Richard Dedekind (1831 - 1916)

Dedekind’s parents had met at the local Collegium - his father was a professor there, and his mother was the daughter of one of his colleagues. He was the youngest of four children, and although he lived to 85, he never married, and instead lived as a bachelor with his unmarried sister.

His first interest at school was science, mainly physics and chemistry. It was not until he was 16 and studying at the Collegium that his life’s work on mathematics began. Dedekind was a talented student, and entered the University of Göttingen in 1850, studying advanced mathematics under Carl Friedrich Gauss, and Number theory, under Wilhelm Weber. Gauss was later to have a strong influence on Dedekind’s doctorate thesis - on Eulerian integrals, which he received in 1852.

Dedekind’s study of number theory lead him to develop a long friendship with another German mathematician - Georg Cantor, whose own highly original and revolutionary concepts had not been accepted by many of his own contemporaries, especially Kronecker.

Dedekind was to receive many honours for his work. He was elected to the Göttingen Academy in 1862, the Berlin Academy in 1880; and both the Academies of Rome and the Académie des Sciences in Paris, in 1900. He also received honorary doctorates from three universities.

Dedekind’s mathematics

It is difficult to describe Dedekind’s work because it is so abstract and so fundamental. What concerned him, and many other mathematicians, was certain properties of the real numbers.

These had hitherto been assumed, no-one had really tried to prove them because it was felt that they just had to be so.

But what if this wasn’t the case? Might mathematics fall apart? That was what motivated Dedekind and others in their work.

We count with the Natural numbers, then comes the need for parts of these so we move on to fractions. These together with the Natural Numbers are called the Rational numbers - numbers made up of ratios. Then we encounter a big problem - unearthed first by the Greeks. The problem concerns numbers like \sqrt2. What sort of number is it?

Is it rational? Let us see what happens if we assume that it is.

So we can write it as a ratio of two Natural numbers

{\sqrt{2}}={a\over{b}}

and we can cancel this down so that we remove all the common factors from the numerator and denominator. Let us square the expression and re-arrange it:

2b^2=a^2

Now the fun and games begin. 2 is a factor of the left hand side - so it must also be a factor on the right. If 2 divided a^2 then it must also divide a - think about it. Let us write this in the form a=2c where c is another Natural number. Now put this back in the expression,

2b^2=4c^2

and so we can cancel a 2 and find that

b^{2}=2c^{2}

But hang on - this means that 2 divides b^2 which in turn means that 2 divides b. But that can’t be - because we have reduced a and b by cancelling them down before we started. This is a contradiction and we conclude that \sqrt2 cannot be written as a Rational number.

So to ‘complete’ our numbers we have to add in all the square roots, cube roots etc. We call these, not unaturally, the Irrationals.

They come from equations of higher and higher degree. In fact this still doesn’t complete the number line. There are still ‘gaps’.

One such is the exponential number e. We don’t find this in any equation of whatever degree. These elusive numbers are called Transcendental.

Now the number system is complete. But what are its properties - do some of the properties of the Natural, Rational and some Irrational numbers carry over - if so, what are they? This important question was what concerned Dedekind and he solved it in a particularly elegant yet practical way. He showed that the number line - the Real numbers - could be constructed, and that when they were they possessed exactly the properties required of them in other areas of mathematics.

Dedekind is best known for his highly significant contributions on infinite and real numbers which was to heavily influence modern day mathematics.