One of the primary motivations for this topic is that it provides an exciting introduction between the interplay of the experimental and theoretical. In particular, we can give some examples showing how formal methods and empirical/experimental processes developed hand-in-hand with probability theory, and how these two can reinforce one another when making practical, epistemic claims. In particular, we trust the examples of this interplay from physics, where the methods are demonstrably successful.
When we describe quantities, particularly in the case of physics physical quantities, we do so by first positing that there is some underlying state space, S, of states s, and a space of observables O, whose elements are (physical) quantities A. We then represent the quantities A, by functions from the state space S to the real-numbers. These representations are typically denoted as f_A, to be read as "f subscript A" (Wix at present does not support math type setting so bear with me).
As a function, for each state s in S, f_A(s) is a real number that is said to represent the value of quantity A at state s. We further suppose that the association of real valued functions is one-to-one in the space of functions between the state space S and the real numbers.
Additionally, relative to each state s in S, a valuation function v_s is a function from the set of observables O (i.e. physical quantities) to the real-numbers. This function is defined on the nose by sending each quantity A to v_s(A):= f_A(s), where v_s(A) represents the physical quantity A at state s.
Crucially, valuation functions must satisfy a functional composition condition (FUNC), such that for all real-valued functions h, all quantities A, and all states s, we have v_s(h(f_A(s))=h(v_s(A)).
One example from physics demonstrating this condition: let E represent the physical quantity of energy. Let h represent the positive square root function. The functional composition condition says then
the value of the positive square root of energy is equal to the positive square root of the value of energy
This may seem tautological, but the equivalence here is not guaranteed! In fact, a consequence of the Kochen-Specker theorem has that (FUNC) fails when working with Hilbert spaces of dimension greater than 2. This in turn indicates that a 'realist' model of physics in the vein of classical logic will fail when describing quantum mechanical systems. In the next post, describing logic and hilbert spaces, we will provide another example of the functional composition principle, which is also relevant to our investigation of the Kochen-Specker theorem.
Finally, the importance of these concepts cannot be oversold. In the empirical (i.e. statistical) sciences, we are working with samples of observables and states, and we are attempting to define representations of f_A and v_s when formulating theories that formally describe the behavior of the state systems. In the statistical and machine learning contexts that many are working in today, we often use formal algebraic techniques to optimize maximum likelihood estimations for approximations of f_A and v_s that happen to minimize some measure of error between the predicted values and the corresponding observed values. These techniques themselves require making sometimes strong and unfounded assumptions about the behavior of the quantities, and sometimes those assumptions lead to broken models which may still have an instrumental utility, but whose valid application cannot be trusted to explain or predict phenomenon.
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