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https://github.com/kai-qu/notation

Collection of quotes on notation design & how it affects thought.
https://github.com/kai-qu/notation

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Collection of quotes on notation design & how it affects thought.

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# 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](http://relprime.com/theunexpected/) (siteswap) and using it to discover new tricks

* Conway’s paper on [powerful knot notation](http://www.maths.ed.ac.uk/~aar/papers/conway.pdf) for knot enumeration (more accessible: a [talk](https://www.youtube.com/watch?v=Wahc9Ocka1g) I gave)

* Channa Horwitz's work on [sonakinatography](http://www.anothermag.com/art-photography/8576/how-channa-horwitz-permeated-las-1960s-art-scene): visual notations for sound, motion, and sculpture

* [bra-ket notation](http://www.physics.umd.edu/courses/Phys374/fall04/files/DiracNotation.pdf) (Dirac notation) in quantum mechanics

* Petre, Green, et al.'s paper ["Cognitive Dimensions of Notations: Design Tools for Cognitive Technology"](https://www.cl.cam.ac.uk/~afb21/publications/CT2001.pdf)

* Knuth's note on [Iverson’s convention and Stirling numbers](http://www.maa.org/sites/default/files/images/images/upload_library/22/Ford/knuth403-422.pdf)

* MathOverflow thread on designing a [unified, visual notation](https://math.stackexchange.com/questions/30046/alternative-notation-for-exponents-logs-and-roots) for exponents, logs, and roots

* An [overview](https://link.springer.com/chapter/10.1007%2F978-3-642-41422-0_2) of other good math notation (the equality sign, algebra, variables, dy/dx (debatable), Einstein notation)

* Sussman’s _[Structure and Interpretation of Classical Mechanics](https://mitpress.mit.edu/sites/default/files/titles/content/sicm/book-Z-H-5.html)_: a book on physics as function composition and code

* Wolfram's [keynote](http://www.stephenwolfram.com/publications/mathematical-notation-past-future/) "Mathematical Notation: Past and Future" (specifically, empirical laws thereof)

* Iverson's notes on [good mathematical notation design and APL](http://www.eecg.toronto.edu/~jzhu/csc326/readings/iverson.pdf)

* Victor's comments on [Roman numerals (a bad notation) vs. Arabic numerals](http://www.fastcodesign.com/1664508/ex-apple-designer-creates-teaching-ui-that-kills-math-using-data-viz)

* Gilles Fauconner's and Mark Turner's book [The Way We Think](http://www.cogsci.ucsd.edu/~nunez/COGS152_S12/TWWT_Ch1.pdf)

* Borges’ short story “[Funes the Memorious](http://www4.ncsu.edu/~jjsakon/FunestheMemorious.pdf)” on memory and number systems

* Chiang’s short story “[The Truth of Fact, the Truth of Feeling](http://subterraneanpress.com/magazine/fall_2013/the_truth_of_fact_the_truth_of_feeling_by_ted_chiang)” on oral culture vs. literacy

* Ong’s book “[Orality and Literacy](https://monoskop.org/images/d/db/Ong_Walter_J_Orality_and_Literacy_2nd_ed.pdf)” on how writing restructures consciousness; writing as a technology; development of writing

* Chiang’s short story “[Story of Your Life](http://www.kameli.net/~raimu/rnd/ted-chiang-story-of-your-life-2000.pdf)” on notation restructuring thought temporally

* Chiang’s article “[Bad Character](http://www.newyorker.com/magazine/2016/05/16/if-chinese-were-phonetic)” on Chinese characters/pictograms (a bad notation) vs. phonetic alphabets, and the [backlash](http://languagelog.ldc.upenn.edu/nll/?p=25876) to this article summarized on _Language Log_

* Heyward's article ["How to Write a Dance"](http://www.theparisreview.org/blog/2015/02/04/how-to-write-a-dance/) on why dance notation remains unused

* Conversation analysis notation (Jefferson transcription notation): [overview](https://www2.le.ac.uk/departments/psychology/research/child-mental-health/cara-1/faqs/jefferson), [examples](https://www.kcl.ac.uk/sspp/departments/education/research/Research-Centres/ldc/knowledge-transfer/DATA/part3.pdf)



## 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](http://relprime.com/theunexpected/)



_(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](http://www.solipsys.co.uk/new/JugglingTalkSummary.html?JugglingTalk)



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](http://www.maths.ed.ac.uk/~aar/papers/conway.pdf)



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](https://www.youtube.com/watch?v=Wahc9Ocka1g), [materials](http://katherineye.com/post/129960474471/strange-loops-capturing-knots-with-powerful))



### Channa Horwitz's sonakinetography 

[source](http://www.anothermag.com/art-photography/8576/how-channa-horwitz-permeated-las-1960s-art-scene)






_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.”



