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Neither in the subjective nor in the objective world can we find a criterion for the reality of the number concept, because the first contains no such concept, and the second contains nothing that is free from the concept. How then can we arrive at a criterion? Not by evidence, for the dice of evidence are loaded. Not by logic, for logic has no existence independent of mathematics: it is only one phase of this multiplied necessity that we call mathematics.
How then shall mathematical concepts be judged? They shall not be judged. Mathematics is the supreme arbiter. From its decisions there is no appeal. We cannot change the rules of the game, we cannot ascertain whether the game is fair. We can only study the player at his game; not, however, with the detached attitude of a bystander, for we are watching our own minds at play.
'He Advanced Him 200 Lambs of Gold': The Pamiers Manuscript
Appendix A: The Land of 'Oc'
The mathematical text described in this article was written in Languedocian, one of a whole family of Romance languages once spoken in southern France. It’s amazing to learn what a variety of tongues could be heard in the recent past, both in the south and in the north of the country.
The peoples of France since the early Middle Ages generally fell into two groups: those of the south, whose word for “yes” was òc, and those of the north, whose word for “yes” was oïl. The latter evolved into the oui of modern French. The term Languedocian derives from the phrase langue d’Òc, meaning “language of Òc.”
Other languages or dialects in this family included Provençal, Niçois, Piemontese, and several more. They were spoken in southern France and in parts of Italy and Spain. Not surprisingly, these languages are more recognizably similar to modern Italian than are those of the north. Some of the greatest literature of the Middle Ages was created in this region by troubadours, the lyric poets who recounted tales in story and song. The word troubadour itself is from the Provençal for “one who finds or invents.”
Language is such an important marker of cultural identity that over time, Languedoc came to mean not only the language family itself, but also all of the French lands where it was spoken. Today, the language family is referred to instead as Occitan.
Today in France, visitors have a hard time finding traces of dialects other than standard Parisian. This reflects the growing political unification of the country after the 1789 Revolution. At that time, there were only about 3 million people who could speak what is now called French. The area around Avignon in the south was absorbed into the country only in 1791, while the city of Nice (homeland of the Niçois language) was ceded to France by Sardinia only in 1860. In 1881, the French prime minister ordered that schoolchildren be punished if they spoke Occitan or other regional languages! By 1893, only about one-quarter of the French population of 30 million didn’t speak the designated official language (Weber 1991, p. 95). The Occitan languages almost became extinct— another sign of a sharp loss of linguistic diversity all over the world because of the impact of modernization and globalization.
Schwartz, Randy K., "'He Advanced Him 200 Lambs of Gold': The Pamiers Manuscript," Loci (July 2012), DOI: 10.4169/loci003888