Biological essentialism? No thanks

P. Godfrey-Smith (left) and M. Devitt (right)

A few weeks ago my CUNY Graduate Center’s colleague Michael Devitt gave a colloquium entitled “Individual essentialism in biology,” which was followed by a response/commentary by Peter Godfrey-Smith by the title “Modality, essence, and biology.” I thought it was a really interesting example of two top notch philosophers going at each other, respectively defending and criticizing a given central thesis, in the best tradition of analytical philosophy (no, I do not mean this as a sneer). It was also, however, a rather surreal experience for a biologist turned philosopher of science such as myself. I mean, essentialism, seriously?

Essentialism is the notion that there are some attributes that are necessary to the identity or function of a given entity, and in the Western tradition is usually traced back to Aristotle and, before him, Plato. Plato’s idealism, his notion of a realm of abstract Forms of which the world as we experience it is a pale reflection, is a form of essentialism. But the kind of essentialism that has been most debated throughout the last couple of millennia is Aristotle’s, which American linguist George Lakoff characterized as the idea that some properties make the thing what it is, and without which it would not be that kind of thing. An essential property of a triangle, for instance, is that the sum of its internal angles amounts to 180 degrees; the essence of the element oxygen is that it has an atomic number of 8 (i.e., its nucleus is made of 8 protons).

In philosophy, however, that sort of essentialism has gone the way of the dodo, because it applies to very few things, which do not even include chemical elements (given the existence of isotopes). Michael, therefore, wasn’t trying to revive Aristotle and apply him to contemporary biology. What, then, was he trying to do? (If you want a lot of details, here is his full 2008 paper on this topic.)

Michael’s opening shot was to say that Linnaean taxa, and especially (but not only) species, have essences that are at least in part intrinsic, underlying, and genetics. He promptly acknowledged that the consensus among biologists is of a rejection of such talk, but he shrug that off (literally, when I pointed it out to him during the q&a) with a “too bad for biologists, then.” (Incidentally, it is also the consensus among philosophers of biology.)

I find this approach rather troubling. I thought that time was long past when philosophers could tell scientists what to think while ignoring the scientific aspects of the question. In this case, for instance there are no Linnaean taxa “out there” (so to speak). Ever since the 1970s and ’80s the consensus among systematists has been that taxa are simply convenient more or less arbitrary containers, useful to orient the limited human mind among the bewildering variety of biological forms. But what biologists recognize as descriptively and explanatory relevant are clades, that is continuously branching lineages resulting from evolutionary processes. Look, for instance, at this simplified view of the cladogenesis that has given rise, among others, to our own species:

Chimpanzees belong to the genus (a Linnaean taxon) Pan, we belong to the genus Homo, and these two species, together with gorillas, orangutans and gibbons belong to the superfamily (another Linnaean taxon) Hominoidea. But there is no biological sense in which the genus Homo is “equivalent” (meaning, it is the same sort of essential thing) as the genus Pan. And even less so is our “genus” anything like a genus of plants, say Achillea (a member of the family Asteraceae), like this individual (from the species Achillea millefolium, also a taxon not at all equivalent, in essence, to the species Homo sapiens, or to Pan troglodytes):

At any rate, Michael defined his concept of essentialism in this way: “A property P is the essence of being an F if and only if anything is an F in virtue of having P.” While logically crystal clear, Michael not once throughout his talk gave an example of an essential biological property, and when I asked him directly he said that none has been discovered. Yet.

Devitt discriminated among intrinsic essences (like the atomic number that identifies the element oxygen), partly intrinsic and partly extrinsic essences (a pencil is defined by the materials of which it is made, internal, as well as by the function that characterizes it, external), and relational essences (being Australian is a matter of being in a certain relation with a particular geographical location).

He is, of course, aware that both biologists and philosophers of biology reject talk of essentialism, maintaining that biological taxa are best thought of as arbitrary breaks in an underlying continuous evolutionary process, as discussed above. Surprisingly, however, he claims that the consensus among his opponents “says strikingly little, and nothing plausible, about what precisely the historical component [to the identification of taxa] is.” I’m not sure what Michael was looking for, here. Cladistic theory is highly precise and quantitative, and so is evolutionary theory — particularly in the mathematized form known as population genetics. So I’m at a loss even imagining why he thinks there is anything amiss in the consensus view.

