Evolving Thoughts

Homology and analogy

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:

Morphism.png

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:

Spitfire8oClock.jpg

The real thing

spitfirefr18bw_title.jpg

The model

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.

References

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.

Epistemology Evolution Metaphysics Natural Classification Philosophy Science Species and systematics Species concept Systematics

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Comments (39)

  1. I’ll bet it’s in the scholastics

    I can’t think of any instance off the top of my head, but I wouldn’t be surprised — it goes back very far indeed:

    “Well, at any rate, he said, justice has some resemblance to holiness; for anything in the world has some sort of resemblance to any other thing. Thus there is a point in which white resembles black, and hard soft, and so with all the other things which are regarded as most opposed to each other; and the things which we spoke of before as having different faculties and not being of the same kind as each other—the parts of the face—these in some sense resemble one another and are of like sort. In this way therefore you could prove, if you chose, that even these things are all like one another. But it is not fair to describe things as like which have some point alike, however small, or as unlike that have some point unlike.” [Plato, Protagoras, 331d-e]

    This was an excellent post, by the way.

    1. Brilliant, Brandon. Straight into the Quotes I Keep To Look Erudite File.

  3. I think you’re a little confused about mappings. An injection is a function that is “one-to-one” in the sense that no two distinct elements of the domain are mapped to the same thing in the range. A surjection is “onto” in the sense that everything in the range is mapped to by something in the domain. A bijection is something that is both an injection and a bijection.

    Your first diagram isn’t an injection. It’s not even a proper function. A proper function has to map everything in its domain. So you need an arrow from “5” to something.

    Your second map isn’t a surjection, since nothing maps to “E”.

    Diagrams of injections, surjections and bijections can be found here

    The comment about the maps of different scales is a little mystifying, since the transformation from one to the other is just a scaling, so it’s straightforwardly bijective. I see what you’re driving at, but it’s not clear that comment makes sense as it stands.

    If there’s a surjection from A to B and from B to A, then there is a bijection between A and B. This is fairly easy to prove. So saying that the bone diagram is surjective in both directions but not bijective or injective is wrong, unless you’re using these terms quite differently.

    1. Thanks. That’s why I put these up – so the educated can teach me.

      1. Is it really mappings or functions you want to talk about? Because the thing about them is that EVERYTHING in the domain has to map to something, which seems kind of restrictive if what you’re talking about is phenotypic traits. You want to allow that two creatures could each have some unique feature that doesn’t map on to anything in the other.

        Maybe the mathematical idea of “relations” is more suited to your purpose (since functions are a particular kind of relation, you don’t lose anything, really.)

  4. I got the HTML tag closing the link wrong. I put the wrong slash in: “” instead of “/”.

    That’s what comes of spending too much time with LaTeX

    1. Fixed. And I’ll fix the post later too. I think in retrospect I don’t need the mapping stuff anyway.

  5. Great post!

    Don’t get rid of all the mapping stuff though: I think including the general concept of mapping is relevant; the different types of mapping relationships not so.

  6. Sorry, but you went a bit too fast here, no?: “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!”

    Wouldn’t you have to know that E is necessary for some bird qua being-a-raptor before you make the inference? (And thus know more about raptoriness, for example that E’s effects are present in all raptors?) E.g., some raptors (may) have very strong stomach acids and enzymes, to dissolve bones. Once you know this and figure out that E causes this trait, you can infer (and eventually test and establish) that all raptors with this trait have E.

    IOW, may E not be lost (or gained) in some raptor species and not others? How would you figure out it’s shared among raptors without actually testing? (That’s what I take you to mean by “infer”.) All raptors may well have had E in the past, and some lost it. Or no raptors had it but some gained it, and you just checked one of those who did.

    Sorry for my confusion here. The recent series is quite instructive to me!

  7. Not to nitpick a good post (ok, nitpicking a good post is exactly what I’m about to do), but your mapping diagrams are not good.

    The first one represents a partial function (i.e. not the whole domain is defined), but is otherwise bijective. The second one is injective, but not surjective! Your definition of surjective is backwards—a surjective function is one which exhausts its range, not its domain (generally functions are always thought of as exhausting their domains, partial functions are the exception). The last one is bijective though.

    No biggie, but I thought I should mention it.

    1. Seamus beat me to the punch. This is what I get for not refreshing the page. Sigh.

  9. Yes, if you discover that Raptor X has enzyme E, you are justified in making an inductive leap to the hypothesis that all Raptors have enzyme E. Probably you have in the back of your mind that maybe enzyme E was invented by the unique common ancestor of Raptors and is thus a synapomorphy for Raptors.

    1. The fact that you might be wrong, or that the identity may have been changed in its form, is part of the inductive nature of the classificatory process. Inductive inferences are defeasible; deductive ones are not (if the premises are true by definition and the conditions obtain).

      1. Ah. I think this takes care of my worry above.

        So determining E offers you two inference paths: one towards genealogical relation of X to other raptors, the other towards the adaptive consequences of E for X (say, bone digestibility). Right?

      2. The inductive (ampliative) nature of the “inference” isn’t what gave me pause — I balk(ed) at the “enormous amount of inductive warrant” being claimed. But now I see that the warrant wasn’t being claimed for the particular inference, since it’s a very weak induction. Rather, it’s an enormous warrant being claimed for making this sort of inference. In other words, taxonomic relatedness warrants a type of inference that analogic relatedness supposedly does not. I say “supposedly” only becasue I suspect there are cases where analogy can support similarly defeasible inductive generalization.

        1. No, I’m going to argue that on its own, analogy does not support ampliative inference, except where it is bracketed by homology, and in that case it is the homology that is doing the amplification.

