# The Geometric Mind…

## Spatial Structures of Thought and Meaning Space

“Geometry is the knowledge of the eternally existent”…

- Pythagoras

“Geometry is the archetype of the beauty of the world” …

- Johannes Kepler

## Geometry

## /dʒɪˈɒmɪtri/

## the branch of mathematics concerned with the properties and relations of points, lines, surfaces, solids, and higher dimensional analogues

## Concepts

## /ˈkɒnsɛpt/

## abstract ideas or general notions that occur in the mind, in speech, or in thought

## the fundamental building blocks of thoughts and beliefs

In a Lecture — *The Geometry of Thinking* [1] — delivered in June 2016 at the *Copernicus Centre for Interdisciplinary Studies[2], *Swedish Professor of Cognitive Scientist* and *Philosopher* Peter Gärdenfors[3] *begins by outlining the two approaches that have been pursued to make *sense *of the nature of the *Human Mind*.

One is an *Explanatory[1] *approach* *— our attempts to understand our *Mental World* through theories about different parts of *cognition*.

The other is a *Constructivist[1] *approach *— *an* *engineering perspective — our attempts to create robots, chess-playing systems, communication and intelligent computational machines.

*Peter *was particularly interested in a critical explanatory question.

How can we explain how people, particularly children, learn new concepts so quickly[1]?

Unlike our current *Deep Learning* models advances that emerged through a combination of more & more data, computation and chip processing power — *Rich Sutton’s *— *Bitter Lesson[4]* — Humans have an innate ability to learn from very few examples.

A capacity to navigate *emergence[5] *&* knightian uncertainty[6] in our complex[7] Material World *through *novelty, open inquiry*, *creativity*,* reflexivity[8] *and *learning*.

The capacity to steer a course through a *quantum world* of *Clocks *and* Clouds.*

“If significant uncertainty is involved, go with the human. They may have inferior pattern recognition capabilities (versus models trained on enormous amounts of data), but they understand what they do, they can reason about it, and they can improvise when faced with novelty[9]”…

— Francois Chollet, Senior AI Researcher, Google and Founder, Keras

This idea was explored in — *Interdependent… — and — The Semantic Mind…*

Our remarkable innate capacity to bring a coherent, rich perspective of *meaning *to a *high dimensional *complex Reality remains one of our greatest Human gifts.

# The Geometric Mind

*Peter Gärdenfors *looks at the same phenomena through a different prism.

*The Geometric Mind*

A lens that sits between a *symbolic* approach to cognition — the *Algorithmic Mind* — the symbolic manipulation through *computation* & *reductionism* — and — *Connectionism *— models associated with relationships & interdependencies & meaning — the *Semantic Mind*.

The *Geometric Mind *is anchored in how we *conceptually *represent our World in *high dimensional* *abstraction*.

*Geometry*, *Ontology,* *Epistemology *and* Axiology.*

Through *Conceptual Spaces[10]* we map *concepts *and *objects*, thereby organising our knowledge of the *World*.

Via notions of *similarity *and principles of *Symmetry *we can bring *coherence *to our Reality.

New *concepts *can be introduced into these *landscapes *through mapping their relationships & connections — *similarity — nearness* — and — *betweenness — convexity — *thereby helping us *learn how to learn.*

*Concepts *— the *How* and *What? *— are not fixed sets of *domains*.

*Categorisations* such as* Ontology *and *Epistemology *at times can be dynamic & fluid — *emergent*.

*Concepts* are represented in* high dimensional geometric structures* enabling us to form *prototypes*.

We learn through *concepts *and *reflecting *on our *Material World experiences.*

“Teaching means creating situations where structures can be discovered”…

— Jean Piaget

“We do not

learnfrom experience… welearnfromreflectingon experience”…— John Dewey

In *Charles* and* Ray Eames* film — *The Power of Ten[11], *completed in 1968 — they present the relative magnitude of the Universe through a *logarithmic scale.*

By changing the dimensionality we changed our perception of Reality.

