The Mysterious Connections Between Irrational Numbers - e, pi, and phi

[Hurkyl]

please give more details, because I do not really understand what do you looking for.

He's looking for you to make a list of statements and say "These are the statements we are assuming to be true", and then for the subject at hand, to only make statements which can be derived from those assumed statements using rules of deduction.

e.g.

if one of the statements was "For any z: If P(z) then Q(z)", and another of the statements was "P(a)", then we can conclude "Q(a)" via:

Forall z: if P(z) then Q(z)
therefore
if P(a) then Q(a)

and

if P(a) then Q(a)
P(a)
therefore
Q(a)


And you should be able to do this (or at least indicate a way this can be done) for any statement you wish to claim true.


This is how mathematics is done. If you don't want to do it this way, then you're doing something other than mathematics.



And if you wish to rewrite logic, then you should list the legal rules of deduction as well. (since logic is simply rules of deduction, then if you want to change logic you have to present new rules of deduction)

 

[matt grime]

As you feel confident in saying that maths can only deal with probability as a higher order object, you must be able to state what you mean by probability.

We may all have some notion about things "possibly" happening and some things being more likely to occur, but, once more, you're confusing vague, fuzzy notions of real life with the abstraction on mathematics and saying they are the same.

You appear to claim that the axiomaitized probability theory of mathematics is inherent as a basic concept in your theory. That indicates that you do not understand the Kolmogorov version of probability theory. Please domonstrate that you somehow have an equivalent theory that is "fundamental". This must contain sets, measures and functions (ie cartesian products), as well as at least some mention of the real numbers.

Do not think that this axiomatic thing *is* the fuzzy concept of likelihood. It is, as tends to be the case, a mathematical construction.

Show in your elemental theory of probability that is as ontological simple as it gets, that the probability of obtaining 3 heads in three throws of an unbiased coin is 1/8.

 

[KingNothing]

I think irrational numbers have a place on the real number line because they do have a real and accurate value, it just can't be represented well as a ratio of two integers.

 

[matt grime]

I think I@ve found a way of expressing what I've been trying to sum up about this for a while now.

You let the properties of the objects and operations in you theory define the theory, whereas you should let the theory define the allowed objects and operations.

Hence your claim to specialize to boolean logic means that if you let the objects be the usual kind of statement and AND and XOR be the usual connectives, then you have Boolean logic. Yet you''ve not offered a generalization properly, becuase you have to redefine all the operations for each specialization; it isn't a genuine generalization. Imagine if you will, and you probably won't, that I am claiming I've got a general theory of Algebra. I don't have any axioms, rules or definitions, just things I call algbraic objects and operations i call algebraic operations. Now I claim that if I let these be groups and group maps I'm doing group theory, ie that is specializes to group theory. But I@ve offered nothing to back that up and it is a completely vacuous theory really. (Incidentally I can offer a generalized theory of algebra which does contain the groups as a special subset: what do you know about cocommutative Hopf algebras?)

 

[WWW]

First, thank you for your positive attitude.

As a first step, I’ll try to explain what is probability through my point of view .

Let us take a piano with 4 possible different notes.

By using the word 'possible' I mean that in the first stage, any key can be anyone of the 4 notes, and we have no way to know what note each key has, before we are using it.

Each time when I press simultaneously on its all 4 keys, I get an accord.

Let us notate each unique note by a different letter, for example: a,b,c,d

Redundancy is (more then one copy of the same value can be found)

Uncertainty is (more than one unique value can be found)

Let XOR be #

Let AND be &

A 4-valued logic is:

              Uncertainty
  <-Redundancy->^
    d  d  d  d  |
    #  #  #  #  |
    c  c  c  c  |
    #  #  #  #  |
    b  b  b  b  |
    #  #  #  #  |
   {a, a, a, a} V
    .  .  .  .
    |  |  |  |
    |  |  |  |
    |  |  |  | <--(First 4-valued logical form)
    |  |  |  |
    |  |  |  |
    |&_|&_|&_|_
    |
    ={x,x,x,x}
[B]In the first case each accord can be one of 4^4 different possibilities.[/B]



