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

[Organic]

do those constants have any relation to each other?

does something like pi-e or pi/e has any significance?

e, pi and phi are irrational numbers, and they are interesting because they are expressing proportions that can be found in many, so called, different systems.

If some proportion is found in many systems, we hope to find through it if there is some deep connection between these systems.

Shortly speaking, we are talking about the signature of some deep symmetry that can be used as a gate between, so called, different systems.

If we find some deep symmetry between, so called, different systems, then this deep point of view, gives us the opportunity to explore these systems from deeper and higher level of understanding.

Because e, pi and phi are irrational numbers, I think we have to start our research by asking ourselves: "what is an irrational number"?

A better answer to this important question can give us a deeper understanding of the connections between e, pi and phi.

By standard Math, irrational number is a number that cannot be expressed by a ratio that exists between at least two integers.

If this is the case, then we have no accurate method to represent an irrational number.

Can somebody have an idea how to represent an irrational number in an accurate way without using the natural numbers notations?

Please be aware that notations like e, pi or phi or pi/e are general notations exactly like oo is for infinity, because they do not give us any accurate but only a trivial information about these irrational numbers.

Another way to think about this problem, is to agree with the idea that redundancy_AND_uncertainty are natural properties of the NUMBER concept right from the level of the natural numbers, for example:

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

 

[matt grime]

That's not true is it. e, pi, phi are well known to be real numbers, and infinity isn't.

 

[Integral]

Any of these numbers can be represented with sufficient precision for any possible use. If you do not need digits of precision then the symbol will do nicely. I suppose you could create a number system where one or all of these numbers are integer or rational. Of course the payback would be that 1 would be irrational.

 

[matt grime]

Notice also the unjustified (unjustifiable?) presumption that writing two numbers as ratios of integers constitutes an exact representation, and is the only possible exact representation, whatever that might mean. Heck, why is even an integer an exact representation of a point in the real numbers?

 

[WWW]

But (by organic) maybe

redundancy_AND_uncertainty are natural properties of the NUMBER concept right from the level of the natural numbers

Is it possible?

 

[The_Imagizer]

Can somebody have an idea how to represent an irrational number in an accurate way without using the natural numbers notations?

if I am not mistaken, or at least for pi, their is not a single way to do so, since it is trancendent, this has been proven... or perhaps I have misunderstood your question, what exacly do you meen with "the natural nimbers notations"?

 

[AntiMagicMan]

All irrational numbers can be represented exactly by infinite series. Defining numbers without using numbers, now you are sounding like someone who has been puffing a bit too much on the old crack pipe.

 

[WWW]

But can an infinite series represented by integers notations can give an exact result of some irrational number?

If yes, then please write sqrt(2) for example.

 

[matt grime]

ok. it's sqrt(2) or \sqrt2 or 2^{1/2} they're all exact 'notations' of that quantity which is the unique positive root of x^2-2

Sorry that you think decimals ARE real numbers, and that terminating decimals constitute the only exact things, which is a not a good way of thinking of these things. I suppose we can understand the idea that only fractions are nice, but this presupposes that it is necessary to talk about the real line as if it were actually physically a line with little notches on it like a ruler. Seeing as you cannot mark any points on a ruler with any certainty I am at a loss to understand where you're comgin from.

They aren't decimals, or rather that is not their defining property, and as you well know the issues of thinking they are, one wonders why you persist in this view.

 

[WWW]

Is the data behind sqrt(2) or 2^(1/2) notations is accurate?

Proove it.

 

[WWW]

Is the data behind sqrt(2) or 2^(1/2) notations is accurate?

Prove it.

 

[matt grime]

Erm, ok. as the only thing that defines it is it is positive and squares to two, let me think... er, yep, it's an accurate notation as far as i can tell. what data behind it? that makes no sense, but you've already adequately demonstrated that you do not accept that the real numbers are cauchy sequences of rational numbers modulo the obvious equivalence, as it 'rapes' something, which is a bizarre choice of words.

 

[WWW]

Cauchy sequences of rationals using rational notations that some of them are finite therefore accurate, some of them has periodic returns therefore they are accurate by periodic returns.

Irrational numbers like sqrt(2) don’t have any of these properties, so by what property they can be accurate?

 

[matt grime]

But you're putting your own particular spin on "accurate" which means: can be specified by a finite number of integers picked from the set 0,1,...,9 with possibly some indication of a recurrence. Fine, but that doesn't stop sqrt(2) being a perfectly well defined real number. I would dismiss your preference for accuracy like this as relatively unimportant. Constructively all numbers are equally hard to indicate on a ruler, and algebraically/analytically you're out of your depth already

 

[WWW]

What depth?

