A history of mathematical notations

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1. breckJan 2023
   This is a project of mine that I take attempts at here and there.

   Another great book on this topic is "History of Mathematical Notations" https://www.amazon.com/History-Mathematical-Notations-Dover-...

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2. breckMay 2022
   I think it's an interesting question.

   Why stick with 2D math? If you're senior and already trained your brain to work in 2D math then it's a no brainer to continue to exploit what you know. If you're young, 2D maths is orthogonal to programming languages and I think very much worth the investment of practice: work on problems from a different perspective; very dynamic, domain and scale free; work offline; utilize large amounts of learning material; read early literature.

   Why does 2D math break naming conventions we have with programming languages? You don't get autocomplete with pen and paper; you have to write out intermediate steps; and there's no backspace key. So offline bytes are expensive today as they were 100 years ago, whereas online bytes in programming languages have dropped exponentially.

   Why does 2D math not stick to 1d linear forms? Notation inventors design their languages to take advantage of what the tools can do. Languages looked different before we had syntax highlighting. The tools for 2D math (pen+paper) do not have many features of the IDE, but they do have 2 dimensions with variable cell sizes. 2D math notations exploit those to make writing (and doing intermediate computations) easier.

   Why not build programming languages for every construct seen in mathematics? People do! You've got symbolic languages like Mathematica to matrix languages like APL. If you've seen it written in 2D math, someone's probably turned it into a language or library.

   What will the future of 2D math look like? I expect much like the present. Maybe someone will discover some brilliant new trick, but I don't think that likely. Perhaps with better hardware and AI we'll have some advanced Remarkable-like Tablets that blend the best of 2D handwritten math with the best features from IDEs. I expect 2D math has probably plateaued and will remain around as a useful orthogonal tool until AI takes over most jobs from humans (In the end both 2d maths and programming languages might not matter—it might just be binary + AIs running the show).

   Recommended reading:

   - A History of Mathematical Notations by Florian Cajori (1929)

   - Syntax-Directed Recognition of Hand-Printed Two-Dimensional Mathematics https://dl.acm.org/doi/abs/10.1145/2402536.2402585 (1967)

   - A Review of Two-Dimensional Programming Languages (1972) https://dl.acm.org/doi/10.1145/942576.807009

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3. breckApr 2022
   If you find this interesting I'd recommend Florian Cajori's "A History of Mathematical Notations" (1930; I have the 2 volume combined one reprinted in 1993).
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4. whatgoodisaroadMar 2021
   I'm slowly reading through a book called the "History of Mathematical Notations" by Florian Cajori and it sounds like just what you're looking for.

   It goes into far more detail than probably anybody would wish for on the histories of various symbols and ways of writing addition, roots, ratios etc. Many of these notations were used only by a few medieval scholars for a few decades but they usually try to optimize for efficiency or clarity in some creative way. Really fascinating stuff.

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5. jacobolusDec 2016
   The most comprehensive source about this kind of thing is Cajori’s History of Mathematical Notations. You should be able to find a used copy of both volumes for a reasonable price.
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6. nemoFeb 2015
   Florian Cajori's A History of Mathematical Notations goes into the history and notes examples of old written manuscripts showing the intermediate form in use, with one example where both were used interchangeably in a letter. He examined the question in depth, showed that the earliest uses in the US had a single stroke, with the double stroke only coming later, and dealt with the other theories showing as conclusively as possible that it's a peso sign composed of a superimposed P/S. It's an old book, and not on a web site, but you could probably find a copy at a library.
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