More is Different[12]…

A similar analogy can be applied to our *Inner World — *our *Mental World*.

Our *Geometric Mind lets *us to *zoom in *and *zoom out* of these *cognitive landscapes*.

*Philipp Otto Runge’s [13] *1810 *Farbenkugel* — colour sphere — and — *Ostwald’s 1916 Colour Solid System[14]* — illustrate that through *Geometry, *we can bring *coherence* to *Complexity*.

The evolution of *Mathematics* — the Science of uncovering *Patterns[15] *in our complex *Material World *— is an example of applying similar principles in areas such as *Graph Theory*, *Group Theory *and* Geometry.*

# The extension of the Geometric Mind beyond the Body — the emergence of Geometry and the application of universal principles of Symmetry to a Classical World

On 27 April 2021, *Michael Bronstein*, *Joan Bruna,* *Taco Cohen* and *Petar Velickovic *published an Academic paper and *proto-book *titled G*eometric Deep Learning: Grids, Groups, Graphs, Geodiscs, and Gauges[16].*

It complimented *Michael Bronstein’s *talk titled — *Geometric Deep Learning: Past, Present, And Future [17] — *at *UCL Centre for Artificial Intelligence *on 11 February 2021, Petar *Velickovic’s [18] talk at Cambridge University on 22 February 2021 and Michael Bronstein’s keynote at ICLR 2021[19].*

*Michael *begins the *UCL Centre for Artificial Intelligence *lecture by providing a brief history of Geometry, beginning with *Euclid[20]*.

An expanded history of *Geometry* leading up to their new frameworks for *Geometric Deep Learning* follows.

**Euclidian Geometry, Algebra & the emergence of a deductive system of Formal Logic**

In ~300BC the Greek Philosopher *Euclid[20] *brought together a work titled — *Elements[21]* — that synthesised and built upon the prior work of *Eudoxus[22]] *and *Plato[23].*

*Euclid’s *method was based on a set of *Axioms* from which a range of *propositional theorems *could be deduced.

In doing so, he demonstrated how these propositions could be integrated into a *deductive system of logic*.

*Plane Geometry[24] *and *mathematical proofs* represented the foundational *Axioms* which, through a *language of abstraction[25] *emerged the basis of* algebra[26]*.

The book & its concepts changed the course of human history.

Providing the foundations for *Human Reason* and* Logic*, which shaped the next 2,000 years of philosophy, art, literature, mathematics and Science.

It provided the seeds for the *Age of Enlightenment*, the *Age of Reason* & drove the *Scientific*,* Industrial *and *Digital Revolutions*.

Providing a capacity to untangle* clocks *from *clouds* in an emergent complex *Material World*.

A capacity to derive o*rder from chaos, *the *regular from the random *and uncover *patterns *and s*ymmetries *in our *World*.

Integrating our *Mental World* and *Material World*.

**The emergence of Non-Euclidian Geometry & the integration of Geometry into the Science of Symmetry — uncovering a universal truth**

In the 19th Century, there was a blossoming in the discipline of new forms of *Geometry*.

New types of relationships of points, lines, surfaces and solids in high dimensional space.

Through thinking of a curved 2 dimensional surface such as a globe or the earth *Carl Friedrich Gauss[27] *uncovered a contradiction in Euclid’s *parallel postulate[28]*.

Up until that time *Euclid’s *parallely postulate stated:

“If a line segment intersects two straight lines forming two interior angles on the same side that sum to less than two right angles, then the two lines, if extended indefinitely, meet on that side on which the angles sum to less than two right angles[29]”…

*Gauss[27], *in thinking about our planet earth, recognised that vertical lines of longitude all make angles of 90 degrees with the equator on the globe’s surface, yet by the time they reach the poles, they have met.

So the parallel postulate is incorrect on curved surfaces.

