Group Operation and True Meaning of Mapping

[Rising Eagle]

Can't find (or maybe recognize when I see it) anything that discusses this question:

A group G is a set of members. We normally assign familiar labels on the members such as a five member group with members labeled as 0, .. , 4. Then, a group operation + is defined as GxG -> G so that a look up table can specify exactly what it means:

+ | 0 1 2 3 4
-----------------
0 | 0 1 2 3 4
1 | 1 2 3 4 0
2 | 2 3 4 0 1
3 | 3 4 0 1 2
4 | 4 0 1 2 3

I have a variable X that is characterized as manifesting (incarnating?) this group. It is said to be able to stand in for or represent any member of this group and any particular value may be substituted in at any time. I think of this variable as a viewport that can zero in on anyone member at a time. Now I create a second variable Y standing in for the same group as X and I want to add them:

X + Y = Z

How is this operation interpreted? Are X and Y each a variable representing separate and independent incarnations of the same group blueprint (essentially twins) where Z would be a new variable representing a third incarnation of the same group? Then the group op is an external mapping that dictates a rule that allows a member from one group G1 to interact with a member of a second group G2 in order to map onto a third group G3.

On the other hand, are the variables X, Y, and Z members of the one and same group G, each being able to take on any value independently from the exact same group (each is an independent viewport with the same vantage point onto the same set of group members and can focus in on anyone desired without influence by the other two variables)? Then the group op is an internal mapping that spells out a rule permitting any pair of members within the same set to interact and combine, and magically become a substitute in (become an alias for?) for a third member. If this is the case, the set is not just 5 members, but also includes every possible combination of two or more members as aliases for the original members. This, then, would be an alternate way of visualizing the structure of the group as specified by the group op. Also, in this case, G is not a blueprint or declaration, but an actual working mathematical entity itself.

If both internal and external mappings are permitted with the same group op mapping rule, then does this point out the difference between an automorphic and homomorphic mapping?

Any further insights about when and why to interpret what the group op means would be very helpful. Thx

Eagle

 

[SteveL27]

If both internal and external mappings are permitted with the same group op mapping rule, then does this point out the difference between an automorphic and homomorphic mapping?

Any further insights about when and why to interpret what the group op means would be very helpful. Thx

You're throwing in groups to make this question unnecessarily complicated.

Ask yourself the same question about regular old numbers. If I'm working with, say, the real numbers, and I say x + y = z, where x, y, and z are real numbers, do you have the same conceptual issues about the use of variables? What exactly is the issue? The variables x, y, and z stand for arbitrary real numbers. Same as with any group ... in fact the real numbers under addition are a group. Right?

Given that the plain old real numbers are group; and that you've been writing x + y = z ever since high school algebra, do you still have the same issues?

 

[Stephen Tashi]

The question "What is a variable?" is a good question and hard to sort out rigorously. We develop a facility in using variables as "unknowns", thinking of them as having definite but unrevealed values. We also come to accept them as being universal symbols that can take any value - for example, a statement such as "x + y = y + x" can be offered as a universal law for numbers. By the time we reach college level math, we are used to statements like "Let c be a constant", which apparently removes the "variable" nature of c, but keeps its status as a universal symbol.

If you want to obtain a clear view of what variables are, I think you must study mathematical logic - in particular, the logic of "quantifiers". This will reveal that the customary ways of speaking about elementary math are informal and unspecific in their use of logical quantifiers.

 

[Rising Eagle]

Thank you both Steve and Stephen for venturing an answer and for your insights. Stephen, I feel like you have truly read my mind. Though the answer is still out there, you have let me know I'm not the only one who sees this issue. And thank you again for pointing me to a good place to look.

Steve, I, of course, reflected on your thought experiment and, unfortunately, found no answers of the kind I am looking for. Please don't think I take your suggestion lightly or with cavalier attitude; my inquiry is a serious one. Rather than complicating the question, my presentation of a group is a very simplified example which I believe highlights the nature of my question clearly and accurately.

