Just think how many different notations there are in linear/multilinear algebra.
In the Earliest Uses siteI noticed this reference:
Perhaps this is the convention referred to?
Yes, that approximately describes the factorial notation I have seen in some old (early 20th century) math books. The versions I’ve seen look more like just the lower left corner of a box.
However, this site on history of factorials says that the n! notation was used as early as 1808 in a text by one Christian Kramp.
Perhaps, but you understood enough to write (an informative post) correcting a point of fact, while re- iterating OP, as pulled here.
I would comment more but I’m in the margins of my iPhone.
Gottfried Wilhelm Leibniz deserves mention for his notation of calculus, about which he wrote:
[QUOTE=Leibniz]
In symbols one observes an advantage in discovery which is greatest when they express the exact nature of a thing briefly and, as it were, picture it; then indeed the labor of thought is wonderfully diminished.
[/QUOTE]
I was going to post exactly this. Newton’s notation didn’t lend itself very well to differential equations. Leibniz’s notation leads to things like separation of variables and integration more easily.
I’ve thought that math students (say, at the elementary algebra levels) would have a greater appreciation for notation if we actually took class time to discuss ideas like this. Here’s a case I’ve thought about:
Logarithmic notation conveys just exactly the same information as exponential notation. That is, the statement:
log[sub]3[/sub] 243 = 5
contains exactly the same information as:
3[sup]5[/sup] = 243
Why then is log notation useful? It seems that we can use it to solve a whole class of problems that couldn’t be handled well with just exponential notation.
But wait! The same could be said even for the square root notation, √
Without this, we could not write a solution to x[sup]2[/sup] = 5
Yet the radical symbol lets us write the solution as x = √5 (well, plus or minus).
But wait! The same could be said even for the use of – to write numbers less than zero.
How would you solve: x + 5 = 0 without it?
The statement x = 0 – 5, which then leads to x = –5, gives us the tools to do it.
We could have written math notation a lot of different ways, but all the notations you can think of don’t necessarily lend themselves equally well to various problems. Finding the right notation for the problem matters!
ETA: So, next time your children complain about how our decimal system of numbers is sooooo complicated, just ask them to multiply LXVII times CXXIV.
Well, a Feynman diagram is essentially a representation of a linear operator, so one could certainly consider them a mathematical notation system. Actually, there’s an argument to be made that much of the modern notation of category theory, especially string diagrams, derives from (or is at least in close analogy to) Feynman diagrams (see John Baez ‘rosetta stone’ article for a bit of an overview of this general sort of thing). Together with Penrose’s graphical notation for tensor calculations, you get what I see as a bit of a revolution in mathematical notation, away from one-dimensional strings of symbols (i.e. equations) to much more elaborate graphical representations of mathematical relationships, like commutative diagrams, the aforementioned string diagrams etc. Via a detour through pure mathematics, this sort of thing is now beginning to have a bit of an impact on physics, as well, with things like Bob Coecke’s ‘quantum picturalism’ approach and in the study of complex quantum systems (e.g. the tensor networks in the multiscale entanglement renormalization ansatz (MERA)).
I think he deserves mention for more than that: with his characteristica universalis, he really was the first one (at least as far as I’m aware) to employ the notion that there could be something like a universal formal language, and notation thereof, that is essentially applicable to formalizing and solving any kind of scientific (and even, on his conception, philosophical) problem. With this and his calculus ratiocinator he can probably be considered to have anticipated much of the essential notions employed today in the study of formal systems and Turing machines (and Frege essentially considered his Begriffsschrift to be a continuation of these ideas).