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WHAT IS LIFE? Homo Sapiens Event – First Outlines

Author: Gerd Doeben-Henisch

Changelog: March 2, 2025 – March 2, 2025

Email: info@uffmm.org

TRANSLATION: The following text is a translation from a German version into English. For the translation I am using the software @chatGPT4o with manual modifications.

CONTENT TREE

This text is part of the TOPIC Philosophy of Science.

CONTEXT

This is not another interim reflection but a continuation in the main thread of the text project ‘What is Life?’

MAIN THREADS: What is Life?

  1.  Jan 17, 2025 : “WHAT IS LIFE? WHAT ROLE DO WE PLAY? IST THERE A FUTURE?”
  2.  Jan 18, 2025 : “WHAT IS LIFE? … DEMOCRACY – CITIZENS”
  3. Jan 21, 2025 : WHAT IS LIFE? … PHILOSOPHY OF LIFE
  4. Feb 10, 2025 : WHAT IS LIFE? … If life is ‘More,’ ‘much more’ …

INSERTIONS SO FAR:

  1. Feb 15, 2025 : INSERTION: A Brief History of the Concept of Intelligence and Its Future
  2. Feb 18, 2025 : INSERTION: BIOLOGICAL INTELLIGENCE NEEDS LEARNING. Structural Analysis 
  3. Feb 20, 2025 : INSERTION : INTELLIGENCE – LEARNING – KNOWLEDGE – MATERIAL CONDITIONS; AI

TRANSITION

In text No. 4, “WHAT IS LIFE? … When Life is ‘More,’ ‘Much More’ …”, there is a central passage that should be recalled here. Following the revelation of the empirically strong acceleration in the development of complexity of life on this planet, it states:

“The curve tells the ‘historical reality’ that ‘classical biological systems’ up to Homo sapiens were able to generate with their ‘previous means.’ However, with the emergence of the ‘Homo’ type, and especially with the life form ‘Homo sapiens,’ entirely new properties come into play. With the sub-population of Homo sapiens, there is a life form that, through its ‘cognitive’ dimension and its novel ‘symbolic communication,’ can generate the foundations for action at an extremely faster and more complex level.”

Following this “overall picture,” much suggests that the emergence of Homo sapiens (that is, us) after approximately 3.5 billion years of evolution, preceded by about 400 million years of molecular development, does not occur randomly. It is hard to overlook that the emergence of Homo sapiens lies almost at the “center of the developmental trajectory.” This fact can—or must?—raise the question of whether a “special responsibility” for Homo sapiens derives from this, concerning the “future of all life” on this planet—or even beyond? This leads to the second quotation from text No. 4:

“How can a ‘responsibility for global life’ be understood by us humans, let alone practically implemented by individual human beings? How should humans, who currently live approximately 60–120 years, think about a development that must be projected millions or even more years into the future?”

Such “responsibility with a view toward the future” would—from the perspective of life as a whole—only make sense if Homo sapiens were indeed the “only currently existing life form” that possesses exactly those characteristics required for “assuming responsibility” in this current phase of life’s development.

PRELIMINARY NOTE

The following text will gradually explain how all these elements are interconnected. At this stage, references to relevant literature will be kept to a minimum, as each section would otherwise require countless citations. Nevertheless, occasional remarks will be made. If the perspective presented in the “What is Life” texts proves fundamentally viable, it would need to be further refined and embedded into current specialized knowledge in a subsequent iteration. This process could involve contributions from various perspectives. For now, the focus is solely on developing a new, complex working hypothesis, grounded in existing knowledge.

THE HOMO SAPIENS EVENT

In modern science fiction novels and films, extraterrestrials are a popular device used to introduce something extraordinary to planet Earth—whether futuristic advancements or adventurous developments from the future appearing on Earth. Of course, these are thought constructs, through which we humans tell ourselves stories, as storytelling has been an essential part of human culture since the very beginning.

Against this backdrop, it is remarkable that the Homo Sapiens Event (HSE) has not yet received a comparable level of empathic attention. Yet, the HSE possesses all the ingredients to surpass even the boldest science fiction novels and films known to us. The developmental timeline on planet Earth alone spans approximately 3.9 billion years.

If we open ourselves to the idea that the biological might be understood as the direct unfolding of properties inherently present in the non-biological—and thus ultimately in energy itself, from which the entire known universe emerged—then we are dealing with a maximal event whose roots are as old as the known universe.

Ultimately—since energy remains more unknown than known to us—the HSE, as a property of energy, could even be older than the known universe itself.

IMAGE 1: Homo Sapiens Event (HSE)

PHILOSOPHICAL APPROACH

In this text, the Homo Sapiens Event (HSE) is discussed or written about because this is the only way in which the author’s brain can exchange thoughts with the brains of readers. This means that—regardless of the content—without some form of communication, there can be no exchange between different brains.

For Homo sapiens, such communication has, from the very beginning, occurred through a symbolic language, embedded in a variety of actions, gestures, facial expressions, vocal tones, and more. Therefore, it makes sense to render this mechanism of symbolic language within a human communication process transparent enough to understand when and what kind of content can be exchanged via symbolic communication.

When attempting to explain this mechanism of symbolic communication, it becomes evident that certain preconditions must be made explicit in advance—without these, the subsequent explanation cannot function.

