Intellectual Trespassing: Math Sciences Edition

A Very Common Fallacy in Quantum Mechanics

Published on November 30, 2011

There is a very common fallacy, here called the separation fallacy, that is involved in the interpretation of quantum experiments involving a certain type of separation such as the:

double-slit experiments,
which-way interferometer experiments,
polarization analyzer experiments,
Stern-Gerlach experiments, and
quantum eraser experiments.

In each case, given an incoming quantum particle, the apparatus creates a labelled or tagged (a type of entanglement) superposition of certain eigenstates (the “separation”). Detectors can be placed in certain positions so that when the evolving superposition state is finally projected or collapsed by the detectors, then only one of the eigenstates can register at each detector (due to the labels or tags). The separation fallacy mistakes the creation of a tagged or entangled superposition for a measurement. Thus it treats the particle as if it had already been projected or collapsed to an eigenstate at the separation apparatus rather than at the later detectors. But if the detectors were suddenly removed while the particle was in the apparatus, then the superposition would continue to evolve and have distinct effects (e.g., interference patterns in the two-slit experiment).

Hence the separation fallacy makes it seem that by the delayed choice to insert or remove the appropriately positioned detectors, one can retro-cause either a projection to an eigenstate or not at the particle’s entrance into the separation apparatus.

The separation fallacy is remedied by:

taking superposition seriously, i.e., by seeing that the separation apparatus created an entangled superposition state of the alternatives (regardless of what happens later) which evolves until a measurement is taken, and
taking into account the role of detector placement (“contextuality”), i.e., by seeing that if a suitably positioned detector, as determined by the tags, can only detect one collapsed eigenstate, then it does not mean that the particle was already in that eigenstate prior to the measurement (e.g., it does not mean that the particle went through one slit, took one path in an interferometer, or was already in a polarization or spin eigenstate).

The separation fallacy will be first illustrated in a non-technical manner for the first four experiments. Then the lessons will be applied in a slightly more technical discussion of quantum eraser experiments where the labels or tags are erased after the separation apparatus and where, due to the separation fallacy, incorrect inferences about retrocausality have been rampant.

A Very Common Fallacy in Quantum Mechanics continued »

Filed in Applications,Philosophy,Quantum Mechanics | 22 replies

Tags: delayed choice, double slit, measurement, quantum eraser, retrocausality, separation fallacy, Stern-Gerlach, superposition

History of the Logical Entropy Formula

Published on February 16, 2010

The logical entropy formula $h(p) = 1-\sum_{i}p_{i}^{2}$

Given a partition $\pi = \{B,B',\ldots\}$ on a finite universe set U, the set of distinctions or dits $dit(\pi)$ is the set $\cup_{B\in \pi}B \times (U-B)$ of ordered pairs of elements in distinct blocks of the partition. The logical entropy $h(\pi)$ of the partition is the normalized cardinality of the dit set: $h(\pi) = \frac{|dit(\pi)|}{|U\times U|}$ . The logical entropy can be interpreted probabilistically as the probability that the random drawing of a pair of elements (with replacement between draws) from U (with equiprobable elements) gives a distinction of the partition. In terms of the probability $p_{B} = \frac{|B|}{|U|}$ , the logical entropy could be computed as:

$h(\pi) = \sum_{B \in \pi}p_{B}(1-p_{B}) = 1-\sum_{B \in \pi}p_{B}^{2}$ .

History of the Logical Entropy Formula continued »

Filed in Applications,Information theory,Partition logic | One reply

Tags: cryptography, Gini index of diversity, logical entropy, numbers-equivalents, Rao's quadratic entropy, relative shares, Simpson's index of diversity, Turing

From Partition Logic to Information Theory

Published on February 6, 2010

Some basic analogies between subset logic and partition logic

A new logic of partitions has been developed that is dual to ordinary logic when the latter is interpreted as the logic of subsets rather than the logic of propositions. For a finite universe, the logic of subsets gave rise to finite probability theory by assigning to each subset its relative cardinality as a Laplacian probability. The analogous development for the dual logic of partitions gives rise to a notion of logical entropy that is related in a precise manner to Claude Shannon’s entropy. In this manner, the new logic of partitions provides a logical-conceptual foundation for information-theoretic entropy or information content. This post continues the earlier post which introduced some of the basic ideas and operations of partition logic.

From Partition Logic to Information Theory continued »

Filed in Applications,Information theory,Partition logic,Philosophy | One reply

Tags: logical entropy, probability theory, Shannon, Shannon entropy

Series-Parallel Duality: Part I: Combating series chauvinism

Published on February 5, 2010

Series chauvinism

This post describes the duality between the usual (series) addition and the dual parallel addition. This duality is normally considered in electrical circuit theory and combinatorics, but it has a much wider applications. In Part I of this post, the focus is on developing series-parallel dual formulas—in contrast to the usual focus on formulas using only the series sum.

From the viewpoint of pure mathematics, the parallel sum is “just as good” as the series sum. It is only for empirical and perhaps even some accidental reasons that so much mathematics is developed using the series sum instead of the equally good parallel sum. There is a whole “parallel mathematics” which can be developed with the parallel sum replacing the series sum. Since the parallel sum can be defined in terms of the series sum (or vice-versa), “parallel mathematics” is essentially a new way of looking at certain known parts of mathematics.

Exclusive promotion of the series sum and prejudice against the parallel sum is series chauvinism. Before venturing further into the parallel universe, we might suggest some exercises to help the politically incorrect reader to combat the heritage of series chauvinism. Anytime the series sum seems to occur naturally in mathematics with the parallel sum being seemingly invisible, it is an illusion due to series chauvinism. The parallel sum has a “parallel” role that has been unfairly neglected.

