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Geometric Dimensions in the ‘Rubber Sheet Geometry’ have a forgotten meaning

                                         (topology of shaping, not just shapes)

                                                                               For applications, see webpage Atelier>Open Letter

The known geometries 2D, 3D, 4D, 0D points of no return

• Video: Mathematician Cédric Villani  explains the reasons for inventing geometries [extracted from lecture (60mn) The Extraordinary Theorems of John Nash (at the Royal Institution, 2016)]

Villani describes the usefulness of these geometries for material endeavours of humans. He does not mention the 1D geometry that the humanities are more familiar with, because it has little relevance to historical technology and mathematics.

 It might have more relevance to prehistoric technology, such as stretching a line of view as an oriented horizontal direction that we might follow in walking and migrating, or for an arrow weapon, or a path for a thrown spear. Humanities are more aware of it because it manifested in anthropology as totem poles in tribal villages. Sometimes called ‘Axis Mundi’ in spiritual philosophies, it is often interpreted in academia as ‘phallic’ symbol. If one eliminates the conventionalisations as physical or spiritual, the 1D geometry relates to the biological creature Homo Sapiens, not so well equipped to thrive in wild environments, and having to activate its survival mechanisms, including physical, material (civilised technology) and human (societies with cultures, social complexities).

The assumed and un-challenged 1D geometry of Up-/Ex-/Out-

However, this vertical 1D geometry is based in the biology of life, but was reduced to its localisation in ‘physical’ bodies or ‘material’ bodies. Automatically, cultures applied it to ‘humans’ (as ‘not animals’). The result is a bifurcation that produces a horizontal symmetry opposing the physical and the human. This has profound implications for life in society and for health, because values are inverted. It is discussed in deep philosophy (generalist philosophy addressing cultures worldwide throughout history) because it rules the human societal behaviour of rising, advancing, expanding, spreading, ‘going’ and other increases. In this reduction to a horizontal symmetry, the ‘going’ acquired a reverse direction, of ‘return’ or shrinking. In this way it has produced two philosohies of ‘life’ for the civilised ‘human social animal’. Now psychologists find that it rules the mind [in the dominant urban culture]. (See Tversky in webpage Atelier>Open Letter to Thinkers and culture makers).

• Video (see at 32mn): Pr Barbara Tversky – Mind In Motion (2020 seminar at Stanford 57mn).     In cultures, “the direction UP is more, better, stronger, more powerful”

The problem is that in the original geometry, ‘up’ is a strategy of ‘rising to the occasion’ for a living Homo Sapiens to survive. In the topologically reduced version to a horizontal symmetry, as a ‘physical’ side to the ‘human’, the same behaviour is used to develop mind, culture, civilisation, but these actions of increase and survival behaviour of ‘selves’ become ego-centred and not so for the living being’s health.  Not to mention the centuries of conflicts due to the philosophies of advancing humans (against nature, separated from wilderness), versus ‘return to nature’,  which then appears as a backward behaviour or mind. This conflict is endemic, for example, in France, where I spent my youth, it is enacted in the eternal conflic between Parisians and country people which they consider ‘bacward’. The conflicts are not inherent in our ‘human nature’ or our ‘physical nature’; they arise from a geometrically reduced representation of the 1D axis.

Another  problem is that the 1D which, topologically is an ‘axis’ that can be ‘oriented’ in 2 ways, in a fixed geometry of shapes without shaping (change or distortion), becomes reduced to a single ‘Di-rection’, breeding single-minded pressures from those who see only 1 direction on all. Collectively, the result is that 21st century advanced cultures in developed countries no longer remember the existence or even the possibility of using the other orientation to steer away from crisis, and those who do and wish to act upon this, do not have a recognised framework to justify doing this without being judged backwords or incapable of societal success. This is absolutely crucial for health, and is an important motivation for my sharing online my work on the Animated Geometry, the more basic form the Rubber Sheet Geometry.

