Reticulate Formal Reasoning
How Formal Sciences Hang Together in a Reticulate Network of Connections.
Friday 12 July 2024
In last week’s newsletter I said that natural science is formalized at one remove because it is formalized by mathematics, and mathematics is formalized by logic. This is an oversimplification. There are distinctively mathematical forms of formalization, making use of distinctively mathematical forms of intuition, which aren’t derived from logic. It could be argued that diagonalization proofs employ both arithmetical and geometrical intuitions, but, either way, diagonalization seems to be a distinctively mathematical form of proof, though in its exposition as a proof, logical elements may be added to it (in a kind of “keeping up of the appearances” of the presumptively logical character of mathematical proof). Gödel’s incompleteness theorems are, essentially, a rigorous formalization of a paradox using means that are specifically mathematical, viz. the arithmetization of syntax.
It is not immediately apparent whether the conceptual framework of axiom systems is distinctively mathematical, distinctively logical, or both, or neither. If we view an axiom system as the paradigmatic form of formalization (which I do), then the ambiguous status of axiomatization infects all forms of thought that fall under the umbrella of formalization. Or one could argue that axiomatization is what is distinctive to formal thought — its differentia — and that axiomatization, as the master category of formalization, is the umbrella under which other forms of formalization fall, such as the mathematical formalization of the natural sciences or the logical formalization of mathematics (when this sequence of formalization obtains, which does occur, but not invariably). In this rational reconstruction of formal thought, distinctively mathematical and logical methods could be understood as fragmentary and imperfect mirrorings of the distant ideal of axiomatic formalization, the various aspects of which trickle down into subordinate logic and mathematics, each seeing the ideal differently because of their respective differences in conceptualization.
The logicist program of “reducing” mathematics to logic, which meant deriving all mathematical concepts from logical concepts so that mathematical concepts can be shown to be dispensable, started with the introduction of mathematical logic in the late nineteenth century, and, after several decades of research, culminated (more or less) in Quine’s claim that mathematics is logic plus set theory. In other words, there was still a need for a primitive mathematical concept, as distinct from logical concepts, for the deduction of mathematics from logic. Quine gave the primitive mathematical concept as set theory (which in turn can be reduced the membership relation), but there are alternative ways to do this, just as there are alternative ways to choose the primitive concepts of logic. And this is another interesting property of the formal sciences, that they admit of a multitude of alternative formulations. Hilary Putnam put it like this at the end of his Philosophy of Logic:
“One group of questions which I might have considered has to do with the existence of what I might call ‘equivalent constructions’ in mathematics. For example, numbers can be constructed from sets in more than one way. Moreover, the notion of set is not the only notion which can be taken as basic; we have already indicated that predicative set theory, at least, is in some sense intertranslatable with talk of formulas and truth; and even the impredicative notion of set admits of various equivalents: for example, instead of identifying functions with certain sets, as I did, I might have identified sets with certain functions. My own view is that none of these approaches should be regarded as ‘more true’ than any other; the realm of mathematical fact admits of many ‘equivalent descriptions’.”
We can see from this pervasive phenomenon of equivalent constructions (which I would prefer to call alternative formulations) how mathematical formalizations vary from discipline to discipline, and an additional effort is required to link up all these equivalent constructions and to prove that they are in fact equivalent, which was one of the underlying themes of last week’s newsletter and its exposition of what I called floating formalisms. These floating formalizations follow from the same primitive concepts, and we can associate these primitive concepts (which are primitive in the sense that they can serve as an axiom in a formal deduction) with distinctive intuitions (which are primitive in the sense that they are readily graspable concepts for the human mind, which is distinct from the previous sense of primitiveness). Now, this “association” between the deductively primitive and the cognitively primitive is another interesting problem. In the above quote from Hilary Putnam he says that a certain group of questions had concerned him in that book, and that he could have spent the book on other equally foundational problems, such as equivalent constructions. My concern with formal reasoning has been bounded by a group of questions that will have been apparent to readers of this newsletter, but I also occasionally work on other questions, and this problem of the association between deductive primitives and cognitive primitives is one of them.
We can imagine ideal conditions of these associations, such as there being a one-to-one correspondence between deductive primitives and cognitive primitives, which would simplify matters, but in light of pervasive equivalent constructions, we have reason to believe that this is not the case. There might be instances when we have a number of equivalent intuitions that correspond to one and the same deductively primitive concept, and we may have several deductively primitive concepts that are conflated in one and the same intuition, so the relationship between deductive primitives and cognitive primitives may be many-one or one-many. Given these considerations, formal thought seems naturally adaptable to a reticulate framework, such as I suggested at the end of last week’s newsletter, and which I had earlier suggested as an historical structure of the natural sciences.
