“The purpose of Logic was to develop a system of thought that could be able to distinguish between Good and Bad Reason “ …
“People think that mathematics is complicated. Mathematics is the simple bit, it’s the stuff we can understand. It’s cats that are complicated”…
- John Conway
Mathematics is the science of uncovering and abstracting Patterns in our emergent complex Material World.
Numbers abstract counts whereas Groups abstract universal structures and relationships — codifying the ideas of Symmetry.
We make sense of the World through these relationships and interdependencies.
The Physical World — Material World — embedded into Metaphysics — Mental World — embedded into a Theory of Logic.
“Bad reasoning as well as good reasoning is possible; and this fact is the foundation of the practical side of logic”…
- Charles Sanders Pierce
2oth Century US Philosopher Charles Sanders Peirce outlined how it is through Abduction that new ideas and theories emerge about our Material World.
“Abduction is the process of forming an explanatory hypothesis. It is the only logical operation which introduces any new idea; for induction does nothing but determine a value and deduction merely evolves the necessary consequences of a pure hypothesis”…
- Charles Sanders Peirce
Rather than embracing a universalist (absolutism — i.e. binary, certain) tradition of Logic Charles Sanders Peirce believed our Human capacity to re-inteprete language and assign different universes to quantifiers — through semantic relationships — hypothesis, abduction and context — enables us to Reason and make sense of the World.
It is through these Mental Structures — Relationships — that we create Mental Models of the World.
When the structure satisfies the defining Axioms of the Theory it becomes a Model — Propositional Theorems — that could be deduced from a series of Axioms.
Building on the core principles of Logic and Algebra outlined by Euclid in his work titled Elements from ~300BC. which explored this geometric methods of Reasoning.
We create mental models of this world — patterns and relational structures — and match these against our Material World.
Concepts and objects — Epistemology, Ontology, Axiology and Geometry — mapped in high dimensional Conceptual Spaces that leverage the unique qualities of our Geometric Mind.
The closer these structures and relationships align with our Actions & Experiences in the Material World the more they provide certainty and simplify complexity.
Narrowing the Reality Gap.
Uncovering Universal Structures, Known Knowns and abstracting Ground Truths.
Reductionism as opposed to emergence.
Symmetries as opposed to breaks in Symmetries.
Reason and Logic require the integration of our Mental World and Material World
Rationalism with Empiricism
Idealism with Pragmatism
The Semantic, Geometric, and Algorithmic Mind
Abduction, Induction, and Deduction
Experience, Perception, and Abstraction
The Reflexive, Transitive and Symmetric.
The principles of Symmetry are at the very core of Good Reason, Logic and our understanding of Reality.
Forming the foundational principles for a range of heuristics in areas such as Geometry, Group Theory, Physics, Ethics and at the same time uncovering order and regularity from an emergent complex Material World.
Symmetry is also a foundational principle of Nature — People , Animals, Plants, Planets, Leaves, Fruit, Flowers, and Spiderwebs are all Symmetrical.
A harmony between our Mental World of abstraction and Material World of experience.
Universal Mental World & Material World Structures of Reality.
Symmetry is a statement on the actions you can take and preserve these universal structures.
A Group is a statement about all the actions you can take and preserve structures.
A structure is a Model if it satisfies the defining axioms of that Theory — Model Theory.
A Periodic Table of Finite Simple Groups — All the ways things can be Symmetrical — atoms of abstract Universal Structures — Algebraic structures of Symmetries — that are the building blocks of patterns and structures of our complex Material World.
From low dimensional simple Groups to high dimensional complex Monster Groups.
Symmetry enables for the compression and simplification of Complexity — Reductionism — with Groups through the abstraction of universal structures being the basis of Algebra, Good Reason and Logic.
It provides the topology to map the basis of Concepts.
Isomorphism enables us to map equivalent forms and structures — equal shapes and forms.
Asymmetry in our Material World highlights the Limits of Logic — the Limits of Knowing — the Limits of Reason.
A break in Symmetry is Emergence.
“Without the application of the Principle of Symmetry to the Quality of Reason — all that is left is a “Marketplace of Ideas” — a relativism of Ground Truths and Ground Motives …
- The Complexity Void
“Intelligence, Data Science, Learning and Reason all require the integration of our Mental World and Material World.
An Embodied Abstraction.
The application of the Principle of Symmetry and the Quality of Reason” …
- The Complexity Void
[To be added shortly ]