### Dirac's bra-ket notation

[source](http://www.physics.umd.edu/courses/Phys374/fall04/files/DiracNotation.pdf)



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.



### "Cognitive dimensions of notations: design tools for cognitive technology"

[source](https://www.cl.cam.ac.uk/~afb21/publications/CT2001.pdf)



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.



### “Two Notes on Notation” by Don Knuth

[source](http://www.maa.org/sites/default/files/images/images/upload_library/22/Ford/knuth403-422.pdf) 



_Note: read the original document for proper math typesetting._



#### Iverson’s convention 



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].



#### Stirling numbers



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.



### "Alternative notation for exponents, logs, and roots?" on MathOverflow

[source](https://math.stackexchange.com/questions/30046/alternative-notation-for-exponents-logs-and-roots)



User **friedo** asks:



![](http://i.imgur.com/OuLIwEP.jpg)



And user **alex.jordan** answers:



![](http://i.imgur.com/JLVkDI6.jpg)



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](https://www.youtube.com/watch?v=sULa9Lc4pck)."



### “Kenneth Iverson: Notation and Thinking” by Lipton and Regan

[source](https://link.springer.com/chapter/10.1007%2F978-3-642-41422-0_2) (available free online too)



Outline of good notations

* The equality symbol (Records, 1557)

* letters as unknowns/variables (Descartes)

* dy/dx derivative notation (Leibniz)—but contentious! 

  * "For example, Spivak's Calculus book devotes a couple pages to describing it and then says that it's actually quite a confusing/fuzzy notation and we're never going to use it again, but you should be able to recognize it." _Alan O'Donnell_

* Congruence notation (Gauss)

* Dirac’s bra-ket notation (1939)

* Einstein notation



### _Structure and Interpretation of Classical Mechanics_ by Gerald Sussman



[source](https://mitpress.mit.edu/sites/default/files/titles/content/sicm/book-Z-H-5.html)



Classical mechanics is deceptively simple. It is surprisingly easy to get the right answer with fallacious reasoning or without real understanding. Traditional mathematical notation contributes to this problem. Symbols have ambiguous meanings that depend on context, and often even change within a given context. For example, a fundamental result of mechanics is the Lagrange equations. In traditional notation the Lagrange equations are written



![Lagrange](https://mitpress.mit.edu/sites/default/files/titles/content/sicm/front-Z-G-1.gif)



The Lagrangian L must be interpreted as a function of the position and velocity components q^i and ![](https://mitpress.mit.edu/sites/default/files/titles/content/sicm/front-Z-G-D-1.gif)^i, so that the partial derivatives make sense, but then in order for the time derivative d/dt to make sense solution paths must have been inserted into the partial derivatives of the Lagrangian to make functions of time. The traditional use of ambiguous notation is convenient in simple situations, but in more complicated situations it can be a serious handicap to clear reasoning. In order that the reasoning be clear and unambiguous, we have adopted a more precise mathematical notation. Our notation is functional and follows that of modern mathematical presentations. An introduction to our functional notation is in an appendix.



Computation also enters into the presentation of the mathematical ideas underlying mechanics. We require that our mathematical notations be explicit and precise enough that they can be interpreted automatically, as by a computer. As a consequence of this requirement the formulas and equations that appear in the text stand on their own. They have clear meaning, independent of the informal context. For example, we write Lagrange's equations in functional notation as follows:



![](https://mitpress.mit.edu/sites/default/files/titles/content/sicm/front-Z-G-4.gif)



The Lagrangian L is a real-valued function of time t, coordinates x, and velocities v; the value is L(t, x, v). Partial derivatives are indicated as derivatives of functions with respect to particular argument positions; ![](https://mitpress.mit.edu/sites/default/files/titles/content/sicm/front-Z-G-D-2.gif)_2L indicates the function obtained by taking the partial derivative of the Lagrangian function L with respect to the velocity argument position. The traditional partial derivative notation, which employs a derivative with respect to a "variable," depends on context and can lead to ambiguity. The partial derivatives of the Lagrangian are then explicitly evaluated along a path function q. The time derivative is taken and the Lagrange equations formed. Each step is explicit; there are no implicit substitutions.