You show nothing but your belief that some irrational number has an accurate place on the real line.

 

[Hurkyl]

Let's reverse the question.

If you're so sure that sqrt(2) is not accurate, then please, tell us the difference between sqrt(2) and sqrt(2).

 

[matt grime]

If you knew about dedekind cuts you wouldn't make such statements about placing things on the real line (you speak as if it were a phyiscal line still), and is this the same meaning in accuracy as in the post before?

 

[WWW]

Dedekind using a Boolean Knife, which means he finds the property of its logical reasoning, which has no vagueness in it.

 

[WWW]

If you're so sure that sqrt(2) is not accurate, then please, tell us the difference between sqrt(2) and sqrt(2).

By your question we can see that you don't understand the meaning of 'not accurate', which is not the difference between accurate things but the self property of an element to be not accurate.

It cannot be understood be false/true reasoning.

 

[matt grime]

then illuminate it for us and tell us what you think accurate means.

 

[WWW]

Accurate means a final state of information.

 

[HallsofIvy]

How about the belief that every irrational number has an exact position on the real number line. That is, after all, pretty much the definition of the real number line!

"You show nothing but your belief that some irrational number has an accurate place on the real line."

In other words, he is showing nothing but his knowledge of the real number system.

 

[PFanalog57]

x^2 + 1 = 0

x = sqrt[-1] = i

i^i = e^[-pi/2]


[1+i^i]*[1-i^i] = [e^pi - 1]/[e^pi]

A physical system is described by a normalized vector[state vector] in Hilbert space. All possible information can be known about the system, since, for every physical observable there corresponds a self adjoint operator in Hilbert space.

The only allowed physical results of measurements of some obervable U, are the elements of the spectrum of the operator which corresponds to U.

So all properties of a number may not be completely known, but that which is known, must be specifiable on logical or analytic grounds.

Now if you are trying to say that mathematics is inherently random at its foundations, you must define what randomness is ...exactly.

Take a coin toss for example, as the number of flips of the coin increase the
HTTTHTHTHHTHT...HTHTHTHTHT...

The probability becomes an "exact" number at an infinite limit. 1/2

So a number becomes an identity in the Platonic aeon.

 

[WWW]

In other words, he is showing nothing but his knowledge of the real number system.

His knowledge is based on the current paradigms of Math Language.

 

[WWW]

So a number becomes an identity in the Platonic aeon.

I prefer to think about simple symmetries that maybe can be found in the basis of, so called, different systems.

 

[Integral]

Using the iterative sequences.

S(n+1)= \frac {x_{n}} 2 + \frac 1 {x_n}

if we let x_1 = 1 we have a sequence which converges to root 2 from below.
If we let x_1=4 we have a second sequence which converges to root 2 from above.
Now I have constructed a set of nested intervals which have length going to zero. By theorems proven in Real Analysis, the intersection of these intervals contains a single Real number, in this case Root 2. Thus we have shown that an irrational number occupies a fixed well defined position on the real number line.

 

[WWW]

What is the meaning of the word 'converges' here?

Prove that above and below really reaching to a final state of information.

 

[PFanalog57]

What is the meaning of the word 'converges' here?

Prove that above and below really reaching to a final state of information.

lim
x->a f(x) = L

If 0 < |x - a| < delta then |f(x) - L| < epsilon

For example:

lim
x->3 (2x-5) = 1

if 0 < |x-3| < delta then |(2x-5)-1| < epsilon

|(2x-5)-1| = |2x-6| = |2(x-3)| = 2|x-3|

If 0 < |x-3| < epsilon/2 then |(2x-5)-1| = 2|x-3| < 2*epsilon/2 = epsilon


Aeon means eternal.

A timeless symmetry?

An infinite number of coin flips gives an equal amount of heads and an equal amount of tails.

[1/2 H and 1/2 T]*n, for n--->oo

A radioactive nucleus decays in accordance with probability P within time t_0 to time t_1

Probability P becomes a timeless mathematical entity governing the future iterations of events at time t. There exists a spectrum of possibilities for the observed quantities. Certain deterministic factors become contingent with respect to uncertainty, DxDp >= h .

An infinite number of observations of the radioactive decay, converges to an exact number for t?

Wave function probability density = |psi (r, t)|^2


The physical meaning of the expectation value appears to be simple. It is the value that would be found by taking the average of many measurements of the observables in question on a large collection of systems all in the state psi. the individual results are weighted by the probability.

