The paper is published as: Ellerman, D. (2021). On Abstraction in Mathematics and Indefiniteness in Quantum Mechanics. Journal of Philosophical Logic. https://doi.org/10.1007/s10992-020-09586-1

## The Logical Theory of Canonical Maps

The purpose of this paper is to show that the dual notions of elements & distinctions are the basic analytical concepts needed to unpack and analyze morphisms, duality, and universal constructions in the Sets, the category of sets and functions.

## Negation and Implication in Partition Logic

Our purpose in this paper is to explore the notions of negation and implication in that other mathematical logic of partitions.

## Probability Theory with Superposition Events

In finite probability theory, events are subsets S⊆U of the outcome set. Subsets can be represented by 1-dimensional column vectors. By extending the representation of events to two dimensional matrices, we can introduce “superposition events.”

## Extending All Boolean Operations to Partitions

The lattice operations of join and meet were defined for set partitions in the nineteenth century, but no new logical operations on partitions were defined and studied during the twentieth century. Yet there is a simple and natural graph-theoretic method presented here to define any n-ary Boolean operation on partitions. An equivalent closure-theoretic method is […]

## Finding the Markets in the Math

This paper shows how to find competitive market prices in the pure mathematics of classical constrained optimization problems.

## New Light on the Objective Indefiniteness or Literal Interpretation of QM

This paper shows how the mathematics of QM is the math of indefiniteness and thus, literally and realistically interpreted, it describes an objectively indefinite reality at the quantum level. In particular, the mathematics of wave propagation is shown to also be the math of the evolution of indefinite states that does not change the degree of indistinctness between states. This corrects the historical wrong turn of seeing QM as “wave mechanics” rather than the mechanics of particles with indefinite/definite properties.

## A Note on Spencer-Brown’s Algebra

George Spencer-Brown in his cryptic book, Laws of Form, started off reasoning about “the Distinction” and ended up with an algebra that later writers showed to be the Boolean algebra of two elements.

## Talk: Hamming distance in classical and quantum logical information theory

This is a set of slides from a talk on introducing the Hamming distance into classical logical information theory and then developing the quantum logical notion of Hamming distance–which turns out to equal a standard notion of distance in quantum information theory, the Hilbert-Schmidt distance.

## Talk: New Foundations for Quantum Information Theory

These are the slides for a talk given at the 6th International Conference on New Frontiers in Physics on Crete in August 2017.