- ... Geometry.
^{1} - For centuries the parallel postulate which
says parallel lines never meet was thought to be a self evident
logical necessity. Eventually geometries were developed such as
that on the surface of a sphere in which this postulate is false.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ... halt
^{3.1} - Many computers have a specific instruction to stop processing
instructions or halt. Today programmers never use such instructions
unless they are writing operating systems but in the early days of
computing there were no operating systems and programmers had to
halt the computer when the program completed. The Halting Problem
need have nothing to do with halting. It is rather the question
will a program
*ever*do some specific action.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ... induction
^{3.2} - To use induction to prove a property is true
for all integers requires two steps. First you prove the property
holds for 0. Then you prove that if the property is true for any
number it is also true for . Having
established these two results the principle of induction allows you
to conclude the property is true for all integers.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ... function
^{3.3} - A function has a domain or set of inputs and a range or set of
outputs. For each possible input there is a unique output. For
example is a function that adds one to its
input . We could limit its domain to the integers
greater than 0 and then its range would be the integers greater
than 1. A more complex example is the function that gives the
payments on a $100,000 mortgage from the interest rate. Such a
function might have a domain of interest rates between 3% and 10%
and a corresponding limited range of payments. Many functions like
these two examples are computable so we could write a computer
program to go from the input to the output. Mathematical functions
need not be computable as long as they are well defined.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ... Paradox
^{4.1} - The barber
paradox concerns a barber who shaves everyone in
the town except those who shave themselves. If the barber shaves
himself then he must be among the exceptions and cannot shave
himself. If he does shave himself that he does not shave himself.
Such a barber cannot exist.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ... distance
^{5.1} - The Planck distance is
or approximately
meters. Where
is the gravitational constant, is Planck's
constant divided by and
is the speed of light.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ... digital.
^{5.2} - The most prominent attempt to reconcile relativity and quantum
mechanics is string theory. This theory establishes minimum
particle sizes to avoid the domain where the two fundamental
theories of physics are incompatible. One cannot know if string
theory is valid because its predictions are impossible to test with
existing or foreseeable technology. String theory is not a branch
of physics. It is mathematical philosophy. Science requires
experiments.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ... equation
^{5.3} - A differential equation describes how a single variable (such
as the level of a lake) changes as a function of all the variables
that affect it such as time and location. A partial differential
equation specifies change relative to other variables such as time
or location. The wave equation relates change relative to time to
change relative to location.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ...
*accelerating*^{5.4} - If you are moving 60 miles an hour and travel for 2 hours you
will go 120 miles. If you are accelerating at 20 miles per hour per
second and go for three seconds from a standing start you will be
going 60 miles an hour. Your car almost certainly cannot accelerate
that fast but if you have a hot motorcycle it might.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ... point
^{5.5} - We understand acceleration in time from driving. Acceleration
is zero when speed is constant neither increasing or decreasing.
Acceleration across distance is similar. A flat plane or a uniform
slope has zero acceleration. It is only when the steepness of the
hill is changing that there is acceleration in space. The hill that
keeps getting steeper or that bottoms out as you approach level
terrain has acceleration.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ... mass
^{5.6} - Any particle has some energy and thus mass. But some particles
like photons that make up light travel at the
speed of light and are said to have no rest mass. No amount of
energy is sufficient to make a particle that has rest mass move at
the velocity of light. In contrast a particle with zero rest mass
must always move at the speed of light.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ... limit
^{5.7} - Consider a sequence
. The limit as
approaches is
. No value in the sequence ever
equals but each
differs from by which gets arbitrarily close to zero as
goes to .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ... differences.
^{5.8} - The second order difference in time is an acceleration. To get
an average velocity we divide distance by time. If you go 100 miles
in two hours your average velocity is 50 miles per hour. To get an
acceleration we divide the change in velocity by time. If you go
from 30 miles per hour to 60 miles per hour in 10 seconds the
acceleration is 3 miles per hour per second.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ... behavior.
^{5.9} - Chaos theory studies the very complex
behavior that can be exhibited by continuous nonlinear systems.
These are usually far more complex than linear systems. Discretized
linear finite difference equations can be made nonlinear by forcing
them to assume only integer values as we did using the truncation
function . This can make the behavior of the
discretized difference equation for more complex than the linear
differential equation from which it was derived although it is not
chaotic because it is a discrete and not continuous system.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ...thermodynamics
^{5.10} - Thermodynamics is the study of heat. Initially heat was thought
of as a liquid that flows. Eventually it was discovered that heat
is a measure of the average random motion of molecules.
Thermodynamics studies the macroscopic aspects of heat as if it
were a fluid. It ignores the motion of individual molecules. Thus
it is a statistical theory.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ... laws
^{5.11} - We have all heard for every action there is an equal an
opposite reaction. This is an informal statement of the law of
conservation of momentum. Momentum is the product of velocity and
mass. Assume a 1000 pound object traveling at 10 miles an hour
smashes head on to a 100 pound object traveling at 100 miles an
hour. The two objects will have equal and opposite momentum. They
will both come to a dead stop. This is required by the conservation
of momentum. If a large truck smashes head on into a massive
concrete building the earth itself (or at least a portion of it
connected to the buildings foundation) will move to conserve the
momentum of the truck. There are many other conservation laws.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ... second
^{6.1} - In contemporary physics the speed of light is assumed to define
locality. In general locality is satisfied if there is any speed
that limits the rate at which effects can propagate.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ... reference
^{6.2} - In special relativity two events are said to
be space-like separated if their separation in
space exceeds the distance light can travel in the time between the
two events. The order that such events seem to occur depends on the
inertial frame of reference. Thus two events
like the measurements in tests of locality in quantum mechanics
will occur in a different order in different frames of reference.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ... explanation
^{6.3} - The observation that the photons in a pair, as used by us, are
always found to have different polarization can not as easily be
understood as the fact that the socks in a pair, as worn by
Bertlmann, are always found to have different color[6].
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ... symmetric
^{6.4} - An equation is symmetric in time if the
solution for is the same as the solution for
. The fundamental laws of physics are
symmetric in time with some exceptions. Time symmetric models are
reversible. Reverse the order in time of the initial conditions and
the sequence of states goes in the opposite direction in time.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ... radians
^{6.5} - Radian is a measurement of angle like degrees. There are
radians in .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ... time
^{6.6} - The Planck time is
where is the gravitational constant, is Plancks constant divided by and
is the speed of light.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ... identical
^{6.7} - The substitution of for
between the two equations has no effect.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ... Joule-seconds.
^{6.8} - A Joule is unit of energy. One Joule is 0.2388 calories.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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