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Foundations of Mathematics

Author: Mark Anthony Higgins
© Mark Anthony Higgins 11 October 2025
Revision Date
27th April 2026
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Introduction

Mathematics begins not with numbers, but with the act of distinction —
the moment a boundary is drawn between this and that.
From this single gesture, coherence is born.
What we call a number is simply the stable record of that distinction:
a symbol that preserves identity across transformation.

The foundations of mathematics are therefore not arbitrary axioms, but expressions of how coherence maintains itself.
Counting, equality, and relation arise from the same principle:
that stability can be repeated, mirrored, and transformed without loss of meaning.

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In this view, arithmetic and logic share a common origin —
each is a reflection of boundary behaviour.
When we say 1+1=2, we do more than add;
we unite two coherent entities under a higher invariant.

When we assert A⇒B, we trace the continuity of form from cause to effect —
the same flow seen in energy, geometry, and thought.

This chapter explores those roots:
how numbers emerge as coherence units,
how sets and mappings define spaces of transformation,
and how logic itself can be seen as recursive boundary preservation.

Here, mathematics returns to its living source —
not a catalogue of rules,
but a field of relations that sustain truth through balance.

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