Notation and thought

This syllabus examines the design of notation. We concern ourselves chiefly with one question: how does working in a particular notational system influence the ways that people think and create in it?

Overview

• Wright's talk on inventing juggling notation (siteswap) and using it to discover new tricks
• Conway’s paper on powerful knot notation for knot enumeration (more accessible: a talk I gave)
• Channa Horwitz's work on sonakinatography: visual notations for sound, motion, and sculpture
• bra-ket notation (Dirac notation) in quantum mechanics
• Petre, Green, et al.'s paper "Cognitive Dimensions of Notations: Design Tools for Cognitive Technology"
• Knuth's note on Iverson’s convention and Stirling numbers
• MathOverflow thread on designing a unified, visual notation for exponents, logs, and roots
• An overview of other good math notation (the equality sign, algebra, variables, dy/dx (debatable), Einstein notation)
• Sussman’s Structure and Interpretation of Classical Mechanics: a book on physics as function composition and code
• Wolfram's keynote "Mathematical Notation: Past and Future" (specifically, empirical laws thereof)
• Iverson's notes on good mathematical notation design and APL
• Victor's comments on Roman numerals (a bad notation) vs. Arabic numerals
• Gilles Fauconner's and Mark Turner's book The Way We Think
• Borges’ short story “Funes the Memorious” on memory and number systems
• Chiang’s short story “The Truth of Fact, the Truth of Feeling” on oral culture vs. literacy
• Ong’s book “Orality and Literacy” on how writing restructures consciousness; writing as a technology; development of writing
• Chiang’s short story “Story of Your Life” on notation restructuring thought temporally
• Chiang’s article “Bad Character” on Chinese characters/pictograms (a bad notation) vs. phonetic alphabets, and the backlash to this article summarized on Language Log
• Heyward's article "How to Write a Dance" on why dance notation remains unused
• Conversation analysis notation (Jefferson transcription notation): overview, examples

Themes

A notation may:

• allow us to enumerate objects by serializing them
• enable us to manipulate objects and perform operations on them more easily
• look beautiful
• lift the one-dimensional to the two-dimensional
• allow searchability
• encode powerful theorems
• make important properties obvious, or encode them
• encourage us to predict and invent new things
• reveal underlying mathematical structure
• perform good bookkeeping
• suggest useful analogies
• be brief and expressive
• avoid ambiguity, or introduce useful ambiguity.

How are notations mapped to objects, how are objects mapped to notations, and what are the properties of that mapping? (e.g. one-to-one, many-to-one, one-to-many?)

Quotes

“Relatively Prime: The Unexpected” by Colin Wright

source

(10:13-15:50) I went around to people and said, “Show me a trick! Show me something interesting to do with three, and people showed me things like “one over the top,” and “one-high,” and “one-high pirouette,” and “behind the back” and “under the leg” and so on. And I wrote all of these down.

And I went up to a guy called Mike Day and I said, “Show me a three ball trick.” And he showed me the most amazing three-ball trick. Anyone who juggles three balls semi-seriously will know of this trick called “Mills’ Mess.” And I was stumped—I could not write down a description of “Mills’ Mess.” It was amazing. But now that I know it really well, it’s not actually that complicated! But back then it was completely, mind-blowingly complex.

And there was no way to write it down! And we thought, “There must be ways of ways of writing down juggling tricks.” There are ways of writing down language, there are ways of writing down music, ways of writing down dance, actually, multiple ways of writing down dance, so there must be a way of writing down juggling tricks. And we looked through all the back issues of the juggling magazines we had—there are, actually, magazines published about juggling—we looked through all the back issues, and none of them had descriptions of juggling tricks. So we decided to invent a notation for juggling. Now this didn’t happen overnight—this took some considerable time—and our early attempts were very poor. They were inadequate to describe many of the tricks we thought a notation should be able to describe. And eventually we hit on a scheme that seemed to work. And we used it to write down loads of different juggling tricks that we knew.

We discovered that if we arranged those tricks in just the right way, they fell into a pattern. There was an underlying, unsuspected structure. As long as you had the courage to leave gaps. And this goes back to things like the Periodic Table, when Mendeley was writing down all the elements—he realized that if you arranged them all according to function, then there were gaps, and that then predicted the existence of chemical elements.

Well, we were predicting the existence of juggling tricks. And it worked! We actually found juggling tricks that no one had ever done before. And when we took these to juggling conventions, people literally sat at my feet for days to try to learn some of these tricks. And months later, at another juggling convention, people from—in particular, I remember going to the European Juggling Convention—and people from America were trying to teach me a juggling trick that I had shown people just a few months earlier at the British Juggling Convention.

See, these were tricks that had gone right round the world suddenly, and people thought they were new. We don’t know for certain that these had never been done before, because there was no written record! But nevertheless, all the evidence is that these were entirely new juggling tricks. Which now form a large part of the canon of early juggling. Some of these tricks are really easy, but some of them are phenomenally difficult. In fact, there’s a two-ball juggling trick that’s pretty much as difficult as juggling five. There’s a whole range of these.

And of course, if you get this kind of thing happening, there’s going to be some kind of structure underneath; there’s going to be mathematics to describe it. And so that’s how we stumbled across unsuspected mathematical structure underlying juggling tricks. And then when I went to the British Maths Colloquium, there was a session that was going to be canceled because there were insufficient speakers, and I offered to give a twenty-minute talk, and I stood up and just sort of rambled on for twenty minutes about the maths of juggling with demonstrations. And afterwards, people invited me to speak at their son’s local school, and to come along to the local maths association meetings. I did three or four talks that year, and that was in 1985, and since then it’s just continued to grow, and for the last eight or ten years, I’ve done between 80 and 100 talks every year, most of which are on the mathematics of juggling.

See also: Wright's juggling talk

source

In 1985 there arose, simultaneously in three places around the world, by groups entirely unconnected and completely ignorant of each others' existence, a notation for juggling tricks. The notation was incomplete, since not every trick could be described, and like many notations, it was not immediately apparent to the uninitiated how to read it, how to use it, or whether it would be of any real use. For those who understood it, however, it was instantly obvious that it was right. Somehow the notation managed to capture the essence of those tricks it described, and the fact that the same notation arose in more than one place at once showed that its time had come, and it was, quite simply, the notation.

"An enumeration of knots and links, and some of their algebraic properties" by John Conway

source

In this paper, we describe a notation in terms of which it has been found possible to list (by hand) all knots of 11 crossings or less, and all links