While G may be dismissive about where our numerals came from, it certainly is a fascinating part of human history. Most accounts agree that numbers originated independently in several different cultures. Notches and other tally marks gave way to knotted cords used by the Incas, and a system of dots and bars used by the Mayans. The Egyptians developed hieroglyphics for a decimal number system, which led to a simpler, non-pictorial script, once papyrus was invented. Greeks used the same symbols used in their alphabet to represent numbers as well, while the Babylonians were perhaps the first to develop a true “positional” system where the quantity denoted by a symbol depended not only on the symbol itself, but also its position relative to other symbols (just like the digit 1 can mean different things in the strings 10, 100 and 1000). It is generally acknowledged that the “Arabic” numerals we now use were developed by the Hindus and brought to the West by Arab traders (so that they are more correctly called “Hindu-Arabic numerals”).
Perhaps the most intriguing part of this history is the invention of zero – something which wasn’t recognized as a full-fledged number in its own right for quite a while. Here’s a provocative piece related to this.
The question of whether mathematics is discovered or invented is a philosophical conundrum which has engendered much debate. Plato believed that all mathematics objects actually exist in their own idealized universe: numbers, perfect circles, perhaps even theorems and their proofs. As a result, humans do no more than discover such pre-existing truths and forms. In contrast, the non-Platonic view is that mathematics is purely a creation of the human mind, that in the absence of humans, it would not exist (or if it did, it might take a very different form). This school believes that all of mathematics is invented. The difference has some very practical monetary consequences, as shown ahead.
The discovery/invention question also pops up in Professor Ray’s construction. She starts with an empy set, but can such a mathematical object really be proven to exist? If so, is it something discovered or invented? As it turns out, defining a set as “a collection of objects” is fraught with some peril, since it leads to logical inconsistencies. This was shown by the English philosopher Bertrand Russell in his famous “Russell’s Paradox” explained ahead. As a result, mathematicians realized that sets could not be defined so naively – rather, various assumptions or “axioms” had to be first put together to create a foundation, and then the rest of the theory of sets deduced step by step. A similar “axiomatic” approach is used for other fields in mathematics as well – the idea being as old as Euclid, who first used it for geometry.
The Big Bang of Numbers (video)