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Chapter 9 : Proof 

Mathematician Bernhard Riemann explains the importance of proofs in mathematics. He proves a theorem on stereographic projection.

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1. Euclid's heritage

This chapter is somewhat special... We could have placed it right after the first chapter, but it can also be viewed independently of the rest. A bonus of some sort! The goal is to explain that proofs are at the heart of mathematics.

Euclidhas clearly laid down the rules of the mathematical game, and this earned him the recognition of mathematicians. Euclid cannot claim any major result in maths, but he had the genius to propose a method for mathematics when he compiled   Elements, one of the greatest mathematical texts of all times,  

This book has remained an uncontested reference for almost 2000 years! Its originality lies in its structure. All statements, theorems, propositions etc. in the book are completely justified on the basis of earlier propositions. However, Euclid understood very well that one cannot prove everything from earlier results: one has to start with something (short of writing a book of infinite length!). Readers must accept a number of facts at the start without proof, and these are called axioms or postulates. So Euclid's idea is to start off with a list of axioms as foundations for a building where each brick will rest solidly on the bricks below it. See an online version of the book here.

All statements except the axioms must be proven, which means explaining why they are true using the rules of logic, statements already proven and of course the axioms that were fixed at the start. This is the axiomatic method. It is clear that not just any set of statements can be chosen as axioms. For instance, the list cannot contain two contradictory axioms! The choice of axioms is not easy. Is it sufficient that axioms are not contradictory? It is obvious that the geometry that is taught in school must contain theorems that are "true" in reality, and so axioms must be chosen to reflect our physical reality. On the other hand, mathematicians can be perfectly happy with a set of non-contradictory axioms that do not reflect the real world at all. A classic example is non-Euclidian geometry which, as the name implies, starts off with different axioms than Euclid's, and is just as coherent as Euclidian geometry, even though its theorems are not valid in physical reality as we know it. There is a lot more to say about this axiomatic method, but let's go on with a specific theorem.

2. A theorem

To illustrate how a mathematical proof works, we have chosen a theorem that is not easy, and certainly not self-evident! We have already stated it in Chapter 1

Theorem :
Stereographic projection transforms a circle drawn on a sphere,
that does not pass through the north pole,
in a circle drawn on the plane tangent to the south pole.

This is an ancient theorem. Did Hipparchus know it already? Did he prove it ? Difficult to say. 

The idea to consider the sphere S2 as a complex line to which a point at infinity has been added is often attributed to Bernhard Riemann (even though the idea surfaced even before him...), and one often speaks of the Riemann sphere. This mathematician is without doubt one of the most creative of all times, and to us he seemed the ideal person to present the proof of this theorem about "his" sphere!

Riemann's works are pure genius: thanks to him, we think differently about a large number of mathematical concepts. Just one example : he taught us how useful it can be to study an algebraic curve in the real plane by considering the complex version in the complex plane, which then becomes a complex curve, or in other words, a surface... This is the theory of  Riemann surfaces. Needless to say that this is a beautiful theory.

So we need to prove that the projection of a circle that does not pass through the north pole is a circle. For a complete proof, we would need to start by explaining the axioms, and little by little prove everything after that. This would be difficult, and above all very long! It would be difficult because the choice of axioms is rather delicate, and one has to say that Euclid's choice was not ideal (but this was 2300 years ago).

An impeccable choice of axioms (until when?) was proposed by Hilbert in the twentieth century, but it is not easy to use, especially in secondary school. In the film, one has to abandon the idea of a complete axiomatic proof, and act "as if" we prove the theorem completely, even if that implies that our proof will be open to a lot of criticism. We also have to assume that the spectator already knows certain theorems, like Pythagoras' theorem for instance, or even that he has understood a proof of it. 

Instead of commenting on the proof of the theorem as Riemann presents it in the film, which we think is clear (if need be, see this page), we prefer to comment on its flaws! Our aim is not to show that the proof is incorrect! We want to explain that a proof often has an implicit character, and that proofs with a complete logical deduction are rare. Proving a theorem, be it in the daily practice of the mathematician, or in the classroom of a secondary school, is essentially convincing the reader or the listener that what one says is true. In doing so, it happens that one uses arguments that remain unjustified because one knows that the reader/listener is perfectly capable of justifying them.

After all, mathematicians are only human (!) and communication amongst human beings cannot (yet) be done totally axiomatically! It is possible to write down a mathematical proof up to the last detail but it is hard to find people who will want to read it. The art of the mathematician or the teacher is to write and present a proof in such a way that it takes into account the experience of readers/listeners, and can convince them with answers to all of their objections.

What are the "flaws" and "implicit items" in this proof? Here are some of them:

- Is it obvious that it is always possible to draw a perpendicular line from a point on to a plane? Was it proved?

- Is it obvious that a line drawn from the north pole to a point on the plane tangent to the south pole will allways cut the sphere at some other point?

- The proof shows that the projection of a circle is contained in a circle, but does it also show that the whole circle is in the projection?

These are just a few examples (that can of course be rigorously proved) but we have shown them here to point out some of the implicit items that are present in almost all proofs. The ideal of the complete mathematical proof is often inaccessible, but the mathematician must keep this in mind in order to avoid errors (...and experience with errors in the past is helpful here!). Today certain proofs can be verified by computer, but this will never replace the deep pleasure that a mathematician or a student experiences when comprehension of a theorem occurs: when he really understands why it is true. This pleasure is often the real motivation of mathematicians.

Doing mathematics is above all proving what one claims!


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