Model - Meaning
Paul Simon in the song “Patterns” on the “Parsley, Sage, Rosemary and Thyme” album writes:
"The night sets softly
With a hush of falling leaves,
Casting shivering shadows
On the houses through the trees,
And the light from the street lamp
Paints a pattern on my wall,
Like the pieces of a puzzle
Or a child’s uneven scroll.
Up a narrow flight of stairs
In a narrow little room,
As I lie upon my bed
In the early evening gloom.
Impaled on my wall
My eyes can dimly see
The patterns of my life
and the puzzle that is me."
Webster’s Third New International Dictionary UNIBRIDGED and Seven Language Dictionary copyright 1976 definition:
1a: The thing one intends to by an act or especially by language. b: the thing that is conveyed or signified esp. by language: the sense in which something (as a statement) is understood.
Like Paul Simon, we are interested in problem solving and will therefore restrict on interest to definition b.
For purposes of this post we shall assume that pattern recognition is a model for meaning.
In Douglas Hofstadter’s book “Godel, Esher, Bach: an Eternal Golden Braid”, Hofstadter introduces something called the p-q system. The system is composed of the letters p, q, and -. In addition he defines a set of rules for combining these symbols. In particular he allows us to write xp-qx- where x is composed of sequences of - only. Finally he adds the rule that if xpyqz, where x, y, and z are sequences of -, is a permitted combination of letters then xpy-qz- is also a permitted combination.
Initially, to most people anyway, this system may seem abstract and without meaning.
However, if we associate the hyphens with numbers, - to one, – to two, and — to three et cetera, and the p to a plus sign and the q to the equals sign, then we can start reading the valid combination of symbols as valid arithmetic calculations such as x plus y equals z.
This association has some specific attributes. First it must associate elements in a foreign setting to unique elements in a personal setting (both - and – can not both associated with one). The personal setting we shall, in keeping with traditional notation, denote as I. In addition it must preserve some property (generally an action, such as p, or relationship, such as q) in the foreign setting with some property (again an action or relationship respectively) in the personal setting. In mathematics an association with these properties is called an isomorphism. The association itself is called a function, denoted by f, the action is generally denoted by F (which can be on multiple elements of the foreign setting which we will denote as set D), and the relationship (which is frequently an equality) is denoted by R. [Compare with homomorphism as defined in Model Theory]
Hofstadter writes that the discovery of this type of association is often “a bolt from the blue”, and a source of wonderment. “And I claim that it is such perceptions of isomorphism which create meanings in the minds of people.” It is with Hofstadter’s slightly more restrictive pattern recognition that I will devote the majority of the text.
Just when you might feel comfortable that you might have discovered something about meaning, Hofstadter then introduces the idea that q might be taken from and p might be =. By checking some examples you can determine that this understanding or meaning is also valid.
At this point the reader might be tempted to simply think that we can not know anything about this foreign structure. Hofstadter leaves us at this point
However, there is actually more to be determined.
You might have already asked yourself if it is really fair to identify these two isomorphisms as different because addition and subtraction are so intimately linked.
Before proceding I will explicitly define some notation. When I write S^n, I will mean the set which consists of n-tupils (x[size=75]1[/size], [size=100]x[/size][size=75]2[/size],…, [size=100]x[/size][size=75]n[/size]) [size=100]such[/size] that x[size=75]i[/size] [size=100]is[/size] an element of some set S for all i where i ranges from 1 to n.
Inverse Operations (Note to readers familiar with mathematics: Inverse Operations are not to be confused with inverse functions. As an example the operation + on R^2 can have +(1,4) = +(2,3) (this is generally read 1 + 4 = 2 + 3) and therefore the inverse function is not uniquely defined.)
The blue copy is my original thought on this matter so the reader should be particularly skeptical. However, the subject matter is simple and probably is already existent. I just don’t know where.
