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Simply put: if 2 cells Mitotically divide, there's almost no difference between them. They're like 2 copied files on my computer. They're identical. Now, doesn't this mean that the law collapsed? if not, how are they (the objects in the aforementioned examples) different? I'm sure I'm missing something so, can you please point out?

Thank you.

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    I'm afraid the premise is wrong: the two daughter cells are not identical. Even their DNA, the only part that's really copied, will most likely have some differences (the copying process is imperfect, and you have epigenetic effects that are not copied), but the rest of the cell's contents are not exact copies either. So no, two cells that have mitotically divided are not equivalent to the perfect copy you get on your computer. Life is dirty.
    – terdon
    Apr 4, 2021 at 16:07
  • Food for thought: ∃z ∣ z ∉ {z} ; It seems possible to my mind that Leibniz's law of identity is founded upon quicksand.
    – Joshua
    Apr 4, 2021 at 16:29
  • @terdon Life is dirty lmao. But, that's interesting to know about cells. thank you for that. Edit: also, @Joshua can you explain to me these 2 symbols: 1- ∃ 2- ∣ thank you.
    – Mostafa Mohamed
    Apr 5, 2021 at 1:31
  • Not in our world, because of differences in minute details and location, but some truly exact copies in sparsely populated and symmetric enough worlds might, see SEP, The Identity of Indiscernibles.
    – Conifold
    Apr 5, 2021 at 4:32
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    Maybe cells don't, but seems like bosons do violate it since as far as we know any number of photons can be in the same exact state.
    – Shufflepants
    Apr 5, 2021 at 10:07

4 Answers 4

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They differ in that they are in different position, just as the two copied files differ by being in different locations. Position is a property and therefore they are not identical.

In fact, the cells will differ in other ways. The DNA is reproduced, but other cellular structures will be split, and it would be immensely improbable for both halves to be the same. Likewise, the files will have different timestamps and such like.

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  • Can you elaborate more on this point about position? My conception of position is that it's a relative thing, so, I have trouble understanding how can a property of a being be relative. Thank you for your answer btw.
    – Mostafa Mohamed
    Apr 5, 2021 at 1:35
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    You could describe it as relative -- this one is on the left, that one on the right -- but you could also give them as absolute positions, such as coordinates.
    – Mary
    Apr 5, 2021 at 2:15
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    Strictly speaking, coordinates need to be relative to some frame of reference. But you can just pick an arbitrary frame. Or if you really want to use a "special" reference frame, use the rest frame of the CMB or some other "interesting" frame.
    – Kevin
    Apr 5, 2021 at 8:30
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    @MostafaMohamed Coordinates are numbers that are chosen relative to a frame of reference. The frame of reference is a physical thing, the coordinates are an abstract description which are assigned meaning in terms of this physical thing. By themselves, co ordinates are not a frame of reference.
    – preferred_anon
    Apr 5, 2021 at 14:55
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    @MostafaMohamed The frame is all you need to specify the position of an object as a vector. Coordinates are needed to create a coordinate vector such as [3, 4, 5] which represents that vector as an array of numbers. That being said, you can also leverage the argument that "for all frames, the positions of the two cells are different."
    – Cort Ammon
    Apr 5, 2021 at 16:22
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Perhaps the biggest achievement of reason is the capability to model the universe as if it would be static.

But the universe is not static. Every atom in the universe changes at any instant. The universe is in continuous change. Yet, you might see the same river twice and give it the same name, or you might look yourself in the mirror and thing you are looking at the same person. The mind is able to recognize a pattern of change as something static. So, while the universe is permanent change, we can perceive patterns and interpret them as if they would be static.

Leibinz's principle of identity is essentially a special case of such phenomenon.

From a different angle, if two cells are identical, how would you know they are not the same? If they are not the same, there will exist some property allowing you to refer them as different entities. If you refer to them as 'the one on the left seems identical to the one on the right', there you have a different property: they have different locations in space.

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  • Uhmm... Can you explain why you said Perhaps the biggest achievement of reason is the capability to model the universe as if it would be static.? Thank you.
    – Mostafa Mohamed
    Apr 5, 2021 at 10:14
  • ..."But the universe is not static. Every atom in the universe changes at any instant". You can keep reading that in the answer.
    – RodolfoAP
    Apr 5, 2021 at 11:04
  • Ah, ok. Thank you.
    – Mostafa Mohamed
    Apr 5, 2021 at 13:06
  • It isn't obvious to me that the universe is constantly changing. A fire burning from logs to ashes can be modelled as a single unchanging object; a worldline, spanning from a mass that looks like logs to a mass that looks like ashes. It isn't obvious to me why the dimension of time stands out from any other.
    – Gershy
    Apr 5, 2021 at 19:25
  • @Gershy Exactly, it is not obvious: "Perhaps the biggest achievement of reason is the capability to model the universe as if it would be static". You can keep reading that in the answer.
    – RodolfoAP
    Apr 6, 2021 at 5:48
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The mitosis example is a bit of a distraction because of the messy realities of DNA copying, positions relative to other objects, etc. What you are trying to get to is the symmetric universe paradox.