The 1D axis of deployment Up-/Ex-/Out manifests in many ways and is described with fundamental parameters of representation

• Bouchon (2020): The Generic axis Up-Ex-Out of deployment [PDF 522kB]

Women describing health changes or pain, and the biological origin of ‘sensed’ geometry

See the webpages  Atelier>Obstacles to Females in Health & In Society and Resource 6: Husserl

Women describe sensations of health inside the body, including pain, with more graphic language than men, typically focusing on the sensory aspect (sensations), whereas men focus on events and emotions (mental constructions about pain).

This can be related to observations that people with autism are more interested in geometric shapes than human faces, or their tendency to describe “how you feel” in terms of physical internal sensations rather than psychological emotions (i.e. the meaning of the sensations, constructed in the mind).These days, theorists trace the cognitive beginnings of geometric abstraction concepts, in the human mind, back to about 7 years old (about the time of adrenarche),  and back to ancient hisotry. But their origination, or why or how we have such images in the mind is still not clarified in scholarly works – they do not take into account the actual biological body apprehended directly (medicine only produces instrumental descriptions of a body-system of matter, not the sensations of what it feels like, not a ‘likeness’ of it). Could it be that listening to women without translating their descriptions in terms of ‘symptoms’ of ‘diseases’ (and then dismissively telling them there is no “real” or “physical” disease and “it’s all in their head”, and “must be” psychological…) could provide pointers to a more primary form of cognition of geometric nature? Small changes at surfaces such as “swelling” are not necessarily inflammation.

This is what the Rubber Sheet Geometry describes: small distortions near or at surfaces – not  large deformed “shapes” (symptoms, describable with mathematical topologies), but a fluid process of “shaping”. This “shaping” is the same fundamental process of appearance of the shapes of living animal bodies in embryology (Pr Vincent Fleury, CNRS, France) – see the webpage Atelier>Open Letter To Thinkers. It can be used to understand the “shaping” of the current global problems and current human behaviour that lets survival drives destroy the basis of viability of both planet and humans; this requires a ‘higher order’ geometry to see that survival behaviours, whethet good or bad, are all already AT LIMITS (that is why everything is now displaying boundary behaviour and the correlate risks of lost integrity in the large Trans-Formations, whose results may or may not be desirable for humans and biosphere). This geometry provides a new understanding what “we” have to do – not push survival and limits even more.

In medicine, the process of appearance of a cyst in a female body is no diffferent that of the appearance of appendages during embryological development; but an appearing cyst may also disappear, be reversible… if what needs to be done is done, rather than disimissed by a doctor. This geometry of small changes can be used as a Dimensional Geometry to describe shifts of “States of Health” – for example from a growth state to a cancerous state, and vice versa. Just 3 orders of “States of Health” could provide a modelling method to understand many immunity problems, and show graphically how to undo them, rather than forever seeking specific cures (with generalised side-effects).

The activated brain also breeds description of the body and the timing of its boundary states in health with the Animated Geometry, expressed in gesture

• Bouchon poster (2008): Thinking in Imaging VS Imagining things  [PDF  288kB]

The Animated Geometry is inherent in human biology, and sensed by some individuals whose autonomic nervous sytem is not operating typically by “balancing” horizontally sympathetic and parasympathetic, but by activating both and changing thresholds. The geometric imaging that expresses it, often in gestures, is very habitually confused with “imagining things”, with imagination, or even lying. But imagination pictures specific-general “things”, naturalistic, realistic, or surrealist or surnatural, not a generic likeness of an unfragmented situation.

The idea that women with syndromes are not believed by doctors, not heard, is directly related to this. The professionals interepret graphic and geometric descriptions of sensations, signals and signs that biological workings are going ‘off track’ (and can develop later into illness) as ‘symptoms’ of ‘disease’. When they do not find ‘physical’ disease, they automatically assume that they are facing ‘mental’ illness, and send women to psycho-therapy, denying not only treatment but even medical investigation.