Computational algorithms are used to communicate precisely some of the methods used in the analysis of dynamical phenomena. Expressing the methods of variational mechanics in a computer language forces them to be unambiguous and computationally effective. Computation requires us to be precise about the representation of mechanical and geometric notions as computational objects and permits us to represent explicitly the algorithms for manipulating these objects. Also, once formalized as a procedure, a mathematical idea becomes a tool that can be used directly to compute results.



Active exploration on the part of the student is an essential part of the learning experience. Our focus is on understanding the motion of systems; to learn about motion the student must actively explore the motion of systems through simulation and experiment. The exercises and projects are an integral part of the presentation.



That the mathematics is precise enough to be interpreted automatically allows active exploration to be extended to it. The requirement that the computer be able to interpret any expression provides strict and immediate feedback as to whether the expression is correctly formulated. Experience demonstrates that interaction with the computer in this way uncovers and corrects many deficiencies in understanding. (...)



When we started we expected that using this approach to formulate mechanics would be easy. We quickly learned that many things we thought we understood we did not in fact understand. Our requirement that our mathematical notations be explicit and precise enough that they can be interpreted automatically, as by a computer, is very effective in uncovering puns and flaws in reasoning. The resulting struggle to make the mathematics precise, yet clear and computationally effective, lasted far longer than we anticipated. We learned a great deal about both mechanics and computation by this process. We hope others, especially our competitors, will adopt these methods, which enhance understanding while slowing research.



### "Mathematical Notation: Past and Future" by Stephen Wolfram (2000)

[source](http://www.stephenwolfram.com/publications/mathematical-notation-past-future/)



#### Empirical laws for mathematical notations



In the study of ordinary natural language there are various empirical historical laws that have been discovered. An example is Grimm's Law, which describes general historical shifts in consonants in Indo-European languages. I have been curious whether empirical historical laws can be found for mathematical notation.



Dana Scott suggested one possibility: a trend towards the removal of explicit parameters.



As one example, in the 1860s it was still typical for each component in a vector to be a separately-named variable. But then components started getting labelled with subscripts, as in a_i. And soon thereafter—particularly through the work of Gibbs—vectors began to be treated as single objects, denoted say by ![](http://www.stephenwolfram.com/publications/mathematical-notation-past-future/Images/revisedtalk.xml_gr_51.gif) or **a**.



With tensors things are not so straightforward. Notation that avoids explicit subscripts is usually called "coordinate free." And such notation is common in pure mathematics. But in physics it is still often considered excessively abstract, and explicit subscripts are used instead.



With functions, there have also been some trends to reduce the mention of explicit parameters. In pure mathematics, when functions are viewed as mappings, they are often referred to just by function names like f, without explicitly mentioning any parameters.



But this tends to work well only when functions have just one parameter. With more than one parameter it is usually not clear how the flow of data associated with each parameter works.



However, as early as the 1920s, it was pointed out that one could use so-called combinators to specify such data flow, without ever explicitly having to name parameters.



Combinators have not been used in mainstream mathematics, but at various times they have been somewhat popular in the theory of computation, although their popularity has been reduced through being largely incompatible with the idea of data types.



### “Notation as a Tool of Thought” by Kenneth Iverson



[source](http://www.eecg.toronto.edu/~jzhu/csc326/readings/iverson.pdf)



The quantity of meaning compressed into small space by algebraic signs, is another circumstance that facilitates the reasonings we are accustomed to carry on by their aid. —_Charles Babbage_



In addition to the executability and universality emphasized in the introduction, a good notation should embody characteristics familiar to any user of mathematical notation: 

* Ease of expressing constructs arising in problems. 

* Suggestivity. 

* Ability to subordinate detail. 

* Economy. 

* Amenability to formal proofs. (...)



If it is to be effective as a tool of thought, a notation must allow convenient expression not only of notions arising directly from a problem but also of those arising in subsequent analysis, generalization, and specialization. (...)



A notation will be said to be suggestive if the forms of the expressions arising in one set of problems suggest related expressions which find application in other problems. 



See also: **_A History of Mathematical Notations_ by Cajori**.



### Bret Victor (after Alan Kay) on Roman numerals



[source](http://www.fastcodesign.com/1664508/ex-apple-designer-creates-teaching-ui-that-kills-math-using-data-viz)



I don't hate math per se; I hate its current representations. Have you ever tried multiplying Roman numerals? It's incredibly, ridiculously difficult. That's why, before the 14th century, everyone thought that multiplication was an incredibly difficult concept, and only for the mathematical elite. Then Arabic numerals came along, with their nice place values, and we discovered that even seven-year-olds can handle multiplication just fine. There was nothing difficult about the concept of multiplication—the problem was that numbers, at the time, had a bad user interface. 



