Chapters

With all these Arabic numbers and strange letters, its very easy to get confused in maths class! Mathematics is the greatest common divisor (in basic arithmetic) of the school curriculum.

That said, you probably will have at some point envied all those little geniuses who see maths problems as a fun game and can rattle off the digits of pi.

To tackle the subject of maths, let us try to follow in their footsteps, to discover mathematical paradoxes, and travel between arithmetic, trigonometry and probability.

You will learn about the culture of maths, you will be won over and, in a short time, you will become your own maths teacher!

Take up helpful math lessons here on Superprof.

## Logical Paradox: A General Definition

You don’t need a Master's degree or be a mathematical genius to get a grasp of mathematics. It's not as complex as pondering philosophical statements - anyone can learn maths!

Does it all seem a bit abstract? The term *paradox* refers to “a statement or proposition which, despite sound reasoning from acceptable premises, leads to a conclusion that seems logically unacceptable or self-contradictory.”

The study of these phenomena is worth a whole class in itself!

There are many paradoxes in the natural world still to be resolved. But there are many others, now understood and even more interestingly, can be useful to our understanding of the world.

Some may be related to physics and chemistry, others to science and technology in general. Mathematical problems and paradoxes fascinate mathematics lovers. A subject just as fascinating as Pi.

## False Paradoxes

### The Paradox of Achilles and the Tortoise

Upon first reading, this would seem an easy paradox to explain away, solving it mathematically however, is another story. If you're able to do so, your achievements are destined for the International Mathematical Olympiad (IMO), the World Championship Mathematics Competition.

The Paradox is a retelling of a fable we're all familiar with, the hare and the tortoise. Back in the 5th century AD, the Greek Philosopher, Zeno of Elea (490 BC 430 BC) proposed that if you gave a turtle a head start in a race against the Trojan War hero, Achilles, he would never be able to overtake the tortoise. To overtake the tortoise Achilles must first catch up it, but every time he caught the tortoise, it would create another gap. No matter how hard Achilles tries to catch up, the tortoise will always open up a new gap, no matter how finite.

Even though this assertion seems completely ludicrous it is rather difficult to explain. The answer lies in how we perceive space, time and motion as well as notions of infinity.

The Missing Dollar riddle falls into the same category of informal fallacy, but it is part of the timeless mathematical puzzles (besides being a way of hiking up the bill ...). It’s great for revising your logic skills!

### The Paradox of the Missing Square

No, it's not a Chinese puzzle! It's a geometry crash course in the land of the absurd.

Simply put, the paradox of the missing square is a logical mathematical hypothesis, that ultimately relies on a visual illusion, thus leading us to an incorrect conclusion.

In constructing a triangle with other geometric shapes, the shapes can be rearranged to create another triangle with the same height and width but an additional mysterious area. So what is going on?

The answer is quite simply... neither triangle is a "true" triangle. There exists a slight curve along the hypotenuse of the constructed shape that the human eye perceives as a triangle. The small empty space is in fact only the result of a small deformation of the perfect triangle with its slightly rounded edges. You don’t need your maths tutor to help you to work that out!

Staying on the same subject, do you know any of the greatest mathematical mysteries?

## Theoretical but Impractical Paradoxes

### The Banach-Tarski Paradox

This pure geometry theory was demonstrated in 1924, relying on the axiom of choice in the construction of non-measurable sets. It can be summarised as follows: one can split a sphere of the usual space R^{3} into a (finite) number of pieces and then reassemble the latter to form two balls, identical to the first, to the nearest displacement.

It is strange to say the least, I’m sure you’ll agree. Indeed, such a thing is only possible if these little bits of sphere are unmeasurable (introducing a volume, for example, would mean a contradiction). The methodology still requires some clarification ... I’ll let you try this out in real life!

### The Plane Geometry of Neumann

In 1929, John von Neumann drove his contemporaries mad.

He departed from the axiom of choice to decompose a square into a (finite) number of 'sets of points'. Then, thanks to polished transformations retaining their surfaces, he obtained ..not two spheres, but two squares.

The problem brought to light by this paradox, allowed Laczkovich, in 2000, to explain this decomposition of the interior of a square unit (equidistant bounded sets).

Hard to follow, right! Learn More about von Neumann's paradox here.

### The Barber's Paradox

Secondary school teachers like using this very much because it makes it easier to teach certain subjects to students.

Imagine a rule that stipulates that a barber must shave all those, and only those, who do not shave themselves." The question is, does the barber shave himself?

If he shaves himself, he breaks the law for he is required to shave only those who do not shave themselves. On the other hand, if he is not shaving his own beard, he would not be doing his job of shaving those who do not shave themselves.

It's a good way to show how to rationalise the absurd, isn’t it?

Russell's antinomy, belonging to set (or class) theory, is slightly different, and takes root in the theoretical field: "In 1905 Bertrand Russell showed that the notion of a “set of sets which are not elements themselves “is contradictory” (Universal Encyclopaedia, 6, 265).

### What if the Earth turned inside out?

Our next stop is differential and linear topology. In 1958, S. Smale formulated “sphere inversion (or reversal)”. What is it? A law that will no doubt amuse students studying for bachelor’s or master’s degree but which will be lost on much of the general public...

With the development of computer animation, we have been able to demonstrate the possibility of turning a ball inside out in our three-dimensional space. But who knows if one day this would cause a real technical revolution?

Learn math online with Superprof!

## Counter-Intuition from Day to Day

### The Simpson Paradox

No, nothing to do with the little yellow animations on TV who, I imagine, would not really be cut out for rational abstraction ...

Statistician Edward Simpson formulated this paradox in 1951. It relates to seemingly contradictory data sets simply because they apply different criteria.

To give an Example: to fight against a certain disease you have a choice of 2 treatments that have both been tested twice. In the first test treatment A cured 63/90 people (70%) and treatment B cured 8/10 people (80%). In the second test, treatment A cured 4/10 people (40%) and treatment B cured 45/90 (50%).

Looking at the the tests individually, it would appear the treatment B has a higher success rate, however if we pool the data we can see that treatment A cured 67/100 people (67%) and treatment B cured 53/100 people (53%), meaning treatment A is the more successful treatment.

Simpson's apparent paradox exists when a trend present in different groups is reversed when those groups are combined.

This paradox has been used in many real life applications to show how pooled data results differ from numerous individual tests.

### Condorcet and Election Methodology

This idea comes from the revolutionary mathematician of the same name.

It is a method applied to a voting system, in which voters rank their preferences rather than vote for a single candidate. Each possible candidate is pitted against every other in this system with the overall winner being the one preferred over and above everyone else. Read more about how Condorcet Voting works here.

Essentially the Condorcet system allows voters to "vote their true preferences without worrying about wasting their vote on a candidate with little or no chance of winning" according to the ElectionMethods website.

In short, between paradoxes and real-life problems, you can really have some fun!

To finish on an even funnier note, did you know that 2 scientists actually used maths to figure out the lead character in GOT? Tell that to your maths tutor!