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Posts Tagged ‘paradox

“One must not think slightingly of the paradoxical”*…



The Building of the Argo, by Antoon Derkinderen, c. 1901. Rijksmuseum.


The thought problem known as the ship of Theseus first appears in Plutarch’s Lives, a series of biographies written in the first century. In one vignette, Theseus, founder-hero of Athens, returns victorious from Crete on a ship that the Athenians went on to preserve.

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…

Of course, the conundrum of how things change and stay the same has been with us a lot longer than Plutarch. Plato, and even pre-Socratics like Heraclitus, dealt in similar questions. “You can’t step in the same river twice,” a sentiment found on inspirational Instagram accounts, is often attributed to Heraclitus. His actual words—“Upon those who step into the same rivers, different and again different waters flow”—might not be the best Instagram fodder but, figuratively at least, provided the waters that the ship of Theseus later sailed.

Two thousand years later the ship is still bobbing along, though some of its parts have been replaced. Now known colloquially as Theseus’ paradox, in the U.S. the idea sometimes appears as “Washington’s ax.” While not as ancient as the six-thousand-year-old stone ax discovered last year at George Washington’s estate, the age-old question remains: If Washington’s ax were to have its handle and blade replaced, would it still be the same ax? The same has been asked of a motley assortment of items around the world. In Hungary, for example, there is a similar fable involving the statesman Kossuth Lajos’ knife, while in France it’s called Jeannot’s knife.

This knife, that knife, Washington’s ax—there’s even a “Lincoln’s ax.” We don’t know where these stories originated. They likely arose spontaneously and had nothing to do with the ancient Greeks and their philosophical conundrums. The only thing uniting these bits of folklore is that the same question was asked: Does a thing remain the same after all its parts are replaced? In the millennia since the ship of Theseus set sail, some notions that bear its name have less in common with the original than do the fables of random axes and knives, while other frames for this same question threaten to replace the original entirely.

One such version of this idea is attributed to Enlightenment philosopher John Locke, proffering his sock as an example. An exhibit called Locke’s Socks at Pace University’s now-defunct Museum of Philosophy serves to demonstrate. On one wall, six socks were hung: the first a cotton sports sock, the last made only of patches. A museum guide, according to a New York Times write-up, asked a room full of schoolchildren, “Assume the six socks represent a person’s sock over time. Can we say that a sock which is finally all patches, with none of the original material, is the same sock?”

The question could be asked of Theseus’ paradox itself. Can it be said that a paradox about a ship remains the same if the ship is replaced with a knife or a sock? Have we lost anything from Theseus’ paradox if instead we start calling it “the Locke’s Sock paradox”?…

Is a paradox still the same after its parts have been replaced?  A consideration: “Restoring the Ship of Theseus.”

* Soren Kierkegaard


As we contemplate change, we might spare a reasoned thought for the Enlightenment giant (and sock darner) John Locke; the physician and philosopher died on this date in 1704.  An intellectual descendant of Francis Bacon, Locke was among the first empiricists. He spent over 20 years developing the ideas he published in his most significant work, Essay Concerning Human Understanding (1690), an analysis of the nature of human reason which promoted experimentation as the basis of knowledge.  Locke established “primary qualities” (e.g., solidity, extension, number) as distinct from “secondary qualities” (sensuous attributes like color or sound). He recognized that science is made possible when the primary qualities, as apprehended, create ideas that faithfully represent reality.

Locke is, of course, also well-remembered as a key developer (with Hobbes, and later Rousseau) of the concept of the Social Contract.  Locke’s theory of “natural rights” influenced Voltaire and Rosseau– and formed the intellectual basis of the U.S. Declaration of Independence.

220px-John_Locke source



Written by LW

October 28, 2019 at 1:01 am

“How wonderful that we have met with a paradox. Now we have some hope of making progress”*…


Consider the simple function Y=1/X:

Take one half and rotate it around X.

It creates the shape you see at the top of this post, known as “Torricelli’s Trumpet” for its discoverer, the 17th century mathematician Evangelista Torricelli.  It’s noteworthy for its peculiar topographical qualities:  while both the volume and the surface area can be calculated, and the volume is a finite number, the surface area is Infinite.  That’s to say that, while one can fill that three dimensional shape with a calculable quantity of paint, one cannot coat the exterior surface, as it would require an infinite amount of paint…  (Supporting math, here.)

(The figure is also known as “Gabriel’s Horn,” a reference to the Archangel Gabriel, who blows his horn to announce Judgment Day– an association of the divine, or infinite, with the finite.)

This contribution (from Pablo Ramos) is just one of the fascinating answers to the question of Quora: “What are the weirdest science paradoxes that are mathematically true but counter-intuitive?

* Niels Bohr


As we rotate our minds around x, we might send post-industrial birthday greetings to Daniel Bell; he was born on this date in 1919.  Bell spent the first twenty years of his adult life as a journalist, exploring sociological issues; in 1960, on the strength of a book he’d written– The End of Ideology: On the Exhaustion of Political Ideas in the Fifties— he was awarded a PhD by Columbia University, where he taught briefly before moving for the rest of his career to Harvard.  One of the leading intellectuals of the Post-War era, Bell is best known for his contributions to the study of “post-industrialism,” and for his acute unpacking of the interactions among science, technology and politics.




Suppose there is a town with just one male barber; and that every man in the town keeps himself clean-shaven: some by shaving themselves, some by attending the barber. It seems reasonable to imagine that the barber obeys the following rule: He shaves all and only those men in town who do not shave themselves. Under this scenario, we can ask the following question: Does the barber shave himself?

From Epimenides’ Paradox to the Omnipotence Paradox, more fun-with-logic at “Brain Twisting Paradoxes.”

As we return to first principles, we might wish a carefully-reasoned Joyeux Anniversaire to Félix-Édouard-Justin-Émile Borel, a mathematician and pioneer of measure theory and its application to probability theory; he was born in Saint-Affrique on this date in 1871.  Borel is perhaps best remembered by (if not for) his thought experiment demonstrating that a monkey hitting keys at random on a typewriter keyboard will– with absolute certainty– eventually type every book in the Bibliothèque Nationale (or, as oft repeated, every play in the works of Shakespeare, or…)– that is, the infinite monkey theorem.

Borel (image source)


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