Scientific Knowledge and Scientific Abstraction

Successful Scientific Research is Predicated upon an Aspect of Reasoning we Scarcely Understand

Jackson Pollock, “There Were Seven in Eight” c. 1945 — for many of us, this is what “abstraction” is.
Georg Cantor formulated a doctrine of mathematical abstraction that was insufficiently rigorous for Frege.

Mathematical Abstraction

The discussion of abstraction in the philosophy of mathematics has been exception to the paucity of discussion about abstraction in science, and this is itself interesting, because mathematics has always had a troubled taxonomic relationship to empirical science. Mathematics has proved to be essential to the development of science, but in so far as mathematical techniques are employed to arrive at a mathematicized science, mathematics itself does not seem to be a part of science, but rather a meta-scientific inquiry, like logic (and, for that matter, the other formal sciences, which today may also include computer science). At other times, science is unproblematically treated as one of the natural sciences. Mathematics tends to be taught in the scientific faculties of universities, and of course it is the “M” in STEM.

Gottfried Wilhelm Leibniz, who first formulated the principle of the identity of indiscernibles.

Scientific Abstraction and the Identity of Indiscernibles

How are we to begin to define, and hence delimit, abstraction? One way that has occurred to me that could be used to define abstractness is anything for which Leibniz’s law holds — Leibniz’s law being the identity of indiscernibles, which is the idea that any two things with all their properties in common are identical. Any existents that have spatial or temporal dimensions would be characterized by spatial and temporal properties that would individuate them from other existents, and only objects with no spatial or temporal properties whatsoever (and therefore entirely abstract) would have the possibility of fulfilling Leibniz’s law.

Wilhelm Windelband

Scientific Abstraction in the Historical Sciences

Scientific historiography — if there is ever to be a discipline that fully deserves to be called as such — will have to make use of scientific abstraction no less than any other branch of scientific inquiry. This is sufficiently obvious to be a truism, except in the case of history it presents certain problems from which the other special sciences do not suffer. History, like mathematics, has always had a troubled relationship to the rest of science. As there has always been a question as to whether mathematics is rightly to be considered a part of science, so too there has always been a question as to whether history is to be considered a science.

Carnap’s classic exposition of taxonomic, comparative, and quantitative concepts from this Philosophical Foundations of Physics: An Outline of the Philosophy of Science.

Scientific Concepts According to Carnap

I would argue that Windelband’s compartmentalization represents a taxonomic approach to epistemology, and if we follow Carnap’s familiar scheme of the development of scientific concepts from the taxonomic to the comparative to the quantitative, we can easily see how an early attempt at a scientific epistemology such as we find in Windelband employs a taxonomic structure. This is the first stage.



One Man Think Tank

Get the Medium app

A button that says 'Download on the App Store', and if clicked it will lead you to the iOS App store
A button that says 'Get it on, Google Play', and if clicked it will lead you to the Google Play store