“Geometry is the knowledge of the eternally existent”…
“Geometry is the archetype of the beauty of the world” …
- Johannes Kepler
the branch of mathematics concerned with the properties and relations of points, lines, surfaces, solids, and higher dimensional analogues
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  — delivered in June 2016 at the Copernicus Centre for Interdisciplinary Studies, Swedish Professor of Cognitive Scientist and Philosopher Peter Gärdenfors begins by outlining the 2 approaches that have been pursued to make sense of the nature of the Human Mind.
One is an Explanatory approach — our attempts to understand our Mental World through theories about different parts of cognition.
The other is a Constructivist approach — an engineering perspective — our attempts to create robots, chess playing systems, communication and intelligent computational machines.
Peter was particularly interested in a key explanatory question.
How can we explain how people, children in particular, learn new concepts so quickly?
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 — Humans have an innate ability to learn from very few examples.
A capacity to navigate emergence & knightian uncertainty in our complex Material World through novelty, open inquiry, creativity, reflexivity 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”…
— 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 we map concepts and objects thereby organising our knowledge of the World.
Via notions of similarity and principles of symmetry we have the capacity to bring coherence to our Reality.
New concepts can be introduced into these landscapes through mapping there relationships & connections — similarity — nearness — and — betweenness — convexity — thereby helping us learn how to learn.
Concepts — the How and What ? — are not a fixed sets of domains.
Categorisations such as Ontology and Epistimology 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 learn from experience… we learn from reflecting on experience” …
— John Dewey
In Charles and Ray Eames film — The Power of Ten 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…
A similar analogy can be applied to our Inner World — our Mental World.
Our Geometric Mind enables us to zoom in and zoom out of these cognitive landscapes.
Philipp Otto Runge’s 1810 Farbenkugel — colour sphere — and — Ostwald’s 1916 Colour Solid System — illustrate that through Geometry we can bring coherence to Complexity.
The evolution of Mathematics — the science of uncovering Patterns 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 the 27 April 2021 Michael Bronstein, Joan Bruna, Taco Cohen and Petar Velickovic published an Academic paper and proto-book titled Geometric Deep Learning: Grids, Groups, Graphs, Geodiscs, and Gauges.
It complimented Michael Bronstein’s talk titled — Geometric Deep Learning: Past, Present, And Future — at UCL Centre for Artificial Intelligence on 11 February 2021, Petar Velickovic’s talk at Cambridge University on 22 February 2021 and Michael Bronstein’s keynote at ICLR 2021.
Michael begins the UCL Centre for Artificial Intelligence lecture by providing a brief history of Geometry beginning with Euclid.
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 brought together a work titled — Elements — that synthesised and built upon the prior work of Eudoxus] and Plato.
Euclid’s method was based off 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 and mathematical proofs represented the foundational Axioms which through a language of abstraction emerged the basis of algebra.
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 order from chaos, the regular from the random and uncover patterns and symmetries 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 uncovered a contradiction in Euclid's parallel postulate.
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”…
Gauss 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 and Gauss’s breakthrough resulted in a range of rapid breakthroughs including Nikolai Lobachevsky and Hyperbolic Geometry, János Bolyai and Absolute Geometry (included both Hyperbolic and Euclidian) and Bernhard Riemann and Elliptic (Riemannian) geometry.
Given the splintering of Geometry into both Euclidian and multiple Non Euclidian 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 that answered this question in 1872 proposing to define Geometry as the study of invariance — symmetries.
Properties that are preserved under some classes of transformation.
A new semantic mathematical language of Group Theory formalised these classes of transformations.
Geometry could now be commonly defined .
Another language of Semantic Abstraction that could be used to uncover regularities & patterns in our Material World in our search for Ground Truths.
At the same time the same principles of Symmetry and Geometry could be also applied as a form of Mental World — 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 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 corrollaries.
