Ellerman, David
(2025)
The Mean and the Variance as Dual Concepts in a
Fundamental Duality.
Axioms, 14.
ISSN 2075-1680
Abstract
A basic duality arises throughout the mathematical and natural sciences. Traditionally,
logic is thought to be based on the Boolean logic of subsets, but the development
of category theory in the mid-twentieth century shows the duality between subsets and
partitions (or equivalence relations). Hence, there is an equally fundamental dual logic
of partitions. At a more basic or granular level, the elements of a subset are dual to the
distinctions (pairs of elements in different blocks) of a partition. The quantitative version of
subset logic is probability theory (as developed by Boole), and the quantitative version of
the logic of partitions is information theory re-founded on the notion of logical entropy. The
subset side of the duality uses a one-sample (or one-element) approach, e.g., the mean of a
random variable; the partition side uses a two-sample (or pair-of-elements) approach. This
paper gives a new derivation of the variance (and covariance) based on the two-sample
approach, which positions the variance on the partition and information theory side of the
duality and thus dual to the mean.
Monthly Views for the past 3 years
Monthly Downloads for the past 3 years
Plum Analytics
Altmetric.com
Actions (login required)
 |
View Item |
![[img]](https://philsci-archive.pitt.edu/style/images/fileicons/text.png)
![[feed]](http://philsci-archive.pitt.edu/style/images/feed-icon-32x32.png){kind=icon}