It has been my experience that the presentation of math and physics, even at college level, is plagued with unanswered questions and sources of ambiguity. We are taught terse, but convincing hand-waving explanations (or is it just very good and distracting slight-of-hand?) and tons of problem solving recipes, so that we gain so much comfort with the procedures, we begin to confuse mastery of process for true knowledge and understanding. We forget that we have unaswered questions at the lowest core level and just try to get the right answers on the exams and then forget about it. I also believe that without true understanding of the mechanical parts that underlie our knowledge of physics, the insights leading to advancement of the science and understanding of the higher theories will escape us. Such capability should not be the exclusive domain of the elite geniuses, but should be accessible to us mere mortals. And it will be if the details are properly presented and explained.

Allow me to present more on why I ask the question. It turns out that, contrary to how regular numbers are taught to us, all counting systems (numerical, algebraic, analytical) are based on sets of abstract members. These members are then assigned lables that allow us to write down and refer to (index into) each member. Then there are axioms, properties, and mapping rules that tell us proper, allowable usage and how these members interact. Mappings internal to the set are used to develop theorems that further characterize the set. Mappings about external interaction with other sets are used to model external processes seen in our world, such as economic or scientific models. Fundamentally, numbers and variables in our models and recipies are abstract members of sets which are assigned labels of various kinds and which are permitted to follow various rules of interaction and substitution. Thus 3 + 5 = 8 is not about numbers of apples, but about a substitution rule that says the squiggle 8 and the squiggle 3 + 5 are one and the same. The fact that we define an apple as a unit is where physics comes in. Then I can say 3 Apples I already had + 5 Apples my mom gave me = 8 Apples I now have that I can share with my brothers and sisters.

This is just the beginning. When we realize an apple unit is an intuitive notion we understand in our heads, and that our brain has become wired to understand and accept this notion, we realize the numbers are actually just numerical label assignments to a physical comparison (measurement) against our notion of a unit Apple which is being used as an abstract standard of reference. 5 Apples is a measure 5 times greater than the abstract, mind-based 1 unit Apple we use as a standard of reference. This is the fundamental beginning of all physics and how physics arises: as comparisons (measurements) between physical effects and standards of reference. The notions in our heads that are the intuitive understanding of these standards of reference are the very foundation of physical knowledge upon which all other physical understanding is built. The math behind these measurements (i.e., sets and rules) are the simplest physics models. Then physics goes beyond measurement models to make predictive models that are very complex. It goes deeper, but we can already see how the use of numbers is guided by mechanics that are fully and completely defined in the abstract and how understanding these mechanics is necessary to understanding physics models.

One source of confusion is the idea of an Affine quantity vs a Vector quantity. Temperature (a measure of average kinetic energy) is an Affine physcial quantity. It has an absolute zero. But its zero point (base point/reference point) can be arbitrarily changed. There is one temperature scale, for example, that uses the freezing point of water as the zero temperature. A point in an Affine space can also be combined with a change to yield another Affine point as in 3 degrees C + 12 degrees delta_C is 15 degrees C. Mass is also an Affine quantity as is any physical coordinate space. I have seen many people take a Vector space and visualize it as a cartesian grid and treat it like it is an Affine coordinate space where particles can be located at various positions inside there. This confusion goes to the heart of my question. Every variable of one type or another (e.g., Affine or Vector) has defined usage and we must learn to distinguish the usage of each type from every other. We must also learn to recognize which type is at play when we see a result, lest we draw wrong conclusions about the behavior it represents. This is why I need to understand what the mapping rules really mean.

