Early students are assumed to know the tools of algebra, geometry, calculus, and parts of linear algebra. After a time, the student usually picks up more knowledge concerning linear and matrix algebra, differential equations, complex numbers and various specialty functions and polynomials.
While each of the added mathematical tool sets greatly expands a student's capabilities for problem solving, they tend to get appended through the working of example problems to which they are powerfully adapted. It is rare for a student to face an unsolved problem to which they must find the appropriate tools until they are at the graduate level of their training. Even then, many never face this task.
This paper focuses upon how we use the mathematical tools we choose. It asks the questions 'How do we choose to represent that which we perceive?' and 'How do we know others are perceiving the same objects we are trying to address?' Which tools we use depends a great deal on our knowledge of them and the depth of our experience with them. This paper assumes the question of why we choose our tools has much to do with simplicity, power, and convention and pays it no further attention.
There is historical evidence of changes to the primary tool sets used for the oldest branches of physics. Newton crafted the contents of his Principia with geometry and his new calculus. Modern students use algebra, calculus, and vectors for the same material and usually find the Principia difficult to decode. Maxwell's description of Electric and Magnetic fields was radically different from what is taught today. With modern vectors and matrices, electromagnetism is hard to recognize for those who put forth the effort to read the older texts.
Whether Newton or the others would have used modern tools had they been available is not the question. The fact is, modern practitioners moved away from older approaches in favor of other means to represent much the same concepts. How they did so is worth some consideration because modern instructors rarely bother to teach or even mention the older approaches. Something important has occurred and the existence of these changes suggests it is worthwhile to consider the foundation of our approaches to object representation.
Imagine a sharp, straight stick on the ground. We wish to represent the length and orientation of the stick mathematically. There is more than one tool that can be used, but modern students will almost always reach for vectors once they have learned how to use them. Two will be shown here.
- Choose a reference frame. Pick an origin for it and then write down some information about the directions of your basis vectors relative to each other. You have quite a bit of freedom, but most people pick right-handed systems and make the basis vectors have a length of one unit and stand at right angles to each other.
- Pick one of the basis vectors. Imagine a light bulb that projects a shadow of the sharp end of the stick onto a line that extends through your basis vector from the origin outward. Put a slit in front of the bulb. Arrange the slit so the light it allows out moves toward the stick parallel to the other basis vector you didn't choose. How long would your basis vector have to have been to be long enough to just reach the projected shadow? Write down that number.
- Do the light trick again swapping your basis vectors and write down that number.
- Multiply your first number by the first basis vector and add it to the product of the second number with the second basis vector. This vector points at the sharp end of the stick.
- If you didn't pick your origin at the blunt end of the stick, do both projections and the addition trick again for the blunt end. This vector points at the blunt end of the stick.
- The length and direction of the stick can be represented by the difference of the two vectors. Take the sharp end vector and subtract the blunt end vector and you are done.
- Choose a coordinate system. Pick an origin for it and then write down some information about the directions and scale of your coordinate axes. You have quite a bit of freedom, but most people pick right-handed systems and make the axes stand at right angles to each other.
- Draw out a coordinate grid using the axes you chose. Find the coordinates of both ends of your stick.
- Find the straight line equation that goes through both of the points you found earlier.
- The stick is represented by the straight line function over a domain and range limited to the span between the two coordinate points. The distance formula between two points gives the length while the domain and range spans start at the blunt end to give the orientation.
These two approaches may sound the same, but they are not. In the first, a vector represents the stick. In the second, a function represents the stick. Both of them have different descriptions for the reference frames used. There are some similarities, of course. This shouldn?t surprise anyone since both techniques are being used to try to describe the same properties of the stick
If the reader is still unsure that these two techniques and their supporting tools are different, imagine expanding the example in the following way. Suppose we also try to describe the size and orientation of the cross-sectional area of the same sharp stick. Vector proponents would be tempted to write another vector oriented normal to the cross-sectional area. Algebraic proponents might be tempted to do the same, but they could also represent the functional equivalent of the cylindrical or conical surface area surrounding the stick and use formulas to demonstrate the cross-sectional area. Both techniques work, so one cannot claim to be superior than the other.