 

[WWW]

lim
x->a f(x) = L

If 0 < |x - a| < delta then |f(x) - L| < epsilon

This proof by contradiction is based on Boolean Logic.

Prove that this proof holds also in a multi-valued logical system.

 

[PFanalog57]

This proof by contradiction is based on Boolean Logic.

Prove that this proof holds also in a multi-valued logical system.

You appear to be implying that Boolean logic is context dependent? ...Interesting.

It seems that many valued logic must be formulated in terms of a stable 2-valued logic background.

Suppose the limit as n-->oo, s_n = s, in the classical sense. It must then be demonstrated that s_n - s is infinitesimal for all infinite n. That is to say, for any epsilon > 0 and for any infinite natural number n , it must be proved that |s - s_n| < epsilon.

For any given epsilon > 0 in R there exists a natural number v in N such that

|s_n - s| < epsilon for n > v, n is an element of N.

For all x, if x is an element of N and x > v then |s_x - s| < epsilon.

Since any infinite natural number is greater than v it can be deduced that |s_n - s| < epsilon for all infinite n.


so if I flip a coin, it will be Heads H, or not-Heads, ~H

So if the coin lands on its side, it is still H or ~H, this being the case that it is ~H. Absolutely true.

So if I go for a more specific multivalued logic it becomes H, ~H, S.

H or ~H is still true.

H or ~H or S is just adding more specification...?

 

[WWW]

It seems that many valued logic must be formulated in terms of a stable 2-valued logic background.

Please Prove that Complementary Logic ( https://www.physicsforums.com/showpost.php?p=192318&postcount=25 ) can be reduced to a false/true logic.

 

[Hurkyl]

I think the best demonstration of this fact can be seen by all of your posts about your theory. While your theory may be many-valued, all statements made about it have been decreed either true or false by you, thus reducing it to binary logic.


In general, any multi-valued logic can be reduced to binary logic by considering statements of the form "P has truth value S" (where P is a multi-valued proposition and S is one of the possible logic values)


I will admit, however, that you've done a nice job of avoiding the "deducible / not deducible" game, upon which mathematics is based, by not presenting any axioms suitible for use in a deduction, and not writing anything that could even be interpreted as an attempt at deduction.

 

[WWW]

I think the best demonstration of this fact can be seen by all of your posts about your theory. While your theory may be many-valued, all statements made about it have been decreed either true or false by you, thus reducing it to binary logic.

Please look again at http://www.geocities.com/complementarytheory/BFC.pdf and see by your self that contradiction or excluded-middle reasoning are trivial private cases of Complementary Logic (which is a symmetrical logical system, unlike Boolean or Fuzzy Logics).

In general, any multi-valued logic can be reduced to binary logic by considering statements of the form "P has truth value S" (where P is a multi-valued proposition and S is one of the possible logic values)

And you lose through this generalization (it is trivialization through my point of view) very interesting included-middle ordered Logical states.

In general, any multi-valued logic can be reduced to binary logic ...

Complementary Logic is a symmetrical logical system, therefore it includes Boolean or Fuzzy Logics as proper sub-systems (some broken-symmetry states) of it.

For better understanding, please read http://www.geocities.com/complementarytheory/ConScript.pdf

 

[matt grime]

You've not proven that it contains, or can be specialized to, those cases. You've claimed it, but not shown it. And when I pointed out that the thing you were claiming was the specialization to boolean logic was ill stated, that it appeared to claim

(a xor b) and (a xor b) was the same as a and b, which it isn't, you said that the things there weren't boolean values anyway. so we're at a slight loss as to know how on Earth it can be boolean, and not be boolean. of course it wasn't clear that the two diagrams for two values were what you claimed was the specialization to boolean logic, because, despite being asked, you once more refused to say if it were or not.
so i f you've a few hours why not clearly explain how all those fuzzy and boolean systems are a subset of whatever this alleged system of yours is.

 

[WWW]

Please first show us how you express redundancy_AND_uncertainty connective in multi-valued ordered logical states, by keeping the excluded-middle Boolean-Logic rule.

Please use only logical connectivies.

 

[WWW]

(a xor b) and (a xor b) was the same as a and b, which it isn't,

I don't understand what do you want to say.

