Ship of Theseus
The ship of Theseus, also known as Theseus's paradox, is a thought experiment 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 each and every one 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". 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
"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, in so much 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.
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, and a third, etc., 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[which?] 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 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.
Writing for ArtReview, Sam Jacob noted that Sugababes, a British band, "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 reunited in 2011 under the name Mutya Keisha Siobhan, with the "original" Sugababes still in existence.
In the popular BBC sitcom Only Fools and Horses, Trigger (a roadsweeper) declares he has won an award for keeping the same broom for 20 years — "17 new heads and 14 new handles". This has become known as the "Trigger's Broom" paradox.
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".
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. die selbe Murmel, "the same [numerical] marble"). Colloquially, the terms are sometimes used interchangeably, however.
Ted Sider and others have proposed that considering objects to extend across time as four-dimensional causal series of three-dimensional 'time slices' could solve the Ship of Theseus problem because, in taking such an approach, each time-slice and all four dimensional objects remain numerically identical to themselves while allowing individual time-slices to differ from each other. The aforementioned river, therefore, comprises different three-dimensional time-slices of itself while remaining numerically identical to itself across time; one can never step into the same river time-slice twice, but one can step into the same (four-dimensional) river twice.
Although no unique "correct" way to make these slices exists in special relativity — speaking of a "point in time" extended in space is meaningless — any way of slicing will do (including no 'slicing' at all) if observers in all reference frames see the boundary of the object change in the same way. Special relativity still ensures that "you can never step into the same river time-slice twice" because even with the ability to change how spacetime is sliced, one is still moving in a timelike fashion.
Essential Element of the thing as the thing per se
Another resolution of this paradox is to designate some element of the thing as the thing itself. For example, in the case of firearms, the lower receiver is considered to be the firearm in and of itself. Thus, every other part can be replaced and it would remain the same object; likewise, if only the lower receiver is replaced, the firearm is a different object, even if the tens of other parts remain the same.
A similar concept would be present in most manufactured goods by looking to what the essential or key part is as being the item itself. In the case of the axe, the axe head would be the essential element; in the case of an automobile, the chassis is the key component.
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