Even as mathematics fissions into ever more narrowly defined specializations, the need is felt for a more comprehensive approach to knowledge, and attempts are made to join together several narrow specializations under more comprehensive programs — Felix Klein’s Erlanger program in the 19th century, logicism and the foundations of mathematics a hundred years ago, and the Langlands program today. Recently formulated programs of mathematical unification in their turn constitute new special areas of mathematical research — this was obviously the case when logicism transitioned into being foundations of mathematics research, which is now its own specialization, with its own conceptual framework and its own techniques — so that mathematics on the whole takes on a reticulate structure in which there are disciplines fissioning into narrower disciplines at the same time as multiple specializations are being collected into more comprehensive research programs. A bird’s-eye view of all these interconnecting mathematical disciplines might be called reticulate mathematics. Moreover, given the close connection between the formal methods of mathematics and the formal methods of logic, this reticulate structure could be said to extend over the whole of both logical and mathematical research, though, again, if we could see it all from a bird’s-eye view, we would see that mathematical methods were tightly-coupled together in a cluster, and logical methods were tightly-coupled together in a separate cluster, and these two clusters were only loosely coupled to each other, though still with a greater number of interconnections between logical and mathematical methods than between, say, these latter classes of methods and the methods of natural science or the methods of the social sciences.
We can continue to iterate this structure, with an internally tightly-coupled class of natural scientific methods, and an internally tightly-coupled class of social science methods (which may or may not include the methods of history and the humanities, depending upon how we define all these terms), each of which constitute their own clusters, and both of which are loosely-coupled to the clusters of logical and mathematical methods. All of these conceptual frameworks taken together constitute a structure larger than most conventional conceptions of human knowledge, but from any point within this reticulate structure one can make one’s way to any other point within this reticulate structure in a finite number of steps. It is this that constitutes the ultimate unity of human knowledge: not that everything can be derived from a single master assumption (which would be one way to construe Cartesianism, but this would be an over-simplification, and wrong in more than one way), and not that all human knowledge is reducible to something more fundamental, but that the whole is interconnected and any one part is finitistically available to any one other part. If we choose to inaugurate a new research program, we could bind together any two or more parts of the entire structure more closely, and this would create a new cluster within the network. But the new cluster would have its own distinctive formal features, which would then become its own research project and a novel specialization that could later be unified in another more comprehensive research program.
How large can this structure become? To put this in a more anthropocentrically relevant way, how far can human beings expand this epistemic structure? These questions are importantly different. Human beings can only finitely expand this finite epistemic structure. The larger the program can become, the more resources that can be fed into it (including human resources, meaning the longer we can sustain civilization, the longer we can grow our epistemic framework), the more it can grow, perhaps indefinitely. But as with our assumptions about the neatness of the formal domain — e.g., the idea discussed above of a one-one correspondence between deductive primitives and cognitive primitives, which is likely wrong — it is natural that we have assumptions about our total epistemic framework, for example, that it is a fragment of some larger whole, and this larger whole is infinite and cannot be exhausted by (finite) human methods. This is something that I have discussed in many newsletters, often indirectly, but it is there in the background. But while this is always in the background of my thinking, it may well be wrong, like the intuition of a one-one correspondence between deductive primitives and cognitive primitives.
So far I have been implicitly probing the extensive expansion of our reticulate structure of knowledge, which is not only extended to greater comprehensivity, but also historically extended insofar as the reticulate structure I have proposed is a reticulate structure of the history of knowledge in which new disciplines continuously appear — some being narrower specializations while others are broader programs of unification, each of which feeds into the other over historical time — thus being an open-ended process, meaning that the elaboration of scientific knowledge is potentially infinitistic. There is another dimension here, and that is the possibility of the intensive expansion of this reticulate framework, which might bring within its fold neglected areas of human experience that have not been successfully assimilated to science. For example, a hundred years ago, when the logicist program was enjoying its heyday, there was also a wide interest among philosophers in extending scientific methods to psychical research, which would mean not merely a scientific study of what we could today call the paranormal, but a folding of psychic and paranormal research into the overall framework of human knowledge.
This never happened, although there is always a fringe interest in such things, and one could argue that the Galileo Project is a contemporary manifestation of this kind of interest. But there are much more interesting and much more problematic examples. One that stands out today is the absence of a science of consciousness. There is cognitive science, which is part of science proper, and psychology of course, but we lack the concepts to study consciousness scientifically, just as before the development of the atomic theory of matter, we lacked the concepts to discuss chemical compounds, which led to Tartuffery like characterizing water in terms of its wetness. This is a dead end, and it is important to recognize dead ends and not to direct a lot of resources to them, which would starve directions of research that are not dead ends. Because there is no science of science, we have no criterion by which to identify a scientific research program as a dead end. We have only our instincts and our intuitions, which can be wrong.