To encompass the broadest possible perspective on the symbolic communication occurring here, the author of this text adopts the term “philosophical perspective”—in the sense that it is intended to include all known and conceivable perspectives.

Three Isolated Perspectives (Within Philosophy)

In addition to the perspective of biology (along with many other supporting disciplines), which has been used to describe the development of the biological on planet Earth up to the Homo Sapiens Event (HSE), some additional perspectives will now be introduced. These perspectives, while grounded in the biological framework, can provide valuable insights:

Empirical Neuroscience: It is concerned with the description and analysis of observable processes in the human brain.

Phenomenology: A subdiscipline of both philosophy and psychology, it serves to describe and analyze subjective experiences.

Empirical Psychology: It focuses on the description and analysis of observable human behavior.

IMAGE 2: (Hand-drawn sketch, illustrating the developmental process) Philosophical Perspective with the subdisciplines ‘Phenomenology,’ ‘(Empirical) Psychology,’ and ‘Neuroscience’

If these three perspectives are arranged side by side, the phenomenological view includes only our own (subjective) experiences, without a direct connection to the body or the world outside the body. This is the perspective with which every human is born and which accompanies them throughout life as the “normal view of things.”

From the perspective of empirical psychology, observable behavior of humans is the central focus (other life forms can also be studied in this way, though this falls more under biology). However, the phenomena of subjective experience are not accessible within the framework of empirical psychology. While the observable properties of the brain as an empirical object, as well as those of the body, are in principle accessible to empirical psychology, the empirical properties of the brain are generally assigned to (empirical) neuroscience, and those of the body to (empirical) physiology.

From the perspective of (empirical) neuroscience, the observable properties of the brain are accessible, but not the phenomena of subjective experience or observable behavior (nor the observable properties of the body).

It becomes clear that in the chosen systematic approach to scientific perspectives, each discipline has its own distinct observational domain, which is completely separate from the observational domains of the other disciplines! This means that each of these three perspectives can develop views of its object that differ fundamentally from those of the others. Considering that all three perspectives deal with the same real object—concrete instances of Homo sapiens (HS)—one must ask: What status should we assign to these three fundamentally different perspectives, along with their partial representations of Homo sapiens? Must we, in the scientific view, divide one material object into three distinct readings of Homo sapiens (HS): the HS-Phenomenal, the HS-Behavioral, and the HS-Brain?

In scientific practice, researchers are, of course, aware that the contents of the individual observational perspectives interact with one another in some way. Science today knows that subjective experiences (Ph) strongly correlate with certain brain events (N). Similarly, it is known that certain behaviors (Vh) correlate both with subjective experiences (Ph) and with brain events (N). In order to at least observe these interactions between different domains (Ph-Vh, Ph-N, N-Vh), interdisciplinary collaborations have long been established, such as Neurophenomenology (N-Ph) and Neuropsychology (N-Vh). The relationship between psychology and phenomenology is less clear. Early psychology was heavily introspective and thus hardly distinguishable from pure phenomenology, while empirical psychology still struggles with theoretical clarity today. The term “phenomenological psychology (Ph-Vh)” appears occasionally, though without a clearly defined subject area.

While there are some interdisciplinary collaborations, a fully integrated perspective is still nowhere to be found.

The following section will attempt to present a sketch of the overall system, highlighting important subdomains and illustrating the key interactions between these areas.

Sketch of the Overall System

The following “sketch of the overall system” establishes a conceptual connection between the domains of subjective experiences (Ph), brain events (N), bodily events (BDY), the environment of the body (W), and the observable behavior (Vh) of the body in the world.

IMAGE 3: (Hand-drawn sketch, illustrating the developmental process) Depicting the following elements: (1) Subjective experiences (Ph), (2) Brain events (N), (3) Bodily events (BDY), (4) Observable behavior (Vh) of the body in the world, (5) The environment of the body (W). In the lower-left corner of the image, a concrete instance of Homo sapiens (HS) is indicated, observing the world (W) along with the various bodies (BDY) of other Homo sapiens individuals. This HS can document its observations in the form of a text, using language (L).

IMAGE 3b: Hand-drawn sketch, illustrating the developmental process – The core idea for the concept of ‘Contextual Consciousness (CCONSC)’

As can be seen, the different domains are numbered from (1) to (5), with number (1) assigned to the domain of subjective experiences (Ph). This is motivated by the fact that, due to the structure of the human body, we perceive ourselves and all other events in the form of such subjective experiences as phenomena. Where these phenomena originate—whether from the brain, the body, or the surrounding world—is not directly apparent from the phenomena themselves. They are our phenomena.

While philosophers like Kant—and all his contemporaries—were still limited to considering the possible world and themselves solely from the perspective of their own phenomena, empirical sciences since around 1900 have gradually uncovered the processes behind the phenomena, localized in the brain, allowing them to be examined more concretely. Over time, increasingly precise correlations in time between brain events (N) and subjective experiences (Ph) were discovered.

One significant breakthrough was the ability to establish a temporal relationship between subjective experiences (Ph) and brain events (N). This suggested that while our subjective experiences cannot be measured directly as experiences, their temporal relationships with brain events allow for the localization of specific areas in the brain whose functioning appears to be a prerequisite for our subjective experience. This also provided a first empirical concretization of the common concept of consciousness, which can be formulated as a working hypothesis:

What we refer to as consciousness (CONSC, 1) corresponds to subjective experiences (Ph) that are enabled by brain events (N) occurring in specific areas of the brain. How exactly this can be understood will be explained further below.