Now consider F = ma. This physical model shows the Vector quantity F (force) is a mapping from the Vector quantity a (acceleration) scaled by the Affine conversion factor m (mass - Force needed to be applied to the mass/unit acceleration to be experienced by the mass - Force/unit acceleration). In this case, the acceleration space is mapped to a new space, the force space, as mediated by the map defined as a scaling of a (acceleration) by m (mass). An alternative way to look at this is that the acceleration space and force space are one and the same Vector space with a simple change of basis where the basis vector unit of one Newton (force unit) replaces the basis vector unit one meter/sec^2 (accel unit) as effected by the conversion factor m (mass - kg). It may seem like splitting hairs, but the disinction is still important.

The m space (mediator of the map to the new space or the conversion factor for change of basis, whichever applies) is an Affine quantity. The basic rules of mapping equipment built into an Affine quantity does not include the ability to scale a Vector. The scaling of Vectors is done by a variable of type Field of Reals with a numerical domain stretching from -infinity to infinity. This is also an unaswered question. Some will argue that what I am calling Affine quantity is a misclassification and that in physics they are all Scalars. In physics, a Scalar is any quantity that is not a Vector (having magnitude and direction). A Scalar physical quantity is not defined rigorously enough to characterize its proper usage (other than the rule: use it any way you want as long as you get the right numerical result). So, maybe the foundations of the algebra for physics needs to be upgraded to include algebraic structures whose defined usage fits properly with the algebraic needs of physical models. Perhaps we need to define something as a Field or Ring or Affine of Reals with external mappings supplied for multiplying/scaling or otherwise interacting with other spaces, such as Affine, Vector, or Field, or Ring. Something that covers all the cases left out or improperly handled by the currently available variable types.

As our third and last example, consider an Affine space coordinate system with two particles, each with trajectory x1(t) = a1*t^2 + b1*t + c1 and x2(t) = a2*t^2 + b2*t + c2. Note that coefficients a, and b are vectors, while the coefficients c are Affine qunatities, all with appropriate physical units (meters/s^2, meters/sec, meter), and yet here again there is no mapping rule for quantities (a,b,c) to be scaled by the Affine quantity (t). We can see that each particle is an Affine quantity as we can show the position x1(t) = (a1*t^2 + b1*t) + c1 is a sum of an initial position combining with a change in position as shown in (a1*t^2 + b1*t), which varies as the Affine parameter t (time) varies. Note that this shows an internal mapping manifested as a point (location) in the space combining with a Vector quantity to yield another point. This internal mapping is well defined for an Affine space. Furthermore, notice that the physical coordinate space specifically accommodates the presence of more than one particle (as incarnated in x1 and x2) simultaneously. In my initial question, I asked about this very property of different variables in same space or different variables in different spaces. In the Force space and acceleration space, there is no accomodation for two different forces or accelerations to exist in the space. I wish to know the rules that determine what is and is not permitted for the various spaces. Now let us define the Vector that translates the location of the first particle to the location of the second (Vd = x2(t)-x1(t)). Vd = (a2-a1)*t^2 + (b2-b1)*t + (c2-c1). This, once again, is an internal mapping when added to the location of the first particle x1(t).

In the Force example, the mapping could be either internal or external. In the coordinate space example, the mappings are clearly internal. And we could show an external mapping in the coordinate space example by showing a warping of the space or some other exotic transformation in the nature of the space. In more complex models, which I wish to master and extend, these distinctions may not be so clear and easily discernable. I hope this explains why this is such an important question to clarify.

Eagle

 

[Rising Eagle]

I would like to invite Mathwonk and other elite members to weigh in on this question.

 

[mathwonk]

qrrrghhhh. ... I'm very sorry but I guess I am am old and cannot quite grasp what is happening here, but I did try, at least briefly to read the question. But it has big words in it. ... I have always wondered what a variable is. But i wanted you to know I did not totally ignore your request.

 

[Diffy]

I don't get it. In your example you say:

have a variable X that is characterized as manifesting (incarnating?) this group.

Isn't it just the case that x is representing a member of your group. And which member it is does not matter.

