Mathematical Paradoxes

As a continuation from my last post, I have decided to discuss paradoxes in maths. The list of mathematical paradoxes is extensive so I will only touch on a few that hopefully you will find interesting! First of all, let’s cover what a paradox actually is; a paradox is a statement that seems self-contradictory or absurd but in reality expresses a possible truth” [Dictionary.com]. 

Zeno’s Paradoxes

The Greek philosopher Zeno of Elea (around 490-430 BC) devised four paradoxes, which deal with the counter-intuitive aspects of continuous space and time.

#1: Dichotomy Paradox

Before an object travels a distance d, it must travel a distance d/2. However, in order to travel a distance d/2, an object must travel a distance of d/4. This sequence could go on forever, and therefore it appears that this distance cannot be traveled.

This paradox has been resolved, however, using calculus and the proof that a geometric series can converge. The resolution highlights that the infinite number of half-steps is balanced by the increasingly short amount of time needed to travel this distance.

File:Zeno Dichotomy Paradox.png

Source: Wikipedia

#2: Achilles and the Tortoise Paradox

In this paradox, Achilles is in a footrace with the tortoise. The tortoise is given a 100 metre head start. Supposing that each racer has a constant speed, then after some time Achilles will have run 100 meters, bringing him to the tortoises starting point. However, during this time the tortoise has moved some distance. This will continue, meaning that there is an infinite amount of points that Achilles must reach where the tortoise has already been and thus he can never overtake the tortoise. However, this is obviously fallacious and has been solved in a similar way to the dichotomy paradox.

File:Zeno Achilles Paradox.png

Source: Wikipedia

#3: Arrow Paradox

An arrow in flight has an instantaneous position at a given time. However, at that instant it is not distinguishable from a motionless arrow in the same position. So the question is, how can the motion of the arrow be perceived?

#4: Stadium Paradox

As stated by Aristotle:

“concerning the two rows of bodies, each row being composed of an equal number of bodies of equal size, passing each other on a race-course as they proceed with equal velocity in opposite directions, the one row originally occupying the space between the goal and the middle point of the course and the other that between the middle point and the starting-post. This…involves the conclusion that half a given time is equal to double that time.

Source: Wikipedia

Gabriel’s Horn

Gabriel’s Horn, studied by Italian physicist and mathematician Evangelista Torricelli in the 17th century, is a geometric figure that has infinite surface area but has finite volume.

Source: qc.edu.hk

When the properties of it were first discovered, it was considered to be paradoxical. It gave rise to the Painter’s Paradox which says that since the Horn has finite volume, but infinite surface area, it could, in principle, be filled with a finite quantity of paint, and yet the paint would not cover the inner surface of the Horn.

However, real paint is made out of molecules. As, after a certain point the radius of the horn is less than the radius of the paint molecules, the molecules cannot pass beyond that point, meaning that the paradoxes collapses when you bring it into the real world.

Russell’s Paradox

Bertrand Russel was a very famous mathematician and philosopher in the 20th century. This paradox was very significant when it was discovered in 1902, as during this time some mathematicians were trying to put maths on a foundation of set theory, however this paradox showed there might be a contradiction within the foundations of set theory.

The paradox proposes a set of all sets that are not members of themselves. Is this set a member of itself? If it is, it isn’t. But if it isn’t, it is. This is an example of a vicious circle paradox.

Hope you enjoyed this post! Let me know if you know any other interesting paradoxes! M x

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