Last updated on 18 Sep 2017
Last time I noted that phylogenetic classification was based on homologies, which I have elsewhere discussed. Now I want to consider how we might generalise it across all the sciences. And in particular I want to consider the other form of classificatory activity, by analogy, might also generalise. This will be a question of epistemology and the warrant for our inferences in science.
Late note: I have edited this to avoid some of the silly misunderstandings of abstract algebra I displayed earlier. I now include other silly misunderstandings.
“Homology”, as a term, arose first in mathematics, where it initially meant a mapping relation between sets of things, such as points in geometry. It was used by many people in various ways, but the founder of the modern use is the much-maligned Richard Owen (see chapters 7 to 10 in Williams and Ebach for the best historical overview of the use of “homology” that I know), who took the prior usage of Geoffroy of “analogy” and the discussions by Macleay and others of “affinity”, and came up with the term in 1843 (page 379), and explicated it in detail in 1848. He wrote:
The corresponding parts in different animals being thus made namesakes, are called technically ‘homologues.’; The term is used by logicians as synonymous with ‘homonyms,’ and by geometricians as signifying ‘the sides of similar figures which are opposite to equal and corresponding angles,’ or to parts having the same proportions: it appears to have been first applied in anatomy by the philosophical cultivators of that science in Germany. [p5]
The term itself is basically just a term that maps a relation from one set of objects to another; the relation is called homology and the things related are called homologs (or, in the Coleridgean spelling, homologues). Such mappings are called isomorphisms (from the Greek of “equal form”), and if an isomorphism is one way only, so that one set has more than is mapped, this is sometimes referred to as a homomorphism. I shall treat homology as any isomorphism. Diagrammatically:
We can think of this more concretely, as the relation between maps. A map at a scale of 1:50,000 maps onto a map at a scale of 1:10,000; this is an injective relation. If there are elements of the 1:50,000 map not on the 1:10,000 map or vice versa, then it is a homomorphic relation.* If it happens that the larger scale map has the same elements as the smaller, only in finer detail, it is isomorphic. Now consider the famous Belon diagram:
Each similarly named element (bone) in the human is mapped to a similarly named element in the pigeon. However, some of the bones are fused in the bird, and some are separated. Homologies are mapping relations of shared elements – traits, or more properly, characters, since the relations apply between abstract objects (trait descriptions). This is critical: the homologies are abstract relations. We shall consider what relations are later, but at the moment it is enough to say they are formal, not concrete, mapping relations, between abstract descriptions.
Now the key aspect of homological relations is that the mapping tells us what are the same elements in each set. Such elements need not be “similar” in some respect, nor must they be in the same “place” in the set. Consider a rotated picture of a Spitfire airplane and a model of that plane:
The real thing
Now the model is shown at a different angle to the real plane, but we could draw mapping relations between them no matter how different each element – say, the wing – looks in the two-dimensional image. If the model were of a later or earlier version, it may lack features on the real plane. It almost certainly lacks some of the features of the real plane (such as the rivets and small elements of the real plane too hard to make at that scale), and vice versa (real Spitfires lacked glue seals and extruded plastic bits inside). And so on. But the relevant elements are there in both, and if we describe each physical object at a suitable coarseness, we can say that they are in a bijective homological relation to each other.
Homology, then, is the relation between abstract objects (descriptions, or representations of real world objects) where the formal description allows a mapping function between them.
What, then, is analogy? It is clear that classifying in any manner involves classifying by mapping from one set (a set of physical objects for our purposes) and so it must involve isomorphisms of some kind. Analogous relations are still a kind of isomorphism, but the mapping is not between sets of objects, but between the form of the objects themselves, and form is a pretty amorphous notion (sorry! It’s really hard to avoid the puns when you talk about these sorts of things; damn, I did it again). The philosophical literature is rife with discussions on “similarity” and “resemblance” and usually starts off with the comment that in some way or another, everything resembles everything else (a point made by Locke, Essay I.XXV.vii, among others. I’ll bet it’s in the scholastics). So we need a formal notion of what “resemblance” or “similarity” consists in. Famously, Nelson Goodman declared similarity to be an ill-defined notion (Goodman 1971), and there is an extensive literature in the psychological and computer science fields as to what counts as similarity. Not surprisingly, there is also a large literature on it in taxonomy. As a rough cut, there are three main approaches I will attend to in another post: Hamming Distance, Edge Number and Tversky Similarity. For now, let us understand classification by analogy as a similarity relation between things, sets, and forms.
Classification by analogy is sensitive to the metric chosen, but also to the representations of the things being analogised. For example, if I classify two organisms as predators, I am representing only a very small number of properties of the two organisms, and they are the properties contained in the definition of “predator” – one species that eats another. We do not represent almost any of the rest of the properties of the taxa being classified this way. One way to make this point, and show the difference between homology and analogy is to ask, what is it that the classification tells us?
If, in biology, I tell you that X and Y are predators, all you know about them is contained in the definition of “predator”, and nothing else. There are exemplary predators, like lions and eagles, of course, but so too are single celled organisms that engulf others, as well as fungi that predate on ants, plants that predate on insects, and so on. You do not even know if they are motile or sedentary, because there are “wait-and-catch” predators.
Contrast my telling you that two organisms are “Raptors“. This it the taxonomic sense of “Raptor”, not the popular sense; it refers to a particular group of birds known as “birds of prey” (in particular, the Falconiformes). I particularly like this group because of the role they played in species concepts, via Frederick II’s book The Art of Hunting With Birds, that I outlined in my book. Now, if I tell you that X is a Raptor, what do you know? An enormous amount. You know that it has a beak (a recurved beak!), claws (recurved also, and very strong), feathers (including flight and tail feathers of a particular structure) and that it has a particular diet (meat) and lifecycle (lays eggs, parents them, builds nests, mates in single pairs, is territorial) and so on. In short, what you know from studying several raptors is generalisable to all others, in a non-grueish kind of way. This also applies to as-yet-undiscovered properties. If I discover that Raptor X has enzyme E, then I can infer that all other members of the Raptor group have E as well! That’s an enormous amount of inductive warrant. Interestingly, if I tell you that a raptor is a predator, you cannot infer that all raptors are (some are scavengers). Homology does not license analogical claims. But it may bracket them, as I will later argue.
We can summarise the difference here by saying that classifications by homology are inductively projectible, while classifications by analogy are deductive only. Moreover, analogies are generally model-based. The choice of what properties to represent usually depend upon some set of “pertinent” properties, and this is not derived from an ignorance of what matters, or some unobtainable theory-neutrality. In order to measure similarity, you need to know what counts. The problem with the phenetics school of classification was that it failed to specify what counted, and so it got inconsistent results depending upon what were the principal component axes used.
That’s enough for today.
Belon, Pierre. 1555. L’histoire de la nature des oyseaux, avec leurs descriptions, & naïfs portraicts retirez du naturel: escrite en sept livres. Paris: G. Cauellat.
Goodman, Nelson. 1970. Seven Strictures on Similarity. In Experience and Theory, edited by L. Foster and J. W. Swanson. Amherst: University of Massachusetts Press:19-29.
Owen, Richard. 1843. Lectures on the comparative anatomy and physiology of the invertebrate animals, delivered at the Royal College of Surgeons, in 1843. By Richard Owen. From notes taken by William White Cooper and revised by Professor Owen. London: Longman, Brown, Green, and Longmans.
Owen, Richard. 1848. The archetype and homologies of the vertebrate skeleton. London: J. van Voorst.
Williams, D. M., Malte C. Ebach, and Gareth Nelson. 2008. Foundations of systematics and biogeography. New York, N.Y.: Springer.