{"id":491,"date":"2016-04-01T23:30:43","date_gmt":"2016-04-02T03:30:43","guid":{"rendered":"http:\/\/aristotle2digital.blogwyrm.com\/?p=491"},"modified":"2016-04-01T22:10:24","modified_gmt":"2016-04-02T02:10:24","slug":"frames-and-systems-of-reference","status":"publish","type":"post","link":"https:\/\/aristotle2digital.blogwyrm.com\/?p=491","title":{"rendered":"Frames and Systems of Reference"},"content":{"rendered":"

Thinking about other points-of-view is a proven strategy for more clearly defining what a concept is and what it isn\u2019t.\u00a0 The heart of the Socratic Method involves repeatedly changing perspectives along this line.\u00a0 Each dialog employs an operational approach roughly rendered as \u2018yes, suppose we look at the matter with this definition, what do we find\u2019.\u00a0\u00a0 Following this line of questioning diligently and in a disciplined manner strips away more of the accidentals and allows a sharper picture to emerge of the essential nature of the idea in question.<\/p>\n

Such an approach is also useful for sharpening the thinking involved in modeling physical or mathematical objects.\u00a0\u00a0 Steps forward in science, particularly physics, comes about often from a cleaner definition of just what some primitive object involves.<\/p>\n

Oddly enough, I had the good fortune to be involved in two separate and unrelated discussions this past week about the essential natures of the points-of-view used to describe the physical world, which, in the physical sciences, are always referred to as reference frames or coordinate systems or some closely similar phrasing.\u00a0 \u00a0The resulting dialogs certainly helped me to see better what the physical sciences really know about frames and systems of reference.<\/p>\n

To set the stage, a disclaimer is in order.\u00a0 As far as I can tell, there is no universal agreement about how to define a reference frame, or how, exactly, it differs from a coordinate system and the associated measurements.\u00a0 This lack of uniform definition points to some deep issue \u2013 either epistemological or ontological \u2013 about the nature of space and time and how humans perceive these things. \u00a0\u00a0One part of the reason seems to be that the operational concepts are so primitive that we have only a basic notion, in many cases, of how to describe it.\u00a0 I liken it to being able to drive a car or ride a bike but yet be unable to describe how to do these things to someone who can\u2019t.\u00a0 But I think that there is an even bigger reason that speaks to how we divide the world up into categories and how we identify the essentials from the accidentals.<\/p>\n

To make this last point clearer, let me concretely discuss my definitions of reference frame and coordinate system and then point out how one may logically use these definitions to come up with something akin to a contradiction.<\/p>\n

A reference frame is a physical object possessing a point-of-view.\u00a0 The prototype is the human being so defined to have the essential parts of a set of limbs to move about, eyes to look, a mind\/brain to process, a mouth to speak the results, and ears to listen.\u00a0 Even when physics speaks about inanimate objects there is, lurking in the background, the notion of what an observer would see were he a disembodied spirit moving along with or sitting upon the object (such is the nature of our imaginations and how we understand the world). \u00a0A convenient abstraction is that a reference frame is any object that has a definite place, which possesses three (independent) directions that it can use, in combination, to point at something, and which has some measure of scale.<\/p>\n

\"frame_of_reference\"<\/a><\/p>\n

Now suppose something of interest comes into this objects field-of-view.\u00a0 As a reference frame it can point towards the object and can denote how far away the thing of interest is.\u00a0 By convention, our primitive reference frame object will adjust the length of the direction to the thing of interest, making the length of the arrow along the direction longer or shorter in proportion to the distance. Thus we have defined a traditional position vector with respect to our primitive reference frame.<\/p>\n

\"vector_in_the_frame\"<\/a><\/p>\n

Note that the notions of direction and distance are also primitive concepts with no easy way to define them in terms of other, simpler things.\u00a0 Also note that there are no names for the directions yet nor is there any developed idea of how to specify these directions or distances mathematically.<\/p>\n

The next step is to remedy this short coming because being able to measure and compute and reproduce values are vital ingredients to understanding the world.\u00a0 The remedy involves giving the primitive reference frame basic measuring tools.\u00a0 For this discussion, the ruler, the clock, and the protractor will suffice and the generalization to more sophisticated modes shouldn\u2019t be too hard.<\/p>\n

Using only rulers, we can decorate the primitive reference frame with a set of planes, each possessing a ruled grid of lines and spaced with a known distance.\u00a0 One such configuration is shown below.<\/p>\n

\"Cartesian_coordinates\"<\/a><\/p>\n

The thing of interest is then specified by the labels specifying on which plane and within which cell is it located.\u00a0 Other objects can be as deftly located and thus we arrive at a coordinate system \u2013 an instance of the Cartesian coordinate system to be precise.<\/p>\n

Two important things are worth noting.\u00a0 First, in this scheme, the reference frame possesses the coordinate system \u2013 we\u2019ll return to this point below.\u00a0 Second, the coordinate system is arbitrary.\u00a0 The planes shown above were oriented so that their edges coincide with the reference frame directions but this choice is no better or worse (at least philosophically) than any other.<\/p>\n

Indeed, the whole idea of using planes, regardless of alignment, can be abandoned altogether.\u00a0 Instead, we could have chosen to use concentric spherical shells of different radii with great circles and latitude lines (taken as primitive notions) drawn on them.\u00a0 The protractor is now our tool of choice and the result is the spherical coordinate system.\u00a0 One such shell of one such instance is shown below.<\/p>\n

\"spherical_coordinates\"<\/a><\/p>\n

In terms of these shells, the location of the thing of interest would be specified by stating on which shell it lies and by giving the great circle and latitude lines on which it lies.\u00a0 Of course the orientation of the great circles and latitude lines are as equally arbitrary as the alignment of the planes pictured above.<\/p>\n

The whole scheme holds up just fine as long as it is being used operationally.\u00a0 The trouble comes when one starts examining it closely with an eye to first principles (yet another point-of-view).\u00a0\u00a0 Several annoying questions come up which bring into doubt the underlying consistency of the scheme.\u00a0 Some of these are:<\/p>\n