A universal language anchored in the Geometric Mind that could now integrate Physics with MetaPhysics and Philosophy with Science.
A language that had the capacity to enable a transition from a Mental Structure of Consciousness to an Integral Consciousness.
An integration 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 further application of universal principles of Symmetry & Geometry to a Quantum World
On the 27 April 2021 Michael Bronstein, Joan Bruna, Taco Cohen and Petar Velickovic published an Academic paper and proto-book titled Geometric Deep Learning: Grids, Groups, Graphs, Geodiscs and Gauges.
Geometric Deep Learning is an attempt for the geometric unification of a broad class of Deep & Machine Learning by extending Flex Klein’s unification of Geometry principles — Symmetry & Invariance.
If Felix Klein’s unification theory for Geometry was achieved in a digital classical 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
 — The Geometry of Thinking — https://youtu.be/Y3_zlm9DrYk
 — Copernicus Center for Interdisciplinary Studies — en
 — Peter Gärdenfors — PeterGardenfors
 — Rich Sutton’s — Bitter Lesson — http://www.incompleteideas.net/IncIdeas/BitterLesson.html
 — Emergence — https://richardschutte.medium.com/emergence-936c096422a5
 — Knightian Uncertainty —https://news.mit.edu/2010/explained-knightian-0602
 — The Complexity Void — https://richardschutte.medium.com/unbundling-complexity-503c77f0b261
 — Reflexivity — https://richardschutte.medium.com/reflexivity-bdd9d0a0fc7d
 — Conceptual Spaces — https://mitpress.mit.edu/books/conceptual-spaces
 — The Power of Ten — https://youtu.be/0fKBhvDjuy0
 — More is Different — https://science.sciencemag.org/content/177/4047/393
 — Philipp Otto Runge — https://www.britannica.com/biography/Philipp-Otto-Runge
 — Ostwald’s 1916 Colour Solid System — https://en.wikipedia.org/wiki/Ostwald_color_system
 — The Semantic Abstraction of our Material World — https://richardschutte.medium.com/the-semantic-abstraction-of-our-material-world-53da868ce53e
 — Geometric Deep Learning: Grids, Groups, Graphs, Geodiscs, and Gauges — https://arxiv.org/pdf/2104.13478.pdf
 — https://youtu.be/LeeUzusWz5g
 — Theoretical Foundations of Graph Neural Networks — Theoretical Foundations of Graph Neural Networks — YouTube
 — Elements — Euclid — https://mathcs.clarku.edu/~djoyce/elements/toc.html
 — Eudoxus — https://www.britannica.com/biography/Eudoxus-of-Cnidus
 — Plato —plato
 — Plane Geometry — epistemology-geometry
 — Semantic Languages — https://richardschutte.medium.com/semantic-languages-de088d48876f
 — Algebra —algebra
 — Carl Friedrich Gauss — https://www.britannica.com/biography/Carl-Friedrich-Gauss
 — Parallel Postulate — https://www.britannica.com/science/parallel-postulate
 — Non-Euclidian Geometry — https://www.britannica.com/science/non-Euclidean-geometry
 — Nikolai Lobachevsky — https://www.britannica.com/biography/Nikolay-Ivanovich-Lobachevsky
 — János Bolyai — https://www.britannica.com/biography/Janos-Bolyai
 — Bernhard Riemann — https://www.britannica.com/biography/Bernhard-Riemann
 — Euclidian Geometry — Euclidean-geometry
 — Felix Klein — https://www.britannica.com/biography/Felix-Klein
 — The idea of Symmetry — https://science.sciencemag.org/content/246/4932/940.2
 — Group Theory — group-theory
 — Ethics — https://en.wikipedia.org/wiki/Ethics_(Spinoza_book)
[38a] — Benedict de Spinoza — spinoza
[38b] — Modal Metaphysics — spinoza-modal
 — Symmetries — Physics — https://plato.stanford.edu/entries/symmetry-breaking/