Max Black has argued against the identity of indiscernibles by counterexample. Notice that to show that the identity of indiscernibles is false, it is sufficient that one provide a model in which there are two distinct (numerically nonidentical) things that have all the same properties. He claimed that in a symmetric universe wherein only two symmetrical spheres exist, the two spheres are two distinct objects even though they have all their properties in common.

Black argues that even relational properties (properties specifying distances between objects in space-time) fail to distinguish two identical objects in a symmetrical universe. Per his argument, two objects are, and will remain, equidistant from the universe's plane of symmetry and each other. Even bringing in an external observer to label the two spheres distinctly does not solve the problem, because it violates the symmetry of the universe.

Mary's answer says that the two objects in this example could be distinguished by "position", but this fails because position can only be defined relative to a coordinate system and it is impossible to choose a coordinate system unambiguously in a symmetric universe without violating the symmetry somehow.

Whether this actually contradict's Leibniz's law is up for debate, but this shows that at least the argument has been considered before.

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  • but this shows that at least the argument has been considered before. oh, cool. Now I just get to read what they said about it. Do you recommend any books or articles on the matter?
    – Mostafa Mohamed
    Apr 5, 2021 at 10:13
  • @MostafaMohamed I must confess that my knowledge on the subject originates from and extends only as far as that Wikipedia section. But you can chase down the reference there, if nothing else.
    – Aaron Rotenberg
    Apr 6, 2021 at 7:03
  • Kant already had similar glovex universe example in his critique of Leibniz, see (britannica.com/topic/philosophy-of-physics/What-is-space). The relationist response to Kant’s argument was essentially to deny that the two symmetric universes (or gloves) are intrinsically different in the way that Kant suggested. The gist of Leibniz's law is right-handedness and left-handedness are not legitimate relationist predicates, handedness itself certainly is. That is, whether or not a certain shape is handed depends only on the distances between its constituent particles.
    – Double Knot
    Apr 6, 2021 at 16:19
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This is an excellent point. When a cell A is not dividing, we want to say that A stays the same cell even as it moves and changes over time. When A does divide into two parts, we have no way to decide which part is the original A. If one part is on the left and the other part is on the right, we may equally well give two versions of events: "A split off its right half and moved to the left," or "A split off its left half and moved to the right." It's like asking if a zebra is black with white stripes or white with black stripes.

Leibniz's law of identity has very little to say about any of this. Even when a cell is not dividing, but merely changing over time, Leibniz's law of identity does not allow us to say it is "the same" cell, as its properties are different. Nor does Leibniz's law of identity allow you to say you are the same person you were this morning.

We might use Leibniz's law in a different way, in which we speak of the whole history or timeline of an object, extending into the past or future, as having a single identity. You are not identical with your self this morning, but you in the present and you in the morning share the same timeline. But this does have difficulty accounting for the cell's mitosis, as we have no rule or principle to tell us which half of the divided cell gets the "original" timeline. The timeline apparently branches; we have a free choice of which to call the original.

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  • Apparently zebras have a dark muzzle, and the skin beneath the coat is uniformly black, which breaks the symmetry; so it seems much more logical to treat them as black with white stripes than vice versa.  (An unfortunate analogy doesn't necessarily spoil your main point, of course.)
    – gidds
    Apr 4, 2021 at 16:06
  • @gidds Most of the muzzle is striped, and that's really a matter of opinion. The skin beneath the coat does not matter. Suppose your wall was initially black, then you paint over it with fresh stripes of white and black. In your second coat you had to paint stripes of both colors, so you cannot say it was definitively black with only white stripes. There are stripes of both colors. Had the wall been initially green, you wouldn't say the wall is now "green with white stripes," would you? At best it is "green with black and white stripes," but really the green shouldn't be mentioned.
    – causative
    Apr 4, 2021 at 18:11
  • @causative interesting. Can we say that the original cell doesn't even exist anymore (in more than just the way my self in the morning doesn't)? This is also one of these questions that hit on the point of the essence and composition, which are quite interesting things.
    – Mostafa Mohamed
    Apr 5, 2021 at 1:41
  • @MostafaMohamed Yes, you can say the original cell doesn't exist, but in my opinion it is unsatisfying to do so. Suppose (counterfactually) that in the mitosis, the cell split off only 1% of its mass to make a new cell - then you would have no problem saying the "original" cell exists as the 99%, wouldn't you? What if the ratio was 5%, 10%, 20%, 30%, 40%? At what point do you stop saying "the original cell split off a piece of itself" and suddenly say "the original cell no longer exists"?
    – causative
    Apr 5, 2021 at 2:03
  • @MostafaMohamed IMO a nicer way to model this is to give yourself the free choice of calling either daughter cell a continuation of the original. This is all a matter of personal preference, because it is up to you how you want to define this term, "the cell," as a system that evolves over time. What "the cell" is depends only on how you define the term, and you can define it however you want. Although some ways of defining it may be simpler and more consistent with other ideas.
    – causative
    Apr 5, 2021 at 2:06

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