### "The Way We Think" by Gilles Fauconnier and Mark Turner



[source](http://www.cogsci.ucsd.edu/~nunez/COGS152_S12/TWWT_Ch1.pdf)



The development of formal systems to leverage human invention and insight has been a painful, centuries-long process. (...)

In the twelfth century, the Hindu mathematician Bhaskara said, "The root of the root of the quotient of the greater irrational

divided by the lesser one being increased by one; the sum being squared and multiplied by the smaller irrational quantity is the

sum of the two surd roots." This we would now express in the form of an equation, using the much more systematically manageable set

of formal symbols shown below. This equation by itself looks no less opaque than Bhaskara's description, but the notation immediately

connects it to a large system of such equations in ways that make it easy to manipulate.



![sqrt((sqrt(n/k)+1)^2 * k) = sqrt(k) + sqrt(n)](http://quicklatex.com/cache3/74/ql_921cefbda62f2dc5dd5465686646b574_l3.png)



### “Funes the Memorious” by Jorge Luis Borges

[source](http://www4.ncsu.edu/~jjsakon/FunestheMemorious.pdf)



_Spoilers follow._



The voice of Funes, out of the darkness, continued. He told me that toward 1886 he had devised a new system of enumeration and that in a very few days he had gone beyond twenty-four thousand. He had not written it down, for what he once meditated would not be erased. The first stimulus to his work, I believe, had been his discontent with the fact that "thirty-three Uruguayans" required two symbols and three words, rather than a single word and a single symbol. Later he applied his extravagant principle to the other numbers. In place of seven thousand thirteen, he would say (for example) Maximo Perez; in place of seven thousand fourteen, The Train; other numbers were Luis Melian Lafinur, Olimar, Brimstone, Clubs, The Whale, Gas, The Cauldron, Napoleon, Agustin de Vedia. In lieu of five hundred, he would say nine. Each word had a particular sign, a species of mark; the last were very complicated. . . . I attempted to explain that this rhapsody of unconnected terms was precisely the contrary of a system of enumeration. I said that to say three hundred and sixty-five was to say three hundreds, six tens, five units: an analysis which does not exist in such numbers as The Negro Timoteo or The Flesh Blanket. Funes did not understand me, or did not wish to understand me. 



Locke, in the seventeenth century, postulated (and rejected) an impossible idiom in which each individual object, each stone, each bird and branch had an individual name; Funes had once projected an analogous idiom, but he had renounced it as being too general, too ambiguous. In effect, Funes not only remembered every leaf on every tree of every wood, but even every one of the times he had perceived or imagined it. He determined to reduce all of his past experience to some seventy thousand recollections, which he would later define numerically. Two considerations dissuaded him: the thought that the task was interminable and the thought that it was useless. He knew that at the hour of his death he would scarcely have finished classifying even all the memories of his childhood. 



The two projects I have indicated (an infinite vocabulary for the natural series of numbers, and a usable mental catalogue of all the images of memory) are lacking in sense, but they reveal a certain stammering greatness. They allow us to make out dimly, or to infer, the dizzying world of Funes. He was, let us not forget, almost incapable of general, platonic ideas. It was not only difficult for him to understand that the generic term dog embraced so many unlike specimens of differing sizes and different forms; he was disturbed by the fact that a dog at three-fourteen (seen in profile) should have the same name as the dog at three-fifteen (seen from the front). His own face in the mirror, his own hands, surprised him on every occasion. Swift writes that the emperor of Lilliput could discern the movement of the minute hand; Funes could continuously make out the tranquil advances of corruption, of caries, of fatigue. He noted the progress of death, of moisture. He was the solitary and lucid spectator of a multiform world which was instantaneously and almost intolerably exact. Babylon, London, and New York have overawed the imagination of men with their ferocious splendor; no one, in those populous towers or upon those surging avenues, has felt the heat and pressure of a reality as indefatigable as that which day and night converged upon the unfortunate Ireneo in his humble South American farmhouse. It was very difficult for him to sleep. To sleep is to be abstracted from the world; Funes, on his back in his cot, in the shadows, imagined every crevice and every molding of the various houses which surrounded him. (I repeat, the least important of his recollections was more minutely precise and more lively than our perception of a physical pleasure or a physical torment) Toward the east, in a section which was not yet cut into blocks of homes, there were some new unknown houses. Funes imagined them black, compact, made of a single obscurity; he would turn his face in this direction in order to sleep. He would also imagine himself at the bottom of the river, being rocked and annihilated by the current. 



