This post describes one problem that someone new to abstract math who is reading mathematical papers might find difficult to understand. Any attempt at machine translation of mathematical prose into computer-readable code would also have to take this phenomenon into account. The math ed literature describes many other problems like this for readers, some of which cause much more trouble than this example.

I referred to the phenomenon described below as a parenthetic assertion in these publications:

• Handbook of Mathematical Discourse.
• Parenthetic assertions (article in abstractmath.org).
• Context-sensitive interpretations (article in abstractmath.org).

A symbol is pivoted if it is embedded just once in an expression that, when spelled out explicitly, requires it appear twice because it play two different roles in two different phrases or clauses. The examples below convince me that “pivoted symbol” is a better name than “parenthetic assertion”.

Example

The expression “$1\lt x\lt 2$” can be spelled out in these ways:

1. “$1$ is less than $x$, which is less than $2$”.
2. “$1$ is less than $x$ and $x$ is less than $2$”.

In both cases, $x$ appears just once in the expression, but any reasonable rendering requires it to appear in two clauses. (The word “which” in the first statement refers to $x$ according to the rules for anaphora in English, so “which” is a homonym for $x$ there.)

Example

“For any $x\gt0$ there is a $y\gt0$ such that $x\gt y$.”

• This is mathematical shorthand for: “For any $x$ that is greater than $0$ there is a $y$ that is greater than $0$ such that $x$ is greater than $y$.”
• The phrase “For any $x$ that is greater than $0$” conveys two pieces of information:
  • $x$ is bound by a universal quantifier.
  • $x$ is a variable in a phrase constraining it.
• The constraint on $x$ to be bigger than $0$ fills the slot of a subordinate clause playing the role of an adjective, with $x$ as head. The word “that” is anaphoric.
• The situation for $y$ in this expression is similar.
• The elimination of “that” (which is one occurrence of $x$) in the expressions “$x$ greater than $0$” and “$x\gt 0$” fit a common pattern in English of omitting “that” or “that is”. Consider: “I saw a house bigger than the White House.” So calling the statement a “parenthetic assertion” doesn’t seem to fit the situation very well.
• That’s why I have titled this post “Pivoted symbols”. The key to this name is that a translation into a formal language is going to have to encode two facts about $x$: It is bound by a quantifier and it is constrained to be greater than $0$. It has to be copied to accomplish this.
• There are similar examples in Contrapositive grammar, by David Butler.

Example

“This infinite series converges to $\zeta(2)=\frac{\pi^2}{6}\approx 1.65$.”

This example can be read in two ways that are different in English grammar but have the same logical content:

• “This infinite series converges to $\zeta(2)$, which is $\frac{\pi^2}{6}$, which is approximately $1.65$.”
• “$\ldots$ converges to $\zeta(2)$, and $\zeta(2)=\frac{\pi^2}{6}$, and $\frac{\pi^2}{6}$ is approximately $1.65$.”

Example

“Let us return for a moment to the circle $S^1\subseteq \mathbb{C}=\mathbb{R}^2$.” (Citation 426 in the Handbook of mathematical discourse.)

Example

\[B(t):=\frac{1+t^2}{2+t^2}\in\mathbb{Q}(t)\]
is a sum of $2n$th powers of elements in $\mathbb{Q}(t)$ for all $n$. (Citation 332 in the Handbook of mathematical discourse.)

Science-fiction fans used to do something similar with words instead of clauses, writing “yed” for “ye editor” and “scientifiction” for “scientific fiction”.

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