Wednesday, February 27, 2013

Fundamental Inequalities

Equality is a cornerstone of the law, but it's not at all a feature of natural law.

Consider Heisenberg's famous Uncertainty Principle (but not in any deeper sense than the nature of the symbols):


On the left, Δχ and Δρ (position and momentum) are physical variables; on the right is a fixed valuei.e., half of ħ, the imponderable Planck constant.

Equations are like sentences with the equal sign being like the verb "to be;" they represent either equality or the more subtle notion of equivalency. Inequalities are also familiar from algebra but an inequality also expresses—more or less—an important direction of inequality. In this case, the observables are greater than or equal to the constants.

Bell's Theorem has been called "the most profound [theorem] in science" and it too reduces to an inequality. There's another: the Clausius Inequality, which embodies the Second Law of Thermodynamics, and which expresses changes in entropy against a value of zero. Comparing all three—without even understanding them—it strikes me that they all specifically pit observables against abstract constants. Is this general?


  1. Jensen's Inequality appears to be an exception.

    1. Jensen's Inequality appears convexed by formal restrictions. :)

  2. "Do you know how fast you were going?"

    No, but I know exactly where I was."