Graphic & Sensory descriptions in women:

The changing “States of Health” they describe are not established “disease” but “syndromes” that are deploying dysfunction and affect the entire basic physiology, producing “somatic” changes (not ‘psycho-somatic’) such as fatigue, pain, muscle tension not well controlled, digestion, circulation, breathing, sleep, etc., and general instability of these functions under sympathetic challenge (not ‘stress’ related Cortisol, but ‘strain’ related Catecholamines: adrenaline, norepinephrine, dopamine in the Autonomic Nervous System).

Not a coincidence if studying physics breeds the topologic mind in the advanced brain

• Bouchon (2021): Geometry & Spatial-Kinesic ability (PPTX 1GB with video clips embedded: must download it for them to play) – See page Atelier>Open Letter To Thinkers

The physical world is full of geometry, and its changes are small or large, experienced as distortion, deformation, disturbance – topology is the most direct way to make sense of this, even before we go through school learning. Physics takes this to a more abstract level, but much of it is still rooted in the physical world and its small modifications and large transformations, and early schooling experiments rely on geometry (e.g. a spring going up and down and an arrow of force). Dreams and insights can be goemetric, or even purely topologic. These slides present a few striking examples of the non-local and inherent nature of the most basic form of topology in human beings.

Edmund Husserl, German philosopher (1859-1938) on the origin of geometry

see webpage Resources 6>Edmund Husserl

Henri Poincaré, French Mathematician (1854-1912) on the ‘hidden geometry’

“Pour faire de la géométrie […] quelque chose d’autre que la logique pure est nécessaire. Pour décrire ce quelque chose, nous n’avons pas d’autre mot qu’ ‘intuition’ .”  [Poincaré] [la logique pure: logique linéarire de raisonnement mathematique et causatif]

• Olga Pombo (2012). Conceptions Of Intuition in Poincaré’s Philosophy Of Mathematics. Center for Philosophy of Science of the University of Lisbon, Faculty of Sciences, University of Lisbon. Extracts.

Pombo comments:
We know that intuition is a very ambiguous word, a word which we use to designate a capacity or a faculty which we are not able to describe with sufficient accuracy.  [Bouchon proposal for the context of mathematics and geometry:  could use the word ‘in-sight’ – see Penrose videos on page Geometry]
As Poincaré states:

« Admettons même que l’on ait établi que toutes les théorèmes peuvent se déduire par des procédés purement analytiques, et que ces axiomes ne sont que des conventions. » (SM : 129).

« Le philosophe conserverait le droit de rechercher les origines de ces conventions, de voir pourquoi elles ont été jugées préférables aux conventions contraires » (SM : 129).

« Parmi toutes les constructions que l’on peut combiner avec les matériaux fournis par la logique, il faut faire un choix; le vrai géomètre fait ce choix judicieusement parce qu’il est guidé par un sûr instinct, ou par quelque vague conscience [‘vague awareness’ ] de je ne sais pas quelle géométrie plus profonde, et plus cachée, qui seule fait le prix de l’édifice construit. » (SM : 129).

« Chercher l’origine de cet instinct, étudier les lois de cette géométrie profonde qui se sentent et que ne s’énoncent pas ».

« Cet instinct dont nous venons de parler est nécessaire à l’inventeur » (SM : 130).

« Il y a une réalité plus subtile, qui fait la vie des êtres mathématiques, et qui est autre chose que la logique » (SM : 110). [autre que la logique linéaire de raisonnement et de  causalité; other than logical linear  reasoning, causal]
The dimensional logic of topology relies on the geometric dimensions, if it is not limited to counting or measuring fixed shapes in a space with a defined dimension (e.g. 3D topographies).
La logique dimensionnelle de la topologie géométrique se sert des dimensions de géométrie, si elle n’est pas limitée à compter ou mesurer des formes fixes dans un espace de dimension définie (ex. topographies en 3D).

Resources relative to flaring syndromes