The brain events (N) localized in the brain (BRAIN, 2) form a complex event space that has been increasingly researched since around 1900. It is generally clear that this space is highly dynamic, manifesting in the fact that all events interact with each other in multiple ways. The brain is structurally distinct from the rest of the body, but at the same time, it maintains exchange processes with the body (BDY, 3) and the brain’s event space (BRAIN, 2). This exchange occurs via interfaces that can (i) translate body events into brain events and (ii) translate brain events into bodily events.

Examples of (i) include our sensory organs (eyes, ears, smell), which transform light, sound, or airborne molecules into brain events. Examples of (ii) include brain events that, for instance, activate muscles, leading to movements, or regulate glandular secretions, which influence bodily processes in various ways.

The body space (BODY, 4) is approximately 450 times larger than the space of brain events. It consists of multiple regions known as organs, which have complex internal structures and interact in diverse ways. Bodily events also maintain a complex exchange with brain events.

With the surrounding world (W,5), there are two types of exchange relationships. First, (i) interfaces where bodily events appear as excretions in the event space of the world (W), and second, (ii) bodily events that are directly controlled by brain events (e.g., in the case of movements). Together, these two forms of events constitute the OUTPUT (4a) of the body into the surrounding world (W). Conversely, there is also an INPUT (4b) from the world into the body’s event space. Here, we can distinguish between (i) events of the world that directly enter the body (e.g., nutrition intake) and (ii) events of the world that, through sensory interfaces of the body, are translated into brain events (e.g., seeing, hearing).

Given this setup, an important question arises:

How does the brain distinguish among the vast number of brain events (N)—whether an event is (i) an N originating from within the brain itself, (ii) an N originating from bodily events (BDY), or (iii) an N originating—via the body—from the external world (W)?

In other words: How can the brain recognize whether a given brain event (N) is (i) N from N, (ii) N from BDY, or (iii) N from W?

This question will be addressed further with a proposed working hypothesis.

Concept of ‘Consciousness’; Basic Assumptions

In the preceding section, an initial working hypothesis was proposed to characterize the concept of consciousness: what we refer to as consciousness (CONSC, 1) pertains to subjective experiences (Ph) that are enabled by brain events (N) occurring in specific regions of the brain.

This working hypothesis will now be refined by introducing additional assumptions. While all of these assumptions are based on scientific and philosophical knowledge, which are supported by various forms of justification, many details remain unresolved, and a fully integrated theory is still lacking. The following additional assumptions apply:

  1. Normally, all phenomena that we can explicitly experience subjectively are classified as part of explicit consciousness (ECONSC ⊆ CONSC). We then say that we are aware of something.
  2. However, there is also a consciousness of something that is not directly correlated with any explicit phenomenon. These are situations in which we assume relationships between phenomena, even though these relationships themselves are not experienced as phenomena. Examples include:
    • Spoken sounds that refer to phenomena,
    • Comparative size relations between phenomena,
    • Partial properties of a phenomenon,
    • The relationship between current and remembered phenomena,
    • The relationship between perceived and remembered phenomena.
      This form of consciousness that exists in the context of phenomena but is not itself a phenomenon will be referred to here as contextual consciousness (CCONSC ⊆ CONSC). Here, too, we can say that we are aware of something, but in a somewhat different manner.
  3. This distinction between explicit consciousness (ECONSC) and contextual consciousness (CCONSC) suggests that the ability to be aware of something is broader than what explicit consciousness alone implies. This leads to the working hypothesis that what we intuitively call consciousness (CONSC) is the result of the way our brain operates.

Basic Assumptions on the Relationship Between Brain Events and Consciousness

Given today’s neuroscientific findings, the brain appears as an exceedingly complex system. For the considerations in this text, the following highly simplified working hypotheses are formulated:

  1. Empirical brain events are primarily generated and processed by specialized cells called neurons (N). A neuron can register events from many other neurons and generate exactly one event, which can then be transmitted to many other neurons. This output event can also be fed back as an input event to the generating neuron (direct feedback loops). Time and intensity also play a role in the generation and transmission of events.
  2. The arrangement of neurons is both serial (an event can be transmitted from one neuron to the next, and so on, with modifications occurring along the way) and hierarchical (layers exist in which events from lower layers can be represented in a compressed or abstracted form in higher layers).

From this, the basic assumptions about the relationship between brain events and conscious events are as follows:

  1. Some brain events become explicitly conscious phenomena (ECONSC).
  2. Contextual consciousness (CCONSC) occurs when a network of neurons represents a relationship between different units. The relationship itself is then consciously known, but since a relationship is not an object (not an explicit phenomenon), we can know these relationships, but they do not appear explicitly as phenomena (e.g., the explicit phenomenon of a “red car” in text and the perceptual object of a “red car”—we can know the relationship between them, but it is not explicitly given).
  3. The concept of consciousness (CONSC) thus consists at least of explicit phenomenal consciousness (ECONSC) and contextual consciousness (CCONSC). A more detailed analysis of both the phenomenal space (Ph) and the working processes of the brain (N) as the domain of all brain events will allow for further differentiation of these working hypotheses.

