I'm interested in alternative systems of notation for mathematics. I've often heard how mathematical notation is illogical, inconsistent, filled with grandfather clauses that serve no purpose, and suggests deprecated ideas (e.g .the $\rm{dx}$ of the integral and derivative).

It seems reasonable to suppose someone has tried to come up with some alternative notation.

My interest in this is mostly curiosity. I'm very interested in what people have come up with to rectify the perceived faults. I vaguely remember seeing a link to some alternative notation of logarithms, but I seem to have lost it.

I know that using highly non-standard notation is a bad idea for many reasons. I highly doubt I will use it.

Also, I don't mean good notation or even consistent notation. I'm just interested in something different, preferably a lot of it, and in one place.

I'm labeling this as a soft question, but I'm not really looking for suggestions or discussions for alternative notation. I just want a good source(s).

  • $\begingroup$ Maybe you were thinking of this posting for logs math.stackexchange.com/questions/30046/… ? Regards $\endgroup$
    – Amzoti
    Jan 15 '13 at 19:31
  • $\begingroup$ Nope, it was actually something else, though I've seen this too. What I remember involved squiggly lines, kind of like a square root. $\endgroup$
    – GregRos
    Jan 15 '13 at 19:37
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    $\begingroup$ One "folklore" alternative notation is write functions in postfix notation, like $xf$ or $(x+3)f$ instead of $f(x+3)$. This way the compositions are naturally read left to right: $(x)fgh$ instead of $h(g(f(x)))$. (NB: This is not to say that LTR is more natural than RTL, just that the rest of mathematical formulas are read LTR). With linear operators, for which parentheses are often omitted, one could write things like $2xAB$, with linearity (or at least homogeneity) built-in. I don't have any references, unfortunately. $\endgroup$
    – user53153
    Jan 15 '13 at 19:37
  • 2
    $\begingroup$ Attend a course taught by an analyst. I guarantee you'll see weird and inconsistent notations. $\endgroup$
    – Git Gud
    Jan 15 '13 at 19:39
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    $\begingroup$ I don't know if this counts as "alternative" enough: In the preface of Structure and Interpretation of Classical Mechanics, Sussman and Wisdom reject the traditional notation for the Euler-Lagrange equations, $\dfrac{\mathrm d}{\mathrm dt}\dfrac{\partial L}{\partial\dot q_i}-\dfrac{\partial L}{\partial q_i}=0$, as ambiguous and inconsistent, opting instead for a computer-algebra-like notation $D(\partial_2 L\circ\Gamma[q])-\partial_1 L\circ\Gamma[q]=0$. All the math in the rest of the book is written in this style. $\endgroup$
    – user856
    Jan 15 '13 at 20:50

Here is a somewhat recent Survey of Notation and has some links for others who have tried.

There is even a book A History of Mathematical Notations: Vol. I & II by Florian Cajori on on the matter, but I think people define, invent and use what they want and it has caused lots of headaches for us all!

Lastly, you can also find some samples in this MSE posting Alternative notation for exponents, logs and roots?

This was added from the comment I made above to close this question out per request.

  • $\begingroup$ Don't forget the references in the other comment of yours! $\endgroup$
    – rschwieb
    Aug 1 '13 at 17:10
  • $\begingroup$ @rschwieb: Thanks - I totally forgot about those, but added and cleaned up. Thanks! Regards $\endgroup$
    – Amzoti
    Aug 1 '13 at 17:38
  • $\begingroup$ Thanks for converting the comment!+1 Visit us someday if you have some time to spare. $\endgroup$
    – Shuhao Cao
    Aug 1 '13 at 22:09
  • $\begingroup$ A late gem, to the rescue...+1 $\endgroup$
    – amWhy
    Aug 2 '13 at 0:15
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    $\begingroup$ (survey of notation) link broken as of Sun 2016-12-11T13:31:40 kwarc.info/publications/papers/kw1_notationSurvey.pdf $\endgroup$
    – dreftymac
    Dec 11 '16 at 21:31

If you want to see what unambiguous math notation looks like, look at an application where it has to be unambiguous: the input languages for proof-checkers (Coq, Mizar) and computer algebra systems (Mathematica, Maple, Maxima). Mathematica (and possibly also the other linked CASs with which I'm less familiar) supports WYSIWYG input mostly resembling traditional textbook notation, so it might be an especially good example.

  • $\begingroup$ See also TeX... $\endgroup$
    – lhf
    Dec 6 '17 at 12:06
  • $\begingroup$ Metamath is a closely related example. $\endgroup$
    – Mark S.
    Sep 14 '18 at 22:49

An interesting list covering notations in several domains is Notes on notation and thought.

Recently, I also stumbled upon G. Spencer-Brown Laws of Form. I recommend the introductory text from L. H. Kauffman Laws of Form - An Exploration in Mathematics and Foundations.


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