Floating Formalisms
Friday 05 July 2024
In newsletters 266 and 280 I quoted something that I first wrote in the summer of 2022, which is an idea to which I keep coming back:
“There is a fundamental intellectual difference between those who get their rigor primarily from mathematics, and those who get their rigor primarily from logic.”
This is framed from a personal point of view, but it is true at a disciplinary scale as well. The empirical sciences get their rigor primarily from mathematics, and this imposes characteristically mathematical forms of rigor and formalism on the empirical sciences. The way in which mathematics is used to formalize the empirical sciences results in many different formalizations — not necessarily mutually exclusive, but distinct in their motivation. Beyond the formalization and elaboration of the discipline being studied, there is an additional effort required to link up these disciplinary-specific formalizations to each other. The Langlands program demonstrates the extent to which formal thought is fragmentary, consisting of a number of islands of formal technique floating in semi-isolation. This image of semi-isolated islands of formalization reminds me of an idea from Archaeological seriation.
One of the most useful ideas I have picked up from archaeology, and which deserves to be better known, is the idea of floating chronologies. When a site is excavated, a site chronology is produced, sometimes by the seriation of pottery decoration, or whatever else in the artifacts which lends itself to being worked out as a temporal sequence, which can be then applied to artifacts from the same context. The site chronology taken in isolation is called a floating chronology until it can be linked up with other chronologies and given dates in an absolute chronology. While the idea of floating chronologies is something I have only found in archaeology, it has applications beyond archaeology. At the largest scales of cosmology — scales of cosmology we have not yet achieved — the chronology of our universe may be a “floating chronology” in a larger context that we cannot now study because we cannot observe beyond the boundaries of the visible universe.
Our absolute chronologies are never really absolute, but they become more comprehensive over time. As we link together larger portions of the past once studied in isolation, we create a more comprehensive past, filling in ellipses and missing details. A famous example of this was the floating dendrochronologies of the desert southwest, where various native peoples built with timbers, which could be dated, but the dating of timbers on one site, or across several related sites, only gives us a floating chronology until we find a dendronchrological sequence that overlaps with some known sequence. The Beam expeditions of 1923 and 1928 assembled floating chronologies for several sites that ranged over 585 years, but the crucial piece was found in 1929 by A. E. Douglass that made it possible to link up the floating chronologies and definitely date the cliff dwellings of the region. (Cf. Douglass, A. E. 1929. “The secret of the Southwest solved by talkative tree rings”. National Geographic Magazine 56(6): 736–770.)
While formalization is not chronology, we can understand partial and parochial formalizations as “floating” until they are systematically integrated into a larger context of formalization and formal knowledge, which we could call absolute formalization. The Langlands program is just such an effort to link up floating formalizations into a larger whole, like connecting logs together to make a log raft. As with chronologies, our formalization is probably never absolute, but they become more comprehensive over time. Since the field of formal research is infinite, our increasingly larger islands of linked together formalizations are still finite assemblages within an infinite context.
The empirical sciences, in their search for greater rigor, seek to integrate their floating formalizations into a wider formal context that will connect formerly isolated disciplines into a larger scientific whole. Often this is successful, but it is not always or inevitably successful, and some sciences with their floating formalizations remain as outliers, unassimilated to the larger body of science and scientific formalization. Partly this is a function of there being no science of science (on which cf. newsletters 287, 288, and 289), making the whole of science systematic and comprehensive, and making the introduction of new sciences and their integration into the whole of science schematic, if not automatic.
In mathematics, with little or no content from the empirical sciences, there is a greater need felt for foundations, so that the logical formalization of mathematical knowledge has gone much further than in the empirical sciences — and while mathematics is the mode of formalization of the empirical sciences, logic is the mode of the formalization of mathematics, which means that the empirical sciences are formalized at one remove. The empirical sciences are likely to view their foundations as belonging in the empirical world, and not in the theoretical framework that makes them possible. Insofar as the justification of the empirical sciences flows from the empirical world, and the empirical world transcends the sciences that schematize it (so that nature is always only partially captured by scientific knowledge, and never exhaustively captured by scientific knowledge), it is understandable that the empirical sciences have been content in deriving their justification from the empirical world while remaining fragmented and multifarious formally. From the complementary perspective, i.e., from the perspective of the formal sciences, the formal sciences transcend the empirical world, so that schematic justification based on formal considerations naturally takes precedence and a fragmentary and multifarious relationship to the empirical world is to be expected.
Must the historical order of formalization require that the formal sciences themselves achieve comprehensive integration with each other before they can move on to the schematic formalization of the empirical sciences? Or might some floating fragment of formality be employed to formally unify the empirical sciences? Or might the integration of both formal knowledge and empirical knowledge be historically complementary, so that developments in the one influence developments in the other, and vice versa? Here we must distinguish between a formal ideal that implies that empirical science will be finished and complete — in which case we cannot look forward to the fulfillment of that formal ideal until after empirical science has fulfilled its empirical ideal, at which point it can be fully formalized — and a formal ideal that leaves empirical science open to further development. While the former may feel more natural for the formal sciences, the latter not only formally unifies the empirical sciences, but facilitates the development of new sciences, so that science itself is understood to be inexhaustible. If humanity extends its sciences through the cosmos, every world is potentially the locus of a plethora of new sciences that would be formulated the more readily in the light of the schematic formalization of the sciences.
That the formal sciences have a history, and unfold (imperfectly) in human experience as a part of human history, suggests a source of the fragmentary, floating formal disciplines that fail to fully exemplify the ideal to which they implicitly point. Any one formal discipline points to its own self-transcendence in a way that the empirical sciences do not. The future of formal thought may involve a systematic formalization of the empirical sciences in which justification follows from formal and not empirical sources of justification. We have seen this in a limited way, as individual special sciences grope their way toward greater rigor in their methods, and we understand how this appears awkward and as an exercise in artificiality (if not also an exercise in futility). These speculations on the nature and the history of the sciences, formal and empirical, could be taken as an example of what I recently talked about in relation to Haskell Fain. At the end of his Between Philosophy and History: The Resurrection of Speculative Philosophy of History Within the Analytic Tradition, Fain wrote:
“…it seems as if modern positivism has been a victim of its own historical tradition. What positivism lacked, I contend, was a penetrating speculative philosophy of the history of science, the encouragement to fashion story-lines upon which better histories of science could be constructed. Just as there is more to history than orthodox political history, so is there more to speculative philosophy of history than Hegel’s philosophy of history, which is, essentially, a speculative philosophy of political history. Each kind of history requires its own kind of speculative philosophy of history. Until that point is fully appreciated by professional analytical philosophers, analytical philosophy of history will consist largely of hand-me-downs from analytical philosophy of science, a subject itself in urgent need of nourishment from the history of science.”
I am sympathetic to the idea that each kind of history requires its own kind of speculative philosophy of history, as I have indulged in this myself. My blog posts on reticulate science — Reticulate Science, Addendum on Reticulate Science, and Infinitistic Epistemic Expansion — present what could be considered a speculative philosophy of the history of science, and the above reflections suggest to me that there may be a parallel formulation of reticulate formal science that I have not yet worked out.
