Foundations of Analysis by Edmund Landau is a great little book which I’ve mentioned before in a couple of my posts; Landau’s book gives a detailed account of the construction of the Real Numbers starting from the Peano Axioms.

It is tempting to take the real numbers for granted to a certain extent but one of the major developments in mathematics towards the end of the 19th Century was that the real numbers can be effectively built up from five basic axioms – the axioms for the natural numbers. These axioms, in a sense, capture the most essential properties of the natural numbers – the essence of the natural numbers – starting from these five axioms and about ninety pages later we arrive at a system of numbers that coincides with our intuitive notion of the real numbers.

The video link in this post is a link to a series of lectures that I have made covering the contents of Landau’s book up to the point where the construction of the real numbers is fully complete. The basic number system is the natural numbers which is extended into the rational numbers. The irrational numbers are added before, finally, defining the real numbers. How this all happens is remarkable (although, it wasn’t without controversy originally) and through reading Landau’s book you will never ever see the real numbers in the same way again.

What follows is a proof of the well known fact that $2 \times 2 = 4$.

Define $2 = 1+1$ and $4=((1+1)+1)+1$

Then $2.2= (1+1).(1+1)$ by definition of $2$ (where $.$ is used instead of $\times$)

                $=1′.1’$ by Theorem 4, since $1+1=1’$

                $=1′.1+1’$ by Theorem 28, with $x=1’$ and $y’=1’$

                $=1’+1’$ by Theorem 28, since $x.1=x$

                $=1’+(1+1)$ by Theorem 4, since $1+1=1’$

                $=(1’+1)+1$ by Theorem 5, associativity of $+$

                $= ((1+1)+1)+1$ by Theorem 4, since $1+1=1’$

                $=4$ by definition of $4$

The theorem and definition numbers refer to the numbering in the book titled Foundations of Analysis by Edmund Landau which can be downloaded at here.

Why do we need to bother proving something as trivial as this? The answer is because we can. Pure mathematics is not really concerned with whether or not something is useful (although it could be argued that the above multiplication could come in quite useful from time-to-time) but takes the approach that if something can be proved then it should be proved. Mathematicians are very strict people and will refuse to accept something as true (or false) unless it has been proved to be the case; a proof allows the mathematician to be 100% certain about something and thus allows the mathematician to use certain facts and theorems, once they have been proved, in proving other facts and theorems and to develop a theory. This process is very strict and there are no exceptions – if something hasn’t been proved in a mathematical sense (which is much stronger than scientific proofs seen in other sciences) then it cannot be trusted.

The above proof is really just a special (and fun) case of the more general theorem (to be found in the book Foundations of Analysis) proving that multiplication works as we expect it to work for all numbers (Theorem 28 and Chapter 4). Note that this is something that we take for granted because we have been told from a very young age that multiplication does work yet no-one has ever checked multiplication explicitly on every pair of numbers so how do we really know that it works? Addition is something that is also considered in the book and proved to work as we expect it to (Theorem 4 and Chapter 2). Verifying that addition and multiplication behave the way that we want them to is an important thing to know – after all if we just blindly accepted that they worked without proving them and things started to go wrong somewhere down the line then we would only have ourselves to blame for being so naive! Imagine if we decided not to bother proving some other mathematical principles such as the ones that stop skyscrapers from falling down, that stop bridges collapsing or even the ones that stop people hacking into your bank account and helping themselves.

Foundations of Analysis is a unique book. This is not the kind of book where you expect to find lots of worked examples and then lots of practice questions for you to have a go at followed by pages of answers and solutions. Nor will you find encouraging and motivating words giving you hints and tips and pitfalls to avoid. The text falls into about four categories – Axiom, Definition, Theorem, Proof and builds up the through various number systems from the Natural Numbers, ${1, ….}$ to the Complex Numbers. The style is extreme and brutally cold with all superfluous writing stripped out – exactly as it is intended to be. This is an example of how efficient mathematical and scientific texts can be. Yet this shows how (necessarily) pedantic mathematicians have to be. I think the author’s attitude would be that if you can’t be bothered to read it then don’t – but I can say that having read the book myself it was certainly worth it!