And Ten Theses on Hilbert's Sixth Problem and Quantum Measurement

This paper has possibly a very different preoccupation than motivated Lucien Hardy in developing his famous paper, 'Quantum Theory from Five Reasonable Axioms' (arxiv.org/ quant-ph/0101012). This paper comes out of a consideration of Hilbert's Sixth Prob-focussed by Wigner [8] in a famous discussion of the Problem of Quantum Measurement (arxiv.org/quant-ph/0502124). Hilbert meant an axiomatisation of the physical concepts analogously to how he and others axiomatised the mathematical concepts of geometry. As he and others had eliminated the need for diagrams or intuition from plane geometry, so he hoped mathematicians could eliminate the need for physical intuition from physics. From this standpoint, which is one close to Wigner's (or at least he shows a sensitivity and awareness of it), Hardy's so-called axiomatisation is quite alien. We will focus on what Hilbert himself singled out as fundamental, the concept of physical probability. Zermelo showed, but has become even more foundational in physics since 1925. ON page 10, Hardy restates his first axiom, which we quote: 'Axiom 1 Probabilities. Relative frequencies (measured by taking the proportion of times a particular outcome is observed) tend to the same value (which we call the probability) for any case where a given measurement is performed on 1