A variable is a tool we use to gain abstraction. Depending on context there are conditions of values that are allowable for the variable to take on. In a sense, variables are sets with conditions.

On the other hand, are the variables X, Y, and Z members of the one and same group G, each being able to take on any value independently from the exact same group (each is an independent viewport with the same vantage point onto the same set of group members and can focus in on anyone desired without influence by the other two variables)?

Well, specifically in this case they aren't able to take on any value independently. There is a dependence if x + y = z isn't there. They can take on any members from the group so long as this equation is satisfied using the group operation.

And you go on..

Then the group op is an internal mapping that spells out a rule permitting any pair of members within the same set to interact and combine, and magically become a substitute in (become an alias for?) for a third member. If this is the case, the set is not just 5 members, but also includes every possible combination of two or more members as aliases for the original members.

Yes the group operation is an internal mapping. I have no idea what you are trying to say beyond that.

The group is 5 members. By definition, it includes every possible combination of those 5 members under the operation. That is closure which is required by groups.

Whether you call the elements 0, 1, 2, 3, 4 or phone, mouse, hair, folder, shirt, it doesn't matter. The group has a structure. It is laid out in your Cauchy table. When you have a variable x, that represents a member of this group. It has 5 possible values. And when you have a second variable y that also represents a member of the group it also has 5 possible elements. And if you are adding your variables you are representing again, a member of this group.

Because if x is in the group and if y is in this group then we know by definition the sum is in the group. It is possible to chose values for x and for y such that we could make the sum any member we want. But we haven't chosen them. All we know is that their sum is some member of the group. So we call that member z.

I hope this helps, but I have a feeling it won't :-)

 

[Rising Eagle]

qrrrghhhh. ... I'm very sorry but I guess I am am old and cannot quite grasp what is happening here, but I did try, at least briefly to read the question. But it has big words in it. ... I have always wondered what a variable is. But i wanted you to know I did not totally ignore your request.

I tend to ramble and my math explanations aren't always clear. I thank you for giving it a try. I had an idea in my head that isn't discussed in textbooks. Let me give the briefest skinny I possibly can.

Background: Defining Internal and External Mappings
To me, a group op is what I would term an Internal Mapping.

It occurred to me that there must also be External Mappings. An example of an External Mapping would be: given an Affine Space A and Vector Space V, we have an External Mapping AxV -> A, the summation of an Affine point with a vector from the External other space V to yield another Affine point that is, in this case, Internal to A.

In physics, the simple law force = m*a is an External Mapping where given a Scalar Set M and Vector Space A maps to the completely new and External Vector Space F: MxA -> F.

Premise:
The Scalar Set (Field of Reals) M has Internal Mappings defined (the + and * ops). The two Vector Sets A and F each has an op which is an External Mapping that allows a scalar from Scalar Set S to scale a Vector to yield another vector that is Internal to the Vector Space: SxV -> V (i.e., SxA -> A and SxF -> F).

Question:
The force law is an External Mapping that is not defined explicitly in the Scalar Set M or the Vector Sets A and F to make this calculation allowable. What makes this calculation legitimate?

This leads to my starting this thread. I wanted to know: if variables are defined to represent a space, is their usage confined to carrying out the Internal Mappings defined within the space, or is there a larger picture that I do not understand? I chose Groups as an example to distill the question to something more precise. I'm afraid my presentation turned out to be vague.

Please come back and try again. Please try to make sense of my first post and see if it makes more sense now.

 

[chiro]

I have to sympathize with Diffy on this one.

A variable is going to be something that takes a range of different values, usually from a set that has been either defined with a mathematical-type definition or with a non-mathematical type definition. The mathematical definition might be pick all real numbers between 0 and 1 inclusive and the non-mathematical definition might be {0,4.5,6.123,10.2}.

Mathematics provides a general way to deal with variation as a general thing by introducing arithmetic and algebra so that we can effectively treat things with variation like numbers and use things like functions and operators to do stuff on those functions.

