Paradox & Resolution I: Theseus’ New Planks

Adam Mehdi
9 min readSep 3, 2021


[source 8]


Does a ship remain the same ship if all its planks are replaced with new planks gradually? Yes. How about if the renovation was sudden? No. The stance may be puzzling, but it is consistent. The locus of identity is in our brains, not the external world. We define the identity of things based on the mental models we use to represent them. In clarifying the origin of identity, we change the question from “Is the ship the same in reality?” to “Do we use the same mental model to represent the ship?”. The new question proves to have teeth.


This may be the most boring article in the series due to its subject: We think more about ships and objects than people. However, don’t let that fool you; the actual subject matter is not ships but the learning process, mental representations and identity. It is a sort of applied preliminary to the subsequent Paradox & Refutations articles, though it’s complete as a philosophical article in its own right. I’ve thought much about these ideas, so you might find the arguments surprising, if reasonable.


Theseus has an old yet seaworthy ship (`original`), and he wants to renovate it into a better one. He hires a maritime company. They order 1,000 new planks and proceed to replace every plank of the ship one-by-one, until the ship has none of the original planks left (`renovated`). Then, the company realizes the old planks are still strong enough to support a new ship, so they construct a ship identical in form to Theseus’ original using those old planks (`reconstructed`).

Which one is the `original` ship, the `renovated` or `reconstructed` ship, both, or neither?

A pared-down version of the paradox conveyed in a superior format. To answer the lady Ariadne, yes, it does remain the Ship of Theseus. [2]


Before we tussle with the real explanation, let me cast light on the facile answer: Neither. “No man ever steps in the same river twice, for it is not the same river and he is not the same man,” as Heraclitus verbigerated. Even an object that does not change whatsoever cannot be identical to itself at two times; they have different locations in spacetime and are therefore “different”. Yeah, yeah. True… but hardly more than trivial. This is to proffer an answer without grasping the lesson of the paradox. Let’s get at the lesson.

Most think the `renovated` ship is the `original`. The reason for this has to do with the learning process and mental models. We understand the world only by creating models of objects in our brains. And there is a difference between the models in our heads and the objects in the world.[3] The strangeness of the Ship of Theseus is born out by exploiting this dichotomy. Here’s how.

The computer at which you are staring can, like our brains, simulate objects. If it is simulating a ship, the properties of the ship correspond to specific movements in the computer. If the ship is on open sea, the wind barraging its sails and waves buffeting its hull, then the computer’s movements corresponding with the sails and hull interface with the movements denoting wind and waves. And the object need not be static; the computer can include the ship’s decay with time and trauma in its simulation. This is what we mean by mental model. There is no faithful simulacrum of a ship inside the computer, only movements that correspond to properties. So it is for our brains: We represent a ship and its properties in our brains, via explanations about the ship — how it withstands waves, how the sails absorb the wind, how it changes with time.

However, in many ways a ship in the world changes unexpectedly, and when that happens the reality of the ship diverges from our mental models. To make the two converge again, we update our mental models with feedback from the ship, by finding conflicts between our model of the ship and observation and guessing a new explanation of the ship that reconciles them.[1] These updates must be discrete, however, while the ship’s unexpected changes are effectively continuous.

We update our mental model of the `original` ship as it appropriates new planks. Everytime we find an error between our model of the sea-worn ship and the new stalwart planks, continue to update our mental model. The question is when the ship is so different that we need to use a different mental model to represent it… but as long as the changes are gradual, we have the capacity to update the same model, over and again. So we end up in a situation where the `original` ship is radically different — entirely new materials — and yet we may still consider it, the `renovated` ship, the same ship. The `reconstructed` ship is more alike to the `original` in even more ways, identical in material and form, but since it was constructed de novo, it seems we construct a new mental model for it.

Identity has to do with the mental models in our heads, not the objects in the world. And whether our mental model of the object is the same depends on the gradualness of a change. If an alien spaceship spirited our `original` ship away, renovated it in a flash and dropped it back on terrestrial soil, would we consider this `renovated` ship the same? I think not. There is a discontinuity between moments — pre-alien and post-alien — so we may mash together a wholly different mental model, making different the identity of the ship at two different times.


The dubious reader may cast a (warranted) skeptical eye on the claim that `original` equals `renovated` with gradual replacement and `original` equals neither with sudden renovation for everyone. Surely there are some who regard the sudden renovation as not enough of a discontinuity to change the ship’s identity, and vice versa for the gradual. My response: Yes. I’m all too ready to bite the bullet on that one. Indeed, some people may disagree that the alien renovation is discontinuous enough to warrant a change of identity. Identity is relative. That’s just the nature of identity, since it — the unique mental models we use to represent physical and abstract objects in the world — is a property of us, not the physical world. More on the implications of this later.