   {a, b, c, d}
    .  .  .  .
    |  |  |  |
    |#_|  |  |
    |     |  | <--(Last 4-valued logical form)
    |#____|  |      
    |        |
    |#_______|
    |
    ={{{{x},x},x},x}
[B]In the last case each accord is a one and only one possibility.[/B]

The ordered possibilities between 4^4 and 1 is:

[b]
============>>>

                Uncertainty
  <-Redundancy->^
    d  d  d  d  |          d  d             d  d
    #  #  #  #  |          #  #             #  #        
    c  c  c  c  |          c  c             c  c
    #  #  #  #  |          #  #             #  #   
    b  b  b  b  |    b  b  b  b             b  b       b  b  b  b
    #  #  #  #  |    #  #  #  #             #  #       #  #  #  #   
   {a, a, a, a} V   {a, a, a, a}     {a, b, a, a}     {a, a, a, a}
    .  .  .  .       .  .  .  .       .  .  .  .       .  .  .  .
    |  |  |  |       |  |  |  |       |  |  |  |       |  |  |  |
    |  |  |  |       |&_|_ |  |       |#_|  |  |       |&_|_ |&_|_
    |  |  |  |       |     |  |       |     |  |       |     |
    |  |  |  |       |     |  |       |     |  |       |     |
    |  |  |  |       |     |  |       |     |  |       |     |
    |&_|&_|&_|_      |&____|&_|_      |&____|&_|_      |&____|____
    |                |                |                |
    {x,x,x,x}       {{x,x},x,x}       {{{x},x},x,x}    {{x,x},{x,x}}     
 
                                      c  c  c
                                      #  #  #      
          b  b                        b  b  b          b  b
          #  #                        #  #  #          #  #         
   {a, b, a, a}     {a, b, a, b}     {a, a, a, d}     {a, a, c, d}
    .  .  .  .       .  .  .  .       .  .  .  .       .  .  .  .
    |  |  |  |       |  |  |  |       |  |  |  |       |  |  |  |
    |#_|  |&_|_      |#_|  |#_|       |  |  |  |       |&_|_ |  |
    |     |          |     |          |  |  |  |       |     |  |
    |     |          |     |          |&_|&_|_ |       |#____|  |
    |     |          |     |          |        |       |        |
    |&____|____      |&____|____      |#_______|       |#_______|
    |                |                |                |
    {{{x},x},{x,x}} {{{x},x},{{x},x}} {{x,x,x},x}      {{{x,x},x},x} 

   {a, b, c, d}
    .  .  .  .
    |  |  |  |
    |#_|  |  |
    |     |  |  
    |#____|  |      
    |        |
    |#_______|
    |    
    {{{{x},x},x},x}
[/b]

r is (more then one copy of the same value can be found)

u is (more than one unique value can be found)

Let XOR be #

Let AND be &


When probability is a first-order property then AND connective is used whenever a no-unique result can be found:

<--[B]r[/B]--> ^ 
 t   t  |
 #   #  [B]u[/B]
 f   f  |
 |   |  v
 |&__|_
 |

When probability is a first-order property then XOR connective is used whenever a unique result can be found:

 f   t   
 |   |   
 |#__| 
 |

If you understand what is probability by me, then try to translate it to the standard excluded-middle reasoning.

 

[matt grime]

Nowhere in there do you state what you mean by probability.

And you're using connectives AND and XOR as if they are the usual objects of boolean logic, when you say they aren't. Moreover it appears that you're saying that a,b,c,d are not events/statements but the possible "truth" values in the system of logic. That is waht you're getting at if you say you have a 4 valued logic system (otherwise you've not define what the logical values may be).

There is also the observation that you're using these & and # connectives (without offering their truth tables) inside these diagrams which we are told are a full set of values between xor and and, so the definition uses the object in its definition. I dont' see any recusive way to make the valid.

