PROPUESTAS DE RESOLUCIÓN PRESENTADAS CON MOTIVO DEL DEBATE DE LA MEMORIA SOBRE EL ESTADO, FUNCIONAMIENTO Y ACTIVIDADES DEL CONSEJO GENERAL DEL PODER

Mathematics is very difficult and mathematicians frequently make mistakes. Sometimes these are trivial misprints and misstatements. Sometimes they are more substantial errors which can, none the less, be corrected. Sometimes the mistakes are so serious that nothing can be saved from the resulting mess. Why should you suppose that what your lecturer says is not mistaken?

The first thing that you will notice about most advanced mathematical pre- sentations is that they consist of a series of numbered Definitions, Proposi- tions, Lemmas, Theorems and so on.2_{This is because the lecturer is using the}

Euclidean method.

Ever since a group of apes descended from the trees, they have been arguing amongst themselves. The first method of argument, which is still very common, involves shouting very loudly and, in extreme cases, throwing large stones. The ape with the loudest voice or the biggest stone is deemed to have won.

Later, people invented the method of advocacy. One side puts forward all the arguments it can think of (‘I was not there. If I was, I did not do it and, if I did it, I am very sorry.’) and then the other side puts forward all of its arguments and then a judge or the electorate decides who had the better arguments. Odd though this system is, it is often the best we have.

In certain circumstances we can use experiment. If two physicists disagree, they can agree on an experiment which distinguishes between their two theories. (If no such experiment exists, they are not discussing physics, but something else.) If experiment decides against a theory, it does not matter how rigorous the mathematics behind it is, the theory must be abandoned. If the theory survives

the test then, again, the rigour or otherwise of the underlying mathematics is essentially irrelevant. There are excellent reasons why engineers and physicists are untroubled by foundational mathematics. On the other hand, we can not argue that the calculus is correct because bridges built using it do not fall down, since this would mean that every time a bridge fell down it would cast doubt on the calculus.

The final type of argument was perfected by Euclid and has been used by mathematicians ever since. It starts with clear definitions and a clear statement of assumptions (that is to say, axioms) and then sets out a series of clear assertions. Each assertion (that is to say, theorem) must be justified by an argument which uses only the initial assumptions and any previously proved assertion. If you fail to prove a particular assertion, then it and any assertions whose purported proof depends on it, remain unproved.

Important note. It is very useful to observe that Euclidean arguments can be

read backwards as well as forwards. We start with the final conclusion Z and observe that it will follow if we can prove Y and Y will follow if we can prove

Xand so on backwards until we reach assumption A. It is often easier to grasp a Euclidean argument in this way.

It is not hard to find eminent people who object to the Euclidean method as unsuitable for the purposes of teaching and research. These are not the purpose of the Euclidean method which is to test correctness and which is a very powerful tool for this purpose.3_{The standard first course in analysis, like}

most advanced courses in mathematics, is set out in Euclidean form so that those attending the course can see the theory being put to a proper test.

A lot of people (including many of the participants) look at a first analysis course and see a lecturer writing notes on a blackboard that the students copy down. The students then take the notes away and store them carefully until the week before the exam. The notes are then revised by highlighting them until at least 25% of their surface area will glow in the dark. There is then the exam itself, after which the papers are marked according to how many of the lecturer’s original words can be glimpsed through the garbled murk. Once the grades have been announced, both lecturer and students can go back to their real lives.

Others see the first analysis course as a joint enterprise of students and teacher in which they examine the justification for the calculus, link by link, looking for weaknesses. If they succeed in finding a serious flaw, then the

3 _{I also believe, contrary to the eminent people just referred to, that the Euclidean method is a} very useful tool both for teaching and research, though not the only one.

participant who has found that flaw will enjoy instant fame throughout the mathematical community. If they find a minor flaw, then it is probably an error of the lecturer and both the lecturer and students will benefit from seeing how it is corrected.

There is an element of truth in both of the views I have recorded. However, you will enjoy and benefit from the course much more if you act as though the second view is correct.

Bertrand Russell thought very little of the mathematics teaching he received in Cambridge,4_{but recorded that:}

I cannot remember any instance of a teacher resenting it when one of his pupils showed him to be in error, though I can remember quite a number of occasions on which pupils succeeded in performing this feat. Once during a lecture on

hydrostatics, one of the young men interrupted to say: ‘Have you not forgotten the centrifugal forces on the lid?’ the lecturer gasped and then said: ‘I have been doing this example this way for twenty years, but you are right.’

Russell Autobiography