If F: D^n => D (read F is such that D^n is mapped to D then an Inverse Operation InvF[size=75]i[/size], [size=100]where[/size] it exists, will be defined as follows:
Let InvF[size=75]i[/size] [size=100]be[/size] the function from D^n => D defined where F(x[size=75]1[/size],…, [size=100]x[/size][size=75]i-1[/size], [size=100]InvF[/size][size=75]i[/size][size=100]( [/size]x[size=75]1[/size],…, [size=100]x[/size][size=75]i-1[/size], [size=100]x[/size],…, x[size=75]n[/size]),…, [size=100]x[/size][size=75]n[/size]) [size=100]=[/size] x
Note: Let us define f[size=75]i[/size]: [size=100]D[/size] =>D and f[size=75]i/size = F(a[size=75]1[/size],…, [size=100]a[/size][size=75]i-1[/size], [size=100]x[/size],…, a[size=75]n[/size]), [size=100]where[/size] a[size=75]i[/size] [size=100]are[/size] constant elements of D, and assume that f[size=75]i[/size] [size=100]is[/size] not uniquely defined, which is to say f[size=75]i/size [size=100]=[/size] f[size=75]i/size [size=100]where[/size] x[size=75]1[/size] [size=100]is[/size] not equal to x[size=75]2[/size]. [size=100]Then[/size] we will arrive at the conclusion that InvF[size=75]i[/size]([size=100]a[/size][size=75]1[/size],…,[size=100]a[/size][size=75]i-1[/size], [size=100]f[/size][size=75]i/size,…,[size=100]a[/size][size=75]n[/size]) [size=100]=[/size] x[size=75]1[/size] [size=100]and[/size] InvF[size=75]i[/size]([size=100]a[/size][size=75]1[/size],…,[size=100]a[/size][size=75]i-1[/size], [size=100]f[/size][size=75]i/size,…[size=100]a[/size][size=75]n[/size]) [size=100]=[/size] x[size=75]2[/size] [size=100]for[/size] that particular i. Therefore we will require that f[size=75]i[/size] [size=100]must[/size] be uniquely defined before we will allow InvF[size=75]i[/size] [size=100]to[/size] exist.
Additionally if InvF[size=75]i[/size]([size=100]x[/size][size=75]1[/size],…, [size=100]x[/size][size=75]n[/size]) [size=100]is[/size] not an element of D then we will say that InvF[size=75]i[/size] [size=100]does[/size] not exist for that particular i. Examples: if F(x[size=75]1[/size],[size=100]x[/size][size=75]2[/size]) [size=100]=[/size] x[size=75]1[/size][size=100]x[/size][size=75]2[/size]² [size=100]and[/size] D = R because f[size=75]i/size =a[size=75]1[/size][size=100]x[/size]² is not uniquely defined (f[size=75]i/size = f[size=75]i/size), [size=100]and[/size] InvF[size=75]i/size [size=100]=[/size] x[size=75]1[/size] [size=100]-[/size] x[size=75]2[/size] [size=100]where[/size] x[size=75]2[/size] [size=100]>[/size] x[size=75]1[/size].[size=100]is[/size] not defined where D is the Counting Numbers.
The reader should view F in Hofstadter’s p-q system as analogous to addition on the Counting Numbers where F: D^n => D is defined by F(x[size=75]1[/size], [size=100]x[/size][size=75]2[/size]) [size=100]=[/size] x[size=75]1[/size] [size=100]+[/size] x[size=75]2[/size] [size=100]and[/size] as example InvF[size=75]2[/size] [size=100]defined[/size] by x[size=75]1[/size] [size=100]+[/size] InvF[size=75]2[/size]([size=100]x[/size][size=75]1[/size], [size=100]x[/size]) = x. Notice here that InvF[size=75]2[/size]([size=100]x[/size][size=75]1[/size], [size=100]x[/size]) = x - x[size=75]1[/size] [size=100]and[/size] InvF[size=75]2[/size] [size=100]is[/size] uniquely defined. i.e. InvF[size=75]2[/size]([size=100]x[/size][size=75]1[/size], [size=100]x[/size][size=75]2[/size]) [size=100]is[/size] not equal to both a and b where a and b are different. This uniqueness would not be the case if f[size=75]i/size was not uniquely defined. The reader should also note that InvF[size=75]1[/size]([size=100]x[/size], x[size=75]2[/size]) [size=100]=[/size] x - x[size=75]2[/size] [size=100]is[/size] also an Inverse Operation. It is clear that InvF[size=75]1[/size]([size=100]x[/size][size=75]1[/size], [size=100]x[/size][size=75]2[/size]) is not equal to InvF[size=75]2[/size]([size=100]x[/size][size=75]1[/size], [size=100]x[/size][size=75]2[/size]) [size=100]because[/size] x - x[size=75]2[/size] [size=100]is[/size] not equal to x - x[size=75]1[/size].
