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

## 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.”

## 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.

## 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.

## Talk: New Foundations for Information Theory

These are the slides for a number of talks on logical information theory as providing new foundations for information theory.

## Logical Entropy: Introduction to Classical and Quantum Logical Information Theory

Logical information theory is the quantitative version of the logic of partitions just as logical probability theory is the quantitative version of the dual Boolean logic of subsets. The resulting notion of information is about distinctions, differences, and distinguishability, and is formalized as the distinctions of a partition (a pair of points distinguished by the partition). This paper is an introduction to the quantum version of logical information theory.

## Quantum Logic of Direct-sum Decompositions

The usual quantum logic, beginning with Birkhoff and Von Neumann, was the logic of closed subspaces of a Hilbert space. This paper develops the more general logic of direct-sum decompositions of a vector space. This allows the treatment of measurement of any self-adjoint operators rather than just the projection operators associated with subspaces.

## From Abstraction in Math to Superposition in QM

This is a draft paper that makes a perhaps surprising connection between the old Platonic notion of a paradigm-universal like ‘the white thing’ and an indefinite superposition state in quantum mechanics.

## On Classical and Quantum Logical Entropy: The analysis of measurement

This paper shows how classical and quantum logical entropy arise out of the logic of partitions, and then it shows how there is a natural connection between the nxn distinctions and indistinctions of a partition and the nxn entries in a density matrix so that the classical and quantum logical entropy can directly register what happens to the density matrix in a projective measurement. The standard notion of von Neumann entropy does nothing of the kind–so the paper is also an indirect critique of von Neumann entropy as the most natural and ‘informative’ notion of entropy to use in quantum information theory.