Metaphors are rampant in both everyday language and in science, and while they are inevitable, readers of this blog also know by now that I’m rather skeptical of their widespread use, both in professional publications and, especially, when addressing the general public. (See here, here, here, and here.) One such problematic metaphor is that of so-called adaptive landscapes, or surfaces, in evolutionary biology, something on which I did a fair amount of research when I was running a laboratory of ecology and evolutionary biology.
My detailed criticism of the way the landscape metaphor has sometimes warped biologists’ thinking is detailed in a chapter that was published back in 2012 as part of a very interesting collection entitled The Adaptive Landscape in Evolutionary Biology, edited by Erik Svensson and Ryan Calsbeek for Oxford University Press. As it often happens, mine was the lone contribution from the token skeptic…
Few metaphors in biology are more enduring than the idea of adaptive landscapes, originally proposed by Sewall Wright in 1932 as a way to visually present to an audience of typically non-mathematically savvy biologists his ideas about the relative role of natural selection and genetic drift in the course of evolution. The metaphor was born troubled, not the least reason for which is the fact that Wright presented different diagrams in his original paper that simply cannot refer to the same concept and are therefore hard to reconcile with each other. For instance, in some usages, the landscape’s non-fitness axes represent combinations of individual genotypes, while in other usages the points on the diagram represent gene or genotypic frequencies, and so are actually populations, not individuals.
Things got even more confusing after the landscape metaphor began to play an extended role within the Modern Synthesis in evolutionary biology and was appropriated by G.G. Simpson to further his project of reconciling macro- and micro-evolution, i.e. to reduce paleontology to population genetics. This time the non-fitness axes of the landscape were phenotypic traits, not genetic measures at all. How one would then translate from one landscape to another (i.e., genes to morphologies) is entirely unaddressed in the literature, except for vague motions to an ill-defined and very rarely calculated “genotype-phenotype mapping function.”
These are serious issues, if we wish to use the landscape metaphor as a unified key to an integrated treatment of genotypic and phenotypic evolution (as well as of micro- and macro-evolution). Without such unification evolutionary biology would be left in the awkward position of having two separate theories, one about genetic change, the other about phenotypic change, and no conceptual bridge to connect them.
To try to clarify things a bit, I went through the available literature and arrived at a typology of four different kinds of “landscapes” routinely used by biologists:
Fitness landscapes. These are the sort of entities originally introduced by Wright. The non-fitness dimensions are measures of genotypic diversity. The points on the landscape are typically population means, and the mathematical approach is rooted in population genetics. (see figure above)
Adaptive Landscapes. These are the non straightforward “generalizations” of fitness landscapes introduced by Simpson, where the non-fitness dimensions now are phenotypic traits. The points on the landscape are populations speciating in response to ecological pressures or even above-species level lineages (i.e., this is about macro-evolution). There is — with very special exceptions discussed in my paper — no known way to move from fitness to adaptive landscapes or vice versa, even though this is usually assumed by authors.
Fitness surfaces.These were introduced by Russell Lande and Steve Arnold back in the ‘80s to quantify the study of natural selection. Here phenotypic traits are plotted against a surrogate measure of fitness, and the landscapes are statistical estimates used in quantitative genetic modeling. The points on the landscape can be either individuals within a population or population means, in both cases belonging to a single species (i.e. this is about micro-evolution).
Morphospaces. These were first articulated by paleontologist David Raup in the mid-’60s, and differ dramatically from the other types for two reasons: (a) they do not have a fitness axis; and (b) their dimensions, while representing phenotypic (“morphological”) traits, are generated via a priori geometrical or mathematical models, i.e. they are not the result of observational measurements. They typically refer to across species (macro-evolutionary) differences, though they can be used for within-species work as well.
The first thing to note is that there are few actual biological examples of fitness landscapes (Wright-style) or Adaptive Landscapes (Simpson-style) available, while there is a good number of well understood examples of morphospaces (Raup-style) and particularly of adaptive surfaces (Lande–Arnold style). These differences are highly significant for my discussion of the metaphor. The paper summarizes examples — both conceptual and empirical — of each type of landscape and the complex, often barely sketched out, relationships among the different types.