Devitt’s next step was to introduce a Kripkean view of individual essentialism. (Saul Kripke was actually sitting in the audience, and when asked about it he said he did not really recognize Devitt’s account of biological essentialism as particularly Kripkean. But I’m no expert on Kripke, so I will sidestep that particular issue.)

Michael’s concern, as he put it, was with “what timeless properties could an object not have failed to have, and what properties could it have lacked while still timelessly existing?” For a biologist, the answer to this question is an empty set: there are no timeless properties in the contingent world of biology, pace recent attempts at mathematical Platonism that I have previously criticized.

Kripke comes into play because according to him — as recounted by Devitt — both the origin and the substance of which an object is made are essential to it (that, and something to do with the idea of modality, see below). In this spirit, Michael added, “it is essential to the Queen’s zygote not just that it is constituted of human genetic material of some sort, but of the particular genetic material of that zygote.”

Let’s set aside that there is no such thing as “the Queen’s zygote,” but at most a zygote that, through a particular combination of historical accidents and cultural influences will, eventually give us a female individual of the human species that some misguided Brits will refer to as “the Queen.” What Michael is saying here presents no problem whatsoever for the anti-essentialist consensus: a particular zygote is the result of a particular series of evolutionary and developmental events, which could have gone otherwise, thus originating another zygote (which in turn may or may not have given rise to a Queen). End of the story, no essence required.

Devitt then continued by providing two arguments, one in favor of intrinsic individual essentialism, the other in favor of historical individual essentialism. Right there, however, we run into yet another problem. Michael is following biologist Ernst Mayr’s sharp distinction between proximate and ultimate causes: the proximate cause of, say, a tiger’s stripes is the particular genetic makeup of tigers; the ultimate cause of the stripes is whatever evolutionary (i.e., historical) series of processes led to that phenotype (e.g., adaptation by natural selection, if it turns out that it is advantageous to have stripes in a typical tiger environment).

But plenty of people have convincingly argued that Mayr’s distinction is far too sharp: genetic-developmental processes affect evolutionary trajectories while at the same time also being a result, generation by generation, of those trajectories. Development is an evolutionary mechanism, the basis of the incredibly fertile field of research known as “evo-devo” (evolution of development). Thus the very distinction between these two types of individual essentialism that Michael makes is, again, based on a shaky appreciation of both biology and philosophy of biology. (Here, incidentally, is my take of the proximate-ultimate distinction.)

At any rate, Devitt considers the case of Benji the (hypothetical) tiger to explain his views on intrinsic and historical individual essentialism. He says that: “the essential nature of Benji, probably unknown, causes him, in his environment, to be striped. What nature? If our concern is structural, an intrinsic nature … The same intrinsic nature or essence that (partly) makes something Benji (partly) explains both why he is striped and being Benji is explanatory.” (The latter part means that the fact that a particular tiger is striped, in Devitt’s account, is due to the fact that he is Benji, a particular individual.)

I find this to be entirely unhelpful. First, notice that Michael admits that he has no clue to what the essential nature of Benji is to begin with. Second, we actually have a pretty darn good account of what makes not just Benji, but most tigers, striped: structurally speaking, it is their particular genetic makeup in interaction with their particular environment, throughout the process of development. While the details may be difficult to fill in, there is absolutely no mystery here. Moreover, this account also explains the exceptions, that is, those individual tigers who are not striped: they either have a mutation that inhibits the formation of stripes during normal development or, less likely in this specific example, have experienced developmental environments that were not suitable to the formation of stripes, given the present genetic background.

What about historical individual essentialism? For Devitt, “if our concern is historical, with what led to there being Benji with those properties, the explanatory nature must be historical. The same history that (partly) makes the organism Benji causes Benji to be striped. That’s why Benji is explanatory.”

Fine, but that “same history” is, in fact, nothing other than a sequence of gene-development-environment interactions that are also explanatory in the structural sense. Once more: no mystery, no need for talk of essences. The latter simply does not add anything to either a biological or a philosophical account of what is going on, it simply obfuscates things by introducing an as yet unknown, and highly vaguely defined “essence.” Why?

Next, Michael moves from individual essentialism to what he calls essential membership. He says: “[consider] Benji and his property of being striped, a property typical of tigers in their environment. A part of the intrinsic component of Benji’s essence explains why he has the property of being striped. That part of the intrinsic component of the essence of tigers also explains why tigers have that property of being striped. The ‘sum’ of all such parts of Benji’s essence, is the intrinsic component of the essence of tigers.” And moving from structural to historical concerns: “organisms of a certain intrinsic kind evolve into organisms of another intrinsic kind, until we reach the taxon in question” (emphasis in the original).

