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Hilbert’s 6 th Problem By Harry Goodhew
In 1900 David Hilbert presented 23 problems which he intended to be the focus of mathematical thought for the coming century and beyond. Since then 17 of the problems have been solved although some solutions have not been fully accepted. The remaining 6 are either too vague to be solved or have resisted solution so far, the latter include the Riemann hypothesis and the one which I will be looking at, the 6 th problem, the axiomatic treatment of physics. This problem is somewhat different to the others that Hilbert proposed as it is not a problem in itself but a general task, which he suggested, following his work on the construction of geometric axioms. So far it has not been fully solved despite exhaustive work from many mathematicians including Hilbert himself. However, some fields within physics have succumbed to such axiomatic treatment, including Mechanics and Statistics. In order to advance the discussion of Hilbert’s 6 th problem it is first necessary to define an axiom, it is one of a set of statements, that are assumed to be true, from which all other true statements within a system of logic can be proved. For instance from the axioms of addition: abൌba ܿ݉݉ ݐݑ ܽ ݐ ݅ ݒ ݅ ݕݐ [1] aሺbcሻൌሺabሻc ܽ ݏݏ ܿ݅ܽ ݐ ݅ ݒ ݅ ݕݐ [2] a0ൌa the existence of an identity [3] aሺെaሻൌ0 the existence of an inverse [4] It can be proved that if abൌca, then bൌc ܾ ൌ 0 ܾ [] ൌ ሺെܽ ܽሻ ܾ [] ൌ ሺെܽሻ ሺܽ ܾሻ[] ൌ ሺെܽሻ ሺܿ ܽሻ ൌ ሺെܽሻ ሺܽ ܿሻ [] ൌ ሺെܽ ܽሻ ܿ [] ൌ 0 ܿ [] ൌ ܿ [] ■(the bracketed numbers indicate which axiom has been used) The usefulness of the existence of an equivalent set of axioms for physics is clear as it would allow physical facts to be proved logically and provide a framework for testing whether or not a proposition is true, as well as a way to ensure that the system did not rely on any assumptions. Furthermore, the nature of such axioms could reveal facts about the universe itself. Unfortunately, due to our lack of a complete understanding of physics, epitomised by the inconsistency between general relativity and the standard model, a solution to this problem remains elusive. However, the developments that have been made are worth discussing. In the years following his presentation of the 23 problems, Hilbert devoted much of his Similarly all other true statements about field addition can be proved from these axioms.
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