After these preliminary considerations regarding the different event spaces in which a Homo sapiens (HS) can participate through different access modalities (W – BDY – N(CONSC)), the following section will provide an initial sketch of the role of language (L) (with further elaborations to follow).

Descriptive Texts

As previously indicated, within each of the listed observational perspectives—observable behavior (Vh), subjective experiences (Ph), and brain events (N) (see IMAGE 2)—texts are created through which actors exchange their individual views. Naturally, these texts must be formulated in a language that all participants can understand and actively use.

Unlike everyday language, modern scientific discourse imposes minimal requirements on these texts. Some of these requirements can be described as follows:

  1. For all linguistic expressions that refer to observable events within the domain of a given perspective, it must be clear how their empirical reference to a real object can be verified intersubjectively. In the verification process, it must be possible to determine at least one of the following: (i) It applies (is true), (ii) It does not apply (is false), (iii) A decision is not possible (undetermined)
  2. It must also be clear: (i) Which linguistic expressions are not empirical but abstract, (ii) How these abstract expressions relate to other abstract expressions or empirical expressions, (iii) To what extent expressions that are themselves not empirical can still be evaluated in terms of truth or falsehood through their relationships to other expressions

How these requirements are practically implemented remains, in principle, open—as long as they function effectively among all participating actors.

While these requirements can, in principle, be fulfilled within the perspective of empirical psychology and neuroscience, a phenomenological perspective cannot fully meet at least the first requirement, since the subjective phenomena of an individual actor cannot be observed by other actors. This is only possible—and even then, only partially—through indirect means.

For example, if there is a subjective phenomenon (such as an optical stimulus, a smell, or a sound) that correlates with something another actor can also perceive, then one could say: “I see a red light,” and the other actor can assume that the speaker is seeing something similar to what they themselves are seeing.

However, if someone says, “I have a toothache,” the situation becomes more difficult—because the other person may never have experienced toothache before and therefore does not fully understand what the speaker means. With the vast range of bodily sensations, emotions, dreams, and other subjective states, it becomes increasingly challenging to synchronize perceptual content.

The Asymmetry Between Empirical and Non-Empirical Perspectives

This indicates a certain asymmetry between empirical and non-empirical perspectives. Using the example of empirical psychology and neuroscience, we can demonstrate that we can engage empirically with the reality surrounding us—yet, as actors, we remain irreversibly anchored in a phenomenological (subjective) perspective.

The key question arises: How can we realize a transition from the inherently built-in phenomenological perspective to an empirical perspective?

Where is the missing link? What constitutes the possible connection that we cannot directly perceive?

Referring to IMAGE 3, this question can be translated into the following format: How can the brain recognize whether a given brain event (N) originates from
(i) another brain event (N from N),
(ii) a bodily event (N from BDY),
(iii) an external world event (N from W)?

This question will be explored further in the following sections.

Outlook

The following text will provide a more detailed explanation of the functioning of symbolic language, particularly in close cooperation with thinking. It will also illustrate that individual intelligence unfolds its true power only in the context of collective human communication and cooperation.

abstract concept, abstract expression, abstraction, abstraction process, Actor, actor - biological, Actor - human, Actor - non-human, behavior, Bourbaki, brain, everyday language, Formal Language, formal theory, formalization, language, language - everyday L0, language - internal, language - ordinary, mathematical language, mathematical objects, mathematical reasoning, mathematics, meta language, metamathematics, object language, theory, theory of sets, Translation

Is Mathematics a Fake? No! Discussing N.Bourbaki, Theory of Sets (1968) – Introduction

eJournal: uffmm.org, ISSN 2567-6458,
6.June 2022 – 13.June 2022, 10:30h
Email: info@uffmm.org
Author: Gerd Doeben-Henisch
Email: gerd@doeben-henisch.de

SCOPE

In the uffmm review section the different papers and books are discussed from the point of view of the oksimo paradigm, which is embedded in the general view of a generalized ‘citizen science’ as a ‘computer aided sustainable applied empirical theory’ (CSAET). In the following text the author discusses the introduction of the book “Theory of Sets” from the series “Elements of Mathematics” by N.Bourbaki (1968) [1b]

CONTEXT

In the foundational post with the title “From SYSTEMS Engineering to THEORY Engineering” [3] the assumptions of the whole formalization approach in logic, mathematics and science are questioned as to narrow to allow a modern sustainable theory of science dealing explicitly with the future. To sharpen some of the arguments in that post it seems to be helpful to discuss one of the cornerstones of modern (formalized) mathematics substantiated in the book ‘Theory of sets’ from the Bourbaki group.[1a] It has to be mentioned that the question of the insufficiency of formalization has been discussed in the uffmm blog in several posts before. (cf. e.g. [2])

