Metaphor Masks Math...

Keith Devlin's four levels of 1:


Energy, as meant outside science, is the sense we have that the food we must eat, the food our beasts of burden consume, and the fuel we must furnish our engines, all  have something essential in common, which we name "energy."  It gives us our go!  It's a metaphorical meaning which also carries over to the need for rest when we work to exhaustion and for sleep when we have been awake for longer than we are accustomed.  The word was coined by Aristotle from the Greek meaning "at work."  It's an abstract pattern everyone notices with ease.
Thermodynamics:  In the 19th century, people who wanted to get the most work possible out of fuel-burning engines applied to that task the best of human reasoning skills they could muster.  They needed to recognize multiple influences, separate and relate the variables ("parameters"), recognize relevance and irrelevance, and avoid at least the most obvious logical inconsistencies and errors.  They recognized and defined several new parameters: "temperature," "heat," "pressure," "entropy," "enthalpy," "work," and and a new, and completely different,  "energy."  This scientific energy is more abstract that Aristotle's by being a pattern within the metaphorical patterns. 
Compared with Aristotle's concept, the energy of science has more power for making things happen the way we want, and it is more free from arbitrary choice we might wish to impose on it.  In the universe of science and math, that makes it "more real" than the metaphor. 

But until we gain some recognition of the math-generated concept--and that takes some hard work for most of us--the metaphorical meaning seems the "more real."  Our more easily-understood sense of meaning easily stands in the way of our awareness that the more abstract sense exists.

The metaphor masks the meaning of the math.

Parameter is today most often used to mean a limit, a boundary, like the circular corral shown above.  This is a "perimeter," and some linguists think that that meaning has crowded out the more abstract mathematical meaning.
Parameter  is a mathematical concept, a defined constant or variable used in a mathematical equation.  The parameters of the circular corral are its radius, "r," it's diameter, "D," its area, pr2, the position of its center, (x, y), etc.
Using letters to represent numbers is an acquired taste.  But it's also a skill, a skill that opens the doors  to the realms of mathematics.  Those parameters, spelled out as "r," "D," pr2, and (x, y), seem like Greek to many.  The metaphor of the corral/boundary/limits/perimeter..., well, it's an easy think and easily seen.

The metaphor easily masks the meaning of the math.



Quantum theory is well-known to be a major change in human thinking about reality.  We made a "leap" in our philosophy!  The epicenter of this humongous reassessment of reality was among the physicists, who went off to make atom bombs, lasers, and other powerful potential weapons of mass death and destruction.  The quantum leap was introduced into our language, not by the physicists, who talked about "quantum jumps," but by the poets and journalists who coined the term "quantum leap!"  The world now sees any humongous advance as being metaphorically related to the scientific revolution which is quantum mechanics.  Physicists have to work very hard to pile up personal publications 
and seldom have time to try to find effective ways to explain what a quantum jump actually is.
"Quantum Jump"  is a milestone in the theory of measurement.  Physicists had always assumed that as their instruments got better and better, they could keep measuring more and more accurately.  Mathematicians insisted on it with their requirement that the math meet the "Cauchy criterion":  no matter how close we get to a given value, say 2.00000000000000, we can always get even closer, and then closer yet, ad infinitum.  The better-and-better instruments disagreed, especially with that criterion.  The instruments told the physicists that the last step wasn't in infinitesimally smaller and smaller steps.  At some humongous level of precision, the familiar math no longer worked: "in-between values" just could not happen.   The quantum jump is the absolutely smallest change a physical value can undergo.
The metaphor masks the meaning of the math.  "Big change" is easy to understand.  However, the meaning of the Uncertainty Principle, the mathematical expression of the quantum limit of measurement, is mired in the mathematics of physics, which takes human thought and insight beyond human perceptions, imagination, and metaphor.  (And to many who use it, much is magic, because it seems beyond human powers.)  Furthermore, its meaning will be missed to the extent our thinking suffers from infection with "The Singles," single components of measurement, single-dimension (rank) ordering when we compare, single cause and single effect, single-mindedness, etc

The Uncertainty Principle always speaks of two measurements taken as a inseverable whole.  Position and momentum.  Energy and  time.  Angular position and  angular momentum.  The quantum uncertainty of one (the uncertainty of the measurement  when the measuring instrument is perfectly precise) is always tied to the uncertainty of the other.  The product of the two uncertainties--say of position and of momentum--is a fundamental constant of nature.  How accurately we choose to measure one of the two is an arbitrary decision on our part, something like the arbitrary decision of which streets are the zero-streets for house numbers.   It affects the numbers, but it doesn't affect the reality represented by the numbers.


A brief glance into the multidimensional, abstract spaces of quantum mechanics
Let's examine one of the tricks of the trade: phase space
(This section will be placed on a separate page when we get more material on these topics.)



The uncertainty principle is important to us because it shows us limits on the number of possible values some parameter can have  within some range of values.  This can let us calculate entropy.  Entropy is a statistical parameter useful for calculating  how much work we can get out of a gallon of fuel or a pound of food.  Our innards are also guided by the statistics of thermodynamics: osmosis, enzyme and hormone actions, ion transfer across membranes, including nerve membranes--all are better understood the better we understand entropy. 


 Meet "phase space."

The vertical dimension represents position--in the x-direction.

The horizontal direction represents momentum (velocity X mass)--in the x-direction, Px.

For a particle, a point in this (blue) space represents: a value of position (x-component only) and a value of  momentum (x-component only). 

X can vary from the value at the bottom edge of the blue to the value at the top edge--It's being held there by impenetrable walls. 

The value of the momentum is limited by the energy the particle has--but it will be constantly  shifting that momentum from direction to direction (x, y, and z): Here we look only at the x-component.

At any instant, the particle has an "uncertainty box" which might be very tall and somewhat narrow, or very wide and not tall at all, or it might be kind of square.  Tall boxes represent relatively uncertain values of position and relatively certain values of momentum.  Wide boxes represent more certainty in position, less in momentum.  Square boxes represent balanced uncertainties.

The uncertainty principle is the observation that all the boxes have the same area no matter what their shape.  That area is 1.0545 X 10-34 joule-seconds.  The orange-ish boxes scattered on this page all have the same area, but different relative uncertainties of position and momentum.

The total number of those boxes which can fit into this blue box is the number of different ways the particle can be in the state we are examining, the more there are the more probable that state.

The calculation of entropy from the uncertatinty principle works like the calculation of the odds of rolling various values, from 2 to 12, in a single roll of a pair of dice.  We count the different ways things can be that are equivalent: with the dice there is only one way to roll a 2 or a 12, and there are many ways to roll a seven.  (How many ways?) 

Calculating entropy involves counting the many possible states that are equivalent.  Before quantum mechanics we would expect the number to be infinite using such an approach.  But  those "uncertainty boxes" are a limit to the number of states we can distinguish between.

The simplest case for entropy is a particle confined to a box of known dimensions, and having an energy of known value
Just like the number of ways of rolling seven with a pair of dice--or rolling a two-- tells us the relative probabilities of getting the various values on a roll, the number of values of position-momentum--or of time-energy--tells us which condition (state) a system of particles has the highest probability of actually being in. 

This is the principle that underlies entropy calculations from the uncertainty principle.  Real calculations are intricate, but the area in phase space is the thing to think about.


Watch for:

Exponential Growth