All of this is a fancy, unclear, and consequently not very helpful way to say what every biologist would say. I’ll rephrase it precisely to make the point: “[consider] Benji and his phenotypic property of being striped, a property typical of tigers in their environment. The genetic component of Benji’s developmental system explains (in part) why he has the property of being striped. That genetic component (variable within the species) also explains (in part) why tigers have that property of being striped. … Organisms of a certain average genetic makeup evolve into organisms of another average genetic makeup, until we reach the (provisional) current state of affairs.”

See? No essences, no mysteries.

Time now finally to turn to Peter Godfrey-Smith’s response to Michael Devitt’s talk.

Peter began by taking on Michael’s reference to Kripke’s philosophy of modality (see this section of a larger essay on modal epistemology in the Stanford Encyclopedia of Philosophy). Peter’s take is that modality — the way or mode in which something exists — plays a role in our talk of causation, and makes sense of our intuition that we see events as “surrounded by a cloud of possibilities.” For instance: the window broke because I accidentally kicked a soccer ball in its direction. But it was possible that I had not kicked the ball in such a manner, and that therefore the window would still be intact. But what does “possible” mean here? If you are a determinist in metaphysics, it cannot mean that I could have done otherwise, at the least not without violating the laws of cause and effect in this universe. But surely there is a logical sense in which I could have done otherwise, meaning that it was not a logical necessity that I thus kicked the ball. Kripkean modal talk then helps us both to organize the actual, as Peter put it, and of making sense of unrealized possibilities: there are nearby worlds (we shall leave their ontological status undecided) in which I did not kick the ball in a manner that led to the breaking of the window.

That, Godfrey-Smith said, is a good use of modality. A bad use, by contrast, includes talk of essences. That, of course, is precisely the use of Kripkean modality made by Devitt (and to which, as I said above, Kripke himself, who was present at the talk, seemed to object).

Peter then moved to consider Michael’s concept of essence. As we have seen, Michael rejects the rigid Aristotelian idea of essences, and would “like to be as thin as possible on the modal front. [For Devitt] essences have a certain explanatory role.” One immediate problem, then, is that “explaratoriness is a matter of degree. A matter of more or less. [It is] also relative — explanatory in this context but not that one. [And] essences are not like this.” Exactly. It seems at times that Devitt’s concept of essence is so “thin” that it is hard to see why it would count as an essence.

[Michael’s refrain here is that he is not wedded to talk of essence, and that something like “nature” (of a thing) would do. But Peter points out that he uses the word “nature” very much in the sense of an essence, and I would further add that if Michael really did drop that talk there would be little left to disagree with him and we could all go out for a beer.]

Peter continued: “If you have a scholastic view of the world, it is a reasonable project to try to sort through the clutter of different sorts of importance of properties, to find essences. If you don’t have a view like that, it makes less sense. You just have various explanatory roles, and facts about how we talk: ‘if it lacked F we would not call it an F.'” This is a rephrasing of my objection above that one could simply drop the word “essence” from what Michael is saying about stripes and tigers, replace it with an account of gene-development-environment interactions, and be done with it, nothing lost in terms of explanation, much gained in terms of clarity. (Also notice that in contemporary philosophy the word “scholastic” is not a complimentary one…)

More from Peter, referring to Michael’s example of Benji the tiger and his stripes: “the tiger’s stripes are not themselves essential. They are conspicuous accidental features. Easy for that tiger to never have had stripes. [This type of] properties are caused by a mix of external and internal, but with a big role for internal specifics — the genome. Why is the cause of those features an essence? What is lost by denying that it is?” Precisely nothing, I’d say.

And more, along the same lines: “In the tiger quote [from Devitt] remove ‘essential’ and the quote is fine. Understand ‘nature’ in terms of ‘its actual make-up.’ … The properties Devitt calls ‘essences’ are good explainers, in certain contexts, mixed in with other properties that explain other things.”

Godfrey-Smith concludes with an analogy between species, like Pan troglodytes (the chimpanzee) and human lineages, say the Churchills.

What sort of things is the Churchill family, Winston & co.? “A scattered particular, it seems. We have a sense of other possibilities surrounding their actual features and activities. Winston might not have been PM. The Churchills left London on Friday but they could have decided to stay in town.” And so on. “Had they lived in another century would they have not been the Churchills? Even if they came out similar looking-ancestors, in that other century?”

