man-Machine Interaction, then  accompanied by   human-computer interaction [HCI] or human-machine interaction [HMI]. While MMI often is discussed in isolation, not as part of engineering, this blog emphasizes a point of view where MMI is understood as an integrated part of systems engineering. The past years have shown, that this integration makes a great difference in the overall layout as well as in the details of the used methods. This integration widened the scope of MMI to the context of engineering in a way which teared down many artificial boundaries in dealing with the subject of MMI. The analysis part of MMI can take into account not only the intended users and a limited set of tasks required for the usage of a system but it can extend the scope to the different kinds of contexts of the intended users as well as the intended service/product as such: cultural patterns, sustainable perspectives, climate relevance, political implications, and more. This triggers the question, whether there are other established scientific disciplines which are sharing this scope with MMI. Traditionally experimental and cognitive psychology has always played an important role as part of the MMI analysis.  Different special disciplines like physiology or neuro-psychology, linguistics, phonetics etc. have played some role too. More recently culture and society have been brought more into the focus of MMI. What about sociology? What about anthropology? The following text discusses a possible role of anthropology in the light of the recent book Why The World Needs Anthropologists?


This review has the addendum ‘Part 1’ pointing to the fact, that this text does not deal with the whole book first, but only with some parts, the introduction and the conclusion.

An Introduction

The introduction of the book is asking, why does the world needs anthropologists?, and the main pattern of the introduction looks back to the old picture of anthropology, and then seeks to identify, what could/is the new paradigm which should be followed.

The roots of anthropology are located in the colonial activities of the British Empire as well as in the federal activities of the USA, which both had a strong bias to serve the political power more than to evolve a really free science. And an enduring gap between the more theoretical anthropology and an applied one is thematised although there existed always  a strong inter-dependency  between both.

To leave the close connection with primarily  governmental interests and to see the relation  between the theory and the different Applications  more positive than negative anthropology is understood  as challenged to rebrand its appearance in the public and in their own practice.

The most vital forces for such a rebranding seem to be rooted in more engagements in societal problems of public interests and thereby challenging the theory to widen their concept and methods.

Besides the classical methods of anthropology (cultural relativism, ethnography, comparison, and contextual understanding)  anthropology has to show that it can make sense beyond pure data, deciphering ambiguity, complexity, and ambivalence, helping with  diversity, investigating the interface between culture, technology, and environment.

What Is Left Out

After the introduction the main chapters of the book  are left out in this text  until later. The chapters in the book are giving examples to the questions, why the world needs anthropology, what have been the motivations for active anthropologists to become one, how they have applied anthropology, and which five tips they would give for practicing and theorizing.


In the conclusion of the book not the five questions are the guiding principle but ‘five axis that matter greatly’, and these five axis are circumscribed as (i) navigate the ethics of change; (ii) own-it in the sense, that an anthropologist should have a self-esteem for his/ her/ x  profession and can co-create it with others; (iii) expand the skill-set; (iv) collaborate, co-create and study-up; (v) recommend as being advisors and consultants.

The stronger commitment with actual societal problems leads anthropology at the crossroads of many processes which require new views, new methods. To gain new knowledge and to do a new practice is  not always accompanied by  known ethical schemata. Doing this induces  ethical questions which have not been known before in this way.  While a new practice is challenging the old knowledge and induces a pressure for change, new versions of knowing can  trigger new forms of practice as well. Theory and application are a dynamic pair where each part learns from the other.

The long-lasting preference of academic anthropology, thinking predominantly  in the mind-setting of   white-western-man, is  more and more resolved  by extending anthropology from academia to application, from man into the diversity of genders, from western culture into all the other cultures, from single persons to assemblies of diverse gatherings living an ongoing discourse with a growing publicity.

This widening of anthropological subjects and methods calls naturally for more interdisciplinarity, transdisciplinarity, and of a constructive attitude  which looks ahead to  possible futures of processes.

Close to this are expressions like collaboration and co-creation with others. In the theory dimension this is reflected by multiperspectivity and a holistic view. In societal development processes — like urban planning — there are different driving forces acting working top-down or acting working bottom-up.

Recommending solutions based on anthropological thinking ending in a yes or no, can be of help and can be necessary because real world processes can not only wait of final answers (which are often not realistic), they need again and again decisions to proceed now.


The just referred texts making a fresh impression of a discipline in a dynamic movement.

General Knowledge Architecture

For the point of view of MMI (Man-Machine Interface, later HMI Human-Machine Interaction, in my theory extended to DAAI Distributed Actor-Actor Interaction) embedded in systems engineering with an openness for the whole context of society and culture arises the question whether such a dynamic anthropology can be of help.

To clarify this question let us have a short look to the general architecture of knowledge.

