This was our first real maths class. It was very important to hear what the students think about mathematics. It was nice to see that at about 1/3 like it, 1/3 are indifferent and „only“ 1/3 hate it. I am used to see that most of the students hate mathematics, even small children in the 4-5 grade.
Our first aim in maths is to see who knows what. I can already say that the children are quite well prepared and very motivated, so we can achieve everything we want to. It will be possible if we manage to concentrate enough in class. That’s why I won’t let anyone to disturb the lessons. We will also have to follow the rhythm that we agreed upon.
The beginning of the first class we devoted on explaining why the morning saying is so important. We’ve spoken about the different natural kingdoms – minerals, plants, animal, humans. We also spoken about the meaning of the sun and the stars. And especially about the space orientation of people and their difference from the animal. We’ve seen that there is a outer and inner world which are both filled with the holly spirit. We’ve also shown a possible mathematical connection on the notion of inner and outer according to some simple construction in a circle. We’ve seen that for every single point outside of a circle there is a different point inside it.
When we speak about mathematics it was first very important to see what are the three different branches that we have – arithmetic, geometry and, let’s say, algebra (according to Rudolf Steiner phoronomie, which is often translated as kinematics, the teaching of motion. More about this teaching we can find in the eurhythmy and the paneurhythmy from Petar Dunov or Master Beinsa Douno, as he is also called).
Our first lesson was about the different kind of numbers and what their meaning is. It is important to realize that there are different aspects of the number. People used to say that, for example, the natural numbers are created from God and all other numbers from the people. We can find some numbers in ourselves and also in the nature (for example the Golden Ratio). We see that there are single numbers like 1, 2, 3, the golden ration and so on in the nature, but there is no such thing as „numbers“. This is an abstraction which people created. According this we have different types of numbers, like natural, rational, real and imaginary (complex) numbers.
We’ve seen that there are the natural number – 1, 2 and so on. First people seem to experience only the 1 (the unity), 2 (the polarity – day/night, man/woman, positive/negative an so on) and many. With the time and because of setting down and changing cultute the other natural numbers came in our consciousness – 3, 4, 5 … We still have to explore them and their meaning in more detail. For the first hour it was enough to see what kind of different numbers the humanity defined and why.
0 is also a natural number, but it was discovered quite late from the indians and came even later through the arabs in the middle and west part of Europe. Leonardo of Pisa, called also Fibonacci, the first „modern“ european mathematician introduced it on our continent. We are going to speak more about hime, because of the famous Fibonacci row which has some very important connection with the Golden Ratio and can be found in the nature. The 0 also prepared the coming of the decimal number system which is the most popular now, but according to Rudolf Steiner not the best one. That’s why and because of computer science student have to learn also about the different number systems.
The german mathematician Dedekind said in 1888 in „What are and What Should Numbers“ that:
Numbers are free creations of the human spirit. They are needed in order to catch the difference of the things and easily. Only through the logic structure of arithmetics (the science for the numbers) and through the continuity of the number realm, we are able to explore our conception of time and space. This happens when we create a connection of the time and space with in out spirit created numbers realm.
Only one year later the italian Peano published the axiomatic system of the natural numbers:
1. 0 is a natural number
2. Every natural number has a successor that is a natural number
3. 0 is not a successor of a natural number.
4. Natural numbers with the same successor are the same.
5. If a set X of natural numbers contains the 0 and for each natural number also its successor, than X is the set of all natural numbers.
Some centuries ago it was impossible for the people to imagine that there are also negative numbers. As some of the student said, this feels quite unnatural, because if we have five apples, we can take three aside, but if we can’t take seven aside. At the same time, if we look at the modern economics and bank accounts system, we will see that there are negative numbers – debts. Also according Rudolf Steiner are negative numbers something quite real, because we have them as powers affecting our physical body from the ether-body. So, the relation between physical and ether body is as positive and negative numbers. in modern physics we also see very exactly what is a positive and negative power, energy or load.
The natural numbers, together with the 0 and the natural numbers with a minus (-) sign build the integer numbers.
Es liegt Dir kein Geheimnis in der Zahl,
allein ein großes in den Brüchen.
Goethe in Urfaust
It was not easy for the people to divide the unity. The Egyptians were the first to introduce the fractions. They used only unit fraction, this means fractions with numerator 1. The Pythagorean (if we have to be really correct here we have to say that most of the knowledge of Pythagoras and his school comes from the legendary musician, poet and prophet Orpheus) discovered the connection between the music and the mathematics. It is a very important notion called the „Musica universalis“ (Sphärenharmonie). We all can be very happy if some of the students experiences some day this Musica universalis. The discovery was that every sound, consisting from two tones (interval), can be described as a proportion of two natural numbers, a fraction.