“Think of the chimps as like the Churchills,” said Peter. “Being a chimp is analogous to being a Churchill. They are the same sort of thing. Whatever sort of modal cloud surrounds the Churchills, the chimps get something similar. … The chimps present at a later time — the slice of the chimps at time t — can be very different with respect to the properties of the individual animals than the chimps at an earlier time. Evolution is open-ended.”

Exactly. And in order to understand evolution, and to describe its products at any given time, we need no talk of essences, at all.

81 thoughts on “Biological essentialism? No thanks

  1. couvent2104

    Hi Coel,

    You suggest that I’m being a bit unfair, but then go on to make a comment which entirely agrees with that I said, namely that “the sum of its internal angles amounts to 180 degrees” is not an essential property of a triangle under the definition: “A property P is the essence of being an F if and only if anything is an F in virtue of having P”.

    Your remark obviously was correct. But I think is was a bit unfair as a criticism of essentialism. “An essential property of a triangle, for instance, is that the sum of its internal angles amounts to 180 degrees” is so obviously wrong (or incomplete) that it can’t be taken seriously as an attempt at stating what the essence of a triangle is.

    Better would be something like this:

    Essential property of being a triangle = “A set of 3 straight line segments XY, YZ and ZX defined by 3 non-collinear points X, Y and Z.”

    This is essentially the definition of a triangle. It’s hard to be more essential, I think.
    But even this doesn’t really work. Suppose I draw a garden-variety triangle, show it to an essentialist and ask him if this drawing has the essence of being a triangle. Strictly speaking – and I think we can ask an essentialist to speak strictly – he’ll have to ask me first in which model I want my answer. Euclidean geometry? Then the answer is yes. The Klein model of hyperbolic geometry? The answer is yes again. The Poincaré model of hyperbolic geometry? Then the answer is no.
    The answer depends on the way the essentialist looks at my drawing. For me, that’s a good reason to question essentialism (in mathematics, at least).
    “If A has P essentially, then necessarily in any world in which A exists, A must have P” or something. But my triangle can live in three worlds and be a triangle in two of them but not in the third one.

    Perhaps there exists something like “context-dependent essentialism”?

    Liked by 1 person

  2. SocraticGadfly

    David P, very interesting comments.

    And, Massimo, might I respectfully suggest at least a Friday link, and perhaps more, for David’s new book, per his Facebook page, about the intersection of sport and philosophy?


  3. Robin Herbert

    I think there is a difference between “P is an essential property of X” and “Property P is the essence of X”. So, (assuming that when we say “triangle” we mean something in a Euclidean space) “the sum of the internal angles is 180 degrees” might be an essential property of a triangle without being it’s essence, under this definition, which says “…P is the essence of…” and not “…P is an essential property of …”.

    I would suggest that, if essence is to be a useful concept at all, the essence of something must be a set of properties. Although, of course, any given set of properties can, strictly speaking, be regarded as a property, so the definition is OK. Whether it is useful in that context, I don’t know.

    Liked by 1 person

  4. brodix

    Can we safely say no definition is ever complete, but clarity requires it be close to the minimum, but no less, for the particular function.


  5. Massimo Post author


    No one denies that there are lots of shared featured among lots of different kinds of objects. The explanation is always going to be some kind of process that falls into the general category of cause-effect / laws of nature. For instance, in the specific case of biological organisms, members of a given species share lots of characteristics because they have similar genetic-developmental systems that are exposed to similar environments. These systems evolved over time as the result of a number of biological processes, including natural selection, genetic drift, etc.. So far, I don’t see the need for either essences or schmessences. Adding talk of either to the above characterization does not seem to add anything at all.

    Millikan’s distinction between eternal and historical kinds is a bit simplistic, and may not even apply to the laws of physics, if people like Lee Smolin are right. But regardless, all molecules of water share the property of being made of H2O because cause-effect relations in this universe, given its laws and regularities, make that the case. To add, “and this is the essence of water” adds nothing from an explanatory point of view, and seems to be metaphysically redundant.

    Biological species: they can’t possibly be eternal kinds. Genomes change over time, gradually. And yes, shared ancestry makes them historical kinds, but “history” is a proxy for a number of well understood mechanisms, none of which is “essential.”

