What are the most philosophical mathematical theorems

The art of math

For the author, mathematics is the most beautiful and unfortunately most misunderstood science there is. And it goes even further with the claim that mathematics is an art form. A declaration of love to the queen of science.

What happens to you when you read here that it will be about math? The first of you may break off the article here because a discomfort arises that is rooted in school:

Anxiety: "I've never understood that before."

Doubts about the usefulness: "I'll never need that again later."

Frustration: "Mathematics can be done or not."

There is even a common nightmare that persists well into old age, when you wake up in the morning before your final math exam but have not studied.

So it is hardly surprising that mathematics no longer plays a conscious role in most people's lives.

Then there are those who use mathematics as a tool. The engineering, natural and some social sciences refer to them as the language in which phenomena can be expressed and worked on.

If you are still with me: Please give me a little more attention on a little trip. Especially if you have school-age children, because besides disaffection and pragmatism, there is another perspective on mathematics, which in my opinion is the prerequisite for being a good mathematics teacher: aesthetics. How one can succeed in opening up the beauty of mathematics so that it becomes an art form.

Before we rush to claim that strict, rule-loving mathematics can be reconciled with pleasurable contemplation or even fantasy, I would like to briefly explain why it can make sense to deal with it apparently purposelessly: prayer-wheel-like is proclaimed everywhere, we are comprehensively surrounded by mathematics. And for every person who has recently pulled out their smartphone, driven a car or simply crossed a traffic light, this statement is correct. What is often not mentioned here: This everyday math is usually very difficult. So difficult that school mathematics is still a long way from grasping it. So difficult that it sometimes took more than 2000 years to develop, even though it would only have required pencil and paper. So, no illusions: math is no walk in the park; not even a marathon; more of an Iron Man 1. And you don't take part in Iron Man untrained. You train for years. This is what happens in school: You start with training and with the acquisition of skills such as discipline, persistence and the constant expansion of frustration tolerance. In summary, mathematics at school is the county league and the mathematics of weather forecasting is the Champions League. That's why the training should be fun on the one hand and feel meaningful on the other.

So here are a few examples of why it is justified to call mathematics "beautiful":

1) Euler's identity

Alternatively, you can write this down even crazier:


The formula is a dictionary that can translate back and forth between trigonometric functions and exponential functions. Even if it looks cryptic, it is visually astonishing simply because there is something seemingly complicated on the left, which on the right goes into a harmonic zero.

2) The Pythagorean Theorem

You already learn this formula for calculating sides in a right triangle in school. It is only when you try to insert numbers into the equation that you realize how unusual it is to create a new square number from the sum of square numbers. That leads us straight to ...

3) The Great Fermatsche Theorem

has no solution though , , and greater than are.

Although mathematics laypeople can intuitively agree with this assertion from the 17th century, the rigorous proof was only provided in 1994. The proof has to take a detour via elliptical curves and uses a wonderful trick of mathematics: the contradiction proof. At first you pretend that the claim is incorrect. After a long calculation, this false assumption leads to the impasse of a contradiction.

Further information in the TIB inventory:

4) The harmonic series

or in smart:

in very smart:

It takes a while before you trust this formula, because the ever-smaller numbers don't seem to make any significant contribution to the total.

Further information in the TIB inventory:
  • Jörn Loviscach, Professor of Engineering Mathematics and Technical Computer Science at the Bielefeld University of Applied Sciences and YouTube star for math-troubled people, explains the divergence of the harmonic series with integral: https://doi.org/10.5446/10331

5) The law of prime twins

Two numbers are called prime twins if one number and the number are both prime numbers. Unproven guess: This combination occurs an infinite number of times.

Prime numbers are a popular playground in mathematics. As the basic building blocks of the world of numbers, they are fascinating exotic species. Especially with very large numbers, it is difficult to believe that two consecutive odd numbers can really be prime numbers. In fact, they are becoming increasingly rare. But no matter how far we go, a new pair of twins always seems to be found. The largest currently known pair was discovered in 2016 and is so large that the representation of the numbers alone requires 388,342 digits.

Further information in the TIB inventory:

6) fractals

There is also beautiful math to look at, sometimes even to touch. Fractals are particularly aesthetic (here nicely explained and with nice examples). This is a geometric pattern as a building block, which is put together in different scales to form a structure (for example a cube made of cubes in the following illustration).

The word describes (like the word Fraktur) a fraction and describes the broken dimension. The mathematician Benoît Mandelbrot dealt with the question of how long the coast of Great Britain was and found that the answer depends on how close he zooms in to the coast, as it always has new little twists and turns, down to the smallest grain of sand, which again with a rough surface has an infinite length. Mandelbrot defined a recursive sequence that creates a beautiful fractal. We encounter fractals in the most banal places: fern leaves unfold fractally, cauliflower has a fractal structure and even in music there are self-similar patterns.

