The 1900 Turn in Bertrand Russell’s Logic, the Emergence of his Paradox, and the Way Out

Russell’s initial project in philosophy (1898) was to make mathematics rigorous reducing it to logic. Before August 1900, however, Russell’s logic was nothing but mereology. First, his acquaintance with Peano’s ideas in August 1900 led him to discard the part-whole logic and accept a kind of intensional predicate logic instead. Among other things, the predicate logic helped Russell embrace a technique of treating the paradox of infinite numbers with the help of a singular concept, which he called ‘denoting phrase’. Unfortunately, a new paradox emerged soon: that of classes. The main contention of this paper is that Russell’s new conception only transferred the paradox of infinity from the realm of infinite numbers to that of class-inclusion. Russell’s long-elaborated solution to his paradox developed between 1905 and 1908 was nothing but to set aside of some of the ideas he adopted with his turn of August 1900: (i) With the Theory of Descriptions, he reintroduced the complexes we are acquainted with in logic. In this way, he partly restored the pre-August 1900 mereology of complexes and simples. (ii) The elimination of classes, with the help of the ‘substitutional theory’, and of propositions, by means of the Multiple Relation Theory of Judgment, completed this process

When is .999... less than 1?

We examine alternative interpretations of the symbol described as nought, point, nine recurring. Is "an infinite number of 9s" merely a figure of speech? How are such alternative interpretations related to infinite cardinalities? How are they expressed in Lightstone's "semicolon" notation? Is it possible to choose a canonical alternative interpretation? Should unital evaluation of the symbol .999 . . . be inculcated in a pre-limit teaching environment? The problem of the unital evaluation is hereby examined from the pre-R, pre-lim viewpoint of the student.Comment: 28 page

Bertrand Russsell's Religion without God

The task of this paper is to reconstruct Bertrand Russell project for religion without God and dogma. Russell made two attempts in this direction, first in the essay “Free Man’s Worship” (1903), and then, in theoretical form, in the paper “The Essence of Religion” (1912). Russell’s explorations of religious impulses run in parallel with his work on technical philosophy. According to Russell from 1903–12, religion is an important part of human pursuits. However, whereas the ordinary man believes in God, the freeman embraces a religion without fear and dogma. He strives for a union with the universe achieved in contemplation made from many perspectives through “impartiality of vision”. For this reason freemen renounce the Self and the Will. Russell abandoned his project for religion without God mainly because of Wittgenstein’s criticism. In his later writings he continued to criticize the religion of the ordinary man, without to further develop a positive philosophy of religion, though

Invariant Set Theory: Violating Measurement Independence without Fine Tuning, Conspiracy, Constraints on Free Will or Retrocausality

Invariant Set (IS) theory is a locally causal ontic theory of physics based on the Cosmological Invariant Set postulate that the universe $U$ can be considered a deterministic dynamical system evolving precisely on a (suitably constructed) fractal dynamically invariant set in $U$'s state space. IS theory violates the Bell inequalities by violating Measurement Independence. Despite this, IS theory is not fine tuned, is not conspiratorial, does not constrain experimenter free will and does not invoke retrocausality. The reasons behind these claims are discussed in this paper. These arise from properties not found in conventional ontic models: the invariant set has zero measure in its Euclidean embedding space, has Cantor Set structure homeomorphic to the p-adic integers ($p \ggg 0$) and is non-computable. In particular, it is shown that the p-adic metric encapulates the physics of the Cosmological Invariant Set postulate, and provides the technical means to demonstrate no fine tuning or conspiracy. Quantum theory can be viewed as the singular limit of IS theory when when $p$ is set equal to infinity. Since it is based around a top-down constraint from cosmology, IS theory suggests that gravitational and quantum physics will be unified by a gravitational theory of the quantum, rather than a quantum theory of gravity. Some implications arising from such a perspective are discussed.Comment: In Proceedings QPL 2015, arXiv:1511.0118

Menorah Review (No. 39, Winter, 1997)

An Interpretive Methodology With Supersessionist Forebodings -- Through a Glass Brightly: Seeing the Unseeable -- The 12th Annual Selma and Jacob Brown Lecture -- Controversy and the Dead Sea Scrolls -- Book Listing -- Jewish Civics -- Leah -- Book Briefing