#foundationalism

René Descartes, Meditations on First Philosophy, Meditation One (p. 59 in Hackett 4th ed.)

The Threshold of Schematization.—The relativistic conception of science we find in Spengler and Danilevsky readily accommodates the interpretation of a shift from a logical to a mathematical science of science, i.e., a shift from logical foundations to mathematical foundations of thought, growing out of the scientific revolution and its mathematization of human knowledge. Apollonian logicism, with its finite, harmonious, and orderly Pythagorean oppositions, has given way to Faustian mathematicism, which is better positioned to schematize the pursuit of the infinite. The shift that has occurred in the epistemic development of Western thought, played out over historical time, could also manifest itself differently in different traditions. In an inversion of the Western experience, some civilization might begin with a mathematical science of science and eventually shift to a logical science of science. A civilization might have a science of science, or a science of technology, or a science of civilization, even while another civilization declares these to be impossible, and a third remains utterly indifferent. And each civilization may be right relative to their conceptual framework. The science of science of a given civilization may be incommensurable with the science of science of any other civilization that has developed its epistemic framework to this level of sophistication. The problem to which a science of science is the solution only appears in those civilizations in which knowledge has passed a given threshold. And with the diachronic agglomeration of civilization that we find in a tradition that passes through discrete stages of civilizational development, as has been the case with Western civilization, in its ancient, medieval, and modern exemplifications, the problem must appear differently, and so too the solution to the problem will be different. Each age will require its own science of science, if indeed if develops epistemically to the point at which the problem appears and demands an answer.

Fulfilling the Destiny of Mathematics.—What unifies modern science, and what distinguishes modern science from its premodern antecedents, is mathematics. This is the common language of the sciences, the language in which its various laws are expressed, thus the syntax for which the special sciences provide the semantics. Is mathematics, then, the science of science? Or will mathematics someday become the science of science that must develop out of modern science? If mathematical logic is to the science of modern science as Aristotle’s Posterior Analytics was to the science of pre-modern science (as I wrote previously), what will mathematical logic become when, at long last, modern science eventually converges on its telos? Logical reasoning is subordinated to mathematical forms in mathematical logic, which has, moreover, already made the transition from logical calculus to foundational research, becoming, in the process, another specialization and not a grand theoretical umbrella that covers the whole of the sciences. And there is no treatise on mathematics that is the equivalent of what Aristotle’s Posterior Analytics was to ancient thought. Euclid’s Elements was thoroughly in the spirit of the Posterior Analytics, so while it is a classic of mathematical thought, it doesn’t represent a counterpart to the logical tradition. The definitive treatise on mathematics has not yet been written because it cannot be written at the present stage of the development of formal thought. The shift from logic as the science of science to mathematics as the science of science remains unfinished as yet. Mathematics has not yet achieved its telos, and we have no assurance that it can achieve this. Like the modern sciences themselves, which multiply and diverge, mathematics has multiplied and diverged, pouring out its substance like a river emptying into a desert, absorbed into the sands and ultimately feeding underground rivers hidden from sight. We have yet to locate the caverns through which these unseen rivers flow; and mathematics has yet further permutations and macroevolutionary stages to pass through before it fulfills its destiny as a science of science.

The Permanent Possibility of Formalization.—Even if most reasoning is not fully formalized, knowing that reasoning can be formalized, and reasoning within the parameters of the possibility of formalization, exercises a formal regulatory function over reason.  There is much that can go wrong, however, if we adopt this attitude too casually. We may imagine that, as we extrapolate our reasoning into unfamiliar theoretical territory that we are all the time reasoning within the parameters of the possibility of formalization even as we incrementally depart from these parameters. Thus it becomes necessary to regularly recur to full formalization under the changed and changing conditions of our reasoning. It is not enough to clarify the foundations of reasoning a single time, and then to move on to further constructions on this foundation. One must return to the foundations time and again to see how they are bearing up under the superstructure that is being erected upon them. 

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Wittgenstein, Philosophical Grammar

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