3 paradoxes that make mathematicians and philosophers lose sleep

3 paradoxes that make mathematicians and philosophers lose sleep


This sentence is false.

This is one of the most popular and illustrative paradoxes: if it is really false, what the sentence enunciates is true, but if the falsehood enunciated is real, the sentence can not be false.

Paradox comes from the words in Latin and Greek that mean ‘the opposite of common opinion  and is, according to the dictionary of the Royal Academy …

2. f. Fact or expression seemingly contrary to logic .

3. f. Ret. Use of expressions or phrases that contain an apparent contradiction among themselves, as in ” look at the miser, in his wealth, poor

There are several types, but what they usually have in common is that they make us stop to think, even for a moment, like “to get there fast, nothing better than to go slowly”.

But others have accompanied us for years, sometimes centuries, and at times have driven important advances in science, philosophy and mathematics.

Is it still your ship?

Ancient greek boat
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Image captionHow much can you change before losing your identity?

Change and identity. This is what the historian, biographer and Greek moral philosopher Plutarch (46 or 50-c.120) has been thinking about for almost 2,000 years with the paradox of Theseus , the mythical founding king of Athens, son of Etra and Eseo, or according to others legends, by Poseidon.

” The ship in which Theseus and the youth of Athens returned from Crete had thirty oars, and was preserved by the Athenians even up to the time of Demetrius of Phalerus , as they removed the old planks as they broke down and introduced new wood and more resistant in its place, so much so that this ship became a permanent example among philosophers, for the logical question of things that grow, one side holds that the ship remains the same, and the other says no . ”

If the ship was preserved by the Athenians until the time of Demetrius of Phalerus, that would mean about 300 years.

With so many replacements, was the ship the same?

And it went beyond. If with the old wood they built another identical boat, which of the two would be the original : the one with the original boards or the one that has been restored?

The movement does not exist

To go anywhere, you have to travel half the distance first, then half the distance you have to travel, then half the distance that you lack, and so on to infinity, so you ‘ll never get there .


Image caption Could it be that the movement is an illusion? This is how this Miranche paradox illustrates in Wikipedia.

This is one of the series of paradoxes of the movement of the Greek philosopher Zeno of Elea created to demonstrate that the Universe is singular and that change, including movement, is impossible , as his teacher Parmenides argued.

If it seems absurd, you are not alone: it was rejected for years.

However, mathematics offered a formal solution in the nineteenth century that was to accept that 1/2 + 1/4 + 1/8 + 1/16 … add 1 .

Although that theoretical solution served certain purposes, it did not respond to what was happening in reality: how something can reach its destination.

That, intuitively understood because we experience it daily, is more complex and to solve it we had to wait until the twentieth century to use theories that show that matter, time and space are not infinitely divisible.

The one that made mathematics stagger

Now that we have warmed up, let’s talk about a paradox that shook the mathematical community at the beginning of the 20th century, including the one who formulated it: the British philosopher, mathematician, logician and writer who won the Nobel Prize for Literature Bertrand Russell.

Bertrand Russell
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Image caption “I would never die for my beliefs because I could be wrong” was one of the many bright things Russell said.

Russell was one of those who were trying to drive logicism, the philosophical thesis that says that mathematics, or most of it, can be reduced to logic.

That project included the Cantor-Frege set theory at its base. Both the German Georg Cantor and his compatriot Gotlob Frege assumed that every predicate defined a set . Thus, the predicate “be of gold” defines the set of all things that are gold.

It sounds more than obvious.

But, Russell discovered that there was a particular predicate that contradicted the theory: “not belong to yourself”

That is the paradox of Russell, and it is complex but luckily we come across one of the clearest explanations, created by M. Carmen Márquez García for a course of SAEM Thales, Distance Training through the Internet, that appears on this site web .


Let us suppose that a well-known expert in works of art decides to classify the paintings of the world in one of two mutually exclusive categories.

A category, of very few paintings, consists of all the paintings that include an image of themselves in the scene presented on the canvas . For example, we could paint a painting, titled “Interior”, of a room and its furniture-hanging in movement, a statue, a grand piano-that includes, hanging above the piano, a small painting of the painting “Interior”. Thus, our canvas would include an image of itself.

The other, much more current, category would consist of all the pictures that do not include an image of themselves . We will call these paintings ” Russell’s paintings “. The Mona Lisa, for example, is a painting by Russell because he does not have inside it a small painting of the Mona Lisa.

Let’s also suppose that our art expert assembles a huge exhibition that includes all Russell’s paintings of the world. After immense efforts, he gathers them and hangs them in a huge room.

Many pictures
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Image captionPictures everywhere in a huge gallery.

Proud of his feat, the expert instructs an artist to paint a picture of the room and its contents.

When the painting is finished, the artist titled it, with all property, “All the paintings of the Russell of the world”.

The gallerist examines the painting carefully and discovers a small flaw: on the canvas, next to the painting of the Mona Lisa there is a representation of “All the paintings of Russell of the world”. This means that “All the paintings of the world” is a picture that includes an image of itself, and therefore, is not a painting of Russell . Consequently, it does not belong to the exhibition and it certainly should not be hanging on the walls.

The expert asks the artist to erase the small representation.

The artist erases it and returns to show the picture to the expert. After examining it, he realizes that there is a new problem: the painting “All the paintings of Russell of the world” now does not include an image of itself and, therefore, it is a painting of Russell that belongs to the exhibition. Consequently, it must be painted as hanging from some part of the walls, lest the work not include all of Russell’s paintings .

The expert calls the artist again and asks him to retouch with a small image the “All the paintings of Russell of the world”.

But once the image has been added, we are again at the beginning of the story. The image must be erased, after which it must be painted, and then deleted, and so on.

Eventually the artist and the expert will realize that something is not working: they have run into Russell’s paradox .


Bearing in mind that what Russell was trying to do was reduce mathematics to logic and what he had discovered was a crack in the foundations of science, his reaction is not surprising.

” I felt about these contradictions what a fervent Catholic must feel about unworthy popes .”

But there was no turning back: the discovered can not be covered again.

Serious with nose of payazo
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Image captionArt is a lie that allows us to realize the truth, said Picasso; Of all the books in the world that should be banned before any other is the catalog of forbidden books, said Lichtenberg. Life is full of paradoxes.

Although the matter left them indifferent to mathematicians and it seemed that they did not deserve so much reflection, others devoted much of the intellectual work of the first half of the 20th century to overcoming Russell’s paradox … until it was decided that a group that contain yourself really is not a set .

The solution did not please many, not even Russell.

M. Carmen Márquez García says that “the intellectual tension and its disheartening conclusion took a terrible toll “.

Russell would recall how after this he “departed from mathematical logic with a kind of nausea.”

He thought about suicide again, although he decided not to do it because, he observed, he would surely live to regret it .

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