When it comes to asking what the metaphor of landscapes in biology is for, we need to distinguish between the visual metaphor, which is necessarily low-dimensional, and the general idea that evolution takes place in some sort of hyper-dimensional space. Remember that Wright introduced the metaphor because his advisor suggested that a biological audience at a conference would be more receptive toward diagrams than toward a series of equations. But of course the diagrams are simply not necessary for the equations to do their work. More to the point, subsequent research by my former University of Tennessee colleague Sergey Gavrilets and his collaborators has shown in a rather dramatic fashion that the original (mathematical) models were far too simple and that the accompanying visual metaphor is therefore not just incomplete, but highly misleading. It turns out that hyper-dimensional dynamics are very much qualitatively different from the low-dimensional ones originally considered by Wright.
In a very important sense Wright’s metaphor of fitness landscapes was meant to have purely heuristic value, to aid biologists to think in general terms about how evolution takes place, not to actually provide a rigorous analysis of, or predictions about, the evolutionary process (it was left to the math to do that work). Seen from this perspective, fitness landscapes have been problematic for decades, generating research aimed at solving problems — like the so-called peak shift one (how do populations stuck on a local fitness peak “shift” to a higher one?) that do not actually exist as formulated, since high-dimensional landscapes don’t have “peaks” at all, as their topology is radically different.
There are problems also with the Lande-Arnold type landscapes (discussed in the paper), but here I want to shift to some good news: the actual usefulness of the fourth type of landscape: Raup-style morphospaces. One of the best examples was produced by Raup himself, with crucial follow-up by one of his graduate students, John Chamberlain. It is a study of potential ammonoid forms that puts the actual (i.e., not just heuristic) usefulness of morphospaces in stark contrast with the cases of fitness and adaptive landscapes. Ammonoids, of course, were beautiful shelled marine invertebrates that existed in a bewildering variety of forms for a good chunk of Earth’s biological history, and eventually went extinct 65 million years ago, together with the dinosaurs. This is going to be a bit technical, but stick with me, it will be worth it.
Raup explored a mathematical-geometrical space of ammonoid forms defined by two variables: W, the rate of expansion of the whorl of the shell; and D, the distance between the aperture of the shell and the coiling axis. Raup arrived at two simple equations that can be used to generate pretty much any shell morphology that could potentially count as “ammonoid-like,” including shells that — as far as we know — have never actually evolved in any ammonoid lineage. Raup then moved from theory to empirical data by plotting the frequency distribution of 405 actual ammonoid species in W/D space and immediately discovered two interesting things: first, the distribution had an obvious peak around 0.3 <D <0.4 and W near 2. Remember that this kind of peak is not a direct measure of fitness or adaptation, it is simply a reflection of the frequency of occurrence of certain forms rather than others. Second, the entire distribution of ammonoid forms was bounded by the W = 1/D hyperbola, meaning that few if any species crossed that boundary on the morphospace. The reason for this was immediately obvious: the 1/D line represents the limit in morphospace where whorls still overlap with one another. This means that for some reason very few ammonites ever evolved shells in which the whorls did not touch or overlap.
Raup’s initial findings were intriguing, but they were lacking a sustained functional analysis that would account for the actual distribution of forms in W/D space. Why one peak, and why located around those particular coordinates? Here is where things become interesting and the morphospace metaphor delivers much more than just heuristic value. John Chamberlain, a student of Raup, carried out experimental work to estimate the drag coefficient of the different types of ammonoid shells. His first result clarified why most actual species of ammonoids are found below the W=1/D hyperbola: shells with whorl overlap have a significantly lower drag coefficient, resulting in more efficiently swimming animals.
However, Chamberlain also found something more intriguing: the experimental data suggested that there should be two regions of the W/D morphospace corresponding to shells with maximum swimming efficiency, while Raup’s original frequency morphospace detected only one peak. It seemed that for some reason natural selection found one peak, but not the other. Four decades had to pass from Raup’s paper for the mystery of the second peak to be cleared up: the addition of 597 new species of ammonoids to the original database showed that indeed the second peak had also been occupied!, a rather spectacular case of confirmed prediction in evolutionary biology, not exactly a common occurrence, particularly in paleontology.
So, is the landscape metaphor in biology useful? It depends. The original versions, those introduced by Sewall Wright to make his math accessible to his colleagues, have been highly influential for decades, and yet have arguably channeled both empirical and theoretical research in unproductive directions, inventing problems (like the peak shift one) that arguably do not exist, at least not as formulated. The Lande-Arnold landscapes, which I have not discussed in this post, but do treat in the paper, have a mixed record. They have been heuristically useful for biologists interesting in quantifying natural selection in the field, but have also arguably brought about a degree of tunnel vision in both the theoretical and empirical study of that most important concept in modern evolutionary theory. Morphospaces, by contrast, have a very good record of being useful in terms of generating insight into the evolution of animal (and plant) form, and yet, they are actually the least commonly deployed version of the landscape idea in the technical literature. And because population genetics, with its mathematical approach, is considered more sophisticated than paleontology, things are unlikely to change in the near future. Unfortunately.
