Scientific Knowledge and Scientific Abstraction

Jackson Pollock, “There Were Seven in Eight” c. 1945 — for many of us, this is what “abstraction” is.

A great deal has been written about the scientific method, but much less about scientific abstraction, that is to say, the particular mode of abstraction employed in the sciences. I would argue that scientific abstraction is equally as important as scientific method, but I think that it is relatively neglected both in science itself and in contemporary analytical philosophy of science, because it is assumed to be so simple as matter as to be unworthy of exposition, while it is, in fact, a much more difficult philosophical problem than scientific method.

Scientific method can be approached in an entirely pragmatic spirit. If a method works, by which I mean if it yields accurate predictions, then it can be adopted, and refinements to the scientific method can be adopted as they prove themselves in the production of scientific knowledge. Method is, by definition, a matter of practice, and practice has practical consequence and can be learned through observation and imitation.

Scientific abstraction is a different matter; in so far as it is a matter of practice, it is a matter of conceptual practice, and it cannot be learned by imitation, example, or observation. At best, scientific abstraction can only be inferred from scientific practice, and different observers will make different inferences because the theoretical process of making this inference has not been formalized. This is, then, a very delicate task, and the various inferences made often correspond with the differing successes of different scientists and research groups.

For science to be successful, the scientific method must operate within the parameters of just the right degree of abstraction — neither too little nor too much. Too little abstraction leads to Pyrrhonian skepticism, as the parsing of empirical details will always lead to a uniqueness that defies schematization and prediction. Too much abstraction leads to circumstances such as the failure of formal models to accurately describe actual human behavior: we have abstracted from too much of the context, something has been lost, and any theory that grows out of the his impoverished context is misleading.

The problem is that science itself is silent on arriving at what exactly constitutes just the right degree of abstraction. This is another way in which theory development in science is unscientific. The greatest scientists who make the greatest discoveries are those who possess an uncanny knack for hitting on just the right degree of scientific abstraction within a body of knowledge heretofore unschematized by science. There is no way to teach having a “knack,” and there is no method that can substitute for it.

Because scientific abstraction hasn’t been given a systematic and explicit exposition, scientists by-and-large follow their intuitions — indeed, it might be said that what distinguishes a great scientist from a mediocre scientist is the quality of their intuition, which means knowing what problems to work on and how to work on them — and these intuitions go largely unexamined if they are successful. Unsuccessful researchers might cast about for a more effective intuition on which to proceed, but science today is so large that there is no shortage of “normal science” problem solving for those who lack the imagination to frame a new intuitive research program.

Science derives much of its predictive power, hence its power as a pragmatic force to extend human agency, from its careful and creative use of scientific abstraction. And even though scientific abstraction is not well understood, and is far from being explicitly delineated, its importance is occasionally recognized. For example:

“…power in science springs from abstraction. Thus, although a feature of nature may be established by close observation of a concrete system, the scope of its application is extended enormously by expressing the observation in abstract terms.” (Peter Atkins, The Laws of Thermodynamics, p. 38)

The expression of scientific truths derived from empirical observation in abstract terms is a problem of formalization: finding the right abstract terms in which to express a theory is the same as finding the right formalization of an empirical theory, and this is much more difficult that designing an experiment, making observations, and recording measurements. Finding a new and more effective formalization is what Einstein did in proposing the theory of relativity, rather than working within the established paradigm of physics.

Here we glimpse one of the underlying difficulties of understanding scientific abstraction, and why it is such a difficult philosophical problem: it demands not merely an extrapolation of formal methods, but self-reflection on formal methods, and the latter is much more difficult, and far more rare, than the former. Logic and mathematics have almost exploded as disciplines over the past hundred years. (Keith Devlin, for example,says that mathematics today is in a “new golden age.”) While the quantity and sophistication of formal thought has grown by leaps and bounds, critical self-reflection on the methods of formalization — almost always taking place at a later stage of intellectual development than the stage of discovery — has not grown, and in some respects has remained notably stagnant over the same time period.

We should not be surprised at the stagnation of critical inquiry into formalization: discovery demands, or at least flourishes best, when it takes place against a uniform and stable background of assumptions, and one of these uniform and stable background conditions is scientific abstraction. The formal background of thought is likely to remain untouched as long as it continues to yield scientific knowledge (i.e., as long as scientific discovery continues). An explicit and thorough critical study of scientific abstraction will probably come later in the development of science, which latter still remains largely within the paradigm of discovery that grew out of the scientific revolution.

