Chapter 9 : Proof
Riemann explains the importance of proofs in mathematics. He proves a
theorem on stereographic projection.
1. Euclid's heritage
This chapter is somewhat special... We could have
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
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
This book has remained an uncontested reference
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,
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
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
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
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
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
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
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
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
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
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
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