[size=100]Relative[/size] to Hofstadter’s p-q system, we have the interpretations:
xpyqz or x + y = z. Here p is plus and q is equals.
xpyqz or x = y taken from z. Here p is equals and q is taken from. Hofstadter could have more easily read it from right to left as z minus y equals x where q is minus and p is equals.
Now we can add
xpyqz or y = z minus x. This is read from the middle to the right and then from the middle to the left. Here q is equals and p is minus.
Next I need to show:
Theorem 1: if InvF[size=75]i[/size] [size=100]exists[/size], and f is the isomorphism with a structure F, then f is also an isomorphism with a structure InvF[size=75]i[/size].
[size=100]Specifically[/size] I must show that there exists F[size=75]Ni[/size] [size=100]where[/size] F[size=75]Ni[/size] mapps I^n => I such that f(InvF[size=75]i/size) = F[size=75]Ni/size.
However we know that f is an isomorphism on F and therefore f(F(x[size=75]1[/size],…,[size=100]x[/size][size=75]n[/size]) [size=100]= [/size]G(f(x[size=75]1[/size]),…, [size=100]f/size) for some G. By simply letting F[size=75]Ni/size [size=100]=[/size] InvG[size=75]i/size, [size=100]we[/size] can, with some effort, get:
f(InvF[size=75]i/size) [size=100]=[/size] F[size=75]Ni/size, [size=100]and[/size] therefore if InvF[size=75]i/size) [size=100]exists[/size] then f is an isomorphism on the foreign set preserving these structures.
At this point we have if f is an isomorphism on a foreign set, D, preserving the function F(x[size=75]1[/size],…,[size=100]x[/size][size=75]n[/size]) [size=100]then[/size] there are up to n + 1 isomorphisms on that foreign set and they are all related (the reader should remember that the original isomorphism F must also be counted).
The question then becomes:
Is there are way to precisely state that these isomorphisms are somewhat equal?
In mathematics there are at least two ways to compare subjects that are not exactly equal. One of the ways is to use measure theory, which was introduced in our model of fractals, and the other way is by introducing the equivalency relationship and equivalency classes.
In measure theory we can say that set A = set B a.e. (which is read: almost everywhere) if and only if the measure m is such that m(A Union B) - m(A Intersection B) = 0. The reader might think of the sets (01) and [01]. (Here (01) is intended to denote the set of Reals between 0 and 1exclusive of 0 and 1. Whereas [01] is intended to denote the set of Reals between 0 and 1 inclusive of 0 and 1. Since the only difference between the sets is the endpoints which have no length, relative to Euclidean measure, we can say that they are almost equal. Somewhat counter intuitively there are infinite sets with a greater number of elements than the integers which have no measure. The Cantor set, which is constructed by successively deleting middle thirds, is such an example.
The other technique is accomplished by the use of the equivalency relation. The equivalency relation ~ compares two objects and states A ~ B if and only if the following relationships hold:
A ~ A
A ~ B implies B ~ A
If A ~ B and B ~ C then A ~ C
The reader should note that ~ is similar to = except that the comparative items need not be identical.
Here we will define an equivalency relation for a given function F as follows:
A ~ B if and only if one of the following conditions is true:
Case A: A = F and B = F
Case B: A = F and B = InvFi for some i
Case C: A = InvFi for some i and B = F
Case D: A = InvFi for some i and B = InvFj for some j
Now I need to show
(1) A ~ A
(2) If A ~ B then B ~ A
(3) If A ~ B and B ~ C then A ~ C
To see that (1) is valid we can look at the possible cases.
If A = F then Case A is valid
If A = InvF[size=75]i[/size] [size=100]for[/size] some i then Case D is valid where i = j
Therefore (1) is valid.
To see that (2) is valid we will consider the following cases:
If A = F and B = F. Then B ~ A due to (1).
If A = F and B is not = F. By definition B = InvF[size=75]i[/size] [size=100]for[/size] some i. By definition from Case C B ~ A
If A is not = F and B = F. Here we write B = F and A = InvF[size=75]i[/size].