In Part II of the post, series-parallel duality is applied to financial arithmetic and is shown to underlie a certain duality of formulas that has long been noted in the literature on valuation and appraisal. Hence this instance of intellectual trespassing applies concepts best known from electrical circuit theory to the operations of financial arithmetic. Moreover in economic theory, the much-used duality of convex functions and their conjugates is the “integral” of series-parallel (SP) duality, or, to put it the other way around, SP duality is the “derivative” of convex duality.

Series-Parallel Duality: Part I: Combating series chauvinism continued »

Filed in Applications,Math economics | One reply

Tags: geometric series, MacMahon, reciprocity, series chauvinism, series-parallel duality

Series-Parallel Duality: Part II: Finanical arithmetic

Published on February 5, 2010

Reciprocal formulas in financial arithmetic

In financial arithmetic and in the appraisal literature, it has been noticed that the basic formulas occur in pairs, one being the reciprocal of the other. For instance, one popular text on real estate appraisal presents the “Basic Functions of Compound Interest and Their Reciprocals” [Friedman, Jack P. and Nicholas Ordway 1988. Income Property Appraisal and Analysis. Englewood Cliffs: Prentice Hall, p. 70]. The functions could be presented as follows to bring out the underlying symmetry.

Function	Reciprocal
Principal Retired by Payment of One $(1+r)^{-n}$	Payment to Retire Principal of One $(1+r)^{n}$
Principal Amortized by Payments of One $a(n,r)=\frac{1}{(1+r)^{1}}+\ldots+\frac{1}{(1+r)^{n}}$	Payments to Amortize a Principal of One $\frac{1}{a(n,r)}=(1+r)^{1}:\ldots:(1+r)^{n}$
Fund Accumulated by One per Period $s(n,r)=(1+r)^{n-1}+\ldots+(1+r)+1$	Payments to Accumulate a Fund of One $\frac{1}{s(n,r)}=\frac{1}{(1+r)^{n-1}}:\ldots:\frac{1}{(1+r)}:1$

The Six Functions of One

This Part II of the series-parallel duality post shows that these reciprocal formulas are an example of the SP duality normally associated with electrical circuit theory.

Series-Parallel Duality: Part II: Finanical arithmetic continued »

Filed in Applications,Math economics | No replies

Tags: amortization payments, financial arithmetic, series-parallel duality, sinking funds

The Math of Double-Entry Bookkeeping: Part I (scalars)

Published on February 4, 2010

Mathematics and Accounting: Two disjoint universes?

Double-entry bookkeeping illustrates one of the most astonishing examples of intellectual insulation between disciplines—the very opposite of intellectual trespassing. Double-entry bookkeeping (DEB) was developed during the fifteenth century and was published in 1494 as a system by the Italian mathematician Luca Pacioli and was anticipated in a 1458 manuscript of the Croatian merchant Benedikt Kotruljević (that was only published in 1573). Double-entry bookkeeping has been used for over five centuries in commercial accounting systems. If the mathematical formulation of any field should be well understood, one would think it might be accounting. Remarkably, however, the mathematical formulation of double entry accounting—algebraic operations on ordered pairs of numbers—was first published only in 1982 and is still largely unknown both in mathematics and accounting.

The Math of Double-Entry Bookkeeping: Part I (scalars) continued »

Filed in Applications,Math economics | 2 replies

Tags: double-entry bookkeeping, Kotruljevic, Pacioli, single-entry accounting

The Math of Double-Entry Bookkeeping: Part II (vectors)

Published on February 4, 2010

Multi-dimensional double-entry accounting?

Although double-entry bookkeeping (DEB) has been used in the business world for 5 centuries, the mathematical formulation of the double entry method is almost completely unknown. In this post, the mathematical treatment of double-entry bookkeeping using scalars given in Part I is generalized to the multi-dimensional case using vectors. The success in maintaining the two-sided accounts, debits and credits, the double-entry principle, and the trial balance in both cases shows that the formulation captures the double-entry method in mathematical form.

The Math of Double-Entry Bookkeeping: Part II (vectors) continued »

Filed in Applications,Math economics | No replies

Tags: double-entry bookkeeping, multi-dimensional accounting, Pacioli, vectors

The implication operation on partitions

Published on February 2, 2010

Partitions and equivalence relations

In a 2001 commemorative volume for my mathematical mentor, Gian-Carlo Rota, three of his associates noted that “the only operations on the family of equivalence relations fully studied, understood and deployed are the binary join $\lor$ and meet $\land$ operations.” This note defines the apparently new operation of implication for partitions, an operation that was key to the development of the logic of partitions that is dual to the usual logic of subsets.

The implication operation on partitions continued »

Filed in Math logic,Partition logic | One reply

Tags: equivalence relations, Gian-Carlo Rota, implication, Partition logic, partitions

From propositional logic to subset logic to partition logic

Published on January 28, 2010

From propositional logic to subset logic

This note outlines the following sequence of ideas. First, ordinary propositional logic is reinterpreted as the logic of subsets of a universe set U, with the propositional case being isomorphic to the special case of U = 1. Then the category-theoretic duality between subsets of a set and partitions on a set is used to broach the idea of a dual logic of partitions. At the end of the note, a link is given to the final draft of my forthcoming paper in the Review of Symbolic Logic which develops the logic of partitions from the basic ideas up through the correctness and completeness theorems for a tableau system of (zeroth order) partition logic.

From propositional logic to subset logic to partition logic continued »