Biological landscapes, surfaces, and morphospaces: what are they good for?

ammonite

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.

typical (hypothetical) fitness landscape

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.

one-peak ammonoid morphospace

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.

two-peak ammonoid morphospace, with representative shell forms

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.

51 thoughts on “Biological landscapes, surfaces, and morphospaces: what are they good for?

  1. brodix

    Massimo,

    At least no one in biology is trying to say the mathematical representations are somehow more real than the activities being modeled.

    I know, it sounds like total nonsense, but apparently it happens. Some in those fields even argue empirical testing isn’t necessary.

    I can probably find some links if you don’t believe me. They aren’t even religions or cults. Well, at least they don’t claim to be.

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  2. SocraticGadfly

    I wonder if, per Shakespeare, “What’s in a name” is part of the issue with these analogies. “Morphospaces” just seems to have less connection with everyday linguistic life than “fitness landscapes” and maybe that’s why, with it, analogies aren’t stretched as far.

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  3. couvent2104

    Very interesting piece, thanks.

    Although I wouldn’t call the Raup-style morphospace you describe here a “metaphor”. For me it’s more a revealing way to present empirical data, based on carefully chosen parameters.

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  4. couvent2104

    A very minor quibble (and perhaps wrong) … I haven’t the articles by Raup etc., but I just made a quick calculation and I think D is not a distance, but a ratio between distances.

    That makes sense when you’re studying morphological aspects (“forms”), because two things can have different dimensions but the same form. Taking the ratio removes the dimension in this case (as far as I understand).
    Making D a ratio also gives you the WD = 1 hyperbola.

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  5. brodix

    Massimo,

    I guess platonism runs deep in the academy.

    We have a brain to process information and a gut to process energy.

    Think about light and vision. We see clear details, but light travels 300 thousand miles a second, yet the forms our minds extract from it are distinct. If we tried to think in terms of the light itself, our minds would liquify. This static form applies to all information. Even when it can’t be reduced to clear references, we quantify it with statistics.

    So if we only see the information and can’t deduce the underlaying dynamic, thoughts tend to harden into a pretty rigid belief system, but the result becomes a detachment from the reality that inexorably moves on.

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  6. davidlduffy

    I guess I have a problem with the description of these as metaphors. If I say that if you turn left here instead of going straight-ahead, it will be quicker because you will cut along the hypotenuse of the triangle, then this is not really a metaphor, is it? As you have documented, Wright is saying that a mathematical model of “mutation, inbreeding, crossbreeding and selection in evolution” has to capture the fact that there are a very large number of possible genotypes, and that in populations of “biparentally reproducing individuals”, there will always be many of these possibilities in play. However, “adaptiveness” of these genotypes is a single outcome variable he can get away with in this setting. So he has particular mathematical models in mind that one applies to these kinds of data. And looking at distributions and (hyper-) surfaces, there are characteristics that are commonly modelled such as how smooth or rough they are, how steep they are at any one point, whether they are multimodal etc. These characteristics are shared with geographical landscapes. So we say things like “steepest descent algorithm”, or “stuck following a deep valley in the likelihood surface”, and these are understood intuitively by listeners, but they are also actual descriptions. And if there are more ridges between fitness peaks than most biologists originally thought, again we are talking rigorously but in a way that is clear to the lay listener. When Wright briefly comments that the heights in the landscape are rising and falling “on the spot” over time, he is not stretching a bad metaphor, he is showing how the listener has to stretch these mathematical models to capture the reality of the processes involved.

    As to Wright’s slipping between different landscapes – from the mutational or sequence space of classes of genotype to that of a collection or population of genotypes, I reckon this is difficult to avoid in a statistically based model. The fitness of the genotype is, as you have shown in earlier articles here, an average behaviour (over multiple realisations) where we usually make the further simplification that the environment and all other genotypes of the individual organisms are held constant or averaged themselves. Sure he has jumped across a giant chasm – Orr [2005] quips that “[t]o an evolutionary geneticist, there is another, simpler way of
    distinguishing between selection and adaptation: we know a lot about the former but little about the latter”.

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  7. Robin Herbert

    Platonism runs deepest in science among some of those who try hardest to avoid it. For example insisting that the evolutionary course of a particular species is a continuum is strictly speaking inaccurate. There are very, very many very small changes and so it is closer to a continuum than if there had been fewer sharp changes, but it is still not a continuum.

    Well of course that is close enough for putting across a concept, but for refuting Platonism it backfires.

    As Plato might put it, we have never observed an actual continuum, only things that approximate continua, and yet we do not doubt that there is such a thing as ‘continuous’. So the ‘continuum’ is the form and we recognise various things as being imperfect instantiations of it. (Plato says this, but uses ‘equality’).

    Thus the biologist who says that evolution represents a continuum in order to avoid the ‘dead hand of Plato’ is playing right into those Plato’s dead hands.

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  8. brodix

    David,

    It does raise the question of what is knowledge, since to know something, we have to define it, which requires distinguishing it from context. Which is necessarily dependent on the context and frame from which we perceive it, then there is the further feedback of using this knowledge to frame further insight and information. Which is where metaphors come in, but they might not be how others are framing/seeing it.

