The Boundless Ocean of Unlimited Possibilities

The View from Oregon – 331: Friday 07 March 2025

Of the vast amount of work done on the philosophy of science during the twentieth century, and so far for the first quarter of the twenty-first century, interdisciplinarity has not been philosophically problematized. No doubt there are philosophers who have worked on this, but in none of the books on philosophy of science I have to hand is interdisciplinarity ever even raised as a question. It’s simply not there. I’ve had to create my own terminology to discuss it. In last week’s discussion of univocal and interdiscipinary sciences I called those sciences that are not interdisciplinary univocal sciences. Some label is needed, so that’s what I’m using until I hit on something better. To this day there remain many potential univocal sciences that have not yet been formulated, perhaps due (if only in part) to my claim of last week that some concepts are intrinsically more suited to scientific schematization than others, or to another claim I made last week that it is not the case that every nascent science can converge on a coherent scientific research program. In several newsletters I’ve mentioned the lack of sciences of civilization, of technology, of consciousness, and, most paradoxically, the lack of a science of science. These are all examples of potential univocal sciences, though I suppose they could also be taken for potential interdisciplinary sciences.

Some putatively interdisciplinary sciences may be misidentified as such. Last week, again, I suggested that a science like astrobiology, which is routinely identified as interdisciplinary, might be a radical re-founding of the univocal discipline of biology as it should have been from the start. But all sciences have their distant origins in folk knowledge, none of which is systematic, so that the long, slow climb to scientific rigor as it should have been means the continual re-founding of a science as scientific knowledge and scientific technique expand. Later more comprehensive univocal disciplines appear as interdisciplinary because they draw on all the knowledge that preceded them, but in reality they are a more comprehensive but still univocal approach to the same subject matter. Thus we discover a fundamental metaphysical idea in our unscientific taxonomy of the sciences (because there is no science of science to furnish a scientific taxonomy of the sciences): the appearance and reality of a given scientific discipline, viz. a discipline may appear to be interdisciplinary when it is, rather, univocal in reality. This also suggests the possibility that a science might appear to be univocal but rather is, in reality, interdisciplinary. We can imagine a case when the development of greater rigor forces us to split a traditional science into two or more sciences to make good on a conflation of which the earlier science is guilty. Must it be an outright conflation? There is a sense in which the fanning out of a discipline into a multiplicity of subdisciplines suggests that the previous unity of the discipline was an interdisciplinary science mistaken for a univocal discipline.

The re-founding of biology as astrobiology (and the implied parallel re-founding of geology as astrogeology, and so on and so forth) has a parallel in the formal sciences. Back in newsletters 303 and 311 I quoted Ian Stewart and David Tall in The Foundations of Mathematics on how mathematics teachers will sometimes tell their students, “Forget all you’ve learned up till now, it’s wrong, we’ll begin again from scratch, only this time we’ll get it right.” This is effectively a re-founding of mathematics, which in a child’s earliest education is usually taught in a way that appeals to intuitions, whereas advanced mathematics requires a more rigorous foundation that is likely to be less tied to anthropogenic intuitions. Logic was similarly re-founded with the advent of mathematical logic. One way to define the re-founding of a science would be to argue that a re-founded science is backward-compatible, in the sense that all (or, at least, most) propositions (or arguments) that could be formulated in the previous iteration of the discipline can be formulated in the re-founded discipline, and the re-founded discipline can also formulate propositions that could not be formulated in the previous iteration. This is clearly the case with logic, as all of the arguments of Aristotle’s syllogistic logic, Stoic propositional logic, and medieval terminist logic can all be formulated in mathematical logic, which can also formulate more subtle and sophisticated propositions that no earlier logic had formalized. In this way the re-founding of a discipline can be a scientific revolution.

Kuhn’s explanation for scientific revolutions was a paradigm shift, and while I have some sympathy for this, in light of what I’ve written recently I am inclined to make a distinction between scientific revolutions that are the result of a paradigm shift and scientific revolutions that are the result of the re-founding of a univocal science, so as to make that univocal science more comprehensive and universal. The continual re-founding of sciences is a Copernican revolution in knowledge. The concepts that were central to the univocal discipline in its prior state are shifted to the periphery, and new concepts of greater generality are substituted at the foundations of the science. This is a de-centering of the previous knowledge in favor of a less parochial perspective on knowledge, and, just as there have been repeated Copernican revolutions that have de-centered human beings from our former Protagorean place as the measure of all things, so too scientific knowledge has been repeatedly de-centered as a univocal discipline has been re-founded in its place, where previously stood the earlier incarnation of knowledge. This Copernican de-centering of knowledge is also exemplified in a quote from Einstein that I’ve had occasion to cite previously:

“ No fairer destiny could be allotted to any physical theory, than that it should of itself point out the way to the introduction of a more comprehensive theory, in which it lives on as a limiting case.”

