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This is the kind of "mathematical thinking" referred to in the title of the book: "the extension of common sense by other means". Ellenberg's talent for finding real-life situations that enshrine mathematical principles would be the envy of any math teacher.
Atheists will generally take offense to having their way of thinking referred to as a "belief".
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"She's trying to make me be a chaperone for Steve Urkel," Ms. Bent recalled thinking, referring to the über-nerd character in the sitcom "Family Matters," with his trademark high-water pants, owlish bifocals and zero sex appeal.
"I've actually done this, I'm in yellow, this is the last day of G.C. and nobody is going to take it away from me," he recalled thinking, referring to the general classification, the group of contenders for the overall victory.
Accordingly, "cathedral thinking" refers to deep dedication to a complex endeavor that will outlive its architects.
"Systems thinking" refers to bringing a perspective to problem solving in which the problem space is conceptualized as a system of interrelated component parts.
The term concept formation describes how a person learns to form classes, whereas the term conceptual thinking refers to an individual's subjective manipulation of those abstract classes.
Concepts are used to sort specific experiences into general rules or classes, and conceptual thinking refers to a person's subjective manipulations of those abstract classes.
Many of his techniques involve manipulating what he terms the "hot" and the "cold" modes of thinking (referring to our powerful emotional and instinctive drives as "hot" and our weaker calculating, rational parts as "cold").
Metacognition, sometimes described as "thinking about your own thinking," refers to knowledge about one's own thoughts and cognitive processes as well as the cognitive regulation involved in directing one's learning.
Here's a technical definition: Visible thinking refers to any kind of observable representation that documents and supports the development of an individual's or group's ongoing thoughts, questions, reasons, and reflections.
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Since I tried Ludwig back in 2017, I have been constantly using it in both editing and translation. Ever since, I suggest it to my translators at ProSciEditing.

Justyna Jupowicz-Kozak
CEO of Professional Science Editing for Scientists @ prosciediting.com