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)

CONTEXT

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.[*]

THE BOOK: THEORY OF SETS

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]

MATHEMATICS WITH ONE LANGUAGE

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] .

METAMATHEMATICS

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)

COGNITIVE-SEMIOTIC TURN

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’.

OUTLINE OF ACTOR THEORY
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?
PART 2

To be continued …

COMMENTS

[*] 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 …

 

 

SUSTAINABLE DEMOCRACIES

eJournal: uffmm.org
ISSN 2567-6458, 28.Aug 2021 – 28.Aug 2021
Email: info@uffmm.org
Author: Gerd Doeben-Henisch
Email: gerd@doeben-henisch.de

CONTEXT

This post is part of the uffmm science blog and will introduce in the new possibility to use the oksimo software for theory development  and theory testing using only everyday language. More options are possible.

PREFACE

What has an engineering blog to do with ‘democracy’ [1]-[5], even with ‘sustainable democracies’?

The answer to this question is embedded in the history of the engineering process which led to this post and in this post to  the idea of defining a process model for an  engineering process with a strong integration of the human-machine interaction perspective. When this has been solved in a first step another requirement showed up: generalize the engineering process for every  kind of solution processes with every  kind of experts. Design a solution process using only normal language, and make the requirements and possible goals to  parts of the process itself. Allow any kind of artificial intelligence and/ or machine learning.

This requirement has been influenced by the application scenario that citizens of a city should be enabled to participate without any restrictions in the communal planning process.

During the elaboration  of this requirement it became clear, that a process model which is able to offer a sufficient environment for citizens is ‘as such’ capable to be a process for every assembly of experts.

Such a process is by design a self-organizing process, enabling dynamic diversity, and produces — by design — empirical theories. If you will apply to such a process, then you will inevitably produce a complete theory.

This is of some interest, but doing this there is no need to think about ‘democracy’.

But, if you start reflecting about the conditions of  a self-organizing processes enabling a dynamic diversity then you can see, that such processes need a certain type of environment without which dynamic diversity is not possible.

The main motivation behind diversity, even dynamic diversity, is the overall aspect of survival on a dynamic planet like the earth embedded in a dynamic solar system, galaxies, the universe. Biological life has succeeded about 3.5 Billion years because it has a dynamic structure enabling by design lots of innovations in advance to be prepared if the situation is changing.  With the advent of homo sapiens and the outstanding capabilities of homo sapiens all processes have been accelerating a lot thereby producing kinds of complexities completely unknown before and faster than the traditional formats of knowledge and learning could cope with. This is a radical challenge to intensify  communication, to enable radical openness and diversity, to improve communication and knowledge technologies, to format the whole society as an intensive living organism.

From all known societal organizational structures so-called democratic structures offer the most of freedom and variety to enable such processes. On the other hand it is well known  that the ‘freedom to act in diversity’ is no guarantee to do it in the right way. Democracies are not protected against failure. But seen from a principal thinking are so-called democracies the best environment for a self-organizing dynamic diversity producing optimal empirical theories. And — this is some flavor of an autopoietic system — having enabled and set up such self-organizing dynamic diversities producing optimal empirical theories is the best support for a sustaining democracy.

This is the logic behind the decision, to see a sustainable democracy as a systemic part for self-organizing dynamic diversities producing optimal empirical theories and vice versa.

The software which is offering application spaces for self-organizing dynamic diversities producing optimal empirical theories is called oksimo and is available on oksimo.com. Ongoing descriptions with examples and theoretical considerations are presented on oksimo.org. At the time of this writing — 28.Aug 2021 — this blog is nearly completely in German only because of lack of resources.

COMMENTS

[1] The website of the V-dem institute, which is a prominent place for empirical research to democracies world-wide: https://www.v-dem.net/en/

[2] Here you can find regular reports: https://www.v-dem.net/en/publications/democracy-reports/

[3] Here you can find interactive graphical user interfaces to work with the data sets : https://www.v-dem.net/en/online-graphing/

[4] Example with the concept  ‚Egalitarian Democracy‘: https://www.v-dem.net/en/news/egalitarian-democracy/

[5] Video about an workshop organized by V-dem: https://www.youtube.com/watch?v=ARBBiMgJT7A