The natural reference frame.
Friday, March 25, 2011 8:04:33 PM
This subjective input is important to understanding some fundamental distinctions in mathematics. Thus one of these is the flip algorithm. The flip algorithm i have described in some earlier posts, and it applies to any measuring system using dimensions and parameters, but it originates in the notion of opposites.
The natural way i look at things is by pointing out a relative orientation and distance from myself. The basic orientations are incredibly common: in front, behind, to my left or right , in the direction(motion) of my pointing finger/ hand as it gesticulated; in the orientation(not moving) of my index or indicating finger; so many steps, measures in that direction.
The really basic left and right, in front or behind involve a flip of orientation/direction, but the flip is not physical it is mental!
This subjective element to measurement owes its formulation to Bomelli, and his piu di meno operator rule.
The flip algorithm i therefore a fundamental subjective determinant of mathematical expectation. It was formally given the name flip by me as a recognition of the flip command in programming languages.
This means that mathematics cannot be done right without an external verifier. Until the advent of automatic machines capable of being programmed to respond cybernetically to conditional situations, this was something only humans could do.
So the flip algorithm has no inherent connection with geometrical structure or motion. Rather it is the product of a human perception of the relative geometry.
Thus when we say a minus times a minus is a plus the first response is bewilderment. We have not been able to adjust our subjective view of the motion or relative positions to perceive the resultant a a plus.
In a sense we have a limited choice for the resultant outcome, so it is a bit of an arm twisting situation, and eventually one accepts the commonly agreed notion. However sometimes one has a clue from the geometry of a situation, and that is when one realises that it is not all just subjective but there i an interelation with geometry whic we recognise as symmetry.
This symmetry is a geometrical entity and what we have found is that our flip algorithm and rules have a real use in describing symmetry, but symmetry is not the basis for the flip algorithm, because again we or i have to perceive symmetry. and so it goes that certain important fundamentals of geometry lie mainly in our perception systems.
Such fundamentals tell us more about our perception than they do about geometry per se, but they do enable us to describe say the effect of a mirror or a rotation etc.so they are necessary.
The flip algorithm is also necessary, but not because it fundamentally alterd the constituents of a geometric structure, but because it fundamentally alters the relationships within a or about a geometrical structure. The flip lgorithm enables us to succinctly describe a motion as left to right, but sometimes we want the long winded method of describing the motion by how much left and how much right , or even how many arc lengths of rotation.
Even with rotation we still flip from one complete rotation to another.
So the natural reference frame is in fact not just an external set of marks, but an internal external symbiosis of measurement and perception, and this seems o gel better with a polar or spherical coordinate system.