View Full Version : Irrationals - e,pi,phi
Organic
Apr19-04, 03:28 AM
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
Apr19-04, 04:08 AM
That's not true is it. e, pi, phi are well known to be real numbers, and infinity isn't.
Integral
Apr19-04, 04:33 AM
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
Apr19-04, 04:43 AM
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?
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
Apr19-04, 11:46 AM
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
Apr19-04, 05:49 PM
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.
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
Apr20-04, 05:28 AM
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.
Is the data behind sqrt(2) or 2^(1/2) notations is accurate?
Proove it.
Is the data behind sqrt(2) or 2^(1/2) notations is accurate?
Prove it.
matt grime
Apr20-04, 06:02 AM
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.
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
Apr20-04, 06:24 AM
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
What depth?
You show nothing but your belief that some irrational number has an accurate place on the real line.
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
Apr20-04, 06:34 AM
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?
Dedekind using a Boolean Knife, which means he finds the property of its logical reasoning, which has no vagueness in it.
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
Apr20-04, 07:18 AM
then illuminate it for us and tell us what you think accurate means.
Accurate means a final state of information.
HallsofIvy
Apr20-04, 11:11 AM
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.
Russell E. Rierson
Apr20-04, 12:46 PM
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.
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.
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
Apr20-04, 06:00 PM
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.
What is the meaning of the word 'converges' here?
Prove that above and below really reaching to a final state of information.
Russell E. Rierson
Apr21-04, 03:30 AM
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.
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.
Russell E. Rierson
Apr21-04, 05:06 AM
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...?
It seems that many valued logic must be formulated in terms of a stable 2-valued logic background.
Please Prove that Complementary Logic ( http://www.physicsforums.com/showpost.php?p=192318&postcount=25 ) can be reduced to a false/true logic.
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.
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
Apr21-04, 08:34 AM
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.
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.
(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
Apr21-04, 09:31 AM
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
. .
| |
|&__|_
|
a b
. .
| | <--- (Standard Math logical system fundamental building-block)
|#__|
|
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?
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
Apr21-04, 10:28 AM
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.
No,
only
a b
. .
| |
|#__|
|
is a Boolean Logic.
Please see http://www.physicsforums.com/showpost.php?p=190983&postcount=1 and http://www.physicsforums.com/showpost.php?p=192318&postcount=25
matt grime
Apr21-04, 11:57 AM
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?
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.
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.
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).
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 http://www.physicsforums.com/showpost.php?p=190983&postcount=1 and http://www.physicsforums.com/showpost.php?p=192318&postcount=25
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)
Russell E. Rierson
Apr22-04, 12:17 AM
Please Prove that Complementary Logic ( http://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?
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
Apr22-04, 05:11 AM
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).
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.
d = dead cat
l = live cat
In Complementary Logic (d & l) is a true statment of dead/live probability (like the wave/particle existence).
Please show what is (d & l) by an excluded-middle logical system.
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).
I used 'v' letter as an arrowhead in my diagram.
I mean that we have a probability of 1:2 and not some accurate value between 1 and 2.
By (f # t) I mean that some single result can be found through a probability of 1:2 .
matt grime
Apr22-04, 09:06 AM
so the and and xor symbols you are using aren't the usual and and xor symbols. so you need to define them. (ie what they do)
note it is not correct to say that the cat in shroedinger's experiment is alive and dead but the the state will take some value in some hilbert space with certain probabilities. and we've done that using boolean logic.. Anyway, your and is some other binary connective.
to some extent the answer is dependent on which school of QM thought you adopt. and you don't know the probability that the cat is alive is 1/2. it depends on how the experiment is set up.
i don't understand how you can say that you can't describe QM with boolean logic seeing as without it you would never have learnt about it in the first place. all the experiments you know of and theory is done in boolean logic.
Let us say that you explore, for example, Mandelbrot farctal only by R members, without using Complex numbers.
In both cases you will be able to research the results, but by using C and R numbers, we can get much more interesting information.
Because Complementary Logic is based on included-middle results of interactions between opposite elements (where Boolean or Fuzzy Logics are proper sub-systems of it) we get a much more powerful tool to explore and understand the QM phenomenon.
so the and and xor symbols you are using aren't the usual and and xor symbols. so you need to define them. (ie what they do)
AND and XOR connectives are independed and can be changed according to the "logical environment" that using them.
In Complementary Logic, probability is a first-order property that changing the results of AND and XOR connectives.
matt grime
Apr22-04, 09:38 AM
what on earth does 'explore the mandelbrot fractal only by R' mean? It doesn't even sound plausible.
you've still not shown that boolean logic is a subsystem of your alleged logic. nor how you would use it in any situation.
for instance what is the truth value, for want of a better phrase, of the proposition: If x, an integer, is divisible by 4, then x is even. I reckon it's true. what does your system say?
Please refresh screen and read again my previous post.
The real numbers can model the complex numbers; thus, anything you can do with complex numbers, you can do (in some fashion) with real numbers.
