Stephen Wolfram:

So, where did all the mathematical notation that we use today come from?

Well, that’s all bound up with the history of mathematics itself, so we have to talk a bit about that. People often have this view that mathematics is somehow the way it is because that’s the only conceivable way it could be. That somehow it’s capturing what arbitrary abstract systems are like.

One of the things that’s become very clear to me from the big science project that I’ve been doing for the past nine years is that such a view of mathematics is really not correct. Mathematics, as it’s practiced, isn’t about arbitrary abstract systems. It’s about the particular abstract system that happens to have been historically studied in mathematics. And if one traces things back, there seem to be three basic traditions from which essentially all of mathematics as we know it emerged: arithmetic, geometry, and logic.   

All these traditions are quite old. Arithmetic comes from Babylonian times, geometry perhaps from then but certainly from Egyptian times, and logic from Greek times.

And what we’ll see is that the development of mathematical notation–the language of mathematics–had a lot to do with the interplay of these traditions, particularly arithmetic and logic.

Mario Livio:

The symbols for the arithmetic operations of addition (plus; “+”) and subtraction (minus; “–”) are so common today we hardly ever think about the fact that they didn’t always exist.  In fact, someone first had to invent these symbols (or at least other ones that later evolved into the current form), and some time surely had to pass before the symbols were universally adopted.  When I started looking into the history of these signs, I discovered to my surprise that they did not have their origin in antiquity.  Much of what we know is based on an impressively comprehensive and still unsurpassed research (in 1928–1929) entitled History of Mathematical Notations by the Swiss-American historian of mathematics, Florian Cajori (1859–1930).

The ancient Greeks expressed addition mostly by juxtaposition, but sporadically used the slash symbol “/” for addition and a semi-elliptical curve for subtraction.  In the famous Egyptian Ahmes papyrus, a pair of legs walking forward marked addition, and walking away subtraction.  The Hindus, like the Greeks, usually had no mark for addition, except that “yu” was used in the Bakhshali manuscript Arithmetic (which probably dates to the third or fourth century).  Towards the end of the fifteenth century, the French mathematician Chuquet (in 1484) and the Italian Pacioli (in 1494) used “” or “p” (indicating plus) for addition and “” or “m” (indicating minus) for subtraction.

There is little doubt that our + sign has its roots in one of the forms of the word “et,” meaning “and” in Latin.  The first person who may have used the + sign as an abbreviation for et was the astronomer Nicole d’Oresme (author of The Book of the Sky and the World) at the middle of the fourteenth century.  A manuscript from 1417 also has the + symbol (although the downward stroke is not quite vertical) as a descendent of one of the forms of et.

The origin of the – sign is much less clear, and speculations range all the way from hieroglyphic or Alexandrian grammar ancestry, to a bar symbol used by merchants to separate the tare from the total weight of goods.

See also: Earliest Uses of Various Mathematical Symbols