By relieving the brain of all unnecessary work, a good notation sets it free to concentrate on more advanced problems, and in effect increases the mental power of the race. —A. N. Whitehead
I'm collecting quotes on interesting notations—both powerful ones and bad ones—and how they influence thought.
This list focuses on notation as "a series or system of written symbols used to represent numbers, amounts, or elements in something such as music or mathematics." This is distinct from a language (computer or natural), interface, diagram, visualization, or tool, but may overlap with them. This list also focuses on examples from math, physics, computer science, and writing systems, though I'm looking to expand it to interesting examples of dance and music notation. General suggestions and pull requests are welcome. (Get in touch!)
- 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
- Bret 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
- Heyward's article "How to Write a Dance" on why dance notation remains unused
- Missing: music notation and the cognitive effects of notation.
A good 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.
Negate these for bad notations. (Most notations are good and bad; they have advantages and disadvantages.)
Consider: do all objects correspond to some notation? does every notation correspond to some object? uniquely?
source | tags: juggling, math, describing hard-to-describe objects, discovery
(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.
source | tags: math, juggling
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.
source | tags: knots, math, describing hard-to-describe objects
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 of 10 crossings or less, and we consider some properties of their algebraic invariants whose discovery was a consequence of this notation. The enumeration process is eminently suitable for machine computation, and should then handle knots and links of 12 or 13 crossings quite readily. Recent attempts at computer enumeration have proved unsatisfactory mainly because of the lack of a suitable notation… Little tells us that the enumeration of the 54 knots of [6] took him 6 years—from 1893 to 1899—the notation we shall soon describe made this just one afternoon’s work!
For an accessible introduction, see my talk at Strange Loop 2015. (video, materials)
source | tags: art, sculpture, 2D, visual, synesthesia
Sonakinatography Composition XVII, 1987-2004. Courtesy Estate of Channa Horwitz, Photography by Timo Ohler
Who? When an LA Times review of her work referred to contemporary artist Channa Horwitz as a housewife, it epitomised everything art historian Linda Nochlin wrestled with in her pioneering essay in 1971, Why Have There Been No Great Women Artists? Despite studying with James Turrell and Allan Kaprow at CalArts in the 1970s, and exchanging letters with Sol LeWitt, Horwitz remained very much an outlier of the California art world until the last few years of her life.
The Los Angeles native created hand-drawn algorithms combining basic principles and strict geometry to generate measured patterns, many of which resemble Aztec prints from a distance. Like her successful male colleagues, she was interested in bringing together colour, movement, sound and light, and introduced unbendable logic into the realm of west coast minimalism with her synaesthetic compositions.
Her breakthrough moment in fact grew out of a rejected proposal for an ambitious kinetic sculpture, as part of LACMA’s innovative Art and Technology exhibition in 1968, which infamously featured no female artists. Diagrams she drew detailing the sculpture’s movement went on to inform her work for the next four decades.
Time Structure Composition III, Sonakinatography I, 1970. Channa Horwitz, Courtesy Estate of Channa Horwitz, Photography by Timo Ohler
What? Horwitz’s Sonakinatography, a colour-coordinated system of notations based solely on the numbers one through eight, was, in particular, an unlikely meeting between new age thought and mathematical reason. The series took shape as a collection of labour-intensive drawings, and as each number corresponds not only to a colour, but also to a duration or beat, these intricately checked and ruled works on paper can function as visual scores or instructions for music or dance.
Drawing was Horwitz’s preferred way of working, mostly on Mylar graph paper with ink and milk-based paint, and she spent the majority of her 50-year career expanding on Sonakinatography – a term of her own invention combining the Greek words for sound, movement and notation – and another group of works, Language Series, first started several years earlier. (...)
Language Series I, 1964-2004. Channa Horwitz, Courtesy Collection Oehmen, Germany
Why? In 1964, casting a glance back over her time studying art at California State University, Horwitz moved on from the programme’s expressionist agenda and instead coined her rigorous, controlled visual language. By confining herself to a few simple rules, she rebelled through discipline and discovered the patterns and shapes that would become a lifelong fixture in her work.
“I have created a visual philosophy by working with deductive logic,” she wrote in Art Flash in 1976. “I had a need to control and compose time as I had controlled and composed two-dimensional drawings and paintings.”
source | tags: math, physics, quantum mechanics, linear algebra, manipulation
Notation can help us substantially in thinking about and manipulating symbolic representations meant to describe complex physical phenomena. The brain’s working memory can only manipulate a small number of ideas at once (“7±2”). We handle complex ideas by “chunking” — binding together many things and manipulating them as a single object. Another way we extend our range is by storing information outside of our brains temporarily and manipulating external objects or symbols, like an abacus or equations written on a piece of paper. (…)
A similar situation pertains for dealing with linear spaces. In some cases, we might want to describe a system of coupled oscillators with the coordinates of the masses. In other cases, we might want to describe them in terms of how much of each normal mode is excited. This change corresponds to a change of coordinates in the linear space describing the state of the system. We would like to have a representation that describes the state without specifying the particular coordinates used to describe them. (…)
The Dirac notation for states in a linear space is a way of representing a state in a linear space in a way that is free of the choice of coordinate but allows us to insert a particular choice of coordinates easily and to convert from one choice of coordinates to another conveniently. Furthermore, it is oriented in a way (bra vs. ket) that allows us to keep track of whether we need to take complex conjugates or not. This is particularly useful if we are in an inner-product space. To take the length of a complex vector, we have to multiply the vector by its complex conjugate — otherwise we won’t get a positive number. The orientation of the Dirac representation allows us to nicely represent the inner product in a way that keeps careful track of complex conjugation.
