Historicism in the math-adjacent sciences
post by mrcbarbier · 2022-10-27T14:38:25.860Z · LW · GW · 0 commentsContents
Why do we use the word “cat” to mean a cat? What about biology? How to make historicism mathematical? References None No comments
I use the word “historicism” to designate one of at least three distinct epistemologies or scientific methods, together with reductionism and functionalism [LW · GW]. These epistemologies have clearly not been developped to the same extent, and most sciences favor only one of them, or have them verbally entangled. Linguistics is a very interesting field to build intuitions, as it has distinct subfields that are neatly dominated by only one such epistemology, with examples of all three.
I will call historical object an object whose current state is best explained, not by dissecting the object, nor by looking at other objects around it, but by looking at its previous states or predecessors.
A very useful archetype is etymology.
Why do we use the word “cat” to mean a cat?
The classic explanation is, to quote etymonline.com, that it comes from:
Old English catt (c. 700) "domestic cat," from West Germanic (c. 400-450), from Proto-Germanic *kattuz (source also of Old Frisian katte, Old Norse köttr, Dutch kat, Old High German kazza, German Katze), from Late Latin cattus.
The near-universal European word now, it appeared in Europe as Latin catta (Martial, c. 75 C.E.), Byzantine Greek katta (c. 350) and was in general use on the continent by c. 700, replacing Latin feles. Probably ultimately Afro-Asiatic (compare Nubian kadis, Berber kadiska, both meaning "cat").
This is a useful explanation, in that it provides a clear reason why the English word is “cat” rather than “glorphle”. It tells you that you could go to any European country and have a reasonable prior expectation for what the word should sound like.
In contrast, we don’t learn much by trying to split “cat” into c, a & t – at least not like we would learn something about water by splitting it into 2 H + O. There is no useful sense in which the meaning of “cat” is a combination of the meanings of each letter.
It’s not much more useful to compare and contrast it with other words in the same language, such as “cattle” and “cadmium”.
There are some weak ‘repulsive’ forces trying to prevent words from sounding too alike, but homophones are still all over the place (consider that in southern French, /le/ can mean ugly, milk, plural the, song, legacy…). There are also much weaker ‘attractive’ forces that sometimes lead words with related meanings to become alike, but whether they apply or not is largely due to chance – to historical accident. All these forces are highly contextual and usually negligible.
All in all, we can treat each word lineage as an independent history of accidental sound changes, that do not impact the “ecological niche” of meaning occupied by that word, and are ultimately unpredictable but understandable in retrospect. Sometimes an entirely unrelated word will come and try to fill that niche instead, or some renegotiation of boundaries will happen, but the outcome of that competition almost never has anything to do with what each word sounds like.
Thus, words, in the sense of sounds associated with certain meanings, are pretty much perfect historical objects, explainable only by their predecessors.
We can see, in that case, that a very important topic of investigation is: which transformations are possible within that lineage, and what rules do they follow? We mostly look within a word and at other words to better understand the kind of historical events that may have happened: 1) sound changes tend to happen simultaneously to many words, and 2) their application depends on within-word context (which sound is next to which other).
Having a clear taxonomy of possible transformations is often essential to being able to correctly identify ancestors, and this can rely on other sciences with different epistemologies.
This situation evokes geology, where proper reductive sciences (chemistry and physics) proved instrumental in providing the taxonomy of pathways transforming one mineral into another, but the proper explanation for why such mineral is found in such mountain range can only be a historical account of its ancestry, i.e. the sequence of transformations that led there.
What about biology?
Genes have some convenient features for historical objects: they live in such a high-dimensional space of possibilities that it is infinitely unlikely to evolve toward the exact same genes from two unrelated starting points. Each one is thus, in principle, a unique fingerprint of a lineage, forever retaining unambiguous traces of its ancestry.
In that respect, they are better than words, which are much shorter and can often converge by chance on the same form. Indeed, the etymology of words must often rely on additional information about meaning to disentangle, say, basil and basal, whereas we almost never use gene function to reconstruct phylogenies.
On the other hand, genes are (more or less) selected; the gene-function relationship is nowhere near as arbitrary as the sound-meaning relationship (at least given the uniqueness of Earth life’s transcription and translation system). Therefore we can learn a lot of very useful things about why a gene is how it is, by looking at other currently existing genes in the organism or in the ecosystem.
Thus biology is messier than language. It does have “neutral” aspects (e.g. genes that appear to be under no selective pressure at all) that behave like pure historical objects, but in many cases:
- a gene contains very clear information about its ancestry,
- yet the best explanation for why that gene is present is not simply that some ancestor organism had a predecessor of that gene.
Since only a very small fraction of possible genes are going to be retained through selection, it seems less important (though it is a growing field) to understand which mutations are possible – we can often ignore the process that generates variants, treating it as random and isotropic, and focus on which ones are selected.
A historical approach also seems appropriate for development, at the organism level: many features of an organism are the results of predecessor states and chance, though many others are selected. Many people have written at length of these questions and I know far too little, so thereof I must remain silent.
It is entirely an open question whether such an approach is possible and relevant for ecosystems: is species composition like a genetic code, open-ended enough to almost surely allow phylogeny? Is it like words, where chance homophones are quite frequent, but lineages can still be distinguished if we take function into account? Or should we instead invoke pure reductionism or pure selectionism or something else?
How to make historicism mathematical?
Phylogeny is probably the biological (or biologically-inspired) problem where mathematics has had the deepest unique contributions. By unique, I mean that this has prompted math that is really different from the (far more developed) math of reductionism inspired by physics, chemistry, etc. By deep, I mean that there are technicalities, or even theorems, that come reasonably close to usefulness.
This situation has no parallel in functionalism, and the applicability of mathematical developments in biology is generally very low – in my life as a theoretical ecologist, I have rarely if ever seen an empirically-upheld conclusion that ultimately required mathematics significantly beyond high school level (occasionally technical aspects of a model might require high virtuosity, but these aspects almost never brought something qualitatively new to the picture, and their particular quantitative predictions were almost never verified).
At the same time, phylogeny is only one facet of the historical epistemology: how exactly do we identify ancestry and kinship, given some realizations and a predefined space of possibilities on which (ideally random) changes occur. I really can’t tell whether that is exactly all that can be made mathematical about history, and everything else is window dressing, but I suspect that many people who think about morphogenesis, for instance, feel in their bones that there is something deeper to say and we just don’t have the tools yet.
At the end of the day, historical explanations are usually verbal narratives, possibly relying on chemistry or physics or population dynamics or some other math-adjacent field to justify some steps in the narrative (such as how a given mineral changed when the atmosphere composition was impacted by oxidation), but pieced out through an investigation that does not possess, at this stage, a native mathematical language.
References
https://montevil.org/publications/articles/2020-Montevil-Historicity-Heart-Biology/
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