Formalization

preface

In the introduction to the ‘Set Theory Book’ the bourbaki group reveals a little bit of their meta-mathematical point of view, which finally belongs to the perspective of philosophy. At the one hand they try to be ‘radically formal’, but doing this they notice themselves that this is — by several reasons — only a ‘regulative idea’, somehow important for our thinking, but not completely feasible. This ‘practical impossibility’ is not necessarily a problem as long as one is conscious about this. The Bourbaki group is conscious about this problem, but different to their ‘rigor’ with the specialized formalization of mathematical ideas, they leave it widely ‘undefined’ what follows from the practical impossibility of being ‘completely rigorous’. In the following text it will be tried to describe the Bourbaki position with both dimensions: the idea of ‘formalization’ and the reality of ‘non-formalized realities’ which give the ‘ground’ for everything, even for the formalization. Doing this it will — hopefully — become clear that the idea of formalization was a great achievement in the philosophical and scientific thinking but it did not really solve our problems of understanding the world. The most important aspects of knowledge are ‘outside’ of this formalization approach, and many ‘problems’ which seem to bother our actual thinking are perhaps only ‘artifacts’ of this simplified formalization approach (somehow similar to the problems which have been induced by the metaphysical thinking of the older philosophy). To say it flatly: to introduce new names for old problems does not necessarily solve problems. It enables new ways of speaking and perhaps some new kinds of knowledge, but it does not really solve the big problems of knowledge. And the biggest problem of knowledge is — perhaps — the primary ‘knowledge machine’ itself: the biological actors which have brains to transform ‘reality’ in ‘virtual models’ in their brains and communication tools to ‘communicate’ these virtual models to enable a ‘collective intelligence’ as well as a ‘collective cooperation’. As long as we do not understand this we do not really understand the ‘process of knowing’.

before formalization

With the advent of the homo sapiens population on the planet earth about 300.000 years ago [4] it became possible that biological systems could transform their perceptions of the reality around their brains into ‘internal’, ‘virtual’ models, which enabled ‘reference points’ for ‘acting’ and a ‘cooperation’ which was synchronized by a ‘symbolic communication’. Those properties of the internal virtual models which have no clear ‘correspondence’ to the ‘reality between the brains’ are difficult to communicate.

Everyday symbolic communication refers to parts of the reality by certain types of expressions, which are ‘combined’ in manners which encode different types of ‘relations’ or even ‘changes’. Expressions which ‘refer’ to ‘concrete’ properties can be ‘overloaded’ by expressions which refer to other expressions, which in turn refer either again to expressions or to ‘concrete meanings’. Those objects which are the targets of a referring relation — concrete objects or other expressions — are here called ‘the meaning’ of the expressions. Thus the ‘meaning space’ is populated by either expressions related to ‘concrete’ properties or by ‘expressions pointing forward’ to other expressions and these ‘pointing-forward’ expressions are here called ‘abstract meaning’. While concrete meanings are usually ‘decidable’ in the everyday world situations as being ‘given’ (being ‘true’) or as ‘not being given’ (‘being false’), abstract meanings are as expressions ‘undefined’: they can lead to some concrete property which in turn perhaps can be decided or not.

The availability of ‘abstract expressions’ in ordinary language can be seen as a ‘problem’ or as a ‘blessing’. Being able to generate and use abstract terms manifests a great flexibility in talking — and thinking! — about possible realities which allow to overcome the dictatorship of the ‘now’ and the ‘individual single’. Without abstraction thinking would indeed be impossible. Thus if one understands that ‘thinking’ is a real process with sequences of different states which reveal eventually more abstract classes, structures, and changes, then abstraction is the ‘opener’ for more reality, the ‘enabler’ for a more broader and flexible knowledge. Only by ‘transcending’ the eternal ‘Now’ we get an access to phenomena like time, changes, all kinds of dynamics, and only thereby are ‘pictures of some possible future’ feasible!

Clearly, the potential of abstraction can also be a source of ‘non-real’ ideas, of ‘fantastic’ pictures, of ‘fake news’ and the like.

But these possible ‘failures’ — if they will be ‘recognized’ as failures! — are inevitable if one wants to dig out some ‘truth’ in the nearly infinite space of the unknown. Before the ‘knowledge that something is true’ one has to master a ‘path of trial and error’ consuming ‘time’ and ‘resources’.

This process of creating new abstract ideas to guide a search in the space of the unknown is the only key to find besides ‘errors’ sometimes some ‘truth’.

Thus the ‘problem’ with abstract ideas is an unavoidable condition to find necessary ‘truths’. Stepping back in the face of possible problems is no option to survive in the unknown future.

the formal view of the world according to bourbaki

Figure 1: Graphical interpretation of N.Bourbaki, Set Theory (1968), Introduction, ‘liberal version’
Language, object language, meta language

Talking about mathematical objects with their properties within an ordinary language is not simple because the expressions of an ordinary language are as such usually part of a network of meanings, which can overlap, which can be fuzzy, which are giving space for many interpretations. Additionally, that which is called a ‘mathematical object’ is not a kind of an object wich is given in the everyday world experience. What can be done in such a situation?

Bourbaki proposes to introduce a ‘specialized language’ constructed out of a finite set of elements constituting the ‘alphabet’ of a new language, together with ‘syntactical rules’, which describe how to construct with the elements of the alphabet chains of elements called ‘(well formed) expressions’, which constitute the ‘language’ LO, which shall be used to talk about mathematical objects.