What the number actually means depends on the context of the variable and what it relates to. It may not relate to anything that is real, but it still can be used in the context where we wish to do computation, algebra, and other related processes on these variables.

The most general form of a function simply maps something to something else. You do not necessarily need to use a mathematical definition that is compact: you can use a table like your example and that is OK.

Also you do not even have to use numbers, but you can use symbols that are not numeric and use a table to represent how you transform pairs of symbols to get another symbol in a group: this is ok as well.

Again the reason for mathematics to use numbers is because you end up, in many situations make things not only easier to describe, but easier to analyze. It's a lot easier to analyze X + Y = Z than having a huge table with a bunch of symbols that are non-numeric and don't really any ordered or ranked information about the symbols. For this reason we often consider how to generalize particular mappings based on compact forms that using the arithmetic operations and also the power operation as well as introducing other complex functions like exponentials, logarithms, and so on.

The other thing ultimately is that for your mapping, a calculable answer for the domain of the function must always exist. If it doesn't then it's kind of pointless. It also must be unique.

Because we use the arithmetic as a basis for this kind of thing, we end up doing things like not allowing division by zero if we resort to using arithmetic as a basis for representing a general mapping.

I really wouldn't think too much beyond this to be honest. As long as you use a way of both representing your variables in terms of proper set definitions, define your mapping in terms of proper set definitions and make sure that there are no contradictions along the way, and are able to genuinely calculate your mapping whether that is using a table, something based on arithmetic and other algebra (or both), then that is the most important thing.

If there are any contradictions though, then this becomes an issue and this is a separate matter.

 

[SteveL27]

This leads to my starting this thread. I wanted to know: if variables are defined to represent a space, is their usage confined to carrying out the Internal Mappings defined within the space, or is there a larger picture that I do not understand? I chose Groups as an example to distill the question to something more precise. I'm afraid my presentation turned out to be vague.

Please come back and try again. Please try to make sense of my first post and see if it makes more sense now.

Let me give this a try. In math, a function is defined as a particular type of association of the elements of two sets. We say f:A -> B, or in words, "f is function from domain A to range B," if three conditions hold:

1) f is a set of ordered pairs (a,b), where a is an element of A; and b is an element of B. Alternately, we can say that f is a subset of the Cartesian product A X B.

2) If (a, b1) and (a, b2) are elements of f (remember, f is a particular set, so we can speak of its elements) then b21 = b2. That is, f sends each element to exactly one thing.

3) If a is an element of A, then there exists some b in B such that (a,b) is in f. That means that everything in the domain A gets sent somewhere by f.

From now on if (a,b) is an element of f, we write f(a) = b.

That's it. That's what a function is in mathematics. We are agnostic about the use or "meaning" other people might give to functions. A Newtonian physicist considers the path of a point through space to be a function of time. An economist considers price to be a function of demand and supply.

So if you can separate the mathematical notion of function as a mapping, from any particular physical application you have in mind, that will very helpful.

Now, there are some special cases of interest. If f:A -> A then the function "stayed inside" as it were. The function just sends each element of a set to some other element of a set. Like sending each real number to its square.

And if f:A X A -> A then we call f a "binary operation." In this face, f takes a pair of elements of A and gives you back an element of A.

+:R X R -> R is a formal way of denoting the functional nature of the usual addition of real numbers. When we say 2 + 3 = 5, we really mean that + is a function that maps the pair (2,3) to 5.

Of course a group operation is an example of a binary operator too. That's your internal mapping.

Now perhaps in your physics example you have three sets A, B, and C, such that you can combine an element of A and an element of B to give you an element of C. So you'd write

f: A X B -> C.

You might call that an external mapping, but there's really less there than meets the eye. All you're saying is that you're calling a mapping internal if the function's a binary operator or the range equals the domain; and external if the range is different than the domain.

