The transcendental deduction

*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.

Synthetic character of human reasoning

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 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 demonstrates that human nature transcends nature; the existence of synthetic logic constitutes a transcendental deduction.

The essays here demonstrate that so far from being certain that the analytic logic of machines can model the human mind, the sources of human transcendance over that analytical logic are vast.

Contemporary Western philosophy is 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 demonstrates that human nature transcends nature; the existence of synthetic logic constitutes a transcendental deduction.

The essays here demonstrate that so far from being certain that the analytic logic of machines can model the human mind, the sources of human transcendance over that analytical logic are vast.

Answer to Hamming

This essay is a refutation of the techno-realist, positivist philosophy of mathematics advanced by R.W. Hamming in his paper

*Mathematics on a Distant Planet*(Published by the American Mathematical Monthly. Vol. 105. No.7). A Platonist reply defends the view that mathematical knowledge gives rise to a transcendental deduction whose conclusion is that the mind is not a material entity. The paper reviews the cultural dominance of Positivism in contemporary philosophy, arguing that on a just estimation of the arguments the dialectical battle between the Gods and Giants is more even than appearances would allow. Ostrich nominalism and the formalist philosophy of mathematics are rejected. The transcendental deduction goes through because of the ontological commitment involved in quantification in mathematical statements. Reviewing the analysis of that commitment in the work of W.V.O. Quine, not only is there ontological commitment in first-order set theory, the paradigm of a Platonist theory, but both number theory and analysis, via the Axiom of Completeness, are irreducibly second-order theories, for which the nominalist’s myth advanced by Quine cannot succeed. The cultural and spiritual significance of the debate is examined: the belief in universals and/or abstract objects is a ground for rational belief in immortality. In an appendix it is argued that limits cannot be constructed on the basis of the potential infinite alone.Rationalist philosophy of mathematics

Proof in mathematics is synthetic in general. Contemporary philosophy expresses an unalysed faith in mechanism 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.

A Neo-Kantian Philosophy of Mathematics

What is missing from the dialectic is a developed neo-Kantian philosophy.

Towards the Universal Characteristic

Liebniz's universal characteristic is an early paradigm of synthetic logic.

Lagrange's Theorem and the Logic of Intensions

Lagrange's theorem is an instance of the logic of intensions, which is non-computable. There is a deep relationship between Lagrange's theorem and the Halting problem.

The Expressive Power of Formal Languages

The power of formal languages, including first-order logic, enables one to define and express notions that refer to objects that are not recursive. It is not sufficient for such an expression to be formally manipulated in a recursively generated language for the object that it denotes also to be effectively computable.