# Metamathematics

This page collects resources that are related to understanding about Mathematics. Many of these have been used in IB Diploma Theory of Knowledge courses around the world. I have written them over the course of the last 30 years.

It Ain’t What You prove, It’s The Way That You Prove It

It is so tempting for non mathematicians and even some mathematicians to portray the subject as one where there is certainty. I have often been told, in my adult life, by those who preferred Maths to Literature at school that “The great thing about Maths is that if you get it right, it is right. Nobody can tell you otherwise”. That this simply isn’t true is disconcerting. This resource is an attempt to explore this in a way that high school kids and non mathematicians can start to grasp.

This is a two act mathematical play, which owes, in style if not in brilliance, a lot to Imre Lakatos’ wonderful book “Proofs and Refutations”. Act Zero opens in a classroom where the teacher is discussing with the class the result of the angle sum of a triangle. At this stage they question each other’s understandings of how we know, what proof is and what the fundamental assumptions of Mathematics are.

Act One is where the fun really starts as the class looks at generalising the rule for polygons. The arguments are about what is allowed, what constitutes a definition and the relationship between understanding the proof to understanding the theorem. There are some great counterexamples at the end to send keen students away pondering. If you like this read Lakatos. He was a genius.

This has been used in a variety of ways in schools. I have used Maths teachers as the actors (if they are sufficiently flamboyant!), and I have used students as the actors. You can also just read it through round a desk with students taking turns at the parts. The first act (zero) was published in Nick Alchin’s book on Theory of Knowledge for the IB Diploma. He did ask my permission,but  don’t think he ever bought me the beer!

A Set Of Dots

The difference between Maths and Science is complex and not well understood by students because they often have to do the same kind of formal calculations in both. The nature of proof is explored more fully in the play above, but this activity is a much shorter introduction. It takes a very simple mathematical activity, perhaps the simplest, and leads the class to understand that our deeply held “truths” about the subject are just not supported by experimental evidence. Hopefully they will come to see the need to find something more than observation, however often repeated, to justify a Mathematical truth. As well as the document with the instructions for the teacher, there is a jgp here, which can be put on the screen at the appropriate moment. For obvious reasons, I do not know how any dots there are!

The Objects of Mathematics

Bertrand Russell’s famous statement that “Mathematics is the subject where we do not know what we are talking about, nor whether what we are saying is true” can be confusing. This resource is to help lead a discussion in class about the things we are talking about. It should be accessible to teachers who do know a great deal of Maths.

If geometry were an experimental science, it would not be an exact science. it would be subject to continual revision … the geometrical axioms are therefore neither synthetic a priori intuitions nor experimental facts. They are conventions. Our choice among all possible conventions is guided by experimental facts; but it remains free, and is only limited by the necessity of avoiding every contradiction, and thus it is that postulates may remain rigorously true even when the experimental laws which have determined their adoption are only approximate. In other words the axioms of geometry (I do not speak of those of arithmetic) are only definitions in disguise. What then are we to think of the question: Is Euclidean geometry true? It has no meaning. We might as well ask if the metric system is true and if the old weights and measures are false; if Cartesian coordinates are true and polar coordinates are false. One geometry cannot be more true than another; it can only be more convenient.” – Henri Poincaré

1. Nicholas Alchin says: