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to the axiomatic treatment of physics? I will discuss here two of the countless important extensionsto probability calculus, the kinetic theory of gases and, as Hilbert called it, “the theory of compensation of errors”, which attempts to calculate the errors associated with the use of measuring equipment in order to reduce it. This study comes from any one of three axioms, the other two being derivable as theorems from the one chosen to be an axiom, these are: “If various values have been obtained from measuring a certain magnitude, the most probableactual value of the magnitude is given by the arithmetical average of the various measurements. … The frequency of error in measuring a given magnitude is given by: න ݁ ି௧ ଶ ఛ … The most probable value of the variables measured is obtained by minimising the squares of the errors involved in each observation.” 7 The fact that these axioms are interchangeable highlights something important about the nature of axioms, that for a system to be logically coherent the choice of axioms is arbitrary, which seems to undermine their importance. However, it merely serves to demonstrate that the proposed axioms are logically equivalent. Furthermore, in many physical theories those statements chosen to be axiomatic are those that directly follow from experimental data. The theorems are predictions that guide future research. This suggests that in many physical systems this problem does not arise as only one set of axioms will follow directly from the data. However, when there is a choice it may allow a problem to be simplified by choosing an alternative equivalent system. The calculus of probabilities is also relevant to the kinetic theory of gases which again requires just a few additional assumptions; that gases are made up of point particles that collide elastically, that their velocities are independent, and that they are uniformly distributed throughout the volume of space that they occupy. These three “axioms” 8 can be used to derive the average potential energy, kinetic energy and the mean free path of a molecule as well as the second law of thermodynamics, and much more besides. Another of the areas within physics that have been successfully axiomatised is vector addition, vital for much of physics, as well as some areas that are more commonly considered mathematics, these axioms are as follows: ݏݏܣ ܿ݅ܽ ݐ ݅ ݒ ݅ ݕݐ ݂ ܽ݀݀݅ ݐ ݅݊: ࢛ ሺ࢜ ࢝ሻ ൌ ሺ࢛ ࢜ሻ ࢝ ܥ ݉݉ ݐݑ ܽ ݐ ݅ ݒ ݅ ݕݐ ݂ ܽ݀݀݅ ݐ ݅݊: ࢛ ࢜ ൌ ࢜ ࢛ ܫ ݀݁݊ ݐ ݅ ݕݐ ݈݁݁݉݁݊ ݐ ݂ ܽ݀݀݅ ݐ ݅݊: 0 א ܸ| ࢜ 0 ൌ ࢜ ࢜ א ܸ ܫ ݊ ݒ ݁ ݏݎ ݁ ݈݁݁݉݁݊ ݐ ݂ ܽ݀݀݅ ݐ ݅݊: ࢜ א ܸ ሺെ࢜ሻ | ࢜ ሺെ࢜ሻ ൌ ܥ ݉ܽ ݐ ܾ݈݅݅݅ ݕݐ ݂ ݏ ݈ܿܽܽ ݎ ݉ ݑ ݈ ݐ ݈݅݅ܿܽ ݐ ݅݊: ܽሺܾ࢜ሻ ൌ ሺܾܽሻ࢜ 7 http://tau.ac.il/~corry/publications/articles/pdf/hilbert.pdf 8 I use the apostrophes here to indicate that these are not usually viewed as such but function in a similar way to axioms
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