Please choose a xor b:

a) ((a xor b) and (a xor b)) = (a and b)

b) ((a xor b) and (a xor b)) not= (a and b)

 

[matt grime]

you're the one who needs to explain how to recover boolean logic from your system because you've not done so as yet. moreover as you've not defined what you mean by uncertainty_and_redeundancy we can't do as you ask.


the second part: you said that these diagrams are the whole collection of somethings between a and b and a xor b

then you drew the cases:

  b   b 
    #   #    
    a   a     
    .   .   
    |   |   
    |&__|_   
    | 
    
    [B]a   b     
    .   .   
    |   |  <--- (Standard Math logical system fundamental building-block) 
    |#__|   
    |[/B]

implying this is boolean logic, and one of these is xor the other and, i#m pointing out that neither of these is an any sense "and".

and everything in there must be boolean so you aren't allowed to cite any other kind of logic.

By the way, you always misuse connectives so why on Earth can you expect anyone to take your things seriously?

 

[WWW]

My reasoning on this is this:

Let # be xor.

Let & be and.

f=false

t=true

u=uncertainty

r=redundancy

By (f # t) I mean that some single result can be found through a probability of 1:2 .


the complementary logical representation of this probability can be expressed in this way:
((f # t)&(f # t)) where all this expresion is under this "cloude of probebility"

<--r--> ^ 
 t   t  |
 #   #  u
 f   f  |
 |   |  v
 |&__|_
 |

So as we see, ((f # t)&(f # t)) is simultaneously (f & f)_(t & f)_(f & t)_(t & t), which is definitely not a Boolean Logic state.


Only (f # t) is an excluded-middle f/t locial state with no probebility, after we find our single result.

 

[matt grime]

but you said it was boolean, that it was a two valued logic of ordinary maths. so were you wrong or lying? seeing as it is supposed to be proper maths then probabilities cannot lies between 1 and 2, unless it's 1 obviously.

 

[WWW]

No,

only

    a   b     
    .   .   
    |   |   
    |#__|   
    |

is a Boolean Logic.

Please see https://www.physicsforums.com/showpost.php?p=190983&postcount=1 and https://www.physicsforums.com/showpost.php?p=192318&postcount=25

 

[matt grime]

so why did you say that both digrams were part of two valued logic? and then this contradicts you assertion that it runs from a and b to a xor b. where's "and" gone then?

 

[Hurkyl]

And you lose through this generalization (it is trivialization through my point of view) very interesting included-middle ordered Logical states.

No, you don't. Any statement is either in a given "included-middle ordered Logical state" or it is not; a binary fact.

 

[WWW]

Really?

No, you don't. Any statement is either in a given "included-middle ordered Logical state" or it is not; a binary fact.

(excluded-middle --> a binary fact) XOR (included-middle --> not a binary fact)

An example of a non-binary system:

Let # be xor.

Let & be and.

f=false

t=true

u=uncertainty

r=redundancy

By (f # t) I mean that some single result can be found through a probability of 1:2 .

the complementary logical representation of this probability can be expressed in this way:
((f # t)&(f # t)) where all this expresion is under this "cloude of probebility"

<--r--> ^ 
 t   t  |
 #   #  u
 f   f  |
 |   |  v
 |&__|_
 |

So as we see, ((f # t)&(f # t)) is simultaneously (f & f)_(t & f)_(f & t)_(t & t), which is definitely not a Boolean Logic state.


Only (f # t) is an excluded-middle f/t locial state with no probebility, after we find our single result.

---------------------------------------------------------------------------------------------------
Now you can say that:

(excluded-middle --> a binary fact) XOR (included-middle --> not a binary fact) in general is a binary fact.

So what. it is a trivial and non-interesting information.
---------------------------------------------------------------------------------------------------

More than that, for example:

f=(excluded-middle --> a binary fact)

t=(included-middle --> not a binary fact)

Let # be xor.

Let & be and.

u=uncertainty

r=redundancy

By (f # t) I mean that some single result can be found through a probability of 1:2 .

the complementary logical representation of this probability can be expressed in this way:
((f # t)&(f # t)) where all this expresion is under this "cloude of probebility"

<--r--> ^ 
 t   t  |
 #   #  u
 f   f  |
 |   |  v
 |&__|_
 |

So as we see, ((f # t)&(f # t)) is simultaneously (f & f)_(t & f)_(f & t)_(t & t), which is definitely not a Boolean Logic state.

 

[Hurkyl]

So as we see, ((f # t)&(f # t)) is simultaneously (f & f)_(t & f)_(f & t)_(t & t), which is definitely not a Boolean Logic state.

This sounds like a statement using binary logic (about your multi-valued logic).

