home Mathematical Nature 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 observes. 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