Nuts and Bolts 002: (numerical) Identity

This is the second instalment in the “nuts and bolts” series of blog posts, where I take some of the “nuts and bolts,” the basic concepts employed within philosophy (and later I suppose I’ll use examples in theology as well) and explain them for those who might not be as familiar with them as people who encounter them a lot.

Recently while I was giving a public talk on the contentious issue of abortion, I made reference to the idea of “numerical identity.” In context, I was explaining that even though the features of a fetus will change considerably over time during gestation, and will continue to change considerably after birth as well, although its qualities at one point are not identical to its qualities at a later point, it is still the same entity. In technical terms, I explained, it remains “numerically identical” the whole time, and so I, an adult, am numerically identical to a fetus that once lived.

This term caused a bit of confusion for a couple of people in attendance. For example, one man thought that “numerically identical” just meant “a set made up of the same number of things.” He objected that my comments summarised above committed me to the claim that I am identical to one of my hairs. After all, there’s just one of me, and if I pluck out a hair, there’s just one of it too, so the two things would be numerically identical (after all, 1 = 1)! So I’ve decided to make this second nuts and bolts blog post all about the concept of numerical identity. It’s not the most riveting of subjects, but a pretty important one in philosophy one nonetheless.

So what is identity? Although it’s a term used in philosophy, it certainly isn’t unique to the field of philosophy. Philosophy isn’t an abstract, arcane discipline unto itself. It’s an approach to concepts and ideas that actually apply to the whole variety of disciplines, subjects and issues that all of us interact with in our lives as we use or employ language, science, medicine, as we engage new beliefs, come up with new ideas about the universe, decide how to evaluate theories, pursue justice and so on. Philosophers have had plenty to say as they have explained and discussed this concept of identity that all of use use in everyday speech and life, whether we realise it or not. For example, it gets used in police line ups (e.g. “looking at these five people, can you identify the man who robbed the bank?”), it gets used in romance novels (e.g. “could this really be the same man I knew all those years ago as a child?”), it gets used in our study of the natural world (e.g. “scientists tagged the salmon so that in the months to come as they tracked its movement, they could identify it as the one they were studying”), it gets used in spy movies (e.g. “my cover was blown. In spite of my changed appearance, the KGB now knew who I really was”), and so on.

Whether we’re aware of it or not, all of these scenarios are taking for granted the most fundamental of all logical laws, namely the law of identity (http://en.wikipedia.org/wiki/Law_of_identity). It is both simple and obviously correct, and is as follows:

A = A

That’s it. In English, it is best stated this way: “everything is identical with itself” (or ?A = A, “necessarily, everything is identical with itself”). This may seem fairly trivial and obvious, but it requires us to distinguish between two important concepts of identity. The law of identity is referring to what is called “numerical identity,” although there is another way that things can be identical, namely by being “qualitatively identical.”

In order for entities to be qualitatively identical, they must share all the same qualities (i.e. their qualities must be identical). Two perfectly manufactured ping pong balls would be qualitatively identical provided they are made exactly the same way. To see the difference between the two kinds of identity, consider this: Imagine that I showed you those two ping pong balls and asked you to point to one of them. Next, imagine that I were to put those ping pong balls behind my back and switch them between my hands a few times. Then imagine that I held them out to you, one in each hand, and asked you “which one is identical to the one you chose?”

You could react in one of two ways, depending on how you interpreted my question. If you thought I was talking about qualitative identity, you might say “they are BOTH identical with the one I pointed to earlier.” And you’d be more or less right if that was what I meant. But that’s not what I meant. What I’m talking about now is numerical identity. Imagine that unbeknownst to you, each of the ping pong balls had a name, X and Y. The one you had pointed at was Y. In terms of numerical identity, the correct answer to my question is “Y. Y is identical with the ball that I originally pointed to.”

Numerical identity is not about the qualities that a thing (or person) has. It has everything to do with whether something is the same object or entity as another. Qualitative identity on the other hand is something that comes in degrees. Two things can be more similar or less similar. Two ping pong balls are very similar. They are not absolutely the same in all qualities (e.g. including even location), or we would be talking about the same ball after all. But two things can be pretty much qualitatively identical while still being not at all numerically identical. Here’s another example to hopefully make this distinction clear: Imagine that you were a witness to a murder on a cold and dark autumn night. You got a good clear look at the killer standing under a street light. He had a menacing scowl on his face, a long beard, and wild woolly red hair. Now you stand in the dock as a witness as this man stands trial. The prosecution lawyer asks you – “is that man the same person you saw at the scene of the murder?” You look over at the accused man. He has had his hair cut short since that terrible night, and now he’s clean shaven as well. From what you’ve heard, he has changed his attitude as well. He felt so terrible because of what he had done that he has really turned his life around, and now he wouldn’t hurt a flea. Because of all these changes, you say to yourself, he’s not the same man anymore. So you say to the lawyer, “No. That’s not the man I saw that night. He’s different from that man.”