I cited this recently in my Today in Philosophy of History episode on Copernicus and the Formal Symmetries of History. There the formal symmetries I had in mind were space here being the same as space there, and what I called the empirical correlates of these symmetries of translation in space, being the idea that our sun is a star, and every star is in turn another sun. Of course, when we get into the details this isn’t quite true, but it isn’t precisely true because stars fall into different classes, and different classes of stars differ in predictable ways. The larger truth here is that we could never have arrived at this knowledge of stellar classifications had we not thought in terms of these symmetries, even if only implicitly. That a red dwarf star here is like a red dwarf star there has been demonstrated by our reconstruction of stellar evolution for stars of different masses. Einstein’s limiting case principle doesn’t work so well with cosmology, as Ptolemaic cosmology does not seem to live on within Copernican cosmology as a limiting case of the former, but this observation allows us to make a distinction among scientific revolutions, such that some leave the former theory in place as a limiting case, and some do not. However, Einstein’s limiting case principle works quite well with my above formulation of the backward-compatibility of re-founded sciences.

One way that we could define a univocal science would be in terms of a foundational concept, or a cluster of tightly related concepts, that are distinctive to the univocal science in question, i.e., not shared with any other univocal science. To say that a science has distinctive concepts of its own is to say that the practitioners of the science have formulated a kind of abstraction that is distinctive to that science and which therefore, in a sense, defines that science, because a concept is abstraction. This definition of a univocal science gives us a picture of the relation to science to world that is significant for what I’ve written recently about whether or not science can be exhausted. If each univocal science is founded on a concept or a cluster of concepts unshared by any other univocal science, then the exhaustive division of concepts gives us the number of sciences there are, and once all concepts have been scientifically schematized, the telos of science has been achieved and scientific knowledge is exhausted. However, if further concepts can be formed, then additional sciences could take these newly formed concepts as their foundation. Thus the ongoing viability of science is a function of concept formation.

In an interdisciplinary science, foundational concepts or clusters of concepts are shared with univocal sciences. We could further distinguish between a strongly interdisciplinary sciences and a weakly interdisciplinary sciences based on whether the science in question has any distinctive concepts of its own (a weakly interdisciplinary science) or whether it is entirely dependent upon concepts borrowed from the conceptual frameworks of the sciences from which it draws (a strongly interdisciplinary science). Among weakly interdisciplinary sciences, we can determine the ratio of distinctive concepts to borrowed concepts and establish the degree of weakness or strength of interdisciplinarity as a continuum that bridges between purely univocal sciences and strongly interdisciplinary sciences with no distinct concepts of their own.

Formal thought has been here already, and there is a defunct distinction we can draw upon that goes back at least to Euclid—the distinction between axioms and postulates. This distinction has fallen away from the formal sciences, especially since the re-foundation of logic after the innovations of mathematical logic, but it remains a legitimate distinction. Because it has fallen out of favor, and never received a formulation in re-founded logic, we have to re-imagine Euclid’s axiomatics implicit in his geometry in contemporary terms. Axioms were common principles of reasoning found in all mathematics, while postulates were narrower assumptions that defined a discipline within mathematics, like plane geometry. In a sense then, axioms are shared concepts, while postulates are concepts of univocal sciences that differentiate one univocal science from another. Axioms, then, are universal borrowed concepts. As the concepts shared by all sciences and necessary to all formal thought, a science uniquely constituted by axioms and only by axioms would constitute the core logic of science, and this would be in fact the only univocal science in a narrow sense, making use of no differentiating postulates. All sciences founded on postulates would also make use of axioms, making them weakly interdisciplinary sciences.

A lot of this kind of thinking was implicit in formal thought up through the nineteenth century, and was largely lost with the re-founding of logic as mathematical logic. Traditional logic was assumed to be the one universal science that was necessary to the practice of all other sciences, so that the logic of science was, in a sense, the science of science. I have been arguing for some time now that there is no science of science, but (I now see) I could just as well argue that a science of science is a function of the conceptual framework of a civilization or an era of history, and that we had a science of science that was slowly and incrementally built up from Aristotle’s Posterior Analytics up through nineteenth century works on logic like Bernard Bolzano’s Theory of Science. The re-founding of logic as mathematical logic upended this entire tradition, ending the science of science that was adequate to the needs to pre-modern science. In this way, mathematical logic is to the science of modern science as Aristotle’s Posterior Analytics was to the science of pre-modern science. We had to abandon our previous science of science and set out upon the uncharted waters of modern science with no foundation or map, in order to inaugurate a new age of science, which might someday converge on its own science of science. Rudolf Carnap, one of the founders of mathematical logic, gave a sense of what was involved in this, and presented it as an adventure:

“The first attempts to cast the ship of logic off from the terra firma of the classical forms were certainly bold ones, considered from the historical point of view. But they were hampered by the striving after ‘correctness.’ Now, however, that impediment has been overcome, and before us lies the boundless ocean of unlimited possibilities.”

All adventures, however, are fraught with peril. The adventure of modern logic, freeing itself from what Carnap called the striving after correctness, has its perils as well, since, without this ancient sense of correctness, we are left with dead reckoning as our only means of navigation.

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