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.

Georg Cantor discussed abstraction as a method by which the mathematician arrives at the sets that are the subject matter of set theory. Gottlob Frege wrote a review criticizing Cantor’s conception of abstraction — an essay that, despite its abstract character, is quite funny. Frege wrote:

“So let us get a number of men together and ask them to exert themselves to the utmost in abstracting from the nature of the pencil and the order in which its elements are given. After we have allowed them sufficient time for this difficult task, we ask the first ‘What general concept (p. 56) have you arrived at?’ Non-mathematician that he is, he answers ‘Pure Being’. The second thinks rather ‘Pure nothingness’, the third — I suspect a pupil of Cantor’s — ‘The cardinal number one’. A fourth is perhaps left with the woeful feeling that everything has evaporated, a fifth — surely a pupil of Cantor’s — hears an inner voice whispering that graphite and wood, the constituents of the pencil, are ‘constitutive elements’, and so he arrives at the general concept called the cardinal number two. Now why shouldn’t one man come out with the answer and another with another?”

Gottlob Frege, Posthumous Writings, “Draft towards a Review of Cantor’s Lehre vom Transfiniten,” pp. 70–71.

Frege’s critique of Cantor on abstraction (and I should point out the the above passage was not included in Frege’s published review of Cantor) points out potentially serious problems with abstraction, but nothing in Frege’s critique demonstrates that even Cantorian abstraction could not be made rigorous if subject to sufficiently strong constraints. Frege was in a particularly good position to appreciate this, because as one of the founding fathers of mathematical logic, responsible for such conceptual innovations as quantification, he knew the shortcomings of traditional logic, and the need to subject logical reasoning to rigor. To this end, in the opening paragraph of his The Foundations of Arithmetic Frege wrote:

“After deserting for a time the old Euclidean standards of rigour, mathematics is now returning to them, and even making efforts to go beyond them. In arithmetic, if only because many of its methods and concepts originated in India, it has been the tradition to reason less strictly than in geometry, which was in the main developed by the Greeks. The discovery of higher analysis only served to confirm this tendency; for considerable, almost insuperable, difficulties stood in the way of any rigorous treatment of these subjects, while at the same time small reward seemed likely for the efforts expended in overcoming them. Later developments, however, have shown more and more clearly that in mathematics a mere moral conviction, supported by a mass of successful applications, is not good enough. Proof is now demanded of many things that formerly passed as self-evident. Again and again the limits to the validity of a proposition have been in this way established for the first time.”

A doctrine of abstraction adequate to mathematics, a fortiori adequate to science, also faces insuperable difficulties, and, in science, a mere moral conviction supported by a mass of successful applications, is not good enough. But if we could establish the limits to the validity of abstraction we would make a good start toward an understanding of abstraction.

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.

Now, in so far as empirical science is concerned, if the objects of empirical science are abstract, then there is no longer anything empirical about the inquiry. Only the objects of mathematics can be abstract in this way. Nevertheless, I have asserted that science makes use of abstraction. That means that we must posit a continuum (or, at least, a distribution) of abstraction that recognizes degrees of abstractness that are more or less abstract.

We can find an implicit continuum of abstraction in the famous quote from Einstein — “As far as the laws of mathematics refer to reality, they are not certain; and as far as they are certain, they do not refer to reality” — that I have discussed in Unpacking an Einstein Aphorism and Einstein on Geometrical Intuition, which points to the abstraction of pure mathematical concepts and contrasts this abstractness to the concreteness of an uncertain reality. Because Einstein formulated this idea in terms of “as far as,” this suggests that laws of mathematics might more or less certain, or more or less applicable to reality, and this more or less would define a continuum of abstraction.

If we posit such a continuum, we can immediately see that mathematics is the idealized end point of the abstract end of such a continuum, and that sciences like physics are somewhere near the mid-point of the continuum of abstraction, while studies such as biology and history are much closer to occupying the ideal end-point antithetical to that of mathematics. Pure experience, if there is such a thing, or what Husserl called pre-predicative experience, would occupy the extreme end point of concreteness on the continuum of abstractness. In this way, we can see the relationship of mathematics to the other sciences, making the claim that mathematics is the result of pursuing scientific abstraction to its non plus ultra of development.