It is usual to try to connect numbers with geometry. Already the old greeks always tried to connect arithmetic with geometry. One of the most important notions was to connect numbers with length of lines. At first it seemed to be quite easy, for example one distance multiplied with another one, gives an area. It’s quite simple, but there is also one problem – the product, the area, is something quite different than the factors, the distances. If we multiply three distances we will obtain a volume, which is also something quite different. And we could not imagine what is the product of four distances!
We have this same difficulty also by division. What is, for example 20 divided by 4? We can think in two different ways:
– If we have a rectangle with an area 20 and one of the lengths of the sides is 4, then die other side’s length is 5. So the result is a distance.
– But we can also ask how many times a line with distance 4 can be found in a line with distance 20. Then the answer is 5 times. And now we have a result with is simply a number, not a distance.
Rene Descartes was the first who managed to solve this difficulties. He is thought to be the father of the analytical geometry (in contrast to the synthetic or projective geometry which I shortly mentioned and which we are going to explore indepth) and so the one who really introduced the abstract way of thinking in mathematics. He managed to give a very simple geometric construction in which we can multiply two distances and obtain again a distance. If AB is 1 and we have to multiply BD with BC, then we have only to connect A and C, construct the DE as a parallel of CA and BE is the product of the multiplication. This follows from the theorem on intersecting lines of Thales. (BE/BC = BD/BA, but BA = 1, so BE = BD*BC).
Later on Bolzano, who invented the numbers set theory, will say that we have first to define the „numbers“ in order to know what is a „number“. As we can see, this is completely the opposite of the truth, what people used to think and what we can see in the nature. But it is very important to get to know the difference between giving meaning, experiencing innerly every single number, and creating an abstract set theory with which we first describe the numbers and after that try to understand a single number.
The abstract thinking leads to a completely different science that the inner lively experience that we will try to have in our maths classes.
But lets go back to the fractions. They have the advantage that if we define them, then we can use all the four simple arithmetic operations without restrictions (please note that we can’t divide through zero). Through the fractions we defined the rational numbers.
But there are also other numbers, that people find out. We had a look at the famous Golden Ratio, which gives us an irrational number. We can construct an irrational number very easy using an equal-sided rectangular triangle with sides 1. Then the hypotenuse will be square root from two,
according to the famous theorem of Pythagoras. It will be nice if everyone of the students will present at least one proof of this famous theorem. And square root of two is an irrational number. We’ve seen that there are also numbers like pi, which describe the proportion between the circumference and the radius of a circle. This number is called transcendental. We also mentioned e, the Euler’s number. It will be nice if a student can explain what is a transcendental number and what is e and how we invented it. It is very important that e is the base of the natural logarithm and so of great importance for the so called exponential function, which has something to do with growth and thus also with our economic development (or better said crisis after crisis). But we will explore this in details some other time, especially in our extra hours in economics.
Back to the numbers. Numbers like square root of two (1,41421 … – endless, nonperiodic decimal fraction) lead us to the real number. But what are the real numbers? They are the numbers which we need in order to have numbers as boundary values of sequences of numbers.
But we can go even further and describe also some other numbers. As we have seen, sometimes, for example by trying to solve a quadratic equation, we have no solutions, because we obtain a negative expression under a square root which is not defined, according to our knowledge about the real numbers. If we want that every quadratic equation has exactly two solutions (and every equation of n-th grade exactly n), then we have to define the complex (or imaginary) numbers. Imaginary numbers have the form a + b*i, where a and b are real numbers and i*i = -1. And again, if we look for a geometric definition of complex numbers, we have one very simple – points in a plane.
We have seen that natural numbers are endless and there are much more real numbers. And there are also much more complex numbers. So much more that we can even say that the number of the real numbers is insignificant, almost 0. Such as the 0 is insignificant in comparison with the positive and negative numbers. If we put the complex numbers in relation to the natural numbers and imagine that the 0 behaves like the real numbers, that there should be also so other numbers which behave in relation to the complex and real numbers, as the negative numbers behave in relation to the natural (positive) numbers and 0.
For further reading I will definitely recommend you: Zahlen – Geschichte, Gesetze, Geheimnisse from Albrecht Beutelspacher. It will be also nice to have a look at the book Das Mysterium der Zahl: Zahlensymbolik im Kulturvergleich from Franz Carl Endres and Annemarie Schimmel. There are also some interesting books about numerology and mysticism of the numbers. As usually, Wikipedia is a good start for such researches: Zahlensymbolik. May be here is the right moment to say once again how important is it to have an account in the city library of Munich, the state library of Bavaria, the libraries of the LMU Munich and TU Munich.