It was the birth of *Non-Euclidian Geometry[30], *and *Gauss’s [27] *breakthrough resulted in a range of rapid breakthroughs, including *Nikolai Lobachevsky[31]* and *Hyperbolic Geometry*, *János Bolyai[32] *and *Absolute Geometry*(included both *Hyperbolic* and *Euclidian*) and *Bernhard Riemann[33] *and *Elliptic* (*Riemannian*) geometry.

Given the splintering of *Geometry* into both *Euclidian[34] *and multiple *Non-Euclidian [30]* forms, a fundamental question arose as to whether these emergent ideas remained consistent with *Euclid’s Elements — *systems of *Logic *and *Reason *which transcended beyond physical shapes in our *Material World*?

Were there unifying principles that could integrate Hyperbolic, Euclidian & Elliptical Geometry?

It was *Felix Klein[35]* that answered this question in 1872, proposing to define *Geometry* as the study of *invariance *— *symmetries[36]*.

Properties are preserved under some classes of *transformation*.

A new *semantic mathematical language* of *Group Theory[37]* formalised these classes of transformations.

*Geometry* could now be commonly defined.

Another language of *Semantic Abstraction[25]* that could be used to uncover *regularitie*s & *patterns* in our *Material World *in our search for *Ground Truths*.

At the same time, the same principles of *Symmetry a*nd *Geometry *could also be applied as a form of *Mental World[38] *— *Good Reasoning.*

17th Century Dutch Philosopher *Benedict De Spinoza[38a] *embraced the *mathematical *and *geometric* ideas of *Rene Descartes *and *Euclid *and applied them to *Meta-Physics *— our *Mental World*.

His book *Ethics[38] *was published posthumously in 1677, and he demonstrated the *Geometric Order[38b]* of *Good Reasoning* and *Logic *as it applied to *Philosophy*.

*Spinoza* puts forward from a small number of definitions and *axioms *hundreds of *propositions* and* corollaries*.

A *universal languag*e anchored in the Geometric Mind that could now integrate *Physics[39] *with *MetaPhysics[38] and Philosophy[38]* with *Science [39].*

A language that had the capacity to enable a transition from a Mental Structure of Consciousness to an Integral Consciousness[40].

An integration[40] of our *Mental World *with our *Material World*.

A recognition that *Reason was a Quality *— a way of achieving a *Harmony *— a *Symmetry *— between *Mental World reductionism* and abstraction and Material World of *Complexity *and *emergence*.

# The further extension of the Geometric Mind beyond the Body — New computational tools for Semantic Abstraction — The emergence of Geometric Deep Learning & the additional application of universal principles of Symmetry & Geometry to a Quantum World

On 27 April 2021, *Michael Bronstein*, *Joan Bruna,* *Taco Cohen* and *Petar Velickovic *published an Academic paper and *proto-book *titled G*eometric Deep Learning: Grids, Groups, Graphs, Geodiscs and Gauges[16].*

*Geometric Deep Learning* is an attempt for the geometric unification of a broad class of* Deep & Machine Learning *by extending *Flex Klein’s [35] *unification of *Geometry* principles — *Symmetry* & *Invariance*.

If *Felix Klein’s [35] *unification theory for *Geometry *was achieved in a classical digital* *World of *Clocks *— Were the co-authors of the April 2021 Geometric Deep Learning paper by applying the same principles to an analog *quantum* world of *Clouds* & *Clocks thereby extending the frontiers of Formal Logic into a world of higher dimensionality and Complexity*?

A World of relationships, interdependencies and entanglement.

A shift from Mathematics to Computation

A shift from Reductionism to Complexity

A shift from low dimensionality to higher dimensionality

New Formal Logic tools for Reason that reflect a Quantum World

Further extending the *Geometric Mind *beyond the Body through new computational tools for the *Semantic Abstraction *of our emergent complex *Material World*.

