# 20.1 Mathematical proof and science

This post was prompted by something written in the art magazine Universal Colours (http://www.eu-man.org/magazine/issue903.df). The article states that, “Modern science uses mathematics … for proving … suppositions in science”. Mathematics is not used to prove scientific ideas; it is used to develop new ideas from ideas that we believe to be true. This is not the same thing as proof – because ideas that we believe to be true may later be shown to be false. For example, if we believe that the earth is flat, we will draw some very strange conclusions about the motion of satellites!

Examples of how mathematics is used in science include: finding the number of bacteria in a colony assuming that the number doubles in some fixed time (post 16.5); predicting the speed at which an object falls from the definitions of velocity and acceleration (post 17.20); predicting the behaviour of a system that converts potential energy into kinetic energy, assuming no energy is dissipated (post 18.6).

Mathematics is used to develop results, called theorems, that follow logically from starting points called axioms (see the appendices of post 16.50). It cannot tell us whether the axioms are true or false; they are simply the starting points for a logical argument. Often axioms can be considered as definitions: for example, “parallel lines are lines that never meet”.

So how can we tell if our ideas are true? We perform experiments to test them. But an experiment cannot prove that an idea is true – only that it is false, for the reasons explained in post 16.3. The number of our bacteria, in post 16.5, will continue to double, if we feed them and remove waste products – but if we don’t then they will die. In post 18.25, we derived the ideal gas equation from Boyle’s law and Charles’ law. Experiments show that the result usually works well. But, at high pressures it doesn’t (post 19.2). This is because Boyle’s and Charles’ laws are usually true – but not always.

So scientific laws can be broken, as in the example of Boyle’s and Charles’ laws. More details on what we mean by “scientific laws”, and why they can be broken, are given in post 16.2. As explained in post 16.2, even a law that can sometimes be broken can often be very useful. This means that we sometimes use mathematics to make predictions about the real world, starting from laws that we know are not always true. Experiments can tell us if the resulting predictions are false.

The article implies that science is flawed because mathematics is flawed – because of Gödel’s incompleteness theorem. But Gödel’s theorem, like all other theorems, was derived logically from axioms. If we believe this process is flawed, we have no reason to believe that Gödel’s theorem is true!

Simply explained, a result of Gödel’s theorem is that there are true mathematical statements that cannot be proved. This implies a weakness in mathematics – that it can’t prove all true statements. But it doesn’t mean that the methods of mathematics can lead to a false conclusion from a starting set of axioms. In other words, Gödel’s incompleteness theorem does not invalidate the way mathematics is used in science.

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