The Infinitistic Structure of Knowledge
Friday 19 July 2024
The reticulate structure of knowledge that I discussed in last week’s newsletter in regard to the formal sciences, and discussed some time ago in blog posts in regard to the natural sciences, isn’t merely about the synchronic interrelations of knowledge. Others have previously spelled out synchronic reticulate structure. The idea of the reticulate structure of knowledge was made fully explicit by Donald Davidson, who discussed logical geography, and can arguably be traced further back to Wittgenstein’s use of “logical space” in the Tractatus. I looked back through past newsletters and I was surprised to see that I hadn’t quoted Davidson several times, so here it is:
“…much of the interest in logical form comes from an interest in logical geography: to give the logical form of a sentence is to give its logical location in the totality of sentences, to describe it in a way that explicitly determines what sentences it entails and what sentences it is entailed by. The location must be given relative to a specific deductive theory; so logical form itself is relative to a theory.” (Donald Davidson, Essays on Actions and Events, p. 119)
There are several remarks on logical space in Wittgenstein’s Tractatus; what follows is 3.4 to 3.42:
“The proposition determines a place in logical space: the existence of this logical place is guaranteed by the existence of the constituent parts alone, by the existence of the significant proposition. The propositional sign and the logical co-ordinates: that is the logical place. The geometrical and the logical place agree in that each is the possibility of an existence. Although a proposition may only determine one place in logical space, the whole logical space must already be given by it. (Otherwise denial, the logical sum, the logical product, etc., would always introduce new elements — in co-ordination.) (The logical scaffolding round the picture determines the logical space. The proposition reaches through the whole logical space.)”
The sense that I am giving to the reticulate structure of knowledge is partly this, but more importantly it is a developmental structure of knowledge, i.e., the diachronic structure of knowledge. As human knowledge develops through historical time, it sometimes spreads out into specialization, and sometimes comes together under schemes of unification. This is true at any one time in history (the synchronic conception), but it is also true through history (the diachronic conception). The historical development of human rationality is an interesting problem both for philosophy of history and the philosophies of the sciences (which includes philosophies of the natural sciences and philosophies of the formal sciences). The way I have phrased this in the past few newsletters has emphasized this as a problem internal to the sciences, which is a relatively “safe” way to talk about the evolution and development of reason, directing the discussion toward technical questions of logical relatedness and derivation. The unsafe way to think about human reason is the big picture that has been philosophy of history.
Hegel notoriously claimed that human history was the development of spirit or reason, and this is the kind of substantive philosophy of history from which philosophers ever since Hegel have sought to distance themselves, but when we see the Hegelian project from the perspective of the attempt to understand the development of science, logic, and mathematics as part of human history (the “safe” narrative of rationality), we can understand the reasonableness of the Hegelian attempt to grapple with the history of reason. Even if one rejects the past paradigm of human beings as rational animals, we must still reckon with the role of knowledge in human history, and its apparent development toward more comprehensive and rigorous forms. If convergence upon rigorous canons of formal reasoning is an arrow of human time, then we should be willing to say so out loud, or at least be willing to examine the claim on its merits.
In several places I have said that there are major disconnects that characterize our understanding. The examples I used typically include the disconnection between philosophy of law and political philosophy, and the disconnection between philosophy of time and the philosophy of history. Another such disconnect is that between the various forms of reason, which we can schematically characterize as the kind of reasoning employed in the natural sciences and the sort of reasoning employed in the formal sciences (and here I say that this is “schematic” because it is an oversimplification for the purpose of brevity). To bring all forms of reasoning together into a single theoretical framework and to see this framework developing over historical time is to pursue the Hegelian project of seeing human history in terms of the development of reason, and, while this is far from the only way to understand human history, it is one valid perspective on human history among others. We can also understand this as the project of unifying epistemic enclaves into a total epistemic framework, which once again puts us in Hegelian territory, as we are here flirting with the concept of absolute knowledge.
Once we see human knowledge in these big picture terms, there are obvious questions we have about the historical structure of knowledge. Is knowledge growing, i.e., is their progress in knowledge, or are we merely lurching from one Kuhnian paradigm to another, without any real progress, even though there are changes? If there is growth of knowledge over historical time, is it growth in knowledge across the board, so that all knowledge expands at the same time, or do we have some knowledge that is relatively static while other areas of knowledge grow, and the roles shift over historical time? And does knowledge expand linearly, adding triumph to triumph, going from strength to strength, or does knowledge advance in fits and starts, with two steps forward followed by one step back?
It seems obvious to me that, given a naturalistic conception of knowledge, we can readily see that knowledge has in fact increased over human history, but that its growth has been highly uneven across the frontier of the unknown, and that it has occurred in fits and starts. Yes, we do appear to have shifted from one paradigm to another many times in the history of knowledge, but even in the midst of paradigm shifts we have retained much of the theoretical framework that we employ for the construction of knowledge. One way to put this would be to say that knowledge has grown, but its growth has been imperfect and suboptimal. In other words, the growth of knowledge has been less than ideal, but it has been growth nevertheless (perhaps ideal growth of knowledge obtains only given a science of science that standardizes research across disciplines).