Within the everyday world philosophy can be understood as the most general point of view of knowing  and thinking.  Traditionally logic and mathematics can be understood as part of philosophy although today this has been changed. But there are no real reasons for this departure: logic and mathematics are not empirical sciences and they are not engineering.

Empirical science can be understood as specialized extension of philosophical thinking with identifiable characteristics which allow to  differentiate to some extend different  disciplines.  Traditionally all the different disciplines of empirical science have a more theoretical part and a more applied part. But systematically they depend from each other. A theory is only an empirical one, if there exists a clear relationship to the everyday world, and certain aspects of the everyday world are only theoretical entities (data) if there exists a relationship to an explicit theory which gives a formal explanation.

Asking for a  systematic place for engineering it is often said, that it belongs to the applied dimension of empirical science.  But engineering has realized processes, buildings, machines long before there was a scientific framework for to do this, and engineering uses in its engineering processes lots of knowledge which is not part of science. On the other side, yes, engineering is using scientific knowledge as far as it is usable and it is also giving back many questions to science which are not yet solved sufficiently. Therefore it is sound to locate engineering besides science, but   being  part of philosophy dealing with the practical dimensions of life.

What About Anthropology?

While philosophy (with logic and mathematics) is ‘on top’ of empirical science and engineering, it is an interesting question where to place anthropology?

While empirical science as well as engineering are inheriting all what philosophy provides remains the question whether  anthropology is more an empirical science or more engineering or some kind of a hybrid system with roots in empirical science as well as in engineering?

Looking back into history it could arise the impression that anthropology is more a kind of an empirical science with strong roots in academia, but doing  fieldwork to feed the theories.

Looking to the new book it could support the image that anthropology should be more like engineering: identifying  open problems in society and trying to transform these problems — like engineers — into satisfying solutions, at least on the level of counseling.

Because in our societies the universities have traditionally a higher esteem then the engineers — although the engineers  are all  trained by highly demanding university courses — it could be a bias in the thinking of  anthropologist not to think of their discipline   as engineering.

If one looks to the real world than everything which  makes human societies livable is realized by engineers. Yes, without science many of the today solutions wouldn’t be possible, but no single scientific theory has ever enabled directly some practical stuff.  And without the engineers there would not exist any of the modern machines used for measurements and experiments for science. Thus both are intimately  interrelated: science inspires engineering and engineering inspires and enables science, but both are genuinely different and science and engineering play their own fundamental role.

Thus if I am reading the new book as engineer (attention: I am also a philosopher and I am trained in the Humanities too!) then I think there are more arguments to understand anthropology  as engineering than as a pure empirical science. In the light of my distributed actor-actor interaction paradigm, which is a ‘spinoff’ of engineering and societal thinking it seems very ‘naturally’ to think of anthropology as a kind of social engineering.

Let us discuss both perspectives a bit more, thereby not excluding the hybrid version.

1) Anthropology as Engineering

The basic idea of engineering is to enable a change process which is completely transparent in all respects: Why, Who, Where, When, How etc. The process starts with explicit preferences turning some known reality into a problem on account of some visions which have been imagined and which have become ranked higher than the given known reality. And then the engineers try to organized an appropriate change process which will lead from the given situation to a new situation until some date in the future where the then given situation — the envisioned goal state — has become real and the situation from the beginning, which has been ranked down, has disappeared, or is at least weakened in a way that one can say, yes, it has changed.

Usually engineers are known to enable change processes which enable the production of everyday things (tools, products, machines, houses, plants, ships, airplanes, …), but to the extend that the engineering is touching the everyday life deeper and deeper (e.g. the global digital revolution absorbing more and more from the real life processes by transforming them into digital realities forcing human persons to act digitally and not any more with their bodies in the everyday world) the sharp boundary between engineering products and the societal life of human persons is vanishing. In such a context engineering is becoming social engineering even if the majority of traditional engineers this doesn’t see yet in this way. As the traditional discipline MMI Man-Machine Interface and then  expanded to HMI Human-Machine Interaction and further morphed into DAAI Distributed Actor-Actor Interaction this  already manifests, that the realm of human persons, yes  the whole of society is already included in engineering.  The border between machines and human actors is already at least fuzzy and the mixing of technical devices and human actors (as well as all other biological actors) has already gained a degree which does not allow any longer a separation.

These ideas would argue for the option to see anthropology as social engineering: thematizing all the important visions which seem to be helpful or important for a good future of modern mankind, and to help to organize change processes, which will support approaching this better future. That these visions can fail, can be wrong is part of the ever lasting battle of the homo sapiens to gain the right knowledge.

2) Anthropology as  an Empirical Science

… to be continued …

3) Anthropology as a Hybrid Couple of Science and Engineering

… to be continued …