This contribution, by a mathematician, to the Common Knowledge symposium “Fuzzy Studies” examines some mechanisms that seem essential for the “ratchet effect” that, in Michael Tomasello's use of the term, refers to the ability of human cultures to preserve their achievements even through serious crises and even where preservation entails substantial loss. By taking the word culture to refer to any group of individuals who closely cooperate over an extended period, this article evaluates mathematicians and mathematics as its main example. The assessment of the enterprise of mathematics as a culture corresponds to the author's personal experience — and mathematics, he argues, is both historically old and simply structured, despite the formidable complexity of mathematical knowledge as it has been compiled over the centuries. The preserving, restructuring, and enlarging of this knowledge may be regarded as the main cultural goals of mathematics and are the objects of study here. This article concentrates first on the techniques used in pursuing the tasks of preservation, enlargement, and restructuring — techniques such as counting methods and geometric diagrams — and it emphasizes their use in communication, when they function as media. The cultural techniques of mathematics, known to and used by all members of that culture, are media that, through intensity of communication, add symbolic functions to mere procedures. (The prototypical example of a cultural technique, in this sense, is language.) Knowledge, as the additional symbolic value, is distributed and at the same time preserved in the process of permanent communication.

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