Without effort, he had learned English, French, Portuguese, Latin. I suspect, nevertheless, that he was not very capable of thought. To think is to forget a difference, to generalize, to abstract. In the overly replete world of Funes there were nothing but details, almost contiguous details. 



### “The Truth of Fact, the Truth of Feeling” by Ted Chiang



[source](http://subterraneanpress.com/magazine/fall_2013/the_truth_of_fact_the_truth_of_feeling_by_ted_chiang)



_Spoilers follow._



It was only many lessons later that Jijingi finally understood where he should leave spaces, and what Moseby meant when he said “word.” You could not find the places where words began and ended by listening. The sounds a person made while speaking were as smooth and unbroken as the hide of a goat’s leg, but the words were like the bones underneath the meat, and the space between them was the joint where you’d cut if you wanted to separate it into pieces. By leaving spaces when he wrote, Moseby was making visible the bones in what he said.



Jijingi realized that, if he thought hard about it, he was now able to identify the words when people spoke in an ordinary conversation. The sounds that came from a person’s mouth hadn’t changed, but he understood them differently; he was aware of the pieces from which the whole was made. He himself had been speaking in words all along. He just hadn’t known it until now.



### “Orality and Literacy” by Walter J. Ong



[source](http://dss-edit.com/prof-anon/sound/library/Ong_orality_and_literacy.pdf) 



#### Chapter 4: Writing restructures consciousness



Most persons are surprised, and many distressed, to learn that essentially the same objections commonly urged today against computers were urged by Plato in the Phaedrus (274–7) and in the Seventh Letter against writing. Writing, Plato has Socrates say in the Phaedrus, is inhuman, pretending to establish outside the mind what in reality can be only in the mind. It is a thing, a manufactured product. The same of course is said of computers. Secondly, Plato’s Socrates urges, writing destroys memory. Those who use writing will become forgetful, relying on an external resource for what they lack in internal resources. Writing weakens the mind. Today, parents and others fear that pocket calculators provide an external resource for what ought to be the internal resource of memorized multiplication tables. Calculators weaken the mind, relieve it of the work that keeps it strong. Thirdly, a written text is basically unresponsive. If you ask a person to explain his or her statement, you can get an explanation; if you ask a text, you get back nothing except the same, often stupid, words which called for your question in the first place. In the modern critique of the computer, the same objection is put, ‘Garbage in, garbage out’. Fourthly, in keeping with the agonistic mentality of oral cultures, Plato’s Socrates also holds it against writing that the written word cannot defend itself as the natural spoken word can: real speech and thought always exist essentially in a context of give-and-take between real persons.Writing is passive, out of it, in an unreal, unnatural world. So are computers. (...)



#### Writing is a technology

Writing is in a way the most drastic of the three technologies. It initiated what print and computers only continue, the reduction of dynamic sound to quiescent space, the separation of the word from the living present, where alone spoken words can exist.



By contrast with natural, oral speech, writing is completely artificial. There is no way to write ‘naturally’. (...)



Writing or script differs as such from speech in that it does not inevitably well up out of the unconscious. The process of putting spoken language into writing is governed by consciously contrived, articulable rules: for example, a certain pictogram will stand for a certain specific word, or a will represent a certain phoneme, b another, and so on. (This is not to deny that the writer-reader situation created by writing deeply affects unconscious processes involved in composing in writing, once one has learned the explicit, conscious rules. More about this later.) (...)



Many scripts across the world have been developed independently of one another (Diringer 1953; Diringer 1960; Gelb 1963): Mesopotamian cuneiform 3500 BC (approximate dates here from Diringer 1962), Egyptian hieroglyphics 3000 BC (with perhaps some influence from cuneiform), Minoan or Mycenean ‘Linear B’ 1200 BC, Indus Valley script 3000–2400 BC, Chinese script 1500 BC, Mayan script AD 50, Aztec script AD 1400. Scripts have complex antecedents. Most if not all scripts trace back directly or indirectly to some sort of picture writing, or, sometimes perhaps, at an even more elemental level, to the use of tokens. 