 

[WWW]

Please forget for a moment the stantard excluded-middle point of view of AND(=&) and XOR(=#) and try to understand it as I wrote it in the previous post.

Can you do that?

Moreover it appears that you're saying that a,b,c,d are not events/statements but the possible "truth" values in the system of logic

Take each note as a "true" statement.

In this 4-notes piano, any given note is "true", because it is not an excluded-middle logical system.

Please look at the Complementary Logic diagram:http://www.geocities.com/complementarytheory/BFC.pdf

In an included-middle reasoning two opposites are simultaneously preventing/defining each other and the result is a middle(=included-middle).

In an excluded-middle reasoning two opposites are simultaneously contradicting each other and the result is no-middle(=excluded-middle).

An excluded-middle system is a private case in Complementary Logic, as you can see in the example of the 2-valued logic in the previous post.

 

[matt grime]

If all the statements must be true you're even omre off beam than you first appear.

 

[Hurkyl]

What if I knew that key #1 plays either A or B, and key #2 playes either B or C, and that key #1 and key #2 play different notes?

 

[WWW]

What if I knew that key #1 plays either A or B, and key #2 playes either B or C, and that key #1 and key #2 play different notes?

1) there is no c but only a XOR b in a 2-valued system.

2) In the first case of a 2-valued system, each accord can be one of 2^2 different possibilities, and we cannot know what an accord we get until we actually pressing simultaneously on both keys (and this is exactly the meaning of probability here).

3) In the last case of a 2-valued system, each accord can be one of 1 different possibilities, and we get only an a,b accord when we are pressing simultaneously on both keys (there is no probability here).

4) In Complementary Logic there is no contradiction but only a simultaneos existencs of at laest two opposite that simultaneously preventing/defining each other, and the result is a middle(=included-middle).

5) In an excluded-middle reasoning two opposites are simultaneously contradicting each other and the result is no-middle(=excluded-middle).

If all the statements must be true you're even omre off beam than you first appear.

Please look again at: http://www.geocities.com/complementarytheory/BFC.pdf

You simply refuse to understand that there is no true XOR false and therefore no truth tables in NATURAL included-middle logical system like Complementary Logic.

Hyrkyl and Matt:

Please read and try to understend post #95 and #97, and if you try again to force an excluded-middle on them, then don't west your time.

---------------------------------------------------------------------------------------------------

If we go back to irrational numbers, then by CL (Complementary Logic) each irrational number is a unique_but_not_accurate element because of a very simple reason:

Each irrational number is a unique path (or cut) along infinitely many different scales, but this path is not accurate because it has no "right side".

 

[matt grime]

"Take each note as a "true" statement"

and now you contradict that...

 

[WWW]

Take each note as a "true" statement

I wrote "true" and not true, which means that there is no true XOR false in my system.

 

[matt grime]

which highlights the fact that you've not explained what the possible truth values are in your systems. it appears that your diagrams just correspond to some constructions involving and and xor in some logic system that you're refusing to explain/ nor have you explained why these (ill-defined - xor is not assiciative so you can't use it without bracketing) diagrams are remotely important or useful.

 

[WWW]

No exluded-middle point of view can understand Complementary Logic, and the reason is very simple:

In CL we have at least two simultaneous levels to a logical expression:

( Its differential side(= a XOR b) / its integral side(= a AND b) ), where a,b are opposites.

If you can't understand that truth tables are not used in Complementary Logic, then don't west your time.

 

[matt grime]

but now you're saying that you can't get boolean logic out of it since it has truth tables that govern it, and there must be some analogous result there. if you're not going to even offer some way of describing the truth value, be it in 0,1, or some fuzzy, or even trivalued F,T,U system then you can't do anything.

what on Earth do you mean by opposites? what is the opposiite of the function sin(x)? remember you've said in the past that anything is allowed to be some 'information form' to be explored.

 

[Hurkyl]

And all of your information forms seem to be "seperable" in some sense; while I still don't think I understand them, I haven't seen anything from you that I could imagine is capable of describing: "I know that key #1 plays either A or B, and key #2 playes either B or C, and that key #1 and key #2 play different notes."