source | tags: cognitive science, evaluation, synthesis
The authors compile a list of dimensions, including the following particularly interesting ones:
- Creative ambiguity. The extent to which a notation encourages or enables the user to see something different when looking at it a second time (based on work by Hewson (1991), by Goldschmidt (1991), and by Fish and Scrivener (1990))
- Free rides. New information is generated as a result of following the notational rules (based on work by Cheng (1998) and by Shimojima (1996))
- Useful awkwardness. It’s not always good to be able to do things easily. Awkward interfaces can force the user to reflect on the task, with an overall gain in efficiency (based on discussions with Marian Petre, and work by O’Hara & Payne (1999))
See the paper for the full list.
source | tags: math, properties
Note: read the original document for proper math typesetting.
Everybody is familiar with one special case of an Iverson-like convention, the “Kronecker delta” symbol δik = 1 , i = k; 0 , i 6= k. (1.16)
Leopold Kronecker introduced this notation in his work on bilinear forms [30, page 276] and in his lectures on determinants (see [31, page 316]); it soon became widespread. Many of his followers wrote δ k j , which is a bit more ambiguous because it conflicts with ordinary exponentiation. I now prefer to write [j = k] instead of δjk, because Iverson’s convention is much more general. Although ‘[j = k]’ involves five written characters instead of the three in ‘δjk’, we lose nothing in common cases when ‘[j = k + 1]’ takes the place of ‘δj(k+1)’.
Another familiar example of a 0–1 function, this time from continuous mathematics, is Oliver Heaviside’s unit step function [x ≥ 0]. (See [44] and [37] for expositions of Heaviside’s methods.) It is clear that Iverson’s convention will be as useful with integration as it is with summation, perhaps even more so. I have not yet explored this in detail, because [15] deals mostly with sums.
It’s interesting to look back into the history of mathematics and see how there was a craving for such notations before they existed. For example, an Italian count named Guglielmo Libri 4 published several papers in the 1830s concerning properties of the function 00 x. (...)
If you are a typical hard-working, conscientious mathematician, interested in clear exposition and sound reasoning—and I like to include myself as a member of that set—then your experiences with Iverson’s convention may well go through several stages, just as mine did. First, I learned about the idea, and it certainly seemed straightforward enough. Second, I decided to use it informally while solving problems. At this stage it seemed too easy to write just ‘[k ≥ 0]’; my natural tendency was to write something like ‘δ(k ≥ 0)’, giving an implicit bow to Kronecker, or ‘τ (k ≥ 0)’ where τ stands for truth. Adriano Garsia, similarly, decided to write ‘χ(k ≥ 0)’, knowing that χ often denotes a characteristic function; he has used χ notation effectively in dozens of papers, beginning with [10], and quite a few other mathematicians have begun to follow his lead. (Garsia was one of my professors in graduate school, and I recently showed him the first draft of this note. He replied, “My definition from the very start was χ(A) = n 1 if A is true 0 if A is false 7 where A is any statement whatever. But just like you, I got it by generalizing from Iverson’s APL. . . . I don’t have to tell you the magic that the use of the χ notation can do.”) If you go through the stages I did, however, you’ll soon tire of writing δ, τ , or χ, when you recognize that the notation is quite unambiguous without an additional symbol. Then you will have arrived at the philosophical position adopted by Iverson when he wrote [21].
I introduced these notations in the first edition of my first book [25], and by now my students and I have accumulated some 25 years of experience with them; the conventions have served us well. However, such brackets and braces have still not become widely enough adopted that they could be considered “standard.” For example, Stanley’s magnificent book on Enumerative Combinatorics [51] uses c(n, k) for (n k) and S(n, k) for {n k}. His notation conveys combinatorial significance, but it fails to suggest the analogies to binomial coefficients that prove helpful in manipulations. Such analogies were evidently not important enough in his mind to warrant an extravagant two-line notation...
Naturally I wondered how I could have been working with Stirling numbers for so many years without having been aware of such a basic fact. Surely it must have been known before? After several hours of searching in the library, I learned that identity (2.4) had indeed been known, but largely forgotten by succeeding generations of mathematicians, primarily because previous notations for Stirling numbers made it impossible to state the identity in such a memorable form. These investigations also turned up several things about the history of Stirling numbers that I had not previously realized.
source | tags: math, notation design
User friedo asks:
And user alex.jordan answers:
Note how visual properties of the notation enabled them to easily write down new identities. The entire thread is interesting as an example of collaborative notation design and debate.
The answer above was later adapted by 3Blue1Brown into the video "The Triangle of Power."