### “Story of Your Life” by Ted Chiang

[source](http://www.kameli.net/~raimu/rnd/ted-chiang-story-of-your-life-2000.pdf)



_Spoilers follow._



“Right. That modulation is applicable to lots of verbs. The logogram for ‘see’ can be modulated in the same way to form ‘see clearly,’ and so can the logogram for ‘read’ and others. And changing the curve of those strokes has no parallel in their speech; with the spoken version of these verbs, they add a prefix to the verb to express ease of manner, and the prefixes for ‘see’ and ‘hear’ are different.



“There are other examples, but you get the idea. It’s essentially a grammar in two dimensions.” 



He began pacing thoughtfully. “Is there anything like this in human writing systems?"



“Mathematical equations, notations for music and dance. But those are all very specialized; we couldn’t record this conversation using them. But I suspect, if we knew it well enough, we could record this conversation in the heptapod writing system. I think it’s a full-fledged, general-purpose graphical language.” (...)



When a Heptapod B sentence grew fairly sizable, its visual impact was remarkable. If I wasn’t trying to decipher it, the writing looked like fanciful praying mantids drawn in a cursive style, all clinging to each other to form an Escheresque lattice, each slightly different in its stance. And the biggest sentences had an effect similar to that of psychedelic posters: sometimes eye-watering, sometimes hypnotic. (...)



Much more interesting were the newly discovered morphological and grammatical processes in Heptapod B that were uniquely two-dimensional. Depending on a semagram’s declension, inflections could be indicated by varying a certain stroke’s curvature, or its thickness, or its manner of undulation; or by varying the relative sizes of two radicals, or their relative distance to another radical, or their orientations; or various other means. These were non-segmental graphemes; they couldn’t be isolated om the rest of a semagram. And despite how such traits behaved in human writing, these had nothing to do with calligraphic style; their meanings were defined according to a consistent and unambiguous grammar. (...)



Comparing that initial stroke with the completed sentence, I realized that the stroke participated in several different clauses of the message. It began in the semagram for ‘oxygen,’ as the determinant that distinguished it om certain other elements; then it slid down to become the morpheme of comparison in the description of the two moons’ sizes; and lastly it flared out as the arched backbone of the semagram for ‘ocean.’ Yet this stroke was a single continuous line, and it was the first one that Flapper wrote. That meant the heptapod had to know how the entire sentence would be laid out before it could write the very first stroke. 



The other strokes in the sentence also traversed several clauses, making them so interconnected that none could be removed without redesigning the entire sentence. The heptapods didn’t write a sentence one semagram at a time; they built it out of strokes irrespective of individual semagrams. I had seen a similarly high degree of integration before in calligraphic designs, particularly those employing the Arabic alphabet. But those designs had required careful planning by expert calligraphers. No one could lay out such an intricate design at the speed needed for holding a conversation. At least, no human could. (...)



I practiced Heptapod B at every opportunity, both with the other linguists and by myself. The novelty of reading a semasiographic language made it compelling in a way that Heptapod A wasn’t, and my improvement in writing it excited me. Over time, the sentences I wrote grew shapelier, more cohesive. I had reached the point where it worked better when I didn’t think about it too much. Instead of carefully trying to design a sentence before writing, I could simply begin putting down strokes immediately; my initial strokes almost always turned out to be compatible with an elegant rendition of what I was trying to say. I was developing a faculty like that of the heptapods. 



More interesting was the fact that Heptapod B was changing the way I thought. For me, thinking typically meant speaking in an internal voice’ as we say in the trade, my thoughts were phonologically coded. My internal voice normally spoke in English, but that wasn’t a requirement. The summer aer my senior year in high school, I attended a total immersion program for learning Russian; by the end of the Summer, I was thinking and even dreaming in Russian. But it was always spoken Russian. Different language, same mode: a voice speaking silently aloud. 



The idea of thinking in a linguistic yet non-phonological mode always intrigued me. I had a iend born of deaf parents; he grew up using American Sign Language, and he told me that he oen thought in ASL instead of English. I used to wonder what it was like to have one’s thoughts be manually coded, to reason using an inner pair of hands instead of an inner voice. With Heptapod B, I was experiencing something just as foreign: my thoughts were becoming graphically coded. There were trance-like moments during the day when my thoughts weren’t expressed with my internal voice; instead, I saw semagrams with my mind’s eye, sprouting like frost on a windowpane. 



As I grew more fluent, semagraphic designs would appear fully-formed, articulating even complex ideas all at once. My thought processes weren’t moving any faster as a result, though. Instead of racing forward, my mind hung balanced on the symmetry underlying the semagrams. The semagrams seemed to be something more than language; they were almost like mandalas. I found myself in a meditative state, contemplating the way in which premises and conclusions were interchangeable. There was no direction inherent in the way propositions were connected, no “train of thought” moving along a particular route; all the components in an act of reasoning were equally powerful, all having identical precedence. (...)



