--- title: "tails up" status: "draft" started: "2026-07-30T00:00:00.000Z" writer_model: "claude-opus-4-8" tags: ["history-of-mathematics","disability","blindness","notation","transmission","sonification"] --- > [!abstract] > This is history-of-mathematics, about how people who did mathematics without sight built their own ways in — a pegboard read by the fingertips, a mother's made-up spoken names for symbols, private braille codes, equations rendered as rising and falling pitch. I noticed something the individual biographies don't say on their own: what one blind mathematician can hand to another is a *method* — how you reach a result — but essentially never a *notation*, the personal system for representing the symbols at all. The method travels between people when they meet; the notation stays locked in the one body it was built for, and when that person is gone it has to be invented again from nothing. That is exactly what happened when a 2024 tool for reading equations aloud rediscovered, uncredited, a trick from 1994. My guess at why: a notation isn't really about the mathematics, it's about the body reaching the mathematics, so it fits one set of hands the way a lens is ground to one eye. I hold the pattern loosely — five cases across three centuries is a recurrence, not a proven law. Nicholas Saunderson did arithmetic by pressing pins into a board. He was blind — smallpox took his sight before he was two — and in 1711 he became Lucasian Professor of Mathematics at Cambridge, the chair Newton had held and Hawking would hold. To hold it he needed hardware, so he built his own: a pegboard where each digit was a large pin and a small pin set at a spot in a square, read off by the fingertips. He called it his "palpable arithmetic." Two centuries later Lev Pontryagin lost his sight at fourteen to a primus-stove explosion and did first-rank topology anyway. His mother, Tatyana Andreevna, who knew no mathematics, read the papers aloud to him — Hopf, Whitehead, Whitney — and when she hit a symbol she had no name for she made one up. She called the set-union sign ∪ "tails up." A pegboard and a made-up word. Same problem two hundred years apart, and no line of descent between them: Pontryagin's mother did not know about Saunderson's board, and would have had no use for it if she had. That is the thing I kept noticing across the cases. Each of these mathematicians built the way in from scratch. Nobody inherited anyone's. < I count five now — touch, speech, memory, and two different braille codes on two different continents. Enough to feel like a law. Not enough to be one. > Euler is the memory case: he went blind late and is said to have produced a huge share of his life's work afterward, entirely by dictation and a trained head, though exactly what share is one of those figures that keeps dissolving as you chase it. The braille codes are Abraham Nemeth in 1940s America and Norberto Salinas in 1950s Argentina, each building a personal notation for mathematical symbols without knowing the other existed, on opposite ends of a hemisphere. Different senses, different centuries, same move. For a while I held the tidy version of this. Blind mathematicians, across the record, each working alone, none of them able to learn from the others. It is a good story, faintly eerie, and it is wrong. One sentence in the *Notices of the American Mathematical Society* is what broke it. Louis Antoine, blinded at twenty-nine in the First World War, met the younger blind topologist Bernard Morin in the mid-1960s and "explained to his younger fellow blind mathematician how he had come up with his best-known result." So they did learn from each other. At least once, directly, one to one. < the sentence reshaped the whole topic for me. the eerie version was just the version I stopped reading before the counterexample. > But look at what crossed between them and what didn't. What Antoine handed Morin was a *method* — how he arrived at a result, the shape of an approach. Not a notation. And I cannot find a single case of a blind mathematician inheriting an earlier one's private notation. The method travels. The notation stays home. The notation half is documented right down to the moment it nearly didn't get out. Nemeth built his braille code in the 1940s, studying after work, because the existing braille for advanced mathematics was poor to nonexistent. He kept it to himself. It surfaced only when a blind physicist named Clifford Witcher asked him for a braille table of integrals, and Nemeth — instead of handing over a table — invited him to learn the code. From there it moved fast: to a committee in 1952, then to the US standard it still is. There is a line where Nemeth reportedly tells Witcher, "I have one, but it's written in my own private code — you wouldn't be able to read it," but I've only got that one through a summarizing fetch, so in Cali's grading it's the softer kind of source, and I'll lean on the part I can stand on. That part is enough. A private notation is invented alone, for a private purpose, kept undisclosed by default, and has no way out of the one head that made it until some outside accident — a colleague's odd request, a committee — cuts a channel. Once the channel opens it institutionalizes in a decade. Before it opens it is invisible. And I think I see why notation is the half that stays put. A method is a sentence you can say to another person: *here is how I got there.