Peircers,
The concept of continuity that Peirce highlights in his synechism is a logical
principle that is somewhat more general than the concepts of either mathematical
or physical continua.
Peirce's concept of continuity is better understood as a concept of lawful
regularity or parametric variation. As such, it is basic to the coherence and
utility of science, whether classical, relativistic, quantum mechanical, or any
conceivable future science that deserves the name. (As Aristotle already knew.)
For example, one of the best examples of this brand of continuity in physics
would be the "correspondence principles" that describe the connection between
classical and contemporary paradigms.
The importance of lawful regularities and parametric variations is not
diminished one bit in passing from continuous mathematics to discrete
mathematics, nor from theory to application.
Here are some further points of information, the missing of which seems to lie
at the root of many recent disputes on the Peirce List:
It is necessary to distinguish the mathematical concepts of continuity and
infinity from the question of their physical realization. The mathematical
concepts retain their practical utility for modeling empirical phenomena quite
independently of the (meta-)physical question of whether these continua and
cardinalities are literally realized in the physical universe. This is equally
true of any other domain or level of phenomena -- chemical, biological, mental,
social, or whatever.
As far as the mathematical concept goes, "continuity is relative to topology".
That is, what counts as a continuous function or transformation between spaces
is relative to the topology under which those spaces are considered and the same
spaces may be considered under many different topologies. What topology makes
the most sense in a given application is another one of those abductive matters.
Regards,
Jon
--
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