For the purpose of further discussion, we propose three postulates to act as an abstraction of the representation techniques and intentions of the users. The postulates make no attempt to describe any physics. Instead, their purpose is to address how we recognize and represent physically interesting things for our physics theories to address. If the abstraction works, it should be possible for the reader to write examples similar to the one above and see their techniques described by the postulates.
1: Completion terms
Complete Objects are represented as a list of observables representing each of the independent properties of an Object.
2: Rendering of representations
Properties are rendered as observables through representations as combinations of components in a suitably complex mathematical tool.
3: Identity terms
Representations of an observable describe the same property of an Object if under an agreed upon family of passive transformations suitably defined for the mathematical tool, all Observers can passively transform their rendition into the renderings of the other Observers. This is not invariance.
An Object is a physically interesting thing. It is the subject of theory and experiment. It is a part of reality singled out for further attention.
A property is a part of an Object describing some definable portion we might observe and test through experimentation.
An observable is related to a property and is the direct result of a rendering of that property into an appropriate mathematical tool. The observable is what actually gets tested in experimentation because the experimenter must make assumptions about the property being tested. Those assumptions are embodied in the observable.
Independence is a concept useful for distinguishing properties of an Object from each other. An example of two properties that are not independent of each other might be color and the reflection coefficient as a function of energy for a body bathed in sunlight.
A component of a math tool is defined relative to that tool. In geometry of two dimensions, line segments, area segments and points make up some basic components. In algebra, polynomials of order N make up an unending list of components. In a geometric algebra, the generators of an algebra and their products make up the available components.
A combination is a sum and products of components scaled by elements from a field such as real or complex numbers. If this definition were limited to sums, it would be equivalent to the standard meaning used for 'linear combination.'
A representation is a particular rendering of an observable within the mathematical tool of choice. There are often many ways to represent the same observable.
A passive transformation is one that alters the representation of an observable without doing anything to the observable itself. These transformations are usually more a measure of what the Observer is doing that changes the representation than anything else.
An Observer is someone or some thing that can definitively answer questions about their representations of observables. The typical observers are the experimenters trying to understand an object.
Postulate Support and Explanation
Postulate one is largely about knowing how to decompose an object to parts that may be rendered and tested separately, but it will most often be used in the other direction. It is possible and quite likely that observers will initially fail to understand the full scope of an object. Without this understanding, initial representations are likely to be partial. A strategy for completing a representation, therefore, is in order.
The completion strategy is to form a list of properties known to be associated with an object. This is what is done today. The length of the list depends on the experience of the observer and in what properties they are interested. The appearance of the list depends on the tool to be used. Most practitioners simply write a natural language list of the representations of the expected observables. To many, initially including this author, this postulate is assumed and goes by without thinking.
Postulate two addresses the issue of how properties are rendered as observables. Each mathematical tool is different enough to make a general discussion of representations quite generic and of little interest here. The proponents of the vector technique in our sharp stick example earlier rendered the length and orientation during steps two through four. The correctly sized components were found in the first two steps while the combination was created in the last one. The proponents of the algebraic technique rendered the length and orientation during step three. The components were suitably scaled monomials (order one and order zero) while the combination was the sum that created the straight line function.
Postulate three forms the basis of a strategy that may be used by multiple observers who wish to know if their renderings describe the same object. The strategy is not fully deterministic. Probabilistic knowledge, therefore, can enter into the concept of object identity long before it enters into the physical theories using the representations.
Consider our earlier example of the sharp stick. Imagine further that many observers create representations of the length and orientation and share them with each other. How does Observer A know that the representation offered by Observer B addresses the same properties of the same stick? Do they simply trust each other? Is there a method for creating identity in the representation that may eliminate the need for trust? In practice, trust is used, but it tends to undermine the purpose of experimentation in the scientific method.
