Is God a Mathematician_ - Mario Livio [112]
Do You Speak Mathematics?
In a previous section I compared the import of the abstract concept of a number to that of the meaning of a word. Is mathematics then some kind of language? Insights from mathematical logic, on one hand, and from linguistics, on the other, show that to some extent it is. The works of Boole, Frege, Peano, Russell, Whitehead, Gödel, and their modern-day followers (in particular in areas such as philosophical syntax and semantics, and in parallel in linguistics), have demonstrated that grammar and reasoning are intimately related to an algebra of symbolic logic. But why then are there more than 6,500 languages while there is only one mathematics? Actually, all the different languages have many design features in common. For instance, the American linguist Charles F. Hockett (1916–2000) drew attention in the 1960s to the fact that all the languages have built-in devices for acquiring new words and phrases (think of “home page”; “laptop”; “indie flick”; and so on). Similarly, all the human languages allow for abstraction (e.g., “surrealism”; “absence”; “greatness”), for negation (e.g., “not”; “hasn’t”), and for hypothetical phrases (“If grandma had wheels she might have been a bus”). Perhaps two of the most important characteristics of all languages are their open-endedness and their stimulus-freedom. The former property represents the ability to create never-before-heard utterances, and to understand them. For instance, I can easily generate a sentence such as: “You cannot repair the Hoover Dam with chewing gum,” and even though you have probably never encountered this sentence before, you have no trouble understanding it. Stimulus-freedom is the power to choose how, or even if, one should respond to a received stimulus. For instance, the answer to the question posed by singer/songwriter Carole King in her song “Will You Still Love Me Tomorrow?” could be any of the following: “I don’t know if I’ll still be alive tomorrow”; “Absolutely”; “I don’t even love you today”; “Not as much as I love my dog”; “This is definitely your best song”; or even “I wonder who will win the Australian Open this year.” You will recognize that many of these features (e.g., abstraction; negation; open-endedness; and the ability to evolve) are also characteristic of mathematics.
As I noted before, Lakoff and Núñez emphasize the role of metaphors in mathematics. Cognitive linguists also argue that all human languages use metaphors to express almost everything. Even more importantly perhaps, ever since 1957, the year in which the famous linguist Noam Chomsky published his revolutionary work Syntactic Structures, many linguistic endeavors have revolved around the concept of a universal grammar—principles that govern all languages. In other words, what appears to be a Tower of Babel of diversity may really hide a surprising structural similarity. In fact, if this were not the case, dictionaries that translate from one language into another might have never worked.
You may still wonder why mathematics is as uniform as it is, both in terms of subject matter and in terms of symbolic notation. The first part of this question is particularly intriguing. Most mathematicians agree that mathematics as we know it has evolved from the basic branches of geometry and arithmetic that were practiced by the ancient Babylonians, Egyptians, and Greeks. However, was it truly inevitable that mathematics would start with these particular disciplines? Computer scientist Stephen Wolfram argued in his massive book A New Kind of Science that this was not necessarily the case. In particular, Wolfram showed how starting from simple sets of rules that act as short computer programs (known as cellular