To sum up: Mathematics does not ascribe any meaning to the notion of function beyond what I've described here. A mathematical function encapsulates what we think of as a "mapping" and it does a good job of it.

Physicists use the idea of functions all the time. But you can get into trouble if you try to apply too much of your physical insight to the underlying math. The variables in a physics equation might have some relationship to each other; but in general, a mathematical function's domain and range are just abstract sets, with no other characteristics at all.

Regarding your internal/external distinction, it seems to me that you're just talking about whether a function f goes from A to A or perhaps A X A to A; versus going from A to B. But the distinction doesn't go any deeper than that. I don't think there's any philosophical significance.

As far as the variables, they just range over the underlying sets, whatever they are for any given application.

Well that's way too many words but I hope some of it was helpful.

 

[Rising Eagle]

Isn't it just the case that x is representing a member of your group. And which member it is does not matter.

Yes that is the presumption. I used careful (though obviously confusing) language because often variables are presented as being from one type of set or another (e.g., some represent groups, some field of reals, some vectors of various kinds). I wanted to generalize the description I use, though I think I failed to be clear. I'll try to refrain from pompous, formal language.

A variable is a tool we use to gain abstraction. Depending on context there are conditions of values that are allowable for the variable to take on. In a sense, variables are sets with conditions.

This is a good concept I have not seen before, but I like it very much. In other words the domain of representation may actually be a subset of the type of set it is assigned to represent.

Well, specifically in this case they aren't able to take on any value independently. There is a dependence if x + y = z isn't there. They can take on any members from the group so long as this equation is satisfied using the group operation . . . Yes the group operation is an internal mapping. I have no idea what you are trying to say beyond that.

What I was driving at is the idea that the internal ops don't do enough to explain usage of the variables in external maps. Let's just say that mass is a measurable physical property that is well modeled by an additive group of reals. Now consider a situation where we have two objects m1 and m2 placed some distance apart. In an elastic collision, they come together and merge and their masses add: m1 + m2 = m3. Are we to view the two masses m1 and m2 as being from the same group, or two masses from two separate, but identical twin groups. If they are from the same group, the mapping is well defined because it is simply the group op of addition. However, if they are from different groups and the m3 is also from a third, but yet again identical group, it is an External Mapping which is not defined within the group. The group is what defines all properties and mathematical uses of the variables m1, m2, and m3, so if the latter case is true, I need to know how the summation (or external mapping) becomes defined and allowable. Where is that usage defined?

The group is 5 members. By definition, it includes every possible combination of those 5 members under the operation. That is closure which is required by groups.

Whether you call the elements 0, 1, 2, 3, 4 or phone, mouse, hair, folder, shirt, it doesn't matter. The group has a structure. It is laid out in your Cauchy table. When you have a variable x, that represents a member of this group. It has 5 possible values. And when you have a second variable y that also represents a member of the group it also has 5 possible elements. And if you are adding your variables you are representing again, a member of this group.

Because if x is in the group and if y is in this group then we know by definition the sum is in the group. It is possible to chose values for x and for y such that we could make the sum any member we want. But we haven't chosen them. All we know is that their sum is some member of the group. So we call that member z.

Yes, thank you. That confirms something I know very well. And when you say "The group has a structure. It is laid out in your Cauchy table.", I visualize the structure as a set in a way that allows me to dispense with the table. Imagine a Set where each original element (in this case the 5 elements) is surrounded by a cluster of every possible additive combination of 2 or more members of the group that add up to that original element. So I have 5 clusters, each containing all expressions that may substitute in for each other in a set that is now countably infinite. This view of the structure works equally well for fields and vector spaces and all other axiomatically described sets.

"now countably infinite": Somehow this all ties into number theory, but I don't even want to touch that one.

I hope this helps, but I have a feeling it won't :-)

Actually it did. Very much so. I do still have some open questions about the usage and interpretation of variables that I would like to get to the bottom of.