The Hellenic cosmos stabilized with Aristotle's synthesis of prior
traditions as a finite, spherical world with nothing beyond and no voids
within (other traditions, like Pythagoreanism, survived, but as a minority).
For good, philosophical reasons, there could never exist an actual infinity
in time or place. Aristotle's tight system remained dominant through the
middle ages, but remixed with the earlier idealistic traditions as well as
Mesopotamian, Egyptian, and Indian indigenous systems (pretty much
everywhere Alexander went). Arab and Islamic scholars preserved and
enriched Aristotle's ideas, and the Latin West was thunderstruck with the
injection of all this during the twelfth century (they had been in their
Monty Python period). Religious and secular university scholars dove into
this material, and found It so compelling and so well-argued, they were
convinced that God must have made the world as the Aristotelians described.
God must have made the world spherical and finite, for not even God himself
could create an actual infinity.
They did this for hundreds of questions and issues, until the more
conservative orders and officials got nervous about all these limitations on
God. They tried censorship, heavy-handed excising of Aristotle's works, and
banned the teaching of hundreds of specific propositions. One of these, was
whether God in his limitless power could create an actual infinity.
University faculty were highly motivated to go beyond this power, and
explore the characteristics and qualities of an existing infinity (which
they privately knew could not possibly exist). By the fourteenth century,
groups of mathematically engaged monks like the Oxford Mertonians were
exploring various sophisticated concepts verbally and inventing novel
graphical formalisms. With the renaissance, a panoply of more faithful
manuscripts opened up the discussion.
Copernicanism was expansive, in a sense, but it still had the spheres and a
finite, closed, spherical universe. Galileo and his telescope set off a
fury of observational astronomy, in which everywhere one looked there were
more stars. The notion of a cosmos boundless in extent became commonplace.
Simultaneously, microscopy revealed worlds within worlds, a universe in a
drop of water. When Newton came on the scene, the concepts of infinity and
boundlessness in the microcosm and the macrocosm were in the air. Voltaire,
in his Letters on England, reported to his French followers how Newton and
Leibniz had grappled with operationalizing these concepts with new
mathematical formalisms ("Letter XVII: On Infinites In Geometry, And Sir
Isaac Newton's Chronology").
It is here that I think attention to mathematical formalism as a language
can help. Speakers of the local natural language had to bend grammar,
syntax, and style to express these new, expansive concepts. Similarly,
their inherited mathematical formalisms were not capable of dealing with an
actual infitinity of either enumerated items or of measure, such as space.
The grammar, syntax, vocabulary, and style of mathematics as a language had
to change.