This [size=100]is[/size] precisely Case B (where A and B are interchanged) and therefore B ~ A
If A is not = F and B is not F. Then A = InvF[size=75]i[/size] [size=100]and[/size] B = InvF[size=75]j[/size]. [size=100]By[/size] interchanging i and j we get Case D and therefore B ~ A.
Since this covers all cases, we can conclude if A ~ B then B ~ A
Finally we need to show that (3) is valid
First if A = B = C = F then A = C by definition.
If A = B = F and C = InvF[size=75]i[/size] [size=100]for[/size] some i then since B ~ C and A = B then A ~ C
If A = F and B = InvF[size=75]i[/size] [size=100]and[/size] C = F then A ~ C
If A = F and B = InvF[size=75]i[/size] [size=100]and[/size] C = InvF[size=75]j[/size] [size=100]then[/size] Case B is valid where i and j are interchanged
If A = InvF[size=75]i[/size] [size=100]B[/size] = F and C = F then Case C is valid and A ~ C
If A = InvF[size=75]i[/size] [size=100]and[/size] B = InvF[size=75]j[/size] [size=100]and[/size] C = F then Case C is still valid and A ~ C
If A = InvF[size=75]i[/size] [size=100]and[/size] B = F and C = InvF[size=75]j[/size] [size=100]then[/size] Case D is valid and A ~ C
If A = InvF[size=75]i[/size] [size=100]and[/size] B = Inv F[size=75]j[/size] [size=100]and[/size] C = InvF[size=75]k[/size] [size=100]then[/size] Case D is still valid where j and k are interchanged.
Therefore in all cases A ~ C and (3) is valid.
Now, by using Theorem 1, we know that ~ is a true equivalency relation and we conclude that any isomorphism which preserves the function F will also preserve the equivalency class of inverse operations associated to that function.
In general we will gain more understanding of meaning if we associate to it the preserved equivalency class of inverse operations associated with the functional structure F.
The reader might get a better feel for the role of equivalency classes by looking at the following equivalency relations.
A ~ B if and only if A = (p,q) and B = (r,s) where p, q, r, and s are Integers and ps = qr. This class looks at ordered pairs that are proportionate.
And finally the reader might look at A ~ B if and only if A and B are composites of the transforms f and g i.e. A(x) = f(g(x)) or A(x) = g(f(x)) where f is such that R^2 is mapped to R^2 is of the form:
x[size=75]2[/size] [size=100]=[/size] x[size=75]1[/size] - [size=100]a[/size]
y[size=75]2[/size] [size=100]=[/size] y[size=75]1[/size] - [size=100]b[/size]
And g is such that R^2 is mapped to R^2 is of the form:
x[size=75]2[/size] [size=100]=[/size] x[size=75]1[/size][size=100]cos[/size]θ - y[size=75]1[/size][size=100]sin[/size]θ
y[size=75]2[/size] [size=100]=[/size] x[size=75]1[/size][size=100]sin[/size]θ + y[size=75]1[/size][size=100]cos[/size]θ
These transforms conserve the eulidean meteric and shapes can be defined as equivalency classes.
What does this model tell us about models in general?
If we look at structures as defined in Model Theory, then we should extend our investigations to include the entire class of Inverse Operations.
Previous observations about models included:
General Conclusions:
The Structures of Model Theory are analogous to grammar. While true statements (which might provide interest, insight, or constraints) can not, generally, be directly decerned from its Structure, we can generally review a given statement with respect to some Structure and accept the statement, reject the statement, or most practically improve upon it.
We also have the option of tinkering with the Structure itself, though, if we are modeling a physical phenomenon, we need to be mindful of the limitations regarding what we can measure.
Some open questions concerning models in general:
Bearing in mind that we are now, generally, looking for a sentence to be true with regard to two frames of reference, one concerning physical phenomena and the other concerning a mathematical Structure, can we have a true statement regarding a physical phenomenon that is not true in any Structure containing its language? Conversely, can we have a true statement with regard to a mathematical Structure, where that structure represents a physical phenomenon, such that the statement is not true about some physical phenomena?
Can we have any sense of meaning if we allow different evaluations (technologies or measures) for comparing models?
Is time somehow equivalent to the Reals? My Physics classes always assumed that this was the case.