    Without further elaboration, I’m not sure how to answer your further questions: how do developmental systems theory and epigenetics bear on the issue? What is the issue, exactly? DST is a theory, epigenetics a field of study. They both add complexity to the causal picture. But they don’t add either essences or schmessences.

    You say that these are complex questions, and they are. But I’m still inclined to dismiss any talk of essence as a medieval leftover unless someone can do the hard (impossible, I think) job of: a) providing some sort of viable example of a biological essence; and b) tell us what work that essence actually does. So far, I haven’t heard anything even approaching this.

    I agree with you and Peter about modal talk being a byproduct of our interest in counterfactuals. I’m not at all sure that modal talk helps a lot when it comes to thinking about counterfactuals, but that’s another issue.

    If modal necessities are, as you (correctly, I think) say, are just all those things we hold fixed in any given counterfactual scenario, then they are not really “necessities,” they don’t do a lot of metaphysical work.

    Liked by 3 people

  6. Robin Herbert

    You don’t need the concept of context dependent essentialism, you just need to ask what is meant by the terms in the definition.

    If, for example the term ‘point’ refers to a two-vector in a Euclidean space then the definition works just fine as the essence of something (and it may be what some people mean by ‘triangle’, but not what others mean by it). If, on the other hand the term ‘point’ refers to any element in any topological space then there is nothing at all that can satisfy the definition since no three such points can possibly be non-collinear.

    Again, what is meant by the term ‘collinear’? If it refers to a specific function (or relation) then any given function has a range and a domain and so the definition of the function involved implies a context. If a collinearity relation or predicate refers to a distance function then does it refer to any kind of distance function on a space, or a specific distance function with a specific range and domain?

    If the latter then the definition works just fine as an essence without needing to supply the context independently. If, on the other hand, it refers to any kind of distance function on any kind of a space then, again, the definition is of something which cannot exist since there will always be a distance function that will make the points collinear.


  7. Robin Herbert

    On the other hand if I define a triangle as the shortest closed path that passes through three non-collinear points on any set closed under a distance function, why would this not be the essence of a triangle just because it is not satisfied in certain sets and distance functions? Incidentally you don’t need to go as fancy as Poincare Hyperbolic Geometry, it is also not satisfied on a number line.


  8. Robin Herbert

    To put it yet another way, the claim that a particular definition of a triangle does not hold in certain geometries contains the premise that there is such a thing as the essence of a “point”, or the essence of a “distance”, so that you could meaningfully say that the expressions “point” and “collinear” in Euclidean geometry is relateable to those expressions in, say, the Poincaré model of hyperbolic geometry, otherwise it would just be a case of using the same word for different concepts, much as if an English person was talking at cross purposes with an American about “suspenders”.


  9. Robin Herbert

    Hi Massimo,

    The “essence” of a point? No, a working definition of a point.

    But then I didn’t say that there was such a thing as the “essence” of a point. I said that the claim “my triangle can live in three worlds and be a triangle in two of them but not in the third one” implies that there is such a thing as the essence of a “point”.

    If there is no such thing as the essence of the point then the claim that the triangle so defined can exist in all three of Euclidean, Poincare Hyperbolic and Klein Hyperbolic geometries is false, since each will have a different definition for “point” and “collinear” and thus you would only using the same words for different concepts. A particular definition of “point” “distance” and “point-line distance” will uniquely imply a particular geometry and therefore that definition is either too vague to be useful or will be about a particular geometry.

    The same goes for the claim that the internal angles of a triangle will not add up to two right angles in some non Euclidean geometry. Again the word “triangle” defined on the basis of “points” and “collinearity” (which is itself defined on the concepts of distance and point-line distance) cannot be said to refer to the same thing in Euclidean geometry as it does in non-Euclidean geometries.

    So the claim would simply be that there are figures in non-Euclidean geometries to which we might arbitrarily apply the label “triangle” where the internal angles do not add up to two right angles – unless that is, you think that “point” and “distance” have some essence that would make them relateable between different geometries where they have different definitions.

    So the claim that the internal angles of a triangle do not necessarily add up to two right angles itself depends upon the premise that there is such a thing as the essence of a “point” or “distance”.

    Liked by 1 person

  10. Markk

    Could essences be a replacement for laws of nature? So instead of saying that water boils at 100 degrees because of a law, you could say it does so because of its essence. The advantage of doing so? “Law” implies causation from outside the water somehow by a Lawgiver, whereas “essence” makes it clear that a thing does what it does because of its internal properties. Not sure how that would translate to biology.


