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A math is a monastic congregation of avout- people who have sworn a vow to have limited contact with the outer world, and to abide by the Discipline. Avout are generally areligious, and comprise most of the 'theors' (scientists and philosophers) of Arbre. Male avout are called Fraas and female avout are called Suurs.
Mathic life is quiet and cloistered, where avout devote their lives to the pursuit of knowledge behind stone walls. This in direct contrast to the extramuros (Saecular) world, which is generally illiterate and driven by praxis (technology).
Avout living in a Mathic society swear a vow upon their entry to a math to observe a highly-prescriptive set of rules called the Discipline. The Discipline, which is periodically reviewed and amended (though by whom is unclear), significantly circumscribes the technologies to which avout may have access. Globally speaking, avout are denied access to a variety of technologies which exist as practical technology in the Saeculum, or non-Mathic, world, including machine tools, computational devices, genetic sequencing tools, and the ability to do practical physics to produce items with material properties differing from 'normal' matter.
In line with the generally monastic character of the mathic world, the Discipline allows only three possessions per avout; interestingly, these are notable exceptions to the rules regarding special matter. The three possessions are the sphere (a ball made of special matter which can vary its physical properties as required to perform a variety of tasks), the bolt (a piece of cloth made of a matter similar to the sphere, used as clothing), and the chord (a rope of the same material as the sphere and bolt, used to fasten clothing and for other purposes at need). These are the only possessions afforded avout, and are made of 'new matter.' 'New Matter' is loosely defined in the text as materials created through the use of ancient technology which have modified atomic properties generated through 'nucleosynthesis', a process by which the constants governing the behavior of matter may be modified to produce items providing capabilities exceeding normal matter.
The reasoning behind the proscription on certain technologies (called "praxis" in the text) is inadequately explained in the book, but appears to derive from Saecular fear of the consequences of unregulated technical access among humans of significant intelligence and technical leanings. Reference is made to various Sacks (generally violent Saecular responses to perceived Mathic wrongdoing which leads to various levels of damage an in the Mathic community) and to a set of significant historical events for which Mathic society has been assigned blame.
There are several different types of maths, with varying degrees of isolation from the non-Mathic (Saecular) world. Multiple maths grouped together are called a Concent. Unarian maths are open to the Saecular world once a year, at a 10-day celebration known as Apert. Avout in Unarian maths are often referred to as 'one-offs', which Stephenson tells us is derogatory usage. Decenarian maths open once every 10 years. Avout of Decenarian maths are commonly referred to as 'tenners'. Centenarian maths open once every 100 years. Avout of centenarian maths are commonly referred to as 'Hundreders'. Millenarian maths open once every 1000 years. Avout of millenarian maths are commonly referred to as 'Thousanders'.
Mathic interaction with the Saecular world is, as might be expected, governed by the Discipline. In general, contact with the Saeculum occurs only during an Apert; however, avout may be extracted from their cloister by the Saecular government for the purposes of performing technical work only avout are capable - this is called "Voco," and recalls the English word "evocation." Avout who violate the Discipline may receive punishment within the concent (see The Book, below) or, for serious offenses, be expelled via a methodology called "Anathem," in which the avout is returned to the Saecular world.
A member of any math may 'pass through the labyrinth' to join the order with the next longest temporal seclusion. This helps to explain how the Centenarian Maths retain membership over their great lengths of time. Millenarian maths are more restrictive still: only newborn babies that still have the umbilical cord remnant attached, often left at the gates of the concents by those extramuros during the time of Apert, may join the Millenarian Math. . They are passed through the labrynth to the Millenarian Math in order to avoid "contaminating" the higher level math. This is somewhat confused in the book as we are told of the umbilical cord requirement for Millenarians, and yet later in the book, Fraa Jad suggests that in time, Fraa Orolo would have come through the Labyrinth and become his (Jad's) peer. Arbran time scales (or Arban lifespans) are apparently different from Earth time scales, as narrator Erasmus notes early on that it is not impossible to live to 130 years (and hence live through > 1 Centenarian Apert).
All the largest concents are centred around a Great Clock, all of which have different aesthetics and operate on different principles, but all of which measure time in the long term, keeping track of time accurately for a duration outside the scope of a single human lifespan.