The notion of identity can be added to the representation through an agreed upon family of passive transformations shared by all observers. Ask an observer to render the observable again and a different outcome may occur. After all, the observers were free to choose a reference frame before finding a suitable linear combination for their vector. Different representations would occur for each choice of reference frame. If one observer is asked to create many different representations of the same observable, another observer could discover the passive transformation family used by the first observer through a careful study of the renderings. If that family is found, and there exists a transformation in that family that turns the representation written by observer B into one written by observer A, observer B may be reasonably confident both renderings refer to the same object.
Postulate three does not lead to a bulletproof concept of identity. If one observer produces a representation that fails the test, the other observers can be certain that particular observer rendered a different property or object. If all representations pass, though, all the observers can say is they are pretty sure the renderings refer to the same object. The higher number of representations that pass, the higher the odds are that they do refer to the same properties of the same object.
Any observable that proves to be invariant under the family of passive transformations used by all observers obviously qualifies to represent an object seen by all observers. The only block to knowing absolutely that a given representation refers to an object is the possibility that the observer might have been looking at a similar object elsewhere. Are you and I looking at the same sharp stick? What if we were instead trying to describe an electron?
Issues of identity are rarely raised if each observer sticks to representations that are invariant under the agreed upon transformations. This suggests that invariance is a suitable stand-in for identity much of the time. It also suggests that observers will choose between two equally powerful tools in favor of the one where invariance is most easily found and demonstrated. This last statement depends largely on the desire of most practitioners to avoid unnecessary work.
Whether the identity of an object is fully agreed upon by all observers or not, the invariance of a representation can be determined with certainty by any one observer and then promised to the others. The mathematical tool offers this certainty, so no trust is needed.
This promise between observers is probably at the root of how we choose our representations. Any supporters of newly developed tools must be able to offer at least this much if they hope to have their work displace older techniques. To test this notion, we can ask if the newer tools offer an equally strong or stronger sense of identity to the objects Newton originally represented with geometry for Principia. If the answer is yes, this premise survives its first test and might be worth consideration by a larger audience.
Consider Proposition 36 of Book 1.
(From Cajori's revision of Motte's translation)
To determine the times of the descent of a body falling from a given place A
Upon the diameter AS, the distance of the body from the centre at the beginning, describe the semicircle ADS, as likewise the semicircle OKH equal thereto, about the center S. From any place C of the body erect the ordinate CD. Join SD and make the sector OSK equal to the area ASD. It is evident (by Prop 35) that the body in falling will describe the space AC in the same time in which another body, uniformly revolving about the centre S, may describe the arc OK.
(Proposition 35 addresses the notion that equal areas are swept out in equal times on a gravitational orbit.)
Newton used proposition 36 in book 3 to show how the rate at which the Moon falls toward the Earth can be directly formulated from our knowledge of how masses accelerate here on the surface of the Earth.
Early physics students learn a similar lesson when they are taught about gravity. These students are taught to replace geometric line segments with distances calculated from a Cartesian or similar coordinate system. The areas swept out (sectors OSK and ASD) can be shown too, but often aren't. The algebraic formula for the force of gravity is used directly in modern texts and the approach is considered sufficient.
Each of the objects represented in Newton's proposition can be translated to modern renderings using algebra. In each case, care can be taken to use the typical coordinate systems taught in the texts and the identities of each object get preserved. The invariants are much the same as they were with the geometric description. Lengths and angles do not get altered between observers. Time ticks at the same rate for all, else proposition 35 would fail. Young students are taught not to violate those invariants whether they realize it or not. In their acceptance, then, the concept of object identity is likely to be preserved and different observers can be confident in their use of each other's renderings.
Anyone who knows special relativity well enough should be able to see the beginnings of how revolutionary the new concepts were. The two postulates of special relativity strike directly at the heart of the invariants used in Newton's work. The representations written in tools capable of supporting special relativity fundamentally alter what a property is, let alone how its observable behaves.
How we choose to represent our objects can be summarized by the creation of a trustable identity for the interesting properties. Invariance provides the root for trust between observers. The rest of the rules concerning representation are technique that varies from one tool to the other.
As long as the practitioner takes steps to represent complete properties for objects whose identity can be trusted by other observers, they have done what the community expects of them. Most everyone except the users of statistical mechanics and its related subjects largely ignore the probabilistic nature of the identification.