Of course, you can see exactly what’s wrong with this answer. The person in the dock is confusing two different understandings of the word “same,” each of which deals with a different type of identity. This man’s qualities have changed over time, so in a qualitative sense he’s different, but it’s still true that he’s the same man as the murderer in a numerical sense. This could have been easily demonstrated if, on the night of the murder, you branded a number into his rump – the number 75 (Why 75? Well, why not!). That way, when standing in the dock, you could have simply asked the man to drop his trousers, and then you could declare – “Yes, that man has the identity of (i.e. he is identical with) the killer I saw that night. You would have established that whatever changes he might have undergone, he is numerically identical with the killer (unless of course there’s another man with the number 75 branded onto his rear, but we won’t go there).

Stated differently, numerical identity means that if everything in the universe had a different number assigned to it (and only one number), the things that I have in mind share that number (meaning that they aren’t different things, but rather the same thing after all). Take for example the fetus that was in my mother’s uterus six months before I was born. Give it a number (let’s pick 498,178, 895, 659). Then look at me, sitting here typing this. What’s my number? It’s 498,178, 895, 659 – the same number as that fetus. The fetus has kept that number for more than 33 years, and now that fetus sits here, typing. I am therefore numerically identical with a fetus that once existed (of course what exists now is not a fetus but an adult).

So there you have it, the concept of numerical identity.

Glenn Peoples

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15 thoughts on “Nuts and Bolts 002: (numerical) Identity

  1. Glenn,
    I thought you were quite clear during your lecture on this topic. I love this idea simply because it’s something (without the technical terms) that I’ve argued for years.

    Might you play devils advocate for some of us and tell us arguments some people might use against it?

    From my non-educated view, I can’t see any.

  2. Hey folks, I have removed the spam filter and replaced it with a mechanism where posters need to type in a code displayed in an image. However, when I deleted spam karma I didn’t check the moderation queue first, and as a result at least one comment was lost. My apologies for this.

  3. Heh, that was mine. I just commented to Gene that one potential problem with the law of identity comes in the form of an internal critique of Christianity, focusing on the Trinity. The persons of the Trinity are all distinct from one another, yet are numerically identical because they are all one God.

    James Anderson has written an excellent examination of this problem, Paradox in Christian Theology, where he argues (I feel decisively) that although the paradox may be insoluble, it doesn’t represent a genuine contradiction. (My own view is similar to his.)

  4. Dominic,

    If you hold that each member of the Trinity is numerically identical to God (with the referent of ‘God’ held fixed) and you hold a classical logic of identity, then you either wind up with modalism or a contradiction. So I think you are right that an orthodox Trinitarian must either deny that each member of the Trinity is numerically to God (provided the referent of ‘God’ is held fixed) or give up a classical logic of identity. I (and many analytic Christian philosophers) opt for the former. Although some Christian philosophers do opt for the latter.

    I think that there are coherent, orthodox models of the Trinity that do not commit one to the claim that each member of the Trinity is identical to God. I think, to be orthodox, we do need to affirm that (in some sense) the Father is God, the Son is God and the Holy Spirit is God, but the ‘is’ of identity is not the only ‘is’ there is 🙂 (I would take the relevant ‘is’ here to be something like an ‘is’ of predication rather than an ‘is’ of identity).

  5. Using the above philosophy, I think it is easy to see how what is commonly called the “soul” need not be an immaterial substance inside someone, but rather, in accordance with some of the connotations of the biblical word “psukee,” the principle of self and identity that allows one to be perpetuated into eternity. The resurrected person is numerically identical to the person who died, even though God will have fashioned that person anew. But that wouldn’t be enough for eternal life to be meaningful; after all, a Buddhist might say that a reincarnated ant is numerically identical to the dog who died, but the ant has no memory of the dog’s existence, so reincarnation might as well not occur. So perhaps soul = numerical identity plus nonphysical aspects of a mortal person, including memory of the past life, personality, and self-awareness.

  6. I think you’re wrong about that Kenny, and I’d direct you to Anderson’s book for why. Models of the Trinity which deny numerical identity all tend to lead straight into heresy. I’d rather be left with an apparent contradiction than a heresy of that magnitude.

  7. Well Dominic, we\’ll have to agree to disagree for now, since doing otherwise would turn this into a discussion of the doctrine of the Trinity rather than a discussion of numerical identity. I do think that you were right to flag the doctrine of the Trinity, though, as a doctrine that is relevant to how Christian philosophers ought to think about the metaphysics of identity.

  8. In my view, we must not say that each individual member of the Trinity is numerically identical to God. This is the modalistic heresy. I’d say that the conjunction of the three persons is numerically identical to God.

  9. While the term numerical identity is likely due to logical identity A = A, there was confusion over the number aspect in at least 1 of your listeners.

    However the term does not describe identical in a pure number sense. Your example of labelling Glenn 498,178,895,659 uses a number but the label is not dependant on a number, any label would be useful (eg letters) as there is no ordering to the labels.

    Perhaps the term “nominal identity” would be more accurate.