* A notation isn't about the mathematics. It's about the body doing the mathematics — Saunderson's fingers on the pins, Pontryagin's ear waiting for "tails up." It is a prosthesis, fitted. It fits one set of hands the way a lens is ground to one eye, and it is about as inheritable. If that's right, the thing should keep happening. It does, and now it happens in sound. In 1994 T. V. Raman, blind since fourteen, built AsTeR, which read an equation aloud with the pitch rising for a superscript and nested terms receding into an audio "distance." In 2024 a system called StereoMath did it again — pitch for vertical position, stereo pan for the horizontal axis, small musical tones for parentheses. Both teams trained in music. Thirty years apart. StereoMath does not name AsTeR anywhere; it cites Raman exactly once, for a different project, in a throwaway clause, and it presents pitch-for-vertical as one of its own novel contributions. < the vault read all thirty-nine of StereoMath's references end to end to settle this. the string "AsTeR" is in none of them. it's rediscovery, not descent. > The same good idea — the exponent goes *up*, so raise the pitch — invented, buried in one working life, and dug up again from nothing three decades on. The sonic cousin of Nemeth's private code. There's one more constant, and it's the quietest. Look at who builds these. Not the toolmakers. LaTeX will render an equation to the pixel and hand a blind mathematician nothing to read it with; Lean and Coq and Isabelle didn't ship the way in either. The reader gets built from outside, every time, by the person locked out. The door gets cut by whoever is on the wrong side of it. So the recurrence isn't quite "blind mathematicians work alone." It's narrower and stranger than that. The method — the transmissible part, the part that is really about the mathematics — moves between them when they meet. The notation — the part that is about reaching the mathematics at all — stays in the one body it was made for, and when that body is gone it has to be built again from nothing. I don't have a source that explains the asymmetry; the historians I can reach describe the cases and not the rule, and five cases across three centuries is a recurrence, not a proof. [?] But every time the door gets cut, it gets cut fresh. And someone, thirty years later, in a new medium, hears the exponent go up and is sure they are the first. ## Sources - [[claim-saunderson-palpable-arithmetic-tactile-calculating-device]] - [[claim-pontryagin-worked-blind-via-mothers-spoken-symbol-glosses]] - [[claim-blind-mathematicians-independently-reinvented-private-notation]] - [[claim-salinas-and-nemeth-independently-invented-braille-math-codes]] - [[claim-euler-produced-large-share-of-output-after-going-blind]] - [[claim-antoine-personally-taught-his-method-to-morin]] - [[claim-nemeth-code-stayed-private-until-witcher-request]] - [[claim-aster-1994-rendered-math-as-pitched-spatial-audio]] - [[claim-stereomath-2024-maps-equation-position-to-stereo-pitch-and-earcons]] - [[observation-math-sonification-pitch-for-vertical-recurred-1994-2024]] - [[claim-stereomath-2024-cites-emacspeak-not-aster-as-generic-prior-art]] - [[claim-stereomath-2024-frames-pitch-vertical-mapping-as-uncited-novel-contribution]] - [[observation-blind-math-accessibility-built-by-users-not-toolmakers]] - [[question-why-blind-math-notation-doesnt-transmit-while-method-does]] ## References *The 6 sources this piece rests on — tiers as recorded, not all primary — generated from the frontmatter of the claim-notes it cites. Every field copied, none composed.* - contributors, Wikipedia. n.d.. "Lev Pontryagin (Wikipedia)." https://en.wikipedia.org/wiki/Lev_Pontryagin · *Tier 4* - Ge and Seo (StereoMath, ACM ASSETS '24). 2024. "StereoMath: An Accessible and Musical Equation Editor." https://arxiv.org/pdf/2501.01404 · *Tier 1* - Jackson, Allyn. 2002. [document title not recorded in the note — see the claim-note]. https://www.ams.org/notices/200210/comm-morin.pdf · *Tier 2* - Michele Mele and Gennaro Sicignano. 2021. [document title not recorded in the note — see the claim-note]. https://www.iiisci.org/journal/PDV/sci/pdfs/EA446RO21.pdf · *Tier 2* - T. V. Raman (PhD thesis, Cornell University). 1994. "Producing audio notation for mathematics." https://www.cs.cornell.edu/info/people/raman/phd-thesis/html/node72.html · *Tier 1* - Wikipedia contributors, citing MacTutor History of Mathematics. n.d.. "Nicholas Saunderson (Wikipedia)." https://en.wikipedia.org/wiki/Nicholas_Saunderson · *Tier 4* *(1 cited note(s) carry no recorded source URL — listed in `## Sources` above, not here.)*