“The mind, once stretched by a new idea, never returns to its original dimensions”…

- Ralph Waldo Emerson

## Footnotes:

[1] — The Geometry of Thinking — https://youtu.be/Y3_zlm9DrYk

[2] — Copernicus Center for Interdisciplinary Studies **— **en

[3] — *Peter Gärdenfors — *PeterGardenfors

[4] — *Rich Sutton’s *— *Bitter Lesson — *http://www.incompleteideas.net/IncIdeas/BitterLesson.html

[5] — Emergence — https://richardschutte.medium.com/emergence-936c096422a5

[6] — Knightian Uncertainty —https://news.mit.edu/2010/explained-knightian-0602

[7] — The Complexity Void — https://richardschutte.medium.com/unbundling-complexity-503c77f0b261

[8] — Reflexivity — https://richardschutte.medium.com/reflexivity-bdd9d0a0fc7d

[9] — https://twitter.com/fchollet/status/1230613828301254657?s=20

[10] — Conceptual Spaces — https://mitpress.mit.edu/books/conceptual-spaces

[11] — The Power of Ten — https://youtu.be/0fKBhvDjuy0

[12] — More is Different — https://science.sciencemag.org/content/177/4047/393

[13] — Philipp Otto Runge* — *https://www.britannica.com/biography/Philipp-Otto-Runge

[14] — Ostwald’s 1916 Colour Solid System *— *https://en.wikipedia.org/wiki/Ostwald_color_system

[15] — The Semantic Abstraction of our Material World — https://richardschutte.medium.com/the-semantic-abstraction-of-our-material-world-53da868ce53e

[16] — G*eometric Deep Learning: Grids, Groups, Graphs, Geodiscs, and Gauges — *https://arxiv.org/pdf/2104.13478.pdf

[17] — https://youtu.be/LeeUzusWz5g

[18] — Theoretical Foundations of Graph Neural Networks — Theoretical Foundations of Graph Neural Networks — YouTube

[19]— Announcing the ICLR 2021 Invited Speakers | by ICLR | Medium

[20] — Euclid — https://www.khanacademy.org/math/geometry/hs-geo-transformations/hs-geo-intro-euclid/v/euclid-as-the-father-of-geometry

[21] — Elements — Euclid — https://mathcs.clarku.edu/~djoyce/elements/toc.html

[22] — *Eudoxus — *https://www.britannica.com/biography/Eudoxus-of-Cnidus

[23] — *Plato *—plato

[24] — *Plane Geometry — *epistemology-geometry

[25] — Semantic Languages — https://richardschutte.medium.com/semantic-languages-de088d48876f

[26] — Algebra —algebra

[27] — *Carl Friedrich Gauss — *https://www.britannica.com/biography/Carl-Friedrich-Gauss

[28] — Parallel Postulate *— *https://www.britannica.com/science/parallel-postulate

[29] — https://en.wikipedia.org/wiki/Playfair%27s_axiom

[30] — *Non-Euclidian Geometry — *https://www.britannica.com/science/non-Euclidean-geometry

[31] — *Nikolai Lobachevsky — *https://www.britannica.com/biography/Nikolay-Ivanovich-Lobachevsky

[32] — *János Bolyai — *https://www.britannica.com/biography/Janos-Bolyai

[33] — *Bernhard Riemann — *https://www.britannica.com/biography/Bernhard-Riemann

[34] — Euclidian Geometry — Euclidean-geometry

[35] — *Felix Klein — *https://www.britannica.com/biography/Felix-Klein

[36] — The idea of Symmetry — https://science.sciencemag.org/content/246/4932/940.2

[37] — Group Theory — group-theory

[38] — Ethics — https://en.wikipedia.org/wiki/Ethics_(Spinoza_book)

[38a] — Benedict de Spinoza — spinoza

[38b] — Modal Metaphysics — spinoza-modal

[39] — Symmetries — Physics — https://plato.stanford.edu/entries/symmetry-breaking/

[40] — Jean Gebser — https://www.institute4learning.com/2020/02/12/the-stages-of-life-according-to-jean-gebser/

.