Ship of Theseus
The ship of Theseus, also known as Theseus's paradox, is a paradox that raises the question of whether an object which has had all its components replaced remains fundamentally the same object. The paradox is most notably recorded by Plutarch in Life of Theseus from the late 1st century. Plutarch asked whether a ship which was restored by replacing all and every of its wooden parts, remained the same ship.
The paradox had been discussed by more ancient philosophers such as Heraclitus, Socrates, and Plato prior to Plutarch's writings; and more recently by Thomas Hobbes and John Locke. There are several variants, notably "grandfather's axe", and in the UK "Trigger's Broom". This thought experiment is "a model for the philosophers"; some say, "it remained the same," some saying, "it did not remain the same".
Variations of the paradox 
Ancient philosophy 
"The ship wherein Theseus and the youth of Athens returned [from Crete] had thirty oars, and was preserved by the Athenians down even to the time of Demetrius Phalereus, for they took away the old planks as they decayed, putting in new and stronger timber in their place, insomuch that this ship became a standing example among the philosophers, for the logical question of things that grow; one side holding that the ship remained the same, and the other contending that it was not the same."—Plutarch, Theseus
Plutarch thus questions whether the ship would remain the same if it were entirely replaced, piece by piece. Centuries later, the philosopher Thomas Hobbes introduced a further puzzle, wondering: what would happen if the original planks were gathered up after they were replaced, and used to build a second ship. Which ship, if either, is the original Ship of Theseus?
Another early variation involves a scenario in which Socrates and Plato exchange the parts of their carriages one by one until, finally, Socrates's carriage is made up of all the parts of Plato's original carriage and vice versa. The question is presented if or when they exchanged their carriages.
Enlightenment era 
John Locke proposed a scenario regarding a favorite sock that develops a hole. He pondered whether the sock would still be the same after a patch was applied to the hole, and if it would be the same sock, would it still be the same sock after a second patch was applied until all of the material of the original sock has been replaced with patches.
George Washington's axe (sometimes "my grandfather's axe") is the subject of an apocryphal story of unknown origin in which the famous artifact is "still George Washington's axe" despite having had both its head and handle replaced.
...as in the case of the owner of George Washington's axe which has three times had its handle replaced and twice had its head replaced!—Ray Broadus Browne, Objects of Special Devotion: Fetishism in Popular Culture, p. 134
The French equivalent is the story of Jeannot's knife, where the eponymous knife has had its blade changed fifteen times and its handle fifteen times, but is still the same knife. In some Spanish-speaking countries, Jeannot's knife is present as a proverb, though referred to simply as "the family knife". The principle, however, remains the same.
In the 1872 story "Dr. Ox's Experiment" by Jules Verne there is a reference to Jeannot's knife apropos of the van Tricasse's family. In this family, since 1340, each time one of the spouses died the other remarried with someone younger, who took the family name. Thus the family can be said to have been a single marriage lasting through centuries, rather than a series of generations. A similar concept, but involving more than two persons at any given time, is described in some detail in Robert Heinlein's novel The Moon Is a Harsh Mistress as a line marriage.
Modern examples 
Writing for ArtReview, Sam Jacob noted that Sugababes, one of the most successful all-female British bands of the 21st century, "were formed in 1998 [..] but one by one they left, till by September 2009 none of the founders remained in the band; each had been replaced by another member, just like the planks of Theseus’s boat." The three original members reformed in 2011 under the name Mutya Keisha Siobhan, with the "original" Sugababes still in existence.
In literature 
In The Wonderful Wizard of Oz (1900) by L. Frank Baum, a lumberjack's cursed axe chopped all his limbs one by one, and each time a limb was cut off, a smith made him a mechanical one, finally making him a torso and a head, thus turning him into the Tin Woodman, an entirely mechanical being, albeit possessing the consciousness of the lumberjack he once was.
Proposed resolutions 
The Greek philosopher Heraclitus attempted to solve the paradox by introducing the idea of a river where water replenishes it. Arius Didymus quoted him as saying "upon those who step into the same rivers, different and again different waters flow". Plutarch disputed Heraclitus' claim about stepping twice into the same river, citing that it cannot be done because "it scatters and again comes together, and approaches and recedes".
Aristotle's causes 
According to the philosophical system of Aristotle and his followers, there are four causes or reasons that describe a thing; these causes can be analyzed to get to a solution to the paradox. The formal cause or form is the design of a thing, while the material cause is the matter that the thing is made of. The "what-it-is" of a thing, according to Aristotle, is its formal cause; so the Ship of Theseus is the same ship, because the formal cause, or design, does not change, even though the matter used to construct it may vary with time. In the same manner, for Heraclitus's paradox, a river has the same formal cause, although the material cause (the particular water in it) changes with time, and likewise for the person who steps in the river.
Another of Aristotle's causes is the end or final cause, which is the intended purpose of a thing. The Ship of Theseus would have the same ends, those being, mythically, transporting Theseus, and politically, convincing the Athenians that Theseus was once a living person, even though its material cause would change with time. The efficient cause is how and by whom a thing is made, for example, how artisans fabricate and assemble something; in the case of the Ship of Theseus, the workers who built the ship in the first place could have used the same tools and techniques to replace the planks in the ship.
Definitions of "the same" 
One common argument found in the philosophical literature is that in the case of Heraclitus' river one is tripped up by two different definitions of "the same". In one sense things can be "qualitatively identical", by sharing some properties. In another sense they might be "numerically identical" by being "one". As an example, consider two different marbles that look identical. They would be qualitatively, but not numerically, identical. A marble can be numerically identical only to itself.
Note that some languages differentiate between these two forms of identity. In German, for example, "gleich" ("equal") and "selbst" ("self-same") are the pertinent terms, respectively. At least in formal speech, the former refers to qualitative identity (e.g. die gleiche Murmel, "the same[qualitative] marble") and the latter to numerical identity (e.g. dieselbe Murmel, "the same[numerical] marble"). Colloquially, the terms are sometimes used interchangeably however.
One solution to this paradox may come from the concept of four-dimensionalism. Ted Sider and others have proposed that these problems can be solved by considering all things as four-dimensional objects. An object is a spatially extended three-dimensional thing that also extends across the fourth dimension of time. This four-dimensional object is made up of three-dimensional time-slices. These are spatially extended things that exist only at individual points in time. An object is made up of a series of causally related time-slices. All time-slices are numerically identical to themselves. And the whole aggregate of time-slices, namely the four-dimensional object, is also numerically identical with itself. But the individual time-slices can have qualities that differ from each other.
The problem with the river is solved by saying that at each point in time, the river has different properties. Thus the various three-dimensional time-slices of the river have different properties from each other. But the entire aggregate of river time-slices, namely the whole river as it exists across time, is identical with itself. So one can never step into the same river time-slice twice, but one can step into the same (four-dimensional) river twice.
A seeming difficulty with this is that in special relativity there is not a unique "correct" way to make these slices — it is not meaningful to speak of a "point in time" extended in space. However, this does not prove to be a problem: any way of slicing will do (including no 'slicing' at all), provided that the boundary of the object changes in a fashion which can be agreed upon by observers in all reference frames. Special relativity still ensures that "you can never step into the same river time-slice twice," because even with the ability to shift around which way spacetime is sliced, one is still moving in a timelike fashion.
See also 
- Identity and change
- Mereological essentialism
- Neurath's boat
- Sorites paradox
- USS Constellation (1854)
- Vehicle restoration
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