## The work of Melampus in logic and the philosophy of mathematics

*Immortality, The Groundwork to the History of Western Consciousness*is an introduction to the work of Melampus in synthetic reasoning, which builds upon the transcendental deduction of Plato and Kant.

## Introduction

Logic is the science of inference. All inference concerns the movement from premises A to conclusions B. The only logic that has been recognised is analytic logic in which the premises A contain more information than the conclusions B and the movement (a "deduction") proceeds from greater to lesser generality. However, there exist necessary inferences in which the conclusions B contain more information than the premises A. Though philosophers have hitherto contrived not to see them, these inferences are found everywhere in discourse, and are exposed in mathematical reasoning. There exists synthetic logic.

Contemporary Western philosophy is collectively falsified by its inability to perceive the synthetic character of much of human reasoning and to investigate its implications.

The synthetic character of human reasoning is high among the valid reasons for believing that human nature transcends nature; the existence of synthetic logic constitutes a transcendental deduction.

Contemporary Western philosophy is collectively falsified by its inability to perceive the synthetic character of much of human reasoning and to investigate its implications.

The synthetic character of human reasoning is high among the valid reasons for believing that human nature transcends nature; the existence of synthetic logic constitutes a transcendental deduction.

## Rationalist philosophy of mathematics

In this essay Melampus presents the argument that proof in mathematics is synthetic. It raises those questions that the formalism of contemporary philosophy must answer in order to remain credible. The essay touches upon the claim that Godel's theorem provides an example of non-algorithmic proof. It also discusses the Turing test and raises important caveats about the manner in which that test is being applied. It implies that contemporary philosophy expresses an unalysed faith in mechanism, and that in the dialectic it ought to be opposed by an equally forceful rationalism.

## On the character of formal analytical logic

This essay explains that all human reasoning is founded on argument by analogy. Arguments by analogy may be exact or inexact; when exact, they are necessary. Analytic logic, also called first-order logic, argues by exact analogy with spatial containment. Models of first-order, analytic logic are lattices; this characterisation demonstrates constraints on what inferences may be represented in analytic logic. For this reason, self-reference is not possible within analytic logic, for a space cannot contain any part of itself. The logic of self-reference illustrates how conceptual thinking departs from what is merely representable in analytic logic.

## What second order concepts tell us about the continuum

The continuum is that object that serves as the foundation of mathematical analysis - the differential and integral calculus. A well-known unsolved problem arising from the work of Cantor is that of the size of the continuum. The reason why this problem has gone unsolved in the C20th is due to the insistence on using first-order set theory. In this essay Melampus explores the phenomenological and empirical foundations of the classical concept of the continuum, and derives the continuum hypothesis from the second-order Axiom of Completeness.

## Solution to the Halting problem

Lagrange's theorem

The halting problem is the problem of determining whether a Turing machine will halt for a given input. One of the results of modern computing theory is that the halting problem cannot be solved. However, this argument, a proof by contradiction, like any other argument has premises. Two of the hidden premises are (a) that there is no distinction between the potential and actual infinite, and (b) that proof by mathematical induction is a form of algorithm. The important point is to realise that there is a distinction between proving that we

*know*something, and having an algorithm that computes a result. The impossibility proof, then, is a proof that any machine that computed an algorithm would need an actually infinite number of parts, and as this is impossible, a computer cannot be built to solve the problem. But this is only an impossiblity proof for computers, not for the human mind. It turns out that a proof that the halting problem for any Turing machine can be solved is a relatively simple induction on the number of states of a Turing machine. Furthermore, as Turing machines decompose into cycles, and these are connected with permutation groups, there is a deep connection between the solvability of the Halting problem and group theory. In a separate essay Melampus analyses Lagrange's Theorem, that the order of a subgroup divides into the order of a group, and explains why Lagrange's theorem cannot be formalised in first-order logic, it having been remarked by Beson that it has so far resisted such a formulation.