(By the way, I think the term you're looking for is 'chord' not 'accord')

 

[matt grime]

Here's a test for your theory. In mathematics the proposition:

If f is a continuous function on a compact subset of R, then it is uniformly continuous.

Is true.


That is to say, if a:={f is a continuous function on a compact subset of R} and b;={f is uniformly continuous} then (not(a))OR(b) is true.

domonstrate the corresponding result and truth value of that proposition in which ever of the subsystems of complementary logic you wish. I'll even let you work it out in the alleged boolean subtype.

 

[WWW]

Here's a test for your theory. In mathematics the proposition:

If f is a continuous function on a compact subset of R, then it is uniformly continuous.

Is true.


That is to say, if a:={f is a continuous function on a compact subset of R} and b;={f is uniformly continuous} then (not(a))OR(b) is true.

domonstrate the corresponding result and truth value of that proposition in which ever of the subsystems of complementary logic you wish. I'll even let you work it out in the alleged boolean subtype.

MY continuous concept is not your concept, therefore there is no meaning to find maps between a and b as you do in an excluded-middle system.

All your results ignoring the inner complexity that existing between (a XOR b/a AND b) mutual relations, which are included-middle results, where all your excluded-middle reasoning including continuous function and compact subset of R, are all limited to true XOR false logical reasoning, which is this CL private case:

 f   t   
 |   |   
 |#__| 
 |

Shortly speaking, no part of CL information forms can be used to get general conclusions on other information form, because each information form has its own unique reasoning that cannot be reduced to structurless_non_complex magnitudes, as you do in excluded-middle reasoning.

By your above test you damonstate again your inability to understand what is an included-middle reasoning.

Shortly speaking, any excluded-middle test is closed under (f XOR t) and there it is stays, as some unique private case of infinitly many ordered and unique(by their internal structures) logical systems.

 

[matt grime]

"MY continuous concept is not your concept, therefore there is no meaning to find maps between a and b as you do in an excluded-middle system."

Sorry, but you can't pick and choose like that. Especially as you've said that this system allows you to explore all information forms such as my definition of continuity. And you've declared boolean logic to be a subsystem of it. It is then up to you to translate statements into your system.

You only appear more crank like if you say 'ah, but I didn't mean you can apply it in that situation' if you refuse to state which situations you are talking about.

 

[WWW]

And all of your information forms seem to be "seperable" in some sense; while I still don't think I understand them, I haven't seen anything from you that I could imagine is capable of describing: "I know that key #1 plays either A or B, and key #2 playes either B or C, and that key #1 and key #2 play different notes."

First thank you for "chord", In Hebrew we call it "accord".

If you have n keys, then in the first stage any n-chord is a one unknown result out of n^n possibilities and you have no way to know what will be the next n-chord.

To this state I call maximum redundancy_AND_uncertainty of n-system.

In the last stage we have a one and only one n-chord, which is constructed of unique well-known n notes (a unique note for each key as we can find in any "normal" piano).

Please read again #95 and #97

 

[WWW]

It is then up to you to translate statements into your system.

I do not have to translate anything because excluded-middle reasoning with all its branches, theorems and proofs of the last 2000 years is already included as a tiny logical sub-system of the infinite grand universe of included-middle reasoning, which includes infinitely many other unique logical systems, exactly as our planet is a sub-system of the solar-system and the solar system is a sub-system of the milky-way... and so on.

If you still don't get it then look at this example:

http://www.geocities.com/complementarytheory/ETtable.pdf

and try to understand that each form in it is a unique logical reasoning.

The excluded-middle reasoning is the (x=2,y=1) form.


-------------------------------------------------------------------------------------

Your response to the idea that what is called logic (f XOR t) is only a tiny part of a gigantic universe of infinitely many different logical forms, is a normal response to unfamiliar new ideas.