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.

One could say that the question of the relationship of history to the sciences is the exact opposite of the question of the relationship of mathematics to the sciences — or one could say that it is essentially the same problem, though seen from a different perspective. The possibility of a continuum of abstraction provides a framework within which we can better understand the relationship both of mathematics and history to the familiar empirical sciences. Most of the sciences are located near the middle of the continuum of abstraction, so that they make use of both abstract and concrete concepts. What distinguishes history and mathematics is that these disciplines are located near the extreme ends of the continuum — though at different ends: mathematics near the abstract end of the continuum, and history near the concrete end of the continuum.

From the perspective of the continuum of abstraction, then, we can relate both mathematics and history to conventional empirical sciences like physics, chemistry, biology, and sociology. Mathematics and history are outliers, but they are outliers as a matter of degree, and not outliers as a matter of kind. Mathematics and history are different perspectives on the same (abstract) state-of-affairs because they are views of abstraction from opposite ends of the continuum of abstraction.

There are any number of traditionalists about historiography that deny the possibility of scientific history on principle. This I find to be uninteresting. Whether or not history is or ever was a science, a hidebound traditionalism in historiography will certainly mean that history will cease to be a science and will become a mere catalog, like anatomy or geography, which have ceased to sciences. There are less traditionalistic arguments that would limit the scope of scientific historiography, and these are more interesting.

While I have not seen this particular argument articulated, I can imagine that someone might argue that history, being at the far end of concreteness on the continuum of abstraction, simply is not and cannot be made sufficiently abstract to enter into any kind of meaningful dialogue with the sciences. Something like this (though not exactly like this) is implicit in the distinction made between nomothetic and the idiographic as made by Wilhelm Windelband in his 1894 rectorial address:

“In their quest for knowledge of reality, the empirical sciences either seek the general in the form of the law of nature or the particular in the form of the historically defined structure. On the one hand, they are concerned with the form which invariably remains constant. On the other hand, they are concerned with the unique, immanently defined content of the real event. The former disciplines are nomological sciences. The latter disciplines are sciences of process or sciences of the event. The nomological sciences are concerned with what is invariably the case. The sciences of process are concerned with what was once the case. If I may be permitted to introduce some new technical terms, scientific thought is nomothetic in the former case and idiographic in the latter case. Should we retain the customary expressions, then it can be said that the dichotomy at stake here concerns the distinction between the natural and the historical disciplines.”

History, then, is a distinct form of knowledge, and the historical sciences partake of a distinct relationship to reality, and this relationship to reality is different from that of the natural sciences. Earlier in the same address Windelband distinguished mathematics from the empirical sciences:

“Although the actual, psychogenetic occasion for research and discovery in philosophy and mathematics may very well lie in empirical motives, the propositions of philosophy and mathematics are never based on single observations or collections of observations. By empirical sciences, on the other hand, we understand disciplines which undertake to establish knowledge of reality which is somehow given and accessible to observation. The formal criterion of the empirical sciences may be described as follows. The validation of the results of these sciences includes not only the general, axiomatic presuppositions and the norms of valid thinking which are necessary conditions for all forms of knowledge; it also requires the verification of facts on the basis of observation.”

This was from a less anti-philosophical era, when scientists could openly acknowledge that science assumed axiomatic presuppositions drawn from philosophy and mathematics. In any case, Windelband’s schema nicely compartmentalizes mathematics, natural science, and history. Is it better that we keep these forms of knowledge isolated from each other, each in an appropriate epistemic category, or are there advantages to stringing out all these forms of knowledge as representing different locations along a continuum of abstraction?

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.

To locate various kinds of knowledge along a continuum of abstraction is to formulate a comparative measure for the role of abstraction in various forms of scientific knowledge: physics is more abstract than biology, and biology is more abstract than history. These are comparative concepts, and this is the second step according to Carnap’s developmental account of scientific concepts.

We could only make the next step to quantitative concepts if a way were found to quantify the degree of abstraction represented by a continuum of abstraction. At this stage of conceptual development any location on the continuum of abstraction would have a precise numerical value, and any form of knowledge could then be given a numerical value representing its degree of abstraction.

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