Intuitinistic Coherentism

This blog aims to explore my thoughts on the philosophy of mathematics. I am fundamentally a Platonist, believing that there is an actual fundamental underlying truth from which our mathematical intuition derives its coherence and practicality.

This is largely a theological statement. Accepting that God exists, and that all actual truths originates from God, I am forced to conclude that any mathematical statement that is actually true must come from God, making God the fundamental underlying truth from which our mathematical intuition derives its coherence and practicality.

The present state of the art in mathematics is to express the objects being studied using the notation of set theory, where it is customary to assume the first order axioms expressed in classical logic known as Zermelo-Fraenkel (ZF) set theory. ZF set theory, however, has several notions that disagree with my personal understanding of the Platonic reality underpinning mathematical reasoning.

In some ways I am an empiricist. Since God is infinite, the Platonic truth mathematics is based upon may be infinite. Since God is a creative God, expressing His infinite nature within His creation, we should expect it should be. In every physical realm of existence we have examined, each new frontier has revealed a reality far more intricate than we had previously imagined. Why should mathematical truth be an exception?

Being finite, we can never grasp the totality of truth. Thus our logical models are mere approximations of the Platonic reality underlying mathematics. As we develop models, weaknesses are found, and alternatives are created to address these weaknesses. ZF set theory is the model that currently has the broadest base of application.

It is my own hubris that convinces me that my personal understanding of the Platonic reality is preferable to the existing structures. Until I have developed a formal model whose consistency, soundness and effectiveness can be examined; I have no reason to expect anyone to ascribe any weight to my understanding.

Further, I expect that the axioms of ZF, or any other set of axioms, can be seen as defining a portion of all mathematical truths for which we wish to discuss. The art of mathematics then becomes a symbol manipulation game to be played among the established paradigms. Pragmatically, then, I am a formalist.

Indeed, to search for an improved model of mathematical truths I must understand the pitfalls of the existing models. This requires that I establish a basic competency in the symbolic manipulation game as it is played in my era. Only then should my considerations be given weight.

Section 1. Introduction

Chapter 1. Intuitionistic Logic
Section 1. Propositional Logic
Section 2. Rules of Inference
Section 3. Predicate Logic
Section 4. Classical Logic

Chapter 2. Sets
Section 1. Membership Operators, Set containing only a set.
Section 2. Set Equivalence
Section 3. Set Builder Notation
Section 4. Union/Intersection
Section 5. Power Sets
Section 6. Complete Sets, ZFC
Section 7. Cardinality of infinite sets
Section 8. Universal Set and Paradoxes

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