The essence of mathematics is "process",
time indexed sequences of recurrent perceptions.
Counting objects exemplifies "time indexing"
using natural numbers. Measuring objects
exemplifies time indexing using real numbers.
Chemistry exemplifies causality and natural order,
too long a pause between aerosol release and
ignition can be catastrophic. Mechanics exemplifies
causality and real measurements, threshold driven
phenomena such as moments activated by whispers
inducing landslides. Order versus quantification
pits logic versus sufficiency. Solutions often
require varying logical order to accommodate such
variants of reality as chance may induce. While
implementing a solution often requires strict
attention to amounts essential to complete a step.
"Stop the leak" exemplifies separation of logical
and quantification components in rational systems:
containment is breached, a normally internal element
is observed beyond some boundary as an external
element. First a breach in some boundary must be
located by causality, second the breach must be
covered...sufficiently by structures able to prevent
a future occurrence of the observation of a normally
internal element in the ambiance of a normally
external element. Leaving a door unlocked changes
reality: for example, "the dog escaped", "intruders
removed...", 'files were lost", "water ruined",
"bacteria entered...",etc.. Whence, the probability
of castastrophic events is increased by preset
conditions in the presence of uncontrolled will (dog)
or turbulence driven perfect gas..laws applied to
analogues of any scale.
"Intuiting reality" means "Process control" becomes
a thematic driver for mathematical development in
which the "process" must be analysed causally for
its logical components, and in which modular logic
characterizing each component is "control"-led by
sequencing perceptions of physical measurements
and "observations". "Walk in woods" and experience
naturally ordered time, "walk thru gardens" and
experience "willfully ordered time". Civilization
is driven by transforming naturally occurring order
and quantities into properly occurring order and the
correct quantities at a relative location in space.
Experimental equations reflect stably occurring order
and stably relating quantifications in reality.
Physics requires the alignment of intuition according
' to the edicts of experimental equations mathematics
Geometry is the link between functional observations
of physics and related observations of mathematics.
Geometry is a property of a force field able to induce
relative accelerations: model Earth's gravity having
an "end" (e.g. at its center of mass) and enveloping
objects which "rest" upon this g-field's end so that
the distance from one point on this object and the
g-field end is fixed (e.g. a fulcrum viewed as the
pedastal of matter from the earth's center of mass to
the fulcrum as a point location maximizing distance)
Nature allows finite extensions to be balanced upon
such distinguished points, yielding scales leading to
Pythagorean Theorems as soon as homogeneous materials
can be mixed and squares of uniform thickness can be
constructed off the three sides of any right triangle.
Quality control in "choice functions" determines the
ability of the individual to utilize geometric tools,
as opposed to their manifestations: convert fulcrums
to balances to compare the force field weight of the
large square off the hypotenuse to the combined weight
of the two smaller squares off the right angle's legs:
balancing when materials have homogenous densities and
the squares have uniform thickness. Faith in life's
experimental regularity is derived from caution and
care in mapping natural numbers into time and then
measuring the real numbers representing lengths and
masses of the reference objects used when we choose
a real manifestation of our mapping as counting.
(c) Eugene Roger Apodaca, PhD
07:11 AM 1/26/03