    Then there isn’t absolute, or universal knowledge, as it would be everything as it is. The noise would remain noise, because any signal we extract would be limited by our definition and less than the whole.

    Then when we start to think we have somehow escaped this feedback loop, say by assuming our frames are universal, say mathematical regularity, or religious conviction, then we really get stuck in a rut, as it is assumed everything must fit these patterns and we become ideologues.

    Knowledge is pattern recognition, but it becomes its own pattern.

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  9. Massimo Post author

    david,

    I guess I have a problem with the description of these as metaphors

    Well, that’s how they are described in the literature. Also, since there are no “landscapes,” “surfaces” and so forth, not to mentions no “hills” or “valleys,” why not call them metaphors?

    These characteristics are shared with geographical landscapes

    Hence the metaphorical analogy.

    Robin,

    For example insisting that the evolutionary course of a particular species is a continuum is strictly speaking inaccurate

    I don’t get what the observability or not of continua has to do with Platonism. In order to bridge the gaps one simply has to apply a common strategy of extrapolation (or interpolation), no need to dream up eternal Forms. And that’s even before one gets to the usual throny questions: what is the relationship between the Forms and reality as we perceived it? Where, exactly, are the Forms? How, precisely, do we perceive them?

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  10. Robin Herbert

    Massimo

    I don’t get what the observability or not of continua has to do with Platonism

    It was a close paraphrase of Plato himself with ‘continuum’ substituted for ‘equality’.

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  11. Robin Herbert

    Massimo

    In order to bridge the gaps one simply has to apply a common strategy of extrapolation (or interpolation), no need to dream up eternal Forms.

    But there is no continuum. There are very many very small steps, which starts to look like a continuum.

    But if someone is going to claim that the reality is the continuum then you need to ask them where exactly this continuum is, since it isn’t in nature.

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  12. Robin Herbert

    davidlduffy,

    It is a metaphor in that we are talking about greater fitness as though it were higher up in three dimensional space and lesser fitness as though it was lower down in three dimensional space.

    We could choose a method of representing it so that greater fitness was the lower number and lesser fitness was the higher number, so that we would be talking about fitness plumbing the valleys and having to climb up across a saddle to get to the next valley.

    Or we could plot it all on a different set of axes and call the protrusions and indents.

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  13. Massimo Post author

    Robin,

    It was a close paraphrase of Plato himself with ‘continuum’ substituted for ‘equality’.

    Yeah, but still not an argument.

    But there is no continuum. There are very many very small steps, which starts to look like a continuum.

    “Continuum” is a question of coarseness of analysis. Regardless, still not an argument for Platonism.

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  14. brodix

    Forms, from words to planets, come into being through positive feedback, ie greater fitness and break down through negative feedback, ie lessor fitness.
    As such they exist in a process, ecosystem, environment, context, etc that is creating new forms and dissolving old ones.
    So forms go from being potential, to actual, to residual, ie future to past. While the process generates new configurations and sheds old, past to future.
    Forms then, are coalescing structure, definition and force ie positive feedback, until they reach a stage of greatest function, after which negative feedback starts to kick in and they dissolve back into context.
    It is an arrow, a narrative arc, for form and a cycle for context.

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  15. Robin Herbert

    Massimo

    “Continuum” is a question of coarseness of analysis.

    Yes, as I said there are no continua, only cases of very many very small changes.

    Regardless, still not an argument for Platonism.

    It wasn’t meant to be. I am not sure even Plato meant that part as an argument for forms.

    But, as I said, someone who insists it is the continuum that is real and that discontinuity is an artifact of mind is operating under the same premises as Plato

    Me I am not a Platonism in that sense – I don’t think of continua and equality as existing as things. I am a Platonist in the sense of thinking that mathematical facts that are true now or will be true in the future were always true, and that the set of all consistent axiomatic systems is not a trivial undifferentiated space like the Library of Babel. Things I didn’t think of as Platonism until recently. But that is another subject.

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  16. Massimo Post author

    Robin,

    as I said, someone who insists it is the continuum that is real and that discontinuity is an artifact of mind is operating under the same premises as Plato

    I don’t mean to be dense, but I still see no connection at all between contunua (or not) and Platonism. What set of premises?

    I am a Platonist in the sense of thinking that mathematical facts that are true now or will be true in the future were always true

    So are things about chess, once one makes up the rules. Or picks the axioms.

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  17. Robin Herbert

    Massimo,

    I don’t mean to be dense, but I still see no connection at all between contunua (or not) and Platonism. What set of premises?

    I already explained the connection when I said “It was a close paraphrase of Plato himself with ‘continuum’ substituted for ‘equality’.”

    Or do you think that Plato would have thought of “equality” as a form, but not “continuity”?

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  18. Robin Herbert

    Massimo,

    So are things about chess, once one makes up the rules. Or picks the axioms.

    You have thrown me here – are you agreeing that the mathematical facts that are true about chess were always true? I thought you disagreed with that.

