Notation as a Tool of Thought

The Technological Turn in Mathematics

Quickly evolving technologies, such as Interactive Theorem Provers (ITPs), Automated Theorem Provers (ATPs), and Large Language Models (LLMs), all falling under the general heading 'AI for mathematics,' are transforming mathematical practice in profound ways. This chapter explores the implications of these innovations, focusing on their impact on how mathematical knowledge is created and shared. It also discusses how they are reshaping the social dimension of mathematics, altering collaboration dynamics, trust relationships, and the collective production of knowledge. For instance, tools like ITPs facilitate large-scale collaborations and make new types of teamwork possible, where trust is not a necessary ingredient. ITPs also help us mitigate our human fallibility, yet they raise questions about the nature of formalization and the relationship between traditional and formal mathematics. Technologies such as LLMs are reshaping the division of epistemic labour between humans and machines and urge philosophers of mathematics to ask questions about the value of their work.

J Notation as a Tool of Thought

Kenneth Iverson’s 1964 language, APL, won him the Turing Award. His award lecture, Notation as a Tool of Thought, argued that better notations would lead people to deeper insights about mathematics. He provided a number of examples ranging across linear algebra, arithmetic, probability, and logic. Unfortunately, most of the mathematics he covers isn’t relevant to programming. However, his core idea still applies, and changing how we describe programs changes how we think about them.

Media for Thinking the Unthinkable

For more information on the context behind this talk, seeAn Ill-Advised Personal Note about "Media for Thinking the Unthinkable".

Kill Math

The power to understand and predict the quantities of the world should not be restricted to those with a freakish knack for manipulating abstract symbols.

Using spaced repetition systems to see through a piece of mathematics

By Michael Nielsen, January 2019

Reinventing Explanation

The Babylonian Map of the World is one of the world's oldest extant maps, dating to 600 BCE. It's a crude map, difficult to read at a glance, but fortunately an accompanying cuneiform text describes the features on the map, including Babylon, seven other cities, a canal, and a mountain: Modern maps are, of course, far better than this early map. They improve on it by taking advantage of the many map-making techniques developed since 600 BCE, such as: surveying to get proportions correct; projections to correct for the curvature of the Earth; methods to depict topographic features; and so on. Even ideas such as showing roads and nautical routes were not a priori obvious, but had to be invented.

Magic Paper

Working notes by Michael Nielsen, November 2017. Followup to (but doesn't require) my notes on Chalktalk.

How can we develop transformative tools for thought?

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Thought as a Technology

Have you ever felt awe and delight upon first experiencing a computer interface? An interface that surprised you with its strangeness, with a sense of entering an alien world?

Notation as a Tool of Thought