Samstag, 19. November 2011

The Origin of the Quantum, Part III: Deviant Logic and Exotic Probability

Classical logic is a system concerned with certain objects that can attain either of two values (usually interpreted as propositions that may be either true or false, commonly denoted 1 or 0 for short), and ways to connect them. Though its origins can be traced back in time to antiquity, and to the Stoic philosopher Chrysippus in particular, its modern form was essentially introduced by the English mathematician and philosopher George Boole (and is thus also known under the name Boolean algebra) in his 1854 book An Investigation of the Laws of Thought, and intended by him to represent a formalization of how humans carry out mental operations. In order to do so, Boole introduced certain connectives and operations, intended to capture the ways a human mind connects and operates on propositions in the process of reasoning.
An elementary operation is that of negation. As the name implies, it turns a proposition into its negative, i.e. from 'it is raining today' to 'it is not raining today'. If we write 'it is raining today' for short as p, 'it is not raining today' gets represented as ¬p, '¬' thus being the symbol of negation.
Two propositions, p and q, can be connected to form a third, composite proposition r in various ways. The most elementary and intuitive connectives are the logical and, denoted by ˄, and the logical or, denoted ˅.
These are intended to capture the intuitive notions of 'and' and 'or': a composite proposition r, formed by the 'and' (the conjunction) of two propositions p and q, i.e. r = p ˄ q, is true if both of its constituent propositions are true -- i.e. if p is true and q is true. Similarly, a composite proposition s, formed by the 'or' (the disjunction) of two propositions p and q, i.e. s = p ˅ q, is true if at least one of its constituent propositions is true, i.e. if p is true or q is true. So 'it is raining and I am getting wet' is true if it is both true that it is raining and that you are getting wet, while 'I am wearing a brown shirt or I am wearing black pants' is true if I am wearing either a brown shirt or black pants -- but also, if I am wearing both! This is a subtle distinction to the way we usually use the word 'or': typically, we understand 'or' to be used in the so-called exclusive sense, where we distinguish between two alternatives, either of which may be true, but not both; however, the logical 'or' is used in the inclusive sense, where a composite proposition is true also if both of its constituent propositions are true.

Samstag, 12. November 2011

Maxwell's Demon, Physical Information, and Hypercomputation

The second law of thermodynamics is one of the cornerstones of physics. Indeed, even among the most well-tested fundamental scientific principles, it enjoys a somewhat special status, prompting Arthur Eddington to write in his 1929 book The Nature of the Physical World rather famously:
The Law that entropy always increases—the second law of thermodynamics—holds, I think, the supreme position among the laws of nature. If someone points out to you that your pet theory of the universe is in disagreement with Maxwell's equations—then so much the worse for Maxwell's equations. If it is found to be contradicted by observation—well these experimentalists do bungle things sometimes. But if your theory is found to be against the second law of thermodynamics I can give you no hope; there is nothing for it but to collapse in deepest humiliation.
But what, exactly, is the second law? And what about it justifies Eddington's belief that it holds 'the supreme position among the laws of nature'?
In order to answer these questions, we need to re-examine the concept of entropy. Unfortunately, one often encounters, at least in the popular literature, quite muddled accounts of this elementary (and actually, quite simple) notion. Sometimes, one sees entropy equated with disorder; other times, a more technical route is taken, and entropy is described as a measure of some thermodynamic system's ability to do useful work. It is wholly unclear, at least at first, how one is supposed to relate to the other.
I have tackled this issue in some detail in a previous post; nevertheless, it is an important enough concept to briefly go over again.