But because mathematics is not restricted to ‘static objects’ but deals also with ‘transformations’ (‘changes’) of objects, one needs ‘successions of objects’ (‘sequences’), which are related by ‘operations with mathematical objects’. In this case the operations are also represented by ‘expressions’ but these expressions are expressions of a ‘higher order’ which have as referenced subject those expressions which are representing objects . Thus, Bourbaki needs right from the beginning two languages: an ‘object language’ (expressions of a language LO representing mathematical objects) and a ‘meta language’ LL (expressions referring to expressions of the object language LO including certain ‘types of changes’ occurring with the object language expressions). Thus a mathematical language Lm consists in the combination of an object language LO with a meta language LL (Lm = (LO,LL)).

And, what becomes clear by this procedure, to introduce such a kind of mathematical language Lm one needs another language talking about the mathematical language Lm, and this is either the everyday (normal) language L, which is assumed to be a language which everybody can ‘understand’ and ‘apply correctly’, or it is a third specialized language LLL, which can talk with special expressions about the mathematical language Lm. Independent of the decision which solution one prefers, finally the ordinary language L will become the meta language for all other thinkable meta languages.

Translating(?) math objects into formal expressions

If the formalized expressions of the mathematical language (Lm = (LO,LL)) would be the mathematical objects themselves, then mathematics would consist only of those expressions. And, because there would be no other criteria available, whatever expressions one would introduce, every expression would claim to be a relevant mathematical expression. This situation would be a ‘maximum of non-sense’ construct: nothing could be ‘false’.

Thus, the introduction of formal expressions of some language alone seems to be not enough to establish a language which is called a ‘mathematical’ language Lm different from other languages which talk about other kinds of objects. But what could it be which relates to ‘specific math objects’ which are not yet the expressions used to ‘refer’ to these specific math objects?

Everybody knows that the main reason for to ‘speak’ (or ‘write’) about math specific objects are humans which are claiming to be ‘mathematicians’ and which are claiming to have some ‘knowledge’ about specific objects called ‘math objects’ which are the ‘content’ which they ‘translate’ into the expressions of a certain language call ‘mathematical language’.[5] Thus, if the ‘math objects’ are not the used expressions themselves then these ‘math objects’ have to be located ‘inside of these talking humans’. According to modern science one would specify this ‘inside’ as ‘brain’, which is connected in complex ways to a body which in turn is connected to the ‘outside world of the body’. Until today it is not possible to ‘observe’ directly math objects assumed to be in the brain of the body of someone which claims to be a mathematician. Thus one mathematician A can not decide what another mathematician B has ‘available in his brain’ at some point of time.

Bourbaki is using some formulations in his introduction which gives some ‘flavor’ of this ‘being not able to translate it into a formalized mathematical language’. Thus at one position in the text Bourbaki is recurring to the “common sense” of the mathematicians [6] or to the “reader’s intuition”. [7] Other phrases refer to the “modes of reasoning” which cannot be formalized [8], or simply to the “experience” on which “the opinion rests”. [9] Expressions like ‘common sense’, ‘intuition’, ‘modes of reasoning’, and ‘experience’ are difficult to interpret. All these expressions describe something ‘inside’ the brains which cannot be observed directly. Thus, how can mathematician A know what mathematician B ‘means’ if he is uttering some statement or writes it down? Does it make a difference whether a mathematician is a man or a woman or is belonging to some other kind of a ‘gender’? Does it make a difference which ‘age’ the mathematician has? How ‘big’ he is? Which ‘weight’ he has?

Thus, from a philosophical point of view the question to the specific criteria which classify a language as a ‘mathematical language’ and not some other language leads us into a completely unsatisfying situation: there are no ‘hard facts’ which can give us a hint what ‘mathematical objects’ could be. What did we ‘overlook’ here? What is the key to the specific mathematical objects which inspired the brains of many many thousand people through centuries and centuries? Is mathematics a ‘big fake’ or is there more than this?

A mathematician as an ‘actor’?
Figure 2: Graphical interpretation of N.Bourbaki, Set Theory (1968), Introduction, ‘Actor view’

This last question “Is mathematics a ‘big fake’ or is there more than this?” can lead o the assumption, that it is not enough to talk about ‘mathematics’ by not including the mathematician itself. Only the mathematician is that ‘mysterious source’ of knowledge, which seems to trigger the production of ‘mathematical expressions’ in speaking or writing (or drawing). Thus a meta-mathematical — and thereby philosophical’ — ‘description’ of mathematics should have at least the ‘components’ (MA, LO,LL,L) with ‘MA’ as abbreviation for the set of actors where each element of the set MA is a mathematician, and — this is decisive ! — it is this math actor MA which is in possession of those ‘criteria’ which decide whether an expression E belongs the ‘mathematical language Lm‘ or not.

The phrase of the ‘mathematician’ as a ‘mysterious source of knowledge’ is justified by an ’empirical observational point of view’: nobody can directly look into the brain of a mathematician. Thus the question of what an expression called ‘mathematical expression’ can ‘mean’ is in such an empirical view not decidable and appears to be a ‘mystery’.

