Science vs Mathematics: The Idea of Proof

Almost all the physics you learn in high school is “wrong”. This is (hopefully) not the fault of your education system, but rather a fundamental flaw in how we describe the universe. Science is essentially based on the following system:

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In this system, a model or theory is developed and then it is tested by comparing the results of the theory to observed phenomena. If the results of the theory match what is observed, then this becomes evidence in favour of the theory. If there is enough supporting evidence, then the theory becomes accepted.

In this system, there is always a possibility that someone will observe something that can’t be explained by the theory or that a better theory will be developed and eventually replace what was once accepted. One of the most familiar examples of this can be found in the history of the atomic model. The atomic model was first introduced by John Dalton and has since undergone several re-developments. In particular, Dalton thought of atoms as the smallest particles in nature. This theory was undone by Thompson’s discovery of the electron – the first subatomic particle to be discovered. Thompson subsequently developed his own incorrect model of an atom (the ‘plum pudding model‘). We now have a very different model of the atom that still explains the observations of Dalton and Thompson, but also accounts for quantum phenomena. This is typical of the scientific process – there is no absolute theory that is guaranteed to explain everything.

Archimedes will be remembered when Aeschylus is forgotten, because languages die and mathematical ideas do not.

G. H. Hardy

In contrast to this, mathematics is based entirely on absolute proof. There is no possibility of Pythagoras’ Theorem being replaced – it is absolutely true and can be rigorously proven. We do not accept that Pythagoras’ Theorem is true because it has worked for every triangle we have tried so far; we accept it because we can show that any right-angled triangle must satisfy Pythagoras’ Theorem. This is fundamentally different to the method of scientific proof. Mathematics is logically built up from fundamental truths: no statement in mathematics is accepted unless it can be rigorously proven to follow from these basic facts.

To illustrate why proof is so important in mathematics, consider the following statement: given any number, , most numbers less than or equal to have an odd number of prime divisors. One way to test this statement is to try some particular examples. If we let , for example, we get the following list of numbers with their prime factorization:

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From this, we see that 1, 4, 6 and 9 have an even number of prime factors (note 1 has zero prime factors), and 2, 3, 5, 7 and 8 have an odd number of prime factors. Thus, most numbers less than or equal to 9 have an odd number of prime factors and the statement is true for . This shows that the statement works in one particular case, but the claim is that this will work no matter what number you start with. If you try some more examples yourself, you will most likely find that this does appear to work no matter what number you start with. Despite all this evidence in favour of the statement, however, it is in fact false. The first number for which it fails is (which is why any numbers you tried probably worked). This is a striking (and famous) example of why mathematicians don’t accept examples as proof that a statement will hold in general.

This presents mathematicians with a difficult problem: how do you show that a statement is true for every number when there are infinitely many numbers? Clearly you can’t check the statement for every possible number. In order to overcome this problem, mathematicians have developed various methods of proof (proof by contradiction and proof by induction, for example). This extra level of rigor is one of the main reasons that mathematics is known as the queen of the sciences. It also gives mathematics a completeness that the other sciences lack.