If there are future paradigm shifts in our scientific knowledge to come, even shifts that constitute major epistemic upheavals, I predict that we will retain that greater part of our theoretical framework, not only mathematics and logic, but also increasingly sophisticated metrology, without which science does not happen, and which seems to be a conditio sine qua non of science. This kind of prediction, however, hangs on one’s interpretation of a paradigm and a paradigm shift, and there are many from which to choose. One can have it either way one wishes, within limits. For example, the nature of formal systems has changed significantly since their introduction in ancient Greece, so that the question of the continuity of this theoretical framework can be called into question.
Once again I can note what I wrote last week, and add a qualification. (I am, in this way, ratcheting my way — hopefully — toward a coherent epistemological position through iterated assertion followed by qualification.) What I wrote last week was, “…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…” The problem — if it is a problem — is that axiomatization has changed over time. Euclid made a distinction between axioms and postulates that has since been abandoned in formal thought, though it remains valid, and in some contexts it can be the clarification that we need. Today we have formation rules and transformation rules, and our conception of what exactly an axiom is has shifted from that which is certain beyond question to a proposition maintained for the sake of hypothetico-deductivism, stripped of any connotation of certainty or even meaningfulness.
The argument could be made that Euclidean axiomatics is sufficiently different from contemporary formal systems that we are doing something different from what the ancient Greeks were doing when we axiomatize a formal system; however, the argument can also be made the contemporary formal systems are derived from earlier axiomatics, through descent with modification, and that they therefore represent the same lineage or genealogy of formal thought as axiomatics. (What is that lineage? Is it logical, or mathematical, or something else?) Like I said, you can have it either way, within limits. One could here employ Quine’s “change of subject, change of logic” argument (chapter 6 of the first edition of Philosophy of Logic), according to which a deviant logic isn’t really a logic at all, but a change of subject. But here we need to specify what falls within the scope of orthodoxy, and what within the scope of deviancy.
The loss of concepts (e.g., postulate), distinctions (e.g., axiom/postulate), and categories of thought (e.g., formal certainty) with the loss of the ancient conception of axiomatics represents an impoverishment of formal thought since antiquity, but, at the same time, formal thought has been greatly enriched by concepts (e.g., quantification), distinctions (e.g., object-language/meta-language), and categories of thought (e.g., hypothetico-deductivism) unknown to antiquity. Overall, formal thought has grown, but it has been uneven growth in fits and starts, and it is not beyond the realm of possibility that future expansions of formal thought may restore losses in a more comprehensive theoretical framework. I was, after all, trying to get at the developmental nature of formal thought, and not to regard it as something static and unchanging. This historical process of development is ongoing, and will remain ongoing as long as human civilization endures and can support it. In the interest of this developmental conception of human reason through history, let me return to a previous point by way of a detour.
It is one of the fascinating peculiarities of the mathematical study of the infinite, i.e., the theory of transfinite numbers, that higher infinities crop up in the most unlikely places, always challenging finitistic human intuition. In an infinitely branching tree structure, for example, the nodes of the tree constitute a countable infinity while the possible pathways through the structure constitute a higher infinity, a non-countable infinity. (It took me years to wrap my head around this.) You cannot put the nodes of the tree into a one-to-one correspondence with the pathways through the structure, because there are simply too many pathways (no matter how many branches at each node, including an infinite number of branchings at each node), and, if you had an infinite amount of time, you could count all the nodes, but you couldn’t count all the pathways (to count all the pathways would require an amount of time of a higher order of infinity).
Suppose that the reticulate structure of science described in the previous newsletter is potentially infinite, so that if you had an infinite amount of time to elaborate science, science would converge on an infinity of disciplines and subdisciplines — we could here imagine a Borgesque catalog of the increasingly bizarre subdisciplines of science of every imaginable kind, which nevertheless fails to do anything more than suggest the infinitude of disciplines. The possible interconnections between the infinite subdisciplines of science, i.e., the possible number of pathways among them, would be an uncountable infinitude, meaning that the reticulate structure of science, far from being finitely exhaustible, and far even from being countably exhaustible, constitutes a higher infinity.
One of the features that disturbs the critics of transfinite reasoning is that it seems to be a big shell game in which we are only infinitely rearranging counters in different silos, not adding any new entities as we build up ever more complex structures of higher infinities. We always start out with the whole numbers (or, if you prefer, with the set a natural numbers), then make sets of the natural numbers, then sets of sets natural numbers, and so on ad infinitum. If one denies that sets are legitimate mathematical entities, then all that is happening is the grouping and re-grouping of the natural numbers, so, in part, this is a question of the ontology of mathematics.
However, in my example of reticulate science, we can imagine a higher (non-countable) infinity as a process that plays out over non-denumerable time, and, given the reticulate history of science, with branching specializations and converging programs of unification, we can imagine these structures not as empty counters infinitely recombined in different ways, but as substantive processes invested with epistemic content always yielding new results. What Hegel postulated as absolute knowledge I thus imagine as an infinitistic structure of knowledge, both synchronically infinite and diachronically infinite.
