From Tim Wilkinson
Richard Elwes reports on Harvey Friedman’s fascinating work on incompleteness in Boolean relation theory, but it is quite a stretch to extend his work to ask “are the rules of arithmetic… unsound?” (14 August, p 34). In logic, as in everyday language, the label “unsound” is generally reserved for arguments which prove false statements or contain faulty reasoning.
As mentioned in the article, Kurt Gödel showed that there are statements in any sufficiently complex theory of arithmetic that can neither be proved nor disproved within the theory. Friedman has found a concrete example. Such theorems are astonishing but they do not imply that our best theories of arithmetic are likely to be harbouring a falsehood or contradiction.
Furthermore, incompleteness does not lie as far away from elementary mathematics as the article suggests. If we extend Pythagoras’s theorem by allowing higher powers and more variables, and consider the set of such equations that cannot be solved in whole numbers, no formal arithmetical system can completely decide which equations are members of this set, no matter how many axioms are added to plug the gaps.
It is remarkable that such apparently straightforward questions can never be answered, even in principle, but this does not mean we should be unduly worried that the tenets of ordinary mathematics might turn out to be false. Gödel’s astounding theorems show that there are some things that mathematics can never prove, but they do not call into question those that it can.
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• It is certainly true that whether or not certain equations can be solved in the whole numbers is undecidable from Peano’s axioms – rules that underlie the logic of arithmetic. However, some arithmetical statements are more undecidable than others, so to speak.
Most known examples of arithmetical statements which are undecidable from Peano’s axioms can be decided when we move up to a stronger system, such as second-order arithmetic or Zermelo-Fraenkel set theory. The most undecidable ones need the use of large cardinal numbers. Friedman has not just found a concrete example of an undecidable statement, he has found a concrete example of a very, very, very undecidable statement: a brand new, and shocking, result.
Andrey Bovykin and Michiel de Smet have recently written on their work translating undecidable statements into statements involving polynomial equations (bit.ly/aydJaL). They even manage to achieve this for Friedman’s recent work.
Richard Elwes asks if we can easily dismiss infinity from mathematics. If he simply means abstaining from setting an upper or positive lower limit to a set or function, then the answer is no. However, this does not mean that mathematical infinity actually exists, or is even a meaningful word within a realistic context.
Unfortunately, pure mathematicians tend to believe in their own, often fantastic, constructions. As far as we know, there is nothing that is infinite in the physical world: all processes involving energy exchange proceed by finite quanta and many physicists are beginning to think that the structure of space-time is grainy as well. As a scientific concept, infinity should be relegated to the has-been category, along with phlogiston and the philosopher’s stone.
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