In what way are the notations Descartes introduced dependent upon his philosophical views?
"And finally we're left with a weird, cobbled-together Popperian/Kuhnian pragmatism that long ago gave up on "certainty" or "clarity" and settled for "less wrong."
What surprises me is that people are so willing to assume that such certainty is desirable. That quantified symbolic logic is both primary and desirable, and that probabilistic reasoning is at best an occasionally necessary but highly suspect tool.
What a sadly limited universe of discourse that is, being forever circumscribed by Godel. It is fortunate that we don't exist trapped in it. I see no reason to intellectually place myself in the same prison.
Sure -- Descartes mathematics in the abstract are surely not dependent on his foundationlism/rationalism. As such they are merely tautological. But the science that deploys that math in pursuit of practical knowledge is.
Agreed that 'certainty,' at least the kind that Descartes was advocating, is an abstraction in search of a definition, and as such not desirable. Godel and incompleteness is just one species of the problem. This is where I would rather see a reformed Baconian science, tempered with (rather than utterly rejecting) an Aristotelian view which had a more sophisticated language for dealing with ambiguity and the fluid borders between substances. Our physics has advanced far beyond simplistic atomism, why can't our philosophy of science?
Popper and falsification is a good rear-guard action against skepticism and making unwarranted philosophical leaps within the Cartesian taxonomy, but why can't we question the taxonomy itself? There are plenty of good starting points -- Whitehead and Bergson come to mind, as well as an open re-examination of medieval and pre-socratic philosophy, as Heidegger does in the Question Concerning Technology. On and on. It's not as if no one is asking these questions or refining philosophical language to deal with them -- it just seems that the scientific community would rather declare its own sovereignty and emancipate itself from first-order questions once and for all.
"And finally we're left with a weird, cobbled-together Popperian/Kuhnian pragmatism that long ago gave up on "certainty" or "clarity" and settled for "less wrong."
What surprises me is that people are so willing to assume that such certainty is desirable. That quantified symbolic logic is both primary and desirable, and that probabilistic reasoning is at best an occasionally necessary but highly suspect tool.
What a sadly limited universe of discourse that is, being forever circumscribed by Godel. It is fortunate that we don't exist trapped in it. I see no reason to intellectually place myself in the same prison.