The Fallibility Of Mathematics

ADAM JASKO

When I was young, and indeed to this day, I loved experimenting to discover interesting results or perhaps to verify results already known to me. In the sciences this involves getting your hands dirty: mixing various chemicals to observe a reaction growing runner beans in jam jars and comparing those placed in cupboards to those under sunlight rolling trolleys down ramps to find the acceleration due to gravity, and so on. Unfortunately this leads to experiments that can`t be done at home, for example imaging complex body tissue or trying to find the Higgs boson. These experiments require very expensive and often very large equipment such as MRI scanners or the Large Hadron Collider at CERN. In maths, however, the only dirtying of one`s hands required is from the ink of a biro.

Often in the sciences one must simply believe a teacher or lecturer when they say that atoms are made of protons, neutrons, and electrons, or that cells contain a double helix structure that encodes all your genetic information. In maths however, you need not take their word for it. If you do not believe some trigonometric identity or the answer to a large sum then you can just check it for yourself. One of the attractive things about maths is that it can be done almost anywhere and the only accessories required are pen and paper.

Another thing that separates maths from the other sciences is that once some result has been proved there is no need to recheck. Scientific theories are constantly under scrutiny and may at any point become inadequate at explaining some newly observed events and need to be replaced with more fitting alternatives. Mathematical truth, however, doesn`t change with time.

Of course, to have this mathematical guarantee we need to be sure our initial reasoning is correct. Take the following proof. Let a and b denote the same arbitrary number. Then

a = b basic assumption a^2 = ab multiply by a 2(a^2) = a^2 + ab add a^2 2(a^2) - 2ab = a^2 - ab subtract 2ab 2(a^2 - ab) = 1(a^2 - ab) rewrite the left hand side 2 = 1 cancel the common factor

Some may look at this and jump to the conclusion that 2 really is equal to 1 (indeed, I once used this argument to convince a friend to doubt all maths) yet it soon becomes apparent that this is no proof at all. Upon inspection it should become clear that the last step contains a fallacy namely dividing by zero. Even this simple trick has the ability to deceive.

Other proofs are far more subtle and mistakes are less noticeable than in the above, so how can we be sure that an error has not been made? The short answer is that we can`t, and many mathematicians have published proofs in which a flaw was found, sometimes at a far later date. The scrutiny of the academic community can be as important to help confirm mathematical theorems as it is to verify scientific theories.

One big problem is big proofs. Some theorems are so large that they become too hard and laborious to follow for most. The perfect example is the classification theorem which I shall not explain here, but which involved finding all finite simple groups (these objects can be thought of as the basic building blocks in an area of mathematics called group theory) and showing that no others exist. This massive proof spans about 500 articles over half a century.

The mathematics is complicated to follow and when one considers that very few have gone through the tens of thousands of pages one can start to question one`s faith in the result.

With good reason too: in 1983 the proof was announced to be complete, but a bit later a mistake was found. This was eventually corrected but it was not until 2008 that a revised version was published. Even now doubts can carry on lurking and Michael Aschbacher (one of those responsible for making those final adjustments to the theorem) said that `The probability of an error in the proof of the classification theorem is virtually 1. On the other hand the probability that any single error cannot be easily corrected is virtually zero, and as the proof is finite, the probability that the theorem is incorrect is close to zero.`

Of course, one could argue from a pedantically philosophical perspective that no matter how many mathematicians go through the proof with a fine tooth comb we can never be sure that there is not some small error that everyone has missed. Perhaps Pythagoras` theorem has some flaw that nobody has yet noticed. This kind of thinking leads us nowhere though, if we desire to seek any truths at all. Such fancies aside there are still important issues with the truth and wholeness of mathematics.

Kurt Gödel`s (first) incompleteness theorem, published in 1931, provides an insight into proof and mathematical systems. This result states that all sufficiently powerful axiomatic systems are either incomplete or inconsistent. An axiomatic system is based on a set of axioms (self-evident truths) and rules to make logical inferences. Gödel`s incompleteness theorem states that if such a system is powerful enough to express the arithmetic of the whole numbers and is free from contradictions, then it can express statements that cannot be proven true or false within the system.

This is rather shocking and you may wonder why Gödel`s result has not wiped out mathematics once and for all. The answer is that the unprovable statements logicians have found so far do not touch on ordinary everyday mathematics. They certainly would not come up in school homework or in the maths used to design aircraft wings. The vast majority of mathematicians leave these logical holes to the logicians and philosophers and get on with their work.

With the dawn of technology, mathematics has evolved down routes that could not have been foreseen. Enormous numerical computations can be performed and computers have begun to play a role in proofs, most notably that of the famous four colour theorem. This old problem asks whether it is always possible to colour a map, using no more than four colours, so that no two bordering countries are coloured the same. After many fallacious proofs, the Four Colour Theorem was eventually proved (four colours do indeed suffice) but the proof relied heavily on computers to check through a large number of possible configurations. This approach has been questioned since it is impossible to check the computer`s calculations by hand. Consequently, some mathematicians remain sceptical of proofs heavily reliant on computers.

In conclusion, complicated (and perhaps unpublished) programs run on super computers do not offer the same accessibility as pen and paper proofs, especially to the interested amateur. The length and technicality of other theorems can make verification difficult and even mathematicians can make mistakes. So although in mathematics infallible truth is in theory achievable, the skills, time, understanding, and in some cases computing power required to verify certain results for oneself can render it as inaccessible as a Large Hadron Collider. There is no need to worry about Pythagoras` theorem though: it can be proved quite easily on pen and paper so Pythagoras, at least, is safe! .................................................................. FURTHER READING [1] Keith Devlin (1998). Mathematics: A new golden age. Penguin (2nd edition). [2] Douglas Hofstadter (1980). Gödel, Escher, Bach: An eternal golden braid. Penguin.