      3. I’ll wait to see that argument. In the meantime, I’ll further question the warrant of inductive generalization based on taxonomic relatedness. Wouldn’t this depend a great deal on what sorts of traits we’re dealing with? For example, suppose I note that my specimen falcoln has a particular pattern of coloration. Would I have an enormous warrant for the inductive generalization that other species of falcoln share this pattern? If not, what accounts for this difference between the projectibility of coloration and the proverbial enzyme E?

  10. For that matter, haven’t we recently learned that falcons are more closely related to parrots than hawks?

    (Dear host, you still haven’t repaired your diagrams. How many of your readers do you think haven’t studied abstract algebra?)

    1. Been a bit busy; I’ll fix them tomorrow.

      Have you got any evidence that Falconiformes are not a monophyletic group? So far as I have heard, they are distantly related to Psittaciformes.

  11. I think a more natural transformation would be based on some idea of “genetic distance.” A simple one would be time, that being the time when two organisms forked from a common ancestor.

    Also I think you need to take genetic history into account, otherwise you won’t be able to distinguish between extinct animals and their “re-evolved” equivalents. (I.e., Old English Bulldog vs Olde English Bulldogge).

  12. Falconiformes was probably a bad choice for an example, since, as bad Jim has noted, falcons and hawks are apparently not a clade. See Hackett et al. 2008. A phylogenomic study of birds reveals their evolutionary history. Science 320:1763-1768. NW vultures are in there with hawks, though. Now, if you meant the new meaning of “Falconiformes” as synonymous with Falconidae, you shouldn’t have referenced Wikipedia, since it preserves the older meaning. How about the Tree of Life instead?

      1. John, you are now being credited with ruining a perfectly good example in the book manuscript.

      1. Please. I published it Science, and I put it up on a prominent web site. Must I also push your face into it?

        Moral: To avoid social disapprobation, keep up to date on avian phylogenetics.

        1. You can’t expect me to read the scientific literature. I’m busy being ignorant in mathematics.

  13. Even after the edit, you are still wrong on the mathematical meaning of “isomorphism” and “homomorphism”. Three problems:
    (1) Both terms refer to mappings between structured sets; mappings that preserve the structure, but your text and diagrams don’t capture this. I think it is important to your points, though, because in this case “preserving structure” means something like “knee bone connected to the thigh bone” in both aves and Homo.
    (2) Homomorphisms should really be considered total functions. So, in your diagram of a homomorphism, you need to include an arrow from 5 to D and (ideally) also remove E from the picture altogether.
    (3) But this of course doesn’t suit your needs because you want to consider cases of partial mapping: cases where there is no arrow from 5. Well, mathematicians can handle this, but now you are no longer dealing with mappings and homomorphisms, but rather with partial mappings and simply “morphisms”.
    I strongly advise you to seek the assistance of an actual mathematician equipped with a whiteboard before proceeding.

  14. Moral: To avoid social disapprobation, keep up to date on avoid any reference to avian phylogenetics.

    1. That, if anything, would be worse. A day without avian phylogenetics is like a day without heartwarming cliches.

      1. I always said, phylogeny is like a box of chocolates. You will get fat on a steady diet of it.

  15. About homomorphisms and isomorphisms, Perplexed in Peoria sort of has it right, but even so I’d say not quite.

    A homomorphism need not be onto. So E can be left in the picture.

    You also write, 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’m having trouble parsing this, but I’d say that an isomorphism, by definition, cannot be “one way only”. There is always an inverse between isomorphisms. That’s the point.

    You might do better to start with a definition of homomorphism: the behavior of elements of one set is identical to the behavior of elements in a subset of the other set. What that behavior is, depends on the structure you are examining (group, ring, module, etc.) So, for example, if there is a homomorphism from G to H, then I can predict how some elements of H behave, based on how their preimages in G behave. However, I can’t do the reverse, because while it’s clear which element(s) of H correspond to any given element of G, the reverse correspondence from G to H is a ambiguous.

    With an isomorphism, by contrast, the two sets are identical with respect to the given structure. I can predict exactly how any element in either G or H behaves based on the corresponding element in the other set — and the correspondence is completely unambiguous in either direction.

    I hope that helps somehow (& that I haven’t given too flawed an analogy — this is pretty hard to give without jargon like “one-to-one”, “onto”, etc.).

    1. That’s very useful Jack. It is clear that what I need is a function for mapping the structure of one set to the structure of another. It sounds like homomorphism is what I need.

      I shall revise the manuscript accordingly.

  16. Have you consider the possibility that homology could be a special type of analogy.
    I took this from Mario Bunge’s work in philosophy of science and try to sum up and simplify his view (which is lightly formalised from a system approach)
    SYSTEM: qualitatively, a concrete system (e.g. atom, molecule, living being, society) can be represented as a quadruple: composition-environment-structure-mechanism and more precisely , a system at a given time may be represented by the ordered quadruple of sets
    (s)= {C (s), E (s), S(s), M (s)}
    Where C(s) is the collection of components of s, E (s) is the set of components (items, things), other than those in C(s) that act on components of s or are acted upon by them and S(s) is the set of relations (connections, couplings) among the components of s as well as among these and the components of the environment. M(s) is the peculiar mechanism (functioning, function) of the system, the process that makes it what it is (e.g. metabolism in a cell). The 4 coordinates of (s) are related to a given level of analysis (molecular, cellular, organs, entire organism) and are obviously embedded into a time/space background
    ANALOGY/HOMOLOGY:
    Accordingly,
    1. two systems s1 and s2 are analogous if one or several of the coordinates are similar but not all (C or S or E or M or combinations)
    2. two systems s1 and s2 are homologus if C,E,S and M are similar
    two systems s1 and s2 are homologous if they satisfy both condition 1 and 2

  17. The last sentence ” two systems…”of my post is a typo, please skip it

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