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  19. Robin Herbert

    Maybe I should have emphasised the operative phrase:

    “I am a Platonist in the sense of thinking that mathematical facts that are true now or will be true in the future, were always true, and that the set of all consistent axiomatic systems is not a trivial undifferentiated space like the Library of Babel. Things I didn’t think of as Platonism until recently.”

    And, yes, I include the mathematical facts that are true about the game of chess.

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  20. brodix

    Robin,

    Math is not so much platonism, in the sense of ideal forms, as it is determinism. The same equation yields the same result, is an example of the same cause yielding the same effect. The view that reality is a computation and if we knew the total input, we could compute the history of the universe.

    Platonism is idealism, in the sense of everything being an expression of some ideal form. Consider how it is foundational to the premise of theology. That there is a spirit of some feature of nature and every example of that feature, say a wild bull, is an expression of the bull god. Then it became a matter of definition, as in defining what it is that makes a chair a chair, some essential qualities it must have to be a chair.

    There is a mathematical platonism, which argues the essential reality, the basis of definition, is this mathematical regularity. Yet that doesn’t explain emergence. The fact that reality isn’t just forms and patterns, but a dynamic process, feeding back on itself, breaking down and rearranging patterns, as much as creating them. So there is no essence, or ideal form, just branches from particular origins, with form emergent from those processes, as patterns.

    There is no God, nor any objective frame. Form doesn’t exist without the energy manifesting it. No fluctuation of the vacuum and all you have is the void. Mathematical operations are verbs.

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  21. Glynn Woodbury

    A small correction to Brodix, the speed of light is, approximately, 300,000 kilometers per second.

    Robin, I’m a raw recruit in the Philosophy wars, and perhaps you use the term continua in a technical sense, however, evolution is surely continuous, in the sense that no parents ever give birth to offspring of a new species. Species are recognized when populations, separated in some way, eventually differ enough to be described; and usually, though not always, become unable to interbreed.

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  22. couvent2104

    How did we get into this discussion about Platonism and continuity?

    Some biologists may believe that the landscape idea is true in a certain sense and can teach us something about evolution. But I don’t think there one single competent biologist who believes that evolution is continuous in the mathematical sense of the word. Even I don’t believe that, and I’m not a competent biologist.

    And what has observing continua to do with it? You don’t need to observe continua to believe the continuum (in the mathematical sense) is a mathematically valid notion. Take segments of straight lines. Do they always have a length? If your answer is “yes”, then you assume the existence of a continuum.

    Of course, nobody has to believe that all line segments have lengths. That’s what is so nice about mathematics. Assume that some don’t, and see how far you get. See how many interesting results you can derive. That’s mathematics.

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  23. Massimo Post author

    Robin,

    I already explained the connection when I said “It was a close paraphrase of Plato himself with ‘continuum’ substituted for ‘equality’.”

    Setting aside that that’s not really an explanation, it certainly is not an argument, which is what I’ve been asking for.

    You have thrown me here – are you agreeing that the mathematical facts that are true about chess were always true? I thought you disagreed with that.

    No, they become true once someone arbitrarily chooses some axioms, just like the rules of chess make true a number of things about chess that were nowhere to be found before someone invented the game.

    I am a Platonist in the sense of thinking that mathematical facts that are true now or will be true in the future, were always true

    Were always true? That’s where we part ways. Since nobody had thought about those “facts” (i.e., nobody had conjured that specific set of axioms) there is no sense in which they were “true” in the past. It’s an optical illusion from the point of view of the present.

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  24. brodix

    Couvent,

    I admit guilt to bringing platonism into the discussion, by observing some, unmentioned, disciplines do have a tendency to see their models as more than models.
    Biology, fortunately, has to conform to a more evident reality.

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  25. Bradley Sherman

    I doubt Darwin would be readable if purged of metaphor. I found the landscape metaphor useful in understanding selection. It became more powerful when I realized that the landscape changes with time and that life, itself, modifies the landscape (or is the landscape to parasites, symbionts and commensals). That’s not to argue that metaphor cannot lead one astray, but then so can reliance on mathematical abstractions and statistical inference.

    How are we doing on the frumious genotype-phenotype mapping function? I spent a fitful few days trying to understand meiosis in wheat (polyploid) and ended up wanting to abandon the dichotomy altogether.

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  26. Robin Herbert

    Massimo

    Setting aside that that’s not really an explanation, it certainly is not an argument, which is what I’ve been asking for.

    You asked for an explanation of what ‘continuity’ had to do with Plato and I tried to explain twice, but obviously I didn’t explain very well as you are not getting it and there doesn’t seem much point in me trying again.

    As I have already said I am not a Platonist in that sense I don’t know why you think I could give an argument for something that possibly even Plato had walked away from by the end of his life.

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  27. Robin Herbert

    Massimo

    Were always true? That’s where we part ways. Since nobody had thought about those “facts” (i.e., nobody had conjured that specific set of axioms) there is no sense in which they were “true” in the past.

    No one has thought of most of them even now, so how are they true now? Smolin seemed to have no trouble with the idea that a large collection of facts which no one has thought of, never mind formulated all became true in one fell swoop.

    I don’t even understand what that event is supposed to be.

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