Further information in the TIB inventory:
  • The father of fractals, Benoît Mandelbrot, speaks at the World Congress of Mathematicians about "roughness" in mathematics, the natural sciences and art and encourages us to be open to beauty in the rough and wild: https://doi.org/10.5446/ 15965.
  • Mandelbrot's autobiography is no less astonishing and unusual than his discovery of fractals: https://tib.eu/bmandelbrot.

7) The Kepler Conjecture

The Kepler Conjecture is an example of how for centuries fruit traders were smarter than any mathematician in the world. Maybe it still is, because so far it has only been proven by a computer. The Kepler conjecture asks how you stack apples in the most space-saving way possible. Expressed mathematically, it looks for the optimal or closest packing of spheres (see figure).

Further information in the TIB inventory:

Computer evidence is one of the great questions of faith in mathematics; some mathematicians categorically reject it. The first famous computer evidence was the solution of ...

8) four-color problem

The claim sounds simple and hard to believe: the countries on every conceivable map can be painted with just four colors, without two adjoining countries being the same color. The conjecture had to be nearly 150 years old before it was proven. The proof is similar to a trial-and-error approach: You categorize the combinations that occur and solve each individual category. In the case of the four-color set, there were an impressive 633 categories that were checked by a computer. Admittedly, the elegance of mathematics is missing here, and it comes across as a bit crude. Therefore, math is not finished with this problem. If we mathematicians find a proof ugly, we keep working on it until we have found an aesthetic solution.

9) Godel's incompleteness theorem

The most philosophical sentence in mathematics says that every formal system (e.g. the system of natural numbers and the four basic arithmetic operations, arithmetic) has limits. So there are not only statements that are true or false, but also statements that cannot be proven. This sentence is a lesson in humility for all righteous who seek their salvation in mathematics. It escapes them in this elegant sentence and leaves us alone with the crucial questions in life.

Further information in the TIB inventory:
  • Harald Sack, computer science professor at the Karlsruhe Institute of Technology, speaks in a logic lecture about Gödel's incompleteness theorem: https://av.tib.eu/media/14265#t=04:01,10:27.
  • “A guided journey through Kurt Gödel's historical evidence” with social and economic embedding, quotes from colleagues at the time and a sensitive introduction to mathematical logic: https://tib.eu/goedelebook.
  • Rebecca Goldstein's biography of Gödel is not only told with empathy, but also illustrates that the mathematician's personality is reflected in his work: https://tib.eu/goldsteingoedel.
  • The natural scientists Manfred Eigen, Hans Frauenfelder, Hermann Haken and Peter Schuster discussed philosophical aspects and the aesthetics of mathematics as an illustrious quartet as early as 1985: https://av.tib.eu/media/11192#t=26:19,28: 20th

10) Last but not least….

... mathematics also beautifully depicts student life:


And what do you need that for now?

The above subjective top ten only has the purpose of enjoyment. If she piqued curiosity or raised a frown, this article wasn't for free. Nevertheless, they are not an end in themselves. Sometimes developments run after the needs of other subjects, sometimes other subjects gratefully take up contemplative thought experiments.

Hopefully I could reconcile you a little with the math. Even if you don't believe me that it isn't art, teaching mathematics is. Empathy, imagination and the instinct to play are prerequisites for enjoying mathematics for yourself. And to pass on this pleasure.

Sources and even more beautiful math (also suitable for giving away):

  • The open interaction platform IMAGINARY shows in its gallery https://imaginary.org/galleries some of the most beautiful mathematical structures that have been created with the software so far. At the same time, it invites you to participate.
  • The book "Pictures of Mathematics" presents the most beautiful visualizations of mathematics. The phenomena are explained, but you can indulge in the pictures even without explanations: https://tib.eu/bilderdermathematik.
  • Since there is no right or wrong way to understand mathematics, here is an example of a very young and successful approach, namely comics, graphic novels and mangas: https://tib.eu/statistikmanga.
  • In Germany Albrecht Beutelspacher has probably made an entertaining contribution to the public dissemination of mathematics like no other. Among other things, he published "How to slip into a soap bubble" with 100 exciting math experiments and "I was always bad in math ...", a glimpse with a wink into the inner workings of mathematics and its servants: https://tib.eu/seifenblase und https://tib.eu/inmathewarichimmerschlecht.
  • Always entertaining is the strange perception of probabilities, which is taken up in the books “The goat problem: Thinking in probabilities” and “The dog who lays eggs”: https://tib.eu/ziegenproblem and https: // tib. eu / dog eggs laid.

Mila Runnwerth with the kind support of Bastian Drees, Christian Hauschke, Jenni Rügge and Stefan Schmeja

1. In terms of hormones, conquering an Iron Man and solving a difficult mathematical problem are identical: the purest dopamine orgy.