I’m not advocating doing away with metaphors in general. That’s pretty close to impossible. But as I detail in the paper, Wright’s metaphor has actually been positively misleading for decades, leading biologists to chase answers to problems that do not, actually, exist, like the peak-shift one.
I guess we just talk past each other in this case, since to me the “explanation” you provided explains nothing. But yeah, best to drop it, since it’s irrelevant to the OP anyway.
I understand it perfectly, I think. And Smolin is exactly right, in my mind. Unless you have a better idea, which would entail a decent account of how we can access Platonic forms.
Yes, as I said there are very many very small changes. But they are definite changes.
And the changes are happening throughout the population and constantly combining.
No problem calling that a continuum, just as long as you don’t say that the continuum is the reality and discontinuities an illusion created by the mind, because that gets it the wrong way round.
Light travels as a continuum. It doesn’t travel from source to reception discontinuously. It is quantified by release and reception.
It from Bit has it the wrong way around.
Think in terms of a wave. The primary information it carries is frequency and amplitude. They are a property of the wave, not the wave an expression of the form of the frequency and amplitude.
That would be platonism. That these characteristics are the universal platonic form that is the essence of the wave, rather than the wave emerging from a dynamic in which the amount of energy the wave carries is expressed in its frequency and amplitude.
blockquote>I understand it perfectly, I think. And Smolin is exactly right, in my mind. Unless you have a better idea, which would entail a decent account of how we can access Platonic forms.</blockquote
As I keep saying I am not a Platonism in that sense, probably Plato wasn’t either in his late period.
You, me and Smolin are all saying that there are many mathematical facts that no one has ever formulated or even thought of (and perhaps never will think of) but which nevertheless are true. In that part we agree.
But Smolin puts in an extra step of them becoming true at some point.
So my ‘bettee idea’ is simply to leave out this ‘becoming true’ event as it adds nothing and leads to all sorts of complications.
If you leave that out you have other complications. Like in what sense these facts are, allegedly, true even though they don’t correspond to a reality out there, and yet have not been conjured by a human mind? How do we access such reality? And so forth. You may call yourself a non-Platonist, but that’s where you ned up if you are not a constructivist or a Smolinist…
Suppose we distinguish between actual and potential continua. The former could be a function defined on RxR, as a pure mathematical construct; and as a mathematician I do not require this to exist in any meaningful way outside this construction anymore than I require Thanos to exist outside the movies and comics in which he is a character. There are still matters of truth pertaining to Thanos – it is true that he acquired the infinity stones, &c. But none of these produce any cause for believing Thanos to exist in any sense, beyond as a character in these fictions.
If I am going to model some dynamic system, giving rise to a discrete number of observables, I may nevertheless assume that it is continuous. This could be justified in a number of ways. It may be that there is no lower bound on the phase space distance between observables, or that I’m not aware of any such. It may even be that I am aware of such a bound, and yet proceed to create a model that ignores it, deeming it, say, to be too small to be of relevance, like we generally ignore Planck’s constant in the study of macrophysical systems, say in order to employ the tools of calculus. And if under such assumptions I come up with a formula that is good fit on the data, then I shall feel justified in saying, informally at least, that the underlying dynamic is continuous (up to isomorphism, and up to resolution) whithout ever anywhere positing an actual continuum existing outside of my construction of it, for my own convenience.
It seems to me – and no offence intended, I assure you – that contemporary Platonism is not so much a legitimate metaphysical stance, as a cognitive deficit; namely a failure to appreciate or even understand how the imagined may constrained, such that definite truths appear, and yet be no more than imagined, or immanent/emergent on such constraints.
You say that you believe that the set of all consistent axiomatic systems is not a trivial undifferentiated space. So do I (but I do not know that it is a set) It is the subject matter of proof theory, category theory, and type theory in computer science – described by one researcher (Robert Harper) as “logic as if people matter”. I would ground this, then, not in a Platonic realm of forms, but in human understanding, discourse, imagination, and ability to create meaningfully constrained models.
Metaphors are OK as long as they serve a purpose and you don’t torture them and are ready to drop them when they cease being useful.
I have read a description where someone talks of a fitness landscape where an organism traverses the valley from one peak to another. Put that way it sounds easy but if you translate that it means that some part of the population of this organism became less successful at completing for a long period of time and then became more successful again.