  11. brodix

    What would be the essence of a point? If it has some minimal dimensionality, it’s not exactly a point, but if it has no dimensionality, i.e., dimensionless, it doesn’t exist, because it is a multiple of zero.

    Isn’t a point an ideal of location? As such, an abstraction without properties and thus without essence?

    Could it really be said a line is made up of infinite dimensionless points, as even infinity multiplied by zero would be zero and if those points, as ideals of location, are not at the same point, they could as easily be a plane or volume and it is only the description of being on a line that makes them a line. So the line would be its own definition, not the points.

    What is a Planck length? If there is no smaller level of measurement to define it, wouldn’t the error bars of its ends be as long as the length?

    Perfection that is not absolute is not perfection and perfection that is absolute does not exist.

    Liked by 1 person

  12. davidlduffy

    “I don’t think you are using the word “essential” in the metaphysical sense, but simply as the everyday meaning of ‘basically'”. Hi Massimo. Those were all examples taken from the literature on essentialism eg

    “Similarly, Socrates is essentially human not just because he couldn’t survive failing to belong to a species — no living thing could do that — but because he couldn’t survive belonging to a different species; he couldn’t be a dog or a marigold or a unicorn. His height, mass or color might vary, but not his species. Mutatis mutandis, for almost all quotidian essential properties.”

    Which is to say that the metaphysical essential is quite akin to the everyday meaning in many cases. As to what work it does, several recent papers I can find online (eg Della Rocca. Soames, Skiles) look to Kripke, claiming his ideas about identity and a posteriori necessity imply some kind of essentialism.


  13. Robin Herbert

    Hi Brodix,

    “Isn’t a point an ideal of location? ”

    Or an element in a set upon which certain functions are defined which evoke the idea of “location” in a human mind.


  14. Massimo Post author


    Again, I don’t see why the essence of a point is implied at all here. Either those spaces do not, in fact, use different definitions of points, or what exists in one type of space is a triangle while what exists in the other two is not a triangle. No matter how you slice it, it comes down to axions and definitions, not essences.


    No essences aren’t laws of nature (which some philosophers and a few scientists consider in and of themselves a problematic concept anyway). Laws simply describe regularities, essences are a kind of metaphysical entity or property.


    The examples you picked may be from a paper on essentialism, but the way you phrased it was clearly not the technical-metaphysical meaning.

    Anyway, Socrates wasn’t essentially human, he was accidentally so, meaning as the result of contingent events and processes of evolution and genetics. To say that he could not have survived as a member of a different species in entirely nonsensical: if he were a member of a different species he wouldn’t be Socrates. Unless your dog’/ name happens to be Socrates. Mine was Galileo…

    Liked by 1 person

  15. Robin Herbert

    Hi Massimo,

    Either those spaces do not, in fact, use different definitions of points, or what exists in one type of space is a triangle while what exists in the other two is not a triangle.

    His claim was that it was a triangle in two of those geometries, but not the third.

    But since those spaces do, in fact, use different definitions of points then you are simply restating what I said, it is a triangle in one space and not a triangle in the other two, as long as we use one definition of triangle throughout.

    So when someone claims that the internal angles of a triangle add up to two right angles in Euclidean geometry but not in some non-Euclidean geometry where the definitions of “point” and “distance” have different definitions between the spaces, then in what sense are they saying that the word “triangle” relates to the same thing in both geometries unless they are assuming that there is some “essence” to a triangle?

    So if what you say is true (and remember I am not saying it is not true) then it is not the case that the angles of a triangle add up to two right angles in Euclidean geometry and not in a non-Euclidean geometry. Someone who says this is simply using the word “triangle” to refer to two different mathematical concepts.

    This follows logically from what you say above, unless you are saying that the definition of a triangle remains constant between Euclidean and non-Euclidean geometries.

    Liked by 1 person

  16. Robin Herbert

    And, keeping the definitions the same throughout, if the definition remains throughout, if it refers to a triangle in Euclidean geometry then none of it refers to anything at all in any non-Euclidean geometry.


  17. Robin Herbert

    It would be no different to me claiming that the internal angles of a triangle might add up to four right angles since I could arbitrarily decide to apply the word “triangle” to a four sided shape.