Consider the sentence “The rabbit is ready to eat.” Interpret “rabbit” to be the object of “eat,” and the sentence was an announcement that dinner would be served shortly. Interpret “rabbit” to be the subject of “eat,” and it was a hint, such as a young girl might give her mother so she’ll open a bag of Purina Bunny Chow. Two very different utterances; in fact, they were probably mutually exclusive within a single household. Yet either was a valid interpretation; only context could determine what the sentence meant.



Consider the phenomenon of light hitting water at one angle, and traveling through it at a different angle. Explain it by saying that a difference in the index of reaction caused the light to change direction, and one saw the world as humans saw it. Explain it by saying that light minimized the time needed to travel to its destination, and one saw the world as the heptapods saw it. Two very different interpretations. 



The physical universe was a language with a perfectly ambiguous grammar. Every physical event was an utterance that could be parsed in two entirely different ways, one casual and the other teleological, both valid, neither one disqualifiable no matter how much context was available. 



When the ancestors of humans and heptapods first acquired the spark of consciousness, they both perceived the same physical world, but they parsed their perceptions differently; the world-views that ultimately across were the end result of that diver- 31 gence. Humans had developed a sequential mode of awareness, while heptapods had developed a simultaneous mode of awareness. We experienced events in an order, and perceived their relationship as cause and effect. They experienced all events at once, and perceived a purpose underlying them all. A minimizing, maximizing purpose. (...)



I finished the last radical in the sentence, put down the chalk, and sat down in my desk chair. I leaned back and surveyed the giant Heptapod B sentence I’d written that covered the entire blackboard in my office. It included several complex clauses, and I had managed to integrate all of them rather nicely. 



Looking at a sentence like this one, I understood why the heptapods had evolved a semasiographic writing system like Heptapod B; it was better suited for a species with a simultaneous mode of consciousness. For them, speech was a bottleneck because it required that one word follow another sequentially. With writing, on the other hand, every mark on a page was visible simultaneously. Why constrain writing with a glottographic straitjacket, demanding that it be just as sequential as speech? It would never occur to them. Semasiographic writing naturally took advantage of the page’s two-dimensionality; instead of doling out morphemes one at a time, it offered an entire page full of them all at once. 



And now that Heptapod B had introduced me to a simultaneous mode of consciousness, I understood the rationale behind Heptapod A’s grammar: what my sequential mind had perceived as unnecessarily convoluted, I now recognized as an attempt to provide flexibility within the confines of sequential speech. I could use Heptapod A more easily as a result, though it was still a poor substitute for Heptapod B.



### “Bad Character” by Ted Chiang

[source](http://www.newyorker.com/magazine/2016/05/16/if-chinese-were-phonetic)



I’m a fan of literacy, and Chinese characters have been an obstacle to literacy for millennia. With a phonetic writing system like an alphabet or a syllabary, you need only learn a few dozen symbols and you can read most everything printed in a newspaper. With Chinese characters, you have to learn three thousand. And writing is even more difficult than reading; when you can’t use pronunciation as an aid to spelling, you have to rely on pure memorization. The cognitive demands are so great that even highly educated Chinese speakers regularly forget how to write characters they haven’t used recently.



The huge number of characters poses other obstacles as well. I’ve flipped through a Chinese dictionary, I’ve seen photographs of a Chinese typewriter, I’ve read about Chinese telegraphy, and despite their ingenuity they are all cumbersome inventions, wheelbarrows for the millstone around Chinese culture’s neck. Computers and smartphones are impossible to use if you’re restricted to Chinese characters; it’s only with phonetic systems of writing, like Bopomofo and Pinyin, that text entry becomes practical. In the past century, there have been multiple proposals to replace Chinese characters with an alphabet, all unsuccessful; the only reform ever implemented was to invent simplified versions of the more complex characters, which solved none of the problems I’ve mentioned and created new ones besides.



So let’s imagine a world in which Chinese characters were never invented in the first place. Given such a void, the alphabet might have spread east from India in a way that it couldn’t in our history, but, to keep this from being an Indo-Eurocentric thought experiment, let’s suppose that the ancient Chinese invented their own phonetic system of writing, something like the modern Bopomofo, some thirty-two hundred years ago. What might the consequences be? Increased literacy is the most obvious one, and easier adoption of modern technologies is another. But allow me to speculate about one other possible effect.