Of course that can happen but it doesn’t seem like such a plausible suggestion put that way and other explanations, such as that the organisms diverged earlier might have more plausibility.
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You and Smolin also have that “complication” since you are saying that there are mathematical facts that are true despite that fact that no one has ever thought of them and maybe no one ever will. .
So in what sense do you and Smolin say these facts which have not have not “been conjured by a human mind” are allegedly true? Smolin even says that these facts come into existence – which is way more Platonic than I would be prepared to go.
So all I have done is to drop a step which has no explanatory power.
Smolin also is faced with the additional task of explaining the nature of this “becoming true” event.
When I first heard this I posed the problem – shortly after the formulation of the rules of the modern game of chess, two players sit down to play, but decide on a small rule change (say “the Queen can jump”). They proceed to play According to Smolin some set of mathematical facts become true at some time around this time.
The next day, three thousand miles away two other people, by coincidence and knowing nothing of the other two players, think up exactly the same rule change and play the game with this new rule.
So, what would Smolin say happens here? Do the mathematical facts become true for a second time? If not then what happened differently the second time that didn’t trigger this “becoming true” event?
If those facts did become true a second time, then was it a complete coincidence that exactly the same set of mathematical rules became true in both instances? If not then there must have already been some fact of the matter.
Hi Massimo, Robin,
Genuinely, I don’t understand why such an account is needed.
We can conceive of mathematical objects because we have rational minds that are capable of thinking about abstract objects. That fact needs an explanation whether or not platonism is true, but that’s a neurological/psychological issue and not a metaphysical one. I don’t think most modern platonists think of this abstract thought as being anything like sensory perception of an otherwise inaccessible world. It’s just a question of whether we ought to regard the objects of this thought as having an independent existence or not.
This is mildly question-begging, in the sense that from a platonist mindset there is no need for mathematical truth to correspond to anything in our physical universe.
The sense in which these facts are true is in the sense that these facts are true of an abstract mathematical object such as the rules of a board game that has never been played. If we discover/invent the board game, we will find these facts to be true. They remain true whether we do this or not.
Robin is right that “becoming true” introduces complications, and to my mind insurmountable ones. It is extremely likely that the game of tic-tac-toe/noughts and crosses was invented independently on multiple occasions, albeit under various names and using various different symbols or counters. What does this say about the idea that the facts about it came true when it was invented? Did they become true more than once, or only once? What if the game was lost for a thousand years and then rediscovered independently? What if it was also discovered on an alien planet on the other side of the universe a billion years ago? How can you make sense of this idea that these facts “became true” when the game comes into existence when even the idea of the game coming into existence is so problematic? If you just assume the game exists platonically along with the truths about it all along, and is discovered and lost and rediscovered at various points, then all these issues go away.
I think I’d like to see more justification of this point, as at first glance you seem to be off-base here. It is true that a very narrow and strict interpretation of “peak” seems to refer specifically to a point on a 3D landscape (1 dimension of fitness, 2 of some other variable), but the concept of local maximum generalises to an arbtitrary number of dimensions. So why is it that peak shift (or local maximum shift) is not a problem? If I can speculate, perhaps it is because with many more dimensions comes many more paths from one local maximum to another, and so a greater chance of avoiding getting stuck. If so, fair enough, but I’d like to hear your thoughts on this as it wasn’t clear from the original article what’s wrong with the peak shift problem.
In any case, I would say there are two approaches to this. You could do what you do and say that the landscape metaphor has led us to be confused by a pseudo-problem with much waste of thought and effort, or you could say that the landscape metaphor has led us to a genuine problem which we have now resolved such that we now have greater understanding than we would have had if we had just blindly assumed that nature would always find the optimum solution. In any case, it’s clear that nature doesn’t. I’m sure you are aware of many situations where nature is stuck on a suboptimal local maximum due to exactly this issue (e.g. the giraffe’s laryngeal nerve).
That’s the peak shift problem. That’s why it is quite likely for an organism to get stuck at a local maximum even when better solutions are possible. So such journeys across the valley ought not happen much unless they are very brief and driven by chance. The person you are talking about may have been discussing the problem rather than getting confused by the metaphor? Or maybe you’re misremembering?
The metaphor works well enough as long as you imagine that the organism is always trying to climb and is unlikely to go down. So maybe it might work better for you if you mentally invert it and imagine a ball rolling down a hill, with the valleys representing good fitness and the peaks representing poor fitness.