  18. Coel

    Hi Markk,

    “Law” implies causation from outside the water somehow by a Lawgiver …

    People did think that way prior to about 1850, but no-one thinks of “laws of nature” that way nowadays,

    Liked by 1 person

  19. Robin Herbert

    In any case, these are side issues. I thought the article was excellent, as far as I am a judge of these things, although admittedly I have a limited knowledge of biology.


  20. brodix


    “Or an element in a set upon which certain functions are defined which evoke the idea of “location” in a human mind.”

    So that element is description, rather than essence, as essence implies irreducible property. A point that is not dimensionless is still reducible, while a dimensionless point has no actual location, being zero, not one.
    Sort of like trying to take a picture with the shutter speed set at zero.
    Description, on the other hand, is context dependent. Everything is only relevant in context.


  21. Massimo Post author


    Than yes, we are saying the same thing: if one changes background definitions (like that of a point) when one goes from Euclidean to other spaces, then one is using the word “triangle” imprecisely or analogically, since it obviously isn’t, and couldn’t be, the same geometrical figure.


  22. couvent2104

    I apologize if I derailed the discussion. It wasn’t my intention. But now that I wrecked it, let me do a bit more damage.


    But since those spaces do, in fact, use different definitions of points then you are simply restating what I said, it is a triangle in one space and not a triangle in the other two, as long as we use one definition of triangle throughout.

    The Klein and the Poincaré model are constructions in the Euclidean plain, so they use the same points. A point in the Poincaré model is a point in the Euclidean plane. There is no difference(*).

    “Line” is another primitive, undefined term of the axioms. The axioms say that for two points X and Y, there’s always a line that goes through X and Y and that this line is unique. This is, I think, the reason why we call the lines of the axioms “straight lines” in the EP. A ruler – a thing we call “straight” – needs only two points to fix it. A circle or a triangle or something wavy needs more points.

    But for two points in de PM there’s also always a unique line that goes through them. Seen from a Euclidean viewpoint these lines are segments of a circle and not straight lines, but they share the relevant properties with Euclidean straight lines. They are for example geodesics (the shortest distance between points). If you call the lines of EG straight lines, I think it’s fair to call the lines of the PM straight lines too(**). I hate to quote Wikipedia, but the articles on mathematics are in general OK as far as I can see, so I’m going to do it anyhow for the Poincaré model:

    “…the straight lines consist of all segments of circles contained within that disk that are orthogonal to the boundary of the disk, plus all diameters of the disk”

    It may sound strange that a segment of a circle can be the shortest distance between points. But it really isn’t. The PM doesn’t have the same distance function as the EP. Mathematically speaking, this is not a problem. A distance function is just a function of two points X and Y that has certain properties. There is no “right” distance function in general. On the surface of a sphere, you don’t measure the distance between two points along a Euclidean straight line. What counts as a distance, depends on the geometrical properties of the model. The important thing is this: he distance function in the PM is just as natural within the model as the usual distance function is within the EP. If you ask someone what the distance is between two dots on a piece of paper, he should – strictly speaking – ask first if these are points in the EP or in the PM.

    In short: I think it must be tough to be an essentialist in mathematics. But philosophers are tougher than I am.

    () I don’t think there can be a different definition of points, because point is an undefined, primitive term in the axioms. There could be a different *model, but that’s not the case here.

    (**) One might decide that only the lines in EG are “really” straight. I don’t see a mathematical reason for it. It only shows Euclidean prejudice.


  23. SocraticGadfly

    Robin, remember that that collinear is a hemisphere, so its angles are more than 180 degrees. But, is a hemisphere a standard unit of geometry?

    Couvent: #IDerailedTheDiscussion is your hashtag for the next 15 minutes of infamy? And, no worry, Massimo will grace us with his Friday links in less than 18 hours.


  24. Robin Herbert

    Hi Couvent2104,

    The Klein and the Poincaré model are constructions in the Euclidean plain, so they use the same points. A point in the Poincaré model is a point in the Euclidean plane. There is no difference(*).

    No difference at all? So you use the same point-point distance function? The same point-line distance function in all cases?

    If that were so then a triangle would indeed be a triangle in all three cases.

    A distance function is just a function of two points X and Y that has certain properties. There is no “right” distance function in general.

    I don’t recall saying or implying that there was a “right” distance function.

    But if the distance function used in each case is different then the definition is different.

    If you are calling two things with different definitions “triangles” then you must be assuming that they share some essential feature that makes them a triangle – ie you are implicitly an essentialist.

    If you reject essentialism and you have two figures which have different definitions then one is a triangle and the other is not.


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