One of the virtues claimed for Chinese characters is that they make it easy to read works written thousands of years ago. The ease of reading classical Chinese has been significantly overstated, but, to the extent that ancient texts remain understandable, I suspect it’s due to the fact that Chinese characters aren’t phonetic. Pronunciation changes over the centuries, and when you write with an alphabet spellings eventually adapt to follow suit. (Consider the differences between “Beowulf,” “The Canterbury Tales,” and “Hamlet.”) Classical Chinese remains readable precisely because the characters are immune to the vagaries of sound. So if ancient Chinese manuscripts had been written with phonetic symbols, they’d become harder to decipher over time.



Chinese culture is notorious for the value it places on tradition. It would be reductive to claim that this is entirely a result of the readability of classical Chinese, but I think it’s reasonable to propose that there is some influence. Imagine a world in which written English had changed so little that works of “Beowulf” ’s era remained continuously readable for the past twelve hundred years. I could easily believe that, in such a world, contemporary English culture would retain more Anglo-Saxon values than it does now. So it seems plausible that in this counterfactual history I’m positing, a world in which the intelligibility of Chinese texts erodes under the currents of phonological change, Chinese culture might not be so rooted in the past. Perhaps China would have evolved more throughout the millennia and exhibited less resistance to new ideas. Perhaps it would have been better equipped to deal with modernity in ways completely unrelated to an improved ability to use telegraphy or computers.



See also the summary of the [backlash](http://languagelog.ldc.upenn.edu/nll/?p=25876) to this article on _Language Log_.



### "How to Write a Dance" by Anna Heyward



[source](http://www.theparisreview.org/blog/2015/02/04/how-to-write-a-dance/) 



Today dance notation is arcane, and mostly inessential to the art of dance. Even the two most prevalent systems, Laban and Benesh, don’t enjoy wide literacy among dancers. Lincoln Kirstein, of the New York City Ballet, wrote in his Ballet Alphabet,



> A desire to avoid oblivion is the natural possession of any artist. It is intensified in the dancer, who is far more under the threat of time than others. The invention of systems to preserve dance-steps have, since the early eighteenth century, shared a startling similarity. All these books contain interesting prefatory remarks on the structure of dancing. The graphs presented vary in fullness from the mere bird’s-eye scratch-track of Feuillet, to the more musical and inclusive stenochoreography of Saint-Léon and Stepanov, but all are logically conceived and invitingly rendered, each equipped with provocative diagrams calculated to fascinate the speculative processes of a chess champion. And from a practical point of view, for work in determining the essential nature of old dances with any objective authority, they are all equally worthless. The systems, each of which may hold some slight improvement over its predecessor, are so difficult to decipher, even to initial mastery of their alphabet, that when students approach the problem of putting the letters together, or finally fitting the phrases to music, they feel triumphant if they can decipher even a single short solo enchaînement. An analysis of style is not attempted, and the problem of combining solo variations with a corps de ballet to provide a chart of an entire ballet movement reduces the complexity of the problem to the apoplectic.



In other words, the very idea of trying to hurry along in the wake of a dance and record its movements is inelegant. But Charlip’s dances show us the fluidity between the dancer and the scribe: they allow us to think of notation as a way to invent movement, rather than just try to preserve and petrify it. One of the chief features of his drawings is their accessibility—they’re like invitations to the audience to join in.



### Conversation analysis notation (Jefferson transcription system)



[overview](https://www2.le.ac.uk/departments/psychology/research/child-mental-health/cara-1/faqs/jefferson), [examples](https://www.kcl.ac.uk/sspp/departments/education/research/Research-Centres/ldc/knowledge-transfer/DATA/part3.pdf)



Conversation analysts and many discourse analysts employ the **Jefferson system of transcription notation**. This is because in conversation analysis the transcripts are designed not only to capture **what** was said, but also the **way** in which it is said. Therefore the transcripts provide a detailed version of the complex nature of interaction.



(...)



_Data example 8_



> M: you'll k [ eep the place ↑ really spotless ↑ ]

> 

> D:          [ i will i'll  make my friends    ]

> 

> M: [ and you'll  ] make

> 

> D: [ i'll make ]         friends



_(Langford 1994: 93)_



Both instances of simultaneous speech can be counted as overlap in this example. In the first case, at line 2, D says ' i will ' and seems to be anticipating M's instruction to keep the place spotless. In the second case, her ' I'll make' begins after an apparent turn transition point by M who appears to have finished her turn after 'spotless'.