 

[WWW]

No it is not, because it is simultaneously (f & f)_(t & f)_(f & t)_(t & t).

And this is only a 2-valued logic + probability proprty.

Now think about n>2-valued logic + probability proprty.

Please read https://www.physicsforums.com/showpost.php?p=190983&postcount=1 and https://www.physicsforums.com/showpost.php?p=192318&postcount=25

 

[Hurkyl]

I said:

"((f # t)&(f # t)) is simultaneously (f & f)_(t & f)_(f & t)_(t & t)" is a boolean statement.

Are you saying that

"((f # t)&(f # t)) is simultaneously (f & f)_(t & f)_(f & t)_(t & t)" is simultaneously "(f & f)_(t & f)_(f & t)_(t & t)"?

And even if you are, isn't this new statement of yours true? (according to you)

 

[PFanalog57]

Please Prove that Complementary Logic ( https://www.physicsforums.com/showpost.php?p=192318&postcount=25 ) can be reduced to a false/true logic.

Your complementary logic system is a proposition that can be proved, P , or not-proved, ~P .

Some excellent ideas regarding symmetry though.

A__~A___A_V_~A

T___F_______T

F___T_______T


A truth table tautology is very much like a symmetry, it is invariant.

If your complementary logic is context dependent, it still must have an invariant structure that gives a meaningful interpretation?

 

[WWW]

I said:

"((f # t)&(f # t)) is simultaneously (f & f)_(t & f)_(f & t)_(t & t)" is a boolean statement.

Are you saying that

"((f # t)&(f # t)) is simultaneously (f & f)_(t & f)_(f & t)_(t & t)" is simultaneously "(f & f)_(t & f)_(f & t)_(t & t)"?

And even if you are, isn't this new statement of yours true? (according to you)

You are mixing between the existence of some system and it’s logical reasoning.

Any consistent system is limited (incomplete) by definition, otherwise it is inconsistent.

Because my system is consistent by it’s internal structure it is also limited by these structures.

The new thing here, if we compare it to the standard excluded-middle system, is that it is naturally using probability right from it’s first-order level.

For example:

Let us examine Schrodinger's Cat experiment.

f=dead cat

t=live cat

Let # be xor.

Let & be and.

u=uncertainty

r=redundancy

By (f # t) I mean that some single result can be found through a probability of 1:2 .

the complementary logical representation of this probability can be expressed in this way:
((f # t)&(f # t)) where all this expression is under this "cloud of probability"

<--r--> ^ 
 t   t  |
 #   #  u
 f   f  |
 |   |  v
 |&__|_
 |

So as we see, ((f # t)&(f # t)) is simultaneously (f & f)_(t & f)_(f & t)_(t & t), which is definitely not a Boolean Logic state.

Please show us (t & f) as a valid(=1=existing) state in an excluded-middle system.

Also through my system the meaning of probability is not some accurate value between 0 and 1 (as we can find in Fuzzy Logic, for example) but an ordered simultaneous associations between redundancy_AND_uncertainty ,which creates “clouds of vagueness” from the most vagueness to the least vagueness, when n > 1 is given.

Shortly speaking, Complementary Logic is based on ordered levels of symmetry breaking, right from its first-order level.

If your complementary logic is context dependent, it still must have an invariant structure that gives a meaningful interpretation?

Yes, because it is consistent it is also incomplete and context depended, but unlike an excluded-middle logical system, it is not looking at vagueness as an enemy that we have to distinct by more and more accurate definitions.

Complementary Logic reasoning is to save and explore the associations between information forms at any given degree of vagueness, where the dynamic process of any research and the explored/explorer interactions are naturally included.

 

[matt grime]

But that makes no sense, doron, unless you tell us what f xor t means. what's xor, your # above? and &? we can only interpret in boolean terms because that's all they are.

what is u, what is r, and for that matter what is v?

and you can't have probabilties between 1 and 2 (unless it is 1).

 

[Hurkyl]

You are mixing between the existence of some system and it?s logical reasoning.

What does that even mean?

I'll take my best guess, and respond that you're the one unable to accept that one can use ordinary "excluded-middle binary logic" to reason about multi-valued logical systems, or those without the excluded middle.


IIRC, synthetic differential geometry is developed by presenting a system where the law of excluded middle is not a tautology, and then using ordinary logic (including the law of the excluded middle) to reason about it "externally".

Please show us (t & f) as a valid(=1=existing) state in an excluded-middle system.

Interpreting your symbols according to their ordinary meaning, t & f = f. Simple as that.