I’ll readily admit that I know next to nothing about this, but it seems to me that underlying all of these landscapes is a more or less clearly realized notion of a phase space, with the landscape emerging as a feature through coarse graining. To what extent may this be true of the models, and to what extent relevant to the actual biology? On that note, how do computational constraints, like Stuart Kaufmann’s ideas of self-organising complexity feature in the literature today, and to what extent do you find such considerations relevant?
I realize it’s bit late, given your three day window, but one day I’d very much like to hear your views on these matters.
But then neither would I. I would not have called it Platonism at all, it would never have occurred to me that it had anything whatsoever to do with Plato It is others that would accuse me of Platonism if I say that the set of all consistent axiomatic systems is not a trivial undifferentiated space like the Library of Babel.
As I have said before, I have written down the entire Library of Babel on a post-it note and have it pinned to my workstation wall right now. That I can do this pretty much shows that the Library of Babel is a trivial undifferentiated space. You can’t do that with formal axiomatic systems.
As I said, it is not me who is calling this observation Platonism. As far as I am concerned Platonism probably did not even last until the end of Plato’s life (the question, as I understand it, depends upon the dating of a couple of the dialogues).
No, Robin, we are saying that those facts become true once people start thinking about those things, not that they are true beforehand. There were no facts about chess before the game was invented.
That’s just a way to restate the problem, not to provide an answer. But I think we’ve had enough discussions about Platonism. Which I still maintain has nothing to do with the OP. I really should be more strict about brodix’s comments…
Indeed, but Gavrilet’s research has shown that hyper-dimensional phase space has radically qualitatively different dynamics from the standard three dimensional one. That’s why the latter has been mislesding.
As for Kaufman, I like his stuff, but my sense is that he has largely been ignored in theoretical biology.
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Though the issue of landscapes in biology does have both advantages and disadvantages, which everyone here seems to agree, so if there had been a controversy, beyond whether they are moderately effective tools, or something more fundamental, it probably would have come out in the discussions.
You yourself seemed to treat them as tools and only tools, so I thought I would snark about how some get sidetracked into thinking they are reading the mind of God, when they only have clever insights.
Are you saying that Smolin does not claim any mathematical fact was true before someone started thinking about it?
But people did not start thinking about those mathematical facts as soon as the rules of chess were formulated. It took years, centuries, and there will be many mathematical facts about chess that no one has thought of yet and some that no one will ever think of. It would probably not even have occurred to the inventors of chess that there were mathematical theorems to be made about it.
He says “all the facts” about it. So that would include all the facts that no one had started thinking about and all the facts that no one will ever think about.
If no mathematical fact become true before people started thinking about them, then it would not be the case that all those facts became true as soon as the rules of chess were formulated.
Suppose in ten years time someone thinks up a theorem about chess and proves it. According to Smolin that theorem currently exists, in some sense, and is already true. But if no one has thought it up yet, then in what sense is it true? In what sense does it exist?
All I say is that I do not require the hypothesis that for any mathematical fact that is true there was a time when it became true. It does not add or explain anything and it leads to complicated questions that do not have answers.
Agreed. Only jumped in because you yourself were posing questions about platonism.
In case you skimmed my post and missed it, I also had a more on-topic question in the last couple of paragraphs…
I suppose it depends upon what the “peak” represents. If it represents the maxima of competitive advantage afforded by various configurations of a particular feature then it does not represent the overall fitness of the organism.
Suppose, for example, some ammonites gain greater sensory acuity, then this might make shell shape less important and thus some of them might develop less than optimal shell shapes and still be the best competitor.
Similarly it might not have outcompeted its ancestor or cousin organisms, so they are still around and still have a chance to evolve into the other body shape. There could be all sorts of ways to ‘traverse the valley’, so to speak
That is not a weakness of the model necessarily, since all models are limited in some sense.
But it makes it potentially misleading, especially when used as a method to explain something to a layman.
Further to Jasper’s query, he might like
Click to access Adaptation_Orr.pdf
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As it turns out, it doesn’t. When Gavrilets simulated thousand-dimension, more realistic landscapes, he found them to be “holy,” meaning that they were characterized by largely flat hyperplanes, occasional “holes” with sudden drops in fitness, and a number of “inter-dimensional bypasses.” In other words, a completely different picture from what one conjures up if one thinks in terms of 3D landscapes. Hence the problem.
When a metaphor has led us down the rabbit hole for seven decades, I don’t find myself in the mood for such charity.
Precisely. I don’t see a problem there. Certainly far less of a problem than imagining all those facts to be mind-independent and yet not corresponding to any physical reality out there. You seem to think you found a third way, but you keep falling right back into Platonism.