    I am not disagreeing with your post and I realise the term is already defined.

  10. Yes, the confusion arose because the person mistook why the term “numerical” was being used. And yes, the label wouldn’t have to be a number, but that’s just how the term has come to be used. I don’t know if any label is perfect. For example “nominal” is usually associated with names, but my name can change. If someone misunderstands numerical identity they would/could also misunderstand nominal identity by thinking that any two people with the same name are nominally identical. So I guess the best we can hope for is to try to get people to understand the existing terminology.

  11. Glenn, these nuts-and-bolts pieces are wonderful! I’m a lay person with an interest in philosophy and apologetics and these pieces help put the abstruse arcana of philosophical jargon on the lower shelf where the common man can access them, though without oversimplification. Keep them coming!

    I’m new to your site, and am catching up on your podcast from the very first episode in a BerettaCast “marathon”, and I’m hooked! I learned of your site from Dee Dee Warren’s podcast (with whom I’m also doing a podcast marathon catch-up).

    I have a little puzzle that has occurred to me which involves the law of identity. If you’ll allow me to indulge here, this involves a thought experiment which has to do with computer building (but the abstract principle could be applied in other areas, for example with biological systems).

    Let’s say there’s a computer nerd named Maynard. Maynard buys a bunch of computer parts from TigerDirect. He assembles the parts, installs his bootleg copy of Windows XP (or a perfectly legal **free** copy of Ubuntu Linux, if you prefer) and he’s got himself a computer.

    Then six months later, after seeing a great bargain online, he decides to upgrade the system board, cpu, and memory. Soon, however, he fills up his tiny hard drive with mp3 files from philosophy podcasts, so he purchases and installs a bigger one. And he keeps upgrading the machine in this way until every single part of the computer has been replaced, including the chassis itself, as well as the monitor and other essential peripherals. But these are replaced one item at a time over the course of maybe two years.

    So, at the end of this process, is the PC he ends up with numerically identical with the one he started out with?

    But wait, there’s more…

    He frequently gets into spats with his wife because she always wants to use the computer while he is using it. They argue about it frequently, and he decides it would be better if she had her own computer. Then he has a brilliant idea: He still has all of the parts that were the cast-offs of upgrades from his computer. All of these parts are still functional, so why not reuse them? He then cobbles them back together into a working machine. In a sense, the old PC is kind-of “resurrected.” However, each part from this “resurrected” machine came off of the original PC a little at a time while the original machine maintained continuous identity as a complete unit. He puts these cast-off parts together, and now his wife can sit in the family room and play stupid simulation and mafia games on Facebook while Maynard reads theology web sites in peace.

    Which PC is numerically identical with the original one?

  12. Hi Jared

    That’s a good question actually. I was mulling it over some years ago when pondering the identity of resurrected persons. Another good example is a sailing ship. One by one, the planks are replaced, at a rate of one plank every two days. In a couple of years time, the whole ship has been replaced, and the remaining wood is used to build a replica, made of the old planks. or better yet, the planks are replaced one by one, not by ordinary planks, but planks made of green jelly! Is the resulting jelly ship really numerically identical to the original ship?

    May answer is yes, it is – and if you replaced the parts on your computer one at a time, the resulting computer with upgraded parts would be numerically identical with the original computer. It’s really no different than an animal body that ingests food and assimilates it to the body, with waste being discarded. I think that becoming integrated into an organism or a system makes something part of it. When you slot that new CPU in, bingo, it immediately becomes part of the computer. So all we really have is a complex object undergoing gradual change. It’s the relationship of the parts to each other that is sufficient to make them part of one thing.

    This means that if you take the leftover parts that were in the original computer and put them together for your wife, strange though it seems, although each of the parts is numerically identical to the original parts of the computer, this computer is not the original computer, just because of the history of their relationship to each other.

    That’s my take anyway.

  13. Thanks, Glen, that makes sense. Actually, though I posed it as a question (and indeed thought of it that way when I started typing), as I was typing it out, the process of expressing it in writing forced me to think it through some more and I was starting to come to a similar conclusion.

    Its interesting that you bring up resurrected bodies. In fact, that sort of association also didn’t occur to me until I was in the process of putting my thoughts into writing (hence the inclusion of the sentence likening the second PC to a “resurrected” one).

    Your response drove me to look something up in scripture I had recently heard referenced somewhere else: Paul writing on the resurrection in I Cor. 15:35-57. “But some man will say, How are the dead raised up? and with what body do they come?” There’s a lot there I’m still contemplating. (I love it how Paul berates us then for asking a “stupid question” [as the REB puts it], but then goes on for a couple dozen verses responding to it!)

    I am waiting with anticipation now for the next podcast episodes on “In Search of the Soul,” since I think this ties in with that discussion (i.e, on non-reductive physicalism,, then where does the numerical identity of the earthly body go until the resurrection?) You suggested you\’d have four episodes on that, but so far we are at only two (I think… unless they’re lurking there and I missed them somehow).

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