Thursday, May 28, 2015

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.  

Sunday, May 24, 2015

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.