You can find the text here.
The main question is “how did contemporary notation develop.” Wolfram’s sprints through the history, making a few important distinctions.
Numerals begin being represented around 25,000 years ago in unary — notches on a bone. Eventually, around the 1800s BC (think: Hammurabi), the Babylonians develop a base-60 positional notation. This is lost until the Greek develop a base-10 positional notation. In the interim, we’re stuck with additive-type systems, where distinct symbols represent values. For instance, the Egyptians represented 1000 with a lotus flower.
Finally, around 1000AD (think: Lief Ericson finds North America; Lions finally go extinct in Europe), Hindu-Arabic notation is in full swing. It gets introduced to Europe in 1202 (Francis of Asisi; Genghis Khan), when Fibonacci publishes Liber Abaci.
So that’s numerals, but we can’t develop much without more, and this is Wolfram’s main other distinction. Numerals, Operators, and Variables.
Variables aren’t really introduced until the 1600s (King James Bible), which in general is when notation really gets going. Leibniz invents the elongated S for integrals and the d for derivatives; an early inventor of a type of slide rule, Oughtred, introduces the ‘X’ for multiplication and argues that algebra should be done by notation, not natural language.
Operators by the way – They didn’t really exist. The plus sign arises in the 1400s.
Things die down again, until the mid-1800s, with the explosion of mathematical logic. The one that sticks around is Peano’s, which is popularized in Russell and Whitehead’s Principia.
Random Notes
- Traditional notation represents objects, not processes.
- Math notation is 2 dimensional (a^2 = 1/10), while natural language is 1
- Pythagorean Triples are right triangles whose sides are whole numbers, like 3, 4, 5 right triangle.
- Sapir-Whorf hypothesis is that the structure of language affects the way we think. A strong formulation is conscious thought is only possible due to language.
Bleg 