1) Hurkyl tried to reduce this gigantic universe of infinitely many different logical forms to (f XOR t) and failed .

2) I gave a lot of examples that based in this included-middle universe of infinitely many different logical forms, and I showed new interpretations to: Natural numbers, sets, logical forms, infinity, irrational numbers, functions, limit, proof, probablility and more things. They can be found in more then 40 short papers here:

http://www.geocities.com/complementarytheory/CATpage.html

and most of them is the result of what I think is the most important thing in any living language, which is a dialog, mostly between Hurkyl you and me.

Recently I discovered that included-middle point of view on Math language is not a one man show.

Shortly speaking, I am not alone and misunderstood as I was in the last 2 years.

Some of the communities that developing an included-middle point of view can be found here:

http://arxiv.org/PS_cache/quant-ph/pdf/0012/0012007.pdf

http://perso.club-internet.fr/nicol/ciret/

http://www.quantonics.com/How_to_Become_A_Student_of_Quantonics.html

http://www.quantonics.com/Acronyms_Used_In_Quantonics.html#SOM

Here is some example that I gave in the past, which clearly shows how two oppsites preventing/defining each other with no-contradiction:

http://www.geocities.com/complementarytheory/BW-BFC.pdf

 

[matt grime]

Care to list any from reputable mathmaticians whose credentials we can check? Care to actually prove any of those statements? You've written lots of things, claimed they are the correct intrepetation of the proper objects despite not actually behaving as the proper objects must do. You cannot multiply 2 by 3 and not get 6. If you do then you've altered the defintions of the objects and the operations. You're entitled to do that all you want but you're not allowed to say they ARE the proper objects because they clearly aren't. Your opinions on what are the important things to consider are very moot since you can't acutally do anything with you system, as you've admitted yourself. As it is you've not even defined what the connectives "and" and "xor" mean. And if your system contains ours as a trivial subsystem then you ought to able to meet all the challenges I've offered since they are part of that trivial subsystem.

Edit: can't believe you've taken this long to come across contructivism.

 

[WWW]

Matt,

You don't want to see the included-middle, because you simply don't have the guts to see things beyond your tiny excluded-middle part.

For more than a year you asked me to reduce a gigantic unexplored (yet) complex universe to your tiny trivial size, without doing even a little step to an included-middle point of view.

I learned a lot, you learned nothing because you can't accept the Idea that any result is system depended.

Any consistent theoretical system is incomplete by definition --> any theoretical system cannot be THE ONE AND ONLY ONE system because any theoretical system is always trivial when it is compared to reality itself, and in my opinion, this is the deep meaning of Godel's incompleteness theorem, that hard logic mind like you simply ignored.

The best we can do is to create theories that including our abilities to define them as part of the theoretical system.

Through this attitude we do not afraid to be opened to changes, because any deep change in our understanding give us more possibilities to be creative living creatures through non-destructive participation.

 

[WWW]

More for Matt,

By keeping Your ONE AND ONLY ONE (f XOR t) point of view you ignored:

http://www.geocities.com/complementarytheory/NewDiagonalView.pdf

http://www.geocities.com/complementarytheory/3n1proof.pdf

http://www.geocities.com/complementarytheory/RiemannsLimits.pdf

http://www.geocities.com/complementarytheory/CompLogic.pdf

http://www.geocities.com/complementarytheory/RealModel.pdf

http://www.geocities.com/complementarytheory/Complex.pdf

and more...

 

[WWW]

ANOTHER TYPICAL EXAMPLE OF YOUR CLOSED SYSTEM ATTITUDE:

Your opinions on what are the important things to consider are very moot since you can't acutally do anything with you system, as you've admitted yourself.

and you reapinitg to write this after I already answered:

https://www.physicsforums.com/showpost.php?p=192318&postcount=25

Shortly speaking, I feel that I am westing my time if you cannot change your trivial attitude to a point of view, which is not your point of view.

Edit: can't believe you've taken this long to come across contructivism.

I don't have to believe in anything to understand that you can only see the shadow of yourself.