360 reviel netz
Because he has a particular task, and particular tools, all refl ecting a
complex historical setting. Everything argued here is tentative but of one
thing I am certain: the history of mathematical symbolism is not linear. Let
us discard the notion of a single linear trajectory from ‘natural language’
to ‘symbolic algebra’, a gradual transition from the concrete to the abstract,
from the less expressive to the more expressive, a simple teleological route
leading to an ever more perfect science. In truth, mathematics never did rely
on natural language: from its very inception it expressed itself, in its various
cultural traditions, through diff erent complicated formulaic languages,
using various specialized traces for numerical values or for diagrams.
History then takes off in a non-linear fashion. Symbolism is invented and
discarded, employing this or that set of cognitive tools, inventing this or
that form of writing, in the service of changing goals: nothing is predeter-
mined. Symbolism – just as mathematics itself – is contingent. Th e same,
fi nally, must be true of our own (various uses of ) symbolism: they should
be seen not as the ‘natural’ achievement of precise abstraction but as a
historical artefact. We should therefore study the precise cognitive tools
our symbolism employs, the precise tasks that such symbols are made to
achieve, and the precise historical route that brought us to the use of such
symbols. Th e modern equation is not the ‘natural’ outcome of a mathemati-
cal history destined to reach its culmination in the nineteenth century; it
is a culturally specifi c form. Th is article, sketching a speculative account
of Diophantus’ symbolism, off ered one chapter from the historical route
leading to that equation.Bibliography
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