Then, the careful reader may notice that I am still presupposing identity, referring to mental models as “the same” or “wholly different”. But what makes them the same or different, or as I phrased it, “so different that we need to use different mental models to represent it”? This question is occasion to clarify my intent. It is an analytic question, but I argued in broader brushstrokes; my intent here is to provide a framework with which to unfurl identity puzzles (and by turns, in next articles, to use it to tackle questions about what it means to live life). In my view, knowing specific criteria of identity is less important than seeing the origin of identity (in our brains, not out there), because, philosophically speaking, the importance of identity is specious. For legal battles and societal order, though, criteria for identity are relevant. So, see the footnote only if you are interested in specific criteria.[5]


Objects in the world and our mental models of them are separate. Identity depends only on our mental models. The explanation seems simple and straightforward, so why was the Ship of Theseus puzzling? I think that is because we are accustomed to conflating mental models and objects in reality.

Most think we see like a video camera, recording reality directly, but most would be wrong. Light comes through our retinas upside-down; our brain has to flip it. And of the light, our brains delete all information not in the visible light spectrum. In fact, we never interact with reality directly; we only ever interface with our mental models (which in turn is in constant feedback with reality). Experience is a virtual reality of sorts, created by our brain using incomplete and flawed sensory cues. We only substantiate that virtual reality with explanations and expectations.[6] So, it is easy to mistake what we see — our mental models — for reality, and that’s why the Ship of Theseus is puzzling.

But if identity is a feature of our mental models, it is not of physical reality. That’s why I have no trouble accepting that identity is relative, not because identity is an abstraction separate from physical reality (abstractions can be real if our best explanations invoke them, like the electromagnetic force [7]) but because they figure in no good philosophical explanations. It is a concept that has trouble accounting for gradual change in objects. Nor does identity bear up to scrutiny when applied to ourselves. I will show that next article (P&R II: Vaporizing and Duplicating Teleporters), when we will consider juicier paradoxes — thought experiments involving our own identities — and come away with a new, more logically sound explanation about what makes us us.

And yet we do think in terms of identity — we name things, we give ourselves names, and we stake objects with our name in legal battles. It is a useful concept that we invoke in our explanations of everyday life. However, in seeing the concept squirm in pain under rational judgment, we have to confront the possibility that the use of ‘identity’ might be just a useful heuristic, no more significant than the reasons we group our code into functions and classes instead of interacting directly with the assembly language.


[1] Our learning process — the process of updating mental models to accord with observed conflicts — parallels that of machine learning models. They consume an input and predict an output (the process that reveals the model’s representation). It compares the predicted output with the real label (finding a conflict between mental model and observation). It then updates itself to make its predictions closer to the real label (guessing new explanations to reconcile the conflict). However, the difference between the two processes is profound: The model updates itself mechanically based on the gradient of a function while we update our mental models creatively. Our knowledge is explanatory; it is predictive. Our knowledge is as good as our thought and can reach into other domains; the machine learning model’s “knowledge” is as good as its data.

[2] Musk, Elon, @elonmusk. (June 12, 2021). Esoteric meme of the day. Twitter.

[3] Following Kant and the metaphysical realists, I am drawing a distinction between things-in-themselves and ideas-of-things. This is quite a fashionable distinction to make nowadays, as it seems to be rising to prominence in theories of cognitive science (See Donald Hoffman’s The Case Against Reality).

[4] I call upon concepts from Popperian epistemology, that we learn by guessing new explanations and reject the ones conflicting with observation. Here, I am suggesting that, to update our conception of the ship, we create new mental models of the ships (which are really just explanations about how the ship behaves) and keep the one with the least error to how we see the ship behaving.

[5] Here’s one appealing criterion for identity from a classic paper. A ship remains the same ship if and only if the form of the ship is preserved over a continuous space-time path. By this criterion, `original` is the same ship as `renovated`. Continuous means here that once an old plank is replaced by a new one, the old plank ceases being and the new one becomes a part of the ship. It can be a vague criterion, but that’s what we get when we ask for criteria for identity. From Smart, B. (1972). “How to Reidentify the Ship of Theseus.” Analysis, 32(5), 145–148.

[6] Deutsch, David. “Why It’s Good to Be Wrong.” 2013. Nautilus Magazine.

[7] This is called Dr. Johnson’s criterion, or the criterion for reality, in David Deutsch, The Fabric of Reality. (Allan Lane, 1997.)

[8] An image created by a neural net optimized for trees. See Gabriel Goh et al. “Multimodal Neurons in Artificial Neural Networks.” (Distill Pub, 2021.)



Adam Mehdi

Thinking about AI & epistemology. Researching CV & ML as published Assistant Researcher. Studying CS @ Columbia Engineering.