The Rule of all Rules

Basically about how "math class" can be different in different parallels.  One ordinarily thinks that there can only one math class and that's probably true.  But, there is a sort of uber math class which completely describes all other math classes as well as itself.

What I mean by "math class" is that, for example, in our system, the one we very much feel we are in right now, math class reveals that 2+1=3.  The question is, do math classes in parallels have to be the same as our math class?  What is the rationale for thinking "yes"?  Thinking the answer is "yes" is precisely like imagining a geocentric universe.  It's logic-centric.  A logic described by human logicians with their own swing on things.  What if human logic is not universal; what if in some parallels, 2+1 is not =3?

The uber math class is the same for all parallels, I think, but in this grand system there are uncountably many systems embedded within.  (Parallel = System.)

I realized that some interesting things happen when analyzing the statement S which stands for "All rules have exceptions."  As it turns out, from just analyzing S, we can conclude that the rules of a system are not more than sets of true statements about that system.  Also, S is NOT in itself a rule.

I wonder if other phrases in English sound very much like rules but, technically, are not rules.

Is my (possibly obscured) definition of "rule" even close to appropriate?  I mean, nothing's wrong with a definition but maybe what I have said has nothing to do with actual rules.

The truth of a system is relative.  No system need work like any other system.  All rules have exceptions.  I don't think we can avoid semantics for very much longer.

As a background, these are what are known as formal systems:


http://en.wikipedia.org/wiki/Formal_system


In Max Tegmark's TOE gif, he cites formal systems as being the basis for all of math, including set theory and first order logic.  You see the circle down there at the bottom:

I believe that within my development of grammatical systems, which are more general than formal systems, I found a way to prove that something is true of all theorems within a formal system.  Could this hint at the nature of truth if we consider theorems to be, in any sense, true?


This is still in the preliminary stages but I think that if we can prove the following about a property P then property P must be true of all theorems in a formal system:


1. P holds for all axioms in the formal system
2. If P is "closed under all inference rules"


then 


P holds for all theorems.


To unpack (2) a bit, it's a little complicated.  A very specific example of an inference rule, one used in first order logic, is modus ponens.  Modus ponens is a binary inference rule, meaning it has two inputs.  Very simply, inference rules are functions that input n inputs, with n at least one, and output a single statement, called the conclusion of the inference rule.  Modus ponens for instance takes two statements of the form 
(a) Statement A
(b) IF A THEN B
and modus ponens outputs
B.


Basically modus ponens is saying that if A is true and if "IF A THEN B" is true, then one can conclude "B".


So for P to be closed under all inference rules in a formal system means that if R is any n-ary inference rule and P is true of all theorems T1, .., Tn, then P is also true of R applied to the n-tuple (T1,...,Tn).
IOW, If P is true of every term in the n-tuple (T1,...,Tn) and if P is true for R(T1,...,Tn) then we say that P is closed under all inference rules.


Again, if we can show that P is true for all axioms and if P is closed under all inference rules, then P is true for all theorems (i.e., true statements) within a formal system.


I am still working on giving some interesting examples of this phenomena.  

Watching a less than glamorous documentary that I've already seen about John Nash while doing math...I know it's kind of a silly question but I'm trying to answer the question WHY is addition commutative, implying things like it won't matter what order you put your groceries on the conveyer belt; you could try putting the cheaper items first or last or in any order and the total will be the same (at least in this parallel--in others, addition need not be commutative).  The question is why.  I am happy to report that the commutativity is just another consequence of the research I have shared with some of you about what I call grammatical systems.  The lattice grid with the taxicab metric is a grammatical system.  I can use the general grammatical system induction principle which works in all grammatical systems to prove that the distance between (p,q) and the origin equals the distance between (q,p) and the origin, hinting that one possible answer to my question lies within the symmetry of rectangles.  Of course, proving the commutativity of addition isn't all that interesting but another consequence of grammatical system induction is that a similar principle applies to all of set theory which forms the basis of the majority of math.  In the back of my mind, the big question I have which I may never solve is whether or not axiom independence can be proven this way.

An ontological feedback loop. Interesting!

Every ring is a group so groups are more general than rings.


Every group is an object within a formal system, so formal systems are more general than groups.


Every formal system is a grammatical system, so grammatical systems are more general than formal systems.


The space of all grammatical systems is a set[*], so sets are more general than grammatical systems.


Every set is an object "within" a grammatical systems, so grammatical systems are more general than sets.


Therefore, grammatical systems are as general as sets.

Since the word set is atomic, given the previous line, the term "grammatical system" is also atomic.




[*] the most questionable step

A brief Analysis of Paradigms

A paradigm can be defined in a few ways though I typically would say that a paradigm is a set of assumptions plus a set of consequences of those assumptions. Less formally, a paradigm is a way of looking at and interpreting the world. What you assume about what you see....

I've been working on a little diagram that illustrates several points, only a couple of which I'll mention in the first post. I post this here because I want to improve upon my diagram; hopefully you the reader can provide useful feedback.

The gray area represents what you can prove based on the number of assumptions you make. If it's a white region, that means it is not provable but something can be non-provable for a couple of reasons: it's false or it's true but unprovable.

If you assume nothing other than the ambient logical axioms (identity, noncontradiction, and law/axiom of excluded middle), the only true statements are tautologies. This is represented by region 1.

On the other extreme, if you assume a statement and that statement's negation (obtained by slapping "NOT" in front of the statement), then this is a contradiction. Assuming a statement and its negation leads us to conclude (through some mathematical tomfoolery) that ALL grammatically-correct utterances are TRUE and provable!! That means, for instance, that all negations of statements are also true and provable. Everything is both true and false simultaneously... This is only if we assume a statement and its negation. So we can't make too many assumptions or else all grammatically-correct utterances are true and false. This corresponds to regions 5, 4, and part of 1.

The "interesting" cases are between when you assume more than nothing but less than assuming a pair of mutually exclusive statements.

The gray area represents all statements that are provable from the assumptions, except for region 5.

More to follow..