But in the reality of everyday life we can observe, that every human actor — not only mathematicians — seems to be able to use expressions of the everyday language with referring to ‘internal states’ of the brain in a way which seems to ‘work’. If we are talking about ‘pain with my teeth’ or about ‘being hungry or thirsty’ or ‘having an idea’ etc. then usually other human actors seem to ‘understand’ what one is uttering. The ‘evidence’ of a ‘working understanding’ is growing up by the ‘confirmation of expectations’: if oneself is hungry, then one has a certain kind of ‘feeling’ and usually this feeling leads — depending from the cultural patterns one is living in — to a certain kind of ‘behavior’, which has — usually — the ‘felt effect’ of being again ‘not hungry’. This functional relation of ‘feeling to be hungry’, ‘behavior of consuming something’, ‘feeling of being not hungry again’ is an association between ‘bodily functions’ common to all human actors and additionally it is a certain kind of observable behavior, which is common to all human actors too. And it seems to work that human actors are able to associate ‘common internal states’ with ‘common observable behavior’ and associate this observable behavior with the ‘presupposed internal states’ with certain kinds of ‘expressions’. Thus although the internal states are directly not observable, they can become ‘communicated by expressions’ because these internal states are — usually — ‘common to the internal experience of every human actor’.

From this follows the ‘assumption’ that we should extend the necessary elements for ‘inter-actor communication’ with the factor of ‘common human actor HA experience’ abbreviated as ‘HAX‘ (HA, HAX, MA, LO,LL,L), which is usually accompanied by certain kinds of observable behavior ‘BX‘, which can be used as point of reference for certain expressions ‘LX‘, which ‘point to’ the associated intern al states HAX, which are not directly observable. This yields the structure (HA, HAX, MA, BX, LO,LL,LX,L).

Having reached this state of assumptions, there arises an other assumption regarding the relationship between ‘expressions of a language’ — like (LO,LL,LX,L) — and those properties which are constituting the ‘meaning’ of these expressions. In this context ‘meaning’ is not a kind of an ‘object’ but a ‘relation’ between two different things, the expressions at one side and the properties ‘referred to’ on the other side. Moreover this ‘meaning relation’ seems not to be a ‘static’ relation but a ‘dynamic’ one, associating two different kinds of properties one to another. This reminds to that what mathematicians call a ‘mapping, a function’, and the engineers a ‘process, an operation’. If we abbreviate this ‘dynamic meaning relation’ with the sign ‘μ’, then we could agree to the convention ‘μX : LX <—> (BX,HAX)’ saying that there exists a meaning function μX which maps the special expressions of LX to the special internal experience HAX, which in turn is associated with the special behavior BX. Thus, we extend our hypothetical structure to the format (HA, HAX, MA, BX, LO,LL,LX,L,μX).

With these assumptions we are getting a first idea how human actors in general can communicate about internal, directly not observable states, with other human actors by using external language expressions. We have to notice that the assumed dynamic meaning relation μX itself has to be located ‘inside’ the body, inside’ the brain. This triggers the further assumption to have ‘internal counterparts’ of the external observable behavior as well as external expressions. From this follows the further assumption that there must exists some ‘translation/ transformation’ ‘τ’ which ‘maps’ the internal ‘counterparts’ of the observable behavior and the observable expressions into the external behavior.(cf. figure 2) Thus, we are reaching the further extended format: (HA, HAX, MA, BX, LO,LL,LX,L,μX,τ).

Mathematical objects

Accepting the basic assumptions about an internal meaning function μX as well an internal translation function τ narrows the space of possible answers about the nature of ‘math objects’ a little bit, but as such this is possibly not yet a satisfying answer. Or, have we nevertheless yet to assume that ‘math objects’ and related ‘modes of reasoning’ are also rooted in internal properties and dynamics of the brain which are ‘common to all humans’?

If one sees that every aspect of the human world view is encoded in some internal states of the brain, and that what we call ‘world’ is only given as a constructed virtual structure in the brains of bodies including all the different kinds of ‘memories’, then there is no real alternative to the assumption that ‘math objects’ and related ‘modes of reasoning’ have to be located in these — yet not completely decoded — inner structures and dynamics of the brain.

From the everyday experience — additionally enlightened by different scientific disciplines, e.g. experimental (neuro) psychology — we know that the brain is — completely automatic — producing all kinds of ‘abstractions’ from concrete ‘perceptions’, can produce any kinds of ‘abstractions of abstractions’, can ‘associate’ abstractions with other abstractions, can arrange many different kinds of memories to ‘arrangements’ representing ‘states’/ ‘situations’, ‘transformations of states’, ‘sequences of states’ and the like. Thus everything which a ‘mathematical reasoning’ HAm needs seems to be available as concrete brain state or brain activity, and this is not only ‘special’ for an individual person alone, it is the common structure of all brains.

Therefore one has to assume that the set of mathematicians MA is a ‘subset’ of the set of human actors HA in general. From this one can further assume that the ‘mathematical reasoning’ HAm is a subset of the general human everyday experience HAX. And, saying this, the meaning function μX as well as the translation function τ should be applicable also to the mathematical reasoning and the mathematical objects as well: (HA, MA, HAX, HAm, BX, LO,LL,LX,L,μX,τ).

These assumptions would explain why it is not only possible but ‘inevitable’ to use the everyday language L to introduce and to use a mathematical language Lm with different kinds of sub-languages (LO,LL, LLL, …). Thus in analogy to the ‘feeling’ ‘to be hungry’ with a cultural encoded kind of accompanying behavior BX we have to assume that the different kinds of internal states and transformations in the case of mathematical reasoning can be associated with an observable kind of ‘behavior’ by using ‘expressions’ embedded (encoded) in certain kinds of ‘behavior’ accepted as ‘typical mathematical’. Introducing expressions like ‘0’, ‘1’, ‘2’, …, ’10’, … (belonging to a language Lo) for certain kinds of ‘objects’ and expressions like ‘+’, ‘-‘ … for certain kinds of operations with these before introduced objects (belonging to a language LL) one can construct combined expressions like ‘1+2=3’ (belonging to a mathematical language Lm). To introduce ‘more abstract objects’ like ‘sets’, ‘relations’, ‘functions’ etc. which have no direct counterpart in the everyday world does not break the assumptions. The everyday language L operates already only with abstract objects like ‘cup’, ‘dog’, ‘house’ etc. The expression ‘cup’ is an abstract concept, which can easily be associated with any kind of concrete phenomena provided by perceptions introducing ‘sets of different properties’, which allow the construction of ‘subsets of properties’ constituting a kind of ‘signature’ for a certain abstract ‘class’ which only exists in the dynamics of the brain. Thus having a set C named ‘cup’ introduces ‘possible elements’, whose ‘interpretation’ can be realized by associating different kinds of sets of properties provided by ‘sensory perception’. But the ‘memory’ as well as the ‘thinking’ can also provide other kinds of properties which can be used too to construct other ‘classes’.

In this outlined perspective of brain dynamics mathematics appears to be a science which is using these brain dynamics in a most direct way without to recur to the the huge amount of everyday experiences. Thus, mathematical languages (today summarized in that language, which enables the so called ‘set theory’) and mathematical thinking in general seems to reflect the basic machinery of the brain processing directly across all different cultures. Engineers worldwide speaking hundreds of different ‘everyday languages’ can work together by using mathematics as their ‘common language’ because this is their ‘common thinking’ underlying all those different everyday languages.

Being a human means being able to think mathematically … besides many other things which characterizes a human actor.

Epiloge

Basically every ordinary language offers all elements which are necessary for mathematics (mathematics is the kernel of every language). But history illustrates that it can be helpful to ‘extend’ an ordinary language with additional expressions (Lo, LL, LLL, …). But the development of modern mathematics and especially computer science shows increasing difficulties by ‘cutting away’ everyday experience in case of elaborating structures, models, theories in a purely formal manner, and to use these formal structures afterwards to interpret the everyday world. This separates the formal productions from the main part of users, of citizens, leading into a situation where only a few specialist have some ‘understanding’ (usually only partially because the formalization goes beyond the individual capacity of understanding), but all others have no more an understanding at all. This has he flavor of a ‘cultural self-destruction’.

In the mentioned oksimo software as part of a new paradigm of a more advanced citizen science this problem of ‘cultural self-destruction’ is avoided because in the format of a ‘computer aided sustainable applied empirical theory (CSAET) the basic language for investigating possible futures is the ordinary language which can be extend as needed with quantifying properties. The computer is not any more needed for ‘understanding’ but only for ‘supporting’ the ‘usage’ of everyday language expressions. This enables the best of both worlds: human thinking as well as machine operations.

COMMENTS

Def: wkp := Wikipedia; en := English

[1a] Bourbaki Group, see: https://en.wikipedia.org/wiki/Nicolas_Bourbaki

[1b] Theory of Sets with a chapter about structures, see: https://en.wikipedia.org/wiki/%C3%89l%C3%A9ments_de_math%C3%A9matique

[2] Gerd Doeben-Henisch, “Extended Concept for Meaning Based Inferences, Version 1”, August 2020, see: https://www.uffmm.org/wp-content/uploads/2020/08/TruthTheoryExtended-v1.pdf

[3] Gerd Doeben-Henisch, “From SYSTEMS Engineering to THEORY Engineering”, June 2022, see: https://www.uffmm.org/2022/05/26/from-systems-engineering-to-theory-engineering/

[4] Humans, How many years ago?, see: wkp (en): https://en.wikipedia.org/wiki/Human#:~:text=Homo%20sapiens,-Linnaeus%2C%201758&text=Anatomically%20modern%20humans%20emerged%20around,local%20populations%20of%20archaic%20humans.

[5] The difference between ‘talking’ about math objects and ‘writing’ is usually not thematised in the philosophy of mathematics. Because the ordinary language is the most general ‘meta language’ for all kinds of specialized languages and the primary source of the ordinary language then ‘speaking’ is perhaps not really a ‘trivial’ aspect in understanding the ‘meaning’ of any kind of language.

[6] “But formalized mathematics cannot in practice be written down in full, and therefore we must have confidence in what might be called the common sense of the mathematician …”. (p.11)

[7] “Sometimes we shall use ordinary language more loosely … and by indications which cannot be translated into formalized language and which are designed to help the reader to reconstruct the whole text. Other passages, equally untranslatable into formalized language, are introduced in order to clarify the ideas involved, if necessary by appealing to the reader’s intuition …”(p.11)

[8] “It may happen at some future date that mathematicians will agree to use modes of reasoning which cannot be formalized in the language described here : it would then be necessary, if not to change the language completely, at least to enlarge its rules of syntax.”(p.9)

[9] “To sum up, we believe that mathematics is destined to survive … but we cannot pretend that this opinion rests on anything more than experience.”(p.13)