For example, I might consider a pair of real numbers, (c, d), and study the pairs of numbers (a_0, b_0) such that the following iteration
\begin{equation*}\begin{split}
a_{n+1} &= a_n^2 - b_n^2 + c \\
b_{n+1} &= 2 a_n b_n + d
\end{split}\end{equation*}
does not diverge to infinity.
And, in this way, one can study Julia sets (and thus the Mandelbrot set) without ever mentioning a complex number.
matt grime
Apr22-04, 04:41 PM
somehow i doubt that was what he had in mind (the unnecessary ontological commitment of the complex numbers...?) , but then i often have no idea what he means.
Hurkyl,
Some times simple thinking can halp us to understand simple examples.
Matt Grime wrote:
don't understand how you can say that you can't describe QM with boolean logic seeing as without it you would never have learnt about it in the first place. all the experiments you know of and theory is done in boolean logic.
My example of Mandelbrot set is this:
If we explore its structures in 1-dim we get 1-dim results.
If we explore its structures in 2-dim we get a 2-dim results and 1-dim results.
Shortly speaking, more dim we have less we are limited in our abilities to explore something.
Because Complementary Logic is naturally included-middle logical system (where Boolean and Fuzzy Logics are proper sub-systems of it) we have the freedom to use its extra logical forms, or not.
so the and and xor symbols you are using aren't the usual and and xor symbols. so you need to define them. (ie what they do)
AND and XOR connectives are independed and can be changed according to the "logical environment" that using them.
In Complementary Logic, probability is a first-order property that changing the results of AND and XOR connectives.
For example:
f=dead cat
t=alive cat
r=redundancy
u=uncertainty
When probability is a first-order property then AND connective is used whenever a no-unique result can be found:
<--r--> ^
t t |
# # u
f f |
| | v
|&__|_
|
When probability is a first-order property then XOR connective is used whenever a unique result can be found:
f t
| |
|#__|
|
Simple as that.
For example:
Let XOR be #
Let AND be &
Let a,b,c,d stands for uniqueness, therefore logical forms of 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}
{a, b, c, d}
. . . .
| | | |
|#_| | |
| | | <--(Last 4-valued logical form)
|#____| |
| |
|#_______|
|
={{{{x},x},x},x}
============>>>
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}
A 2-valued logic is:
b b
# #
a a
. .
| |
|&__|_
|
a b
. .
| | <--- (Standard Math logical system fundamental building-block)
|#__|
|
matt grime
Apr23-04, 03:07 AM
still not defined uncertainty and redundancy, non-standard terms.
mandelbrot's set doesn't have integer dimension....
so your logical theory trivially encompasses all others, yet you've not shown it has any other non-extant models.
mandelbrot's set doesn't have integer dimension....
I know it, but in 1-dim all you can get is the shadow of what you can find between 1-dim and 2-dim, isn't it?
still not defined uncertainty and redundancy, non-standard terms.
Please explain Why do you think they are not defined?
Russell E. Rierson
Apr23-04, 03:38 AM
Using the concept of "invariance/symmetry" :
T|F = F|T = T
The | represents a "choice" between T or F
Some might question the "equals sign".
Here is someone explaining a type of complementary logic also :eek:
http://users.erols.com/igoddard/gods-law.html
"choice" is a key-word in Complementary Logic because explored/explorer interacions cannot be ignored through its point of view, for example:
http://www.geocities.com/complementarytheory/Moral.pdf
http://users.erols.com/igoddard/gods-law.html is very interesting and supports Complementary Logic main point of view.
But in Complementary Logic 100%A is some unique result that we can get out of x1 xor x2 xor x3 xor ...
Russell E. Rierson
Apr23-04, 04:51 AM
http://users.erols.com/igoddard/gods-law.html is very interesting and supports Complementary Logic main point of view.
But in Complementary Logic 100%A is some unique result that we can get out of x1 xor x2 xor x3 xor ...
Of course! One asks oneself the question "What the heck does it mean for a wave function to collapse?"
According to Einstein, there is no instantaneous action at a distance!
What is 'distance' from your point of view?
For me 'distance' is the preventing side of some perevent/complement system for example: http://www.geocities.com/complementarytheory/4BPM.pdf
By Complementary Logic any existing element that can be chaneged, is the result of at least two opposites that simultaneously pereventing/defining each other.
Some times simple thinking can halp us to understand simple examples.
Take your own advice. :smile:
I demonstrated how, using logic and a "1-dim system", we are able to fully "explore" a "2-dim system".
The lesson I'm trying to demonstrate is:
Even if one system is a special case of another system, the first system can still be just as powerful as the second system.
AND and XOR connectives are independed and can be changed according to the "logical environment" that using them.
Incorrect. The representations may be the same (e.g. they're both called AND and XOR), but your AND and XOR are certainly very different from the AND and XOR from boolean logic.
Please explain Why do you think they are not defined?
Because you have not defined them.
I demonstrated how, using logic and a "1-dim system", we are able to fully "explore" a "2-dim system".
No you did not, because in 1-dim(=x-dim) universe no point can be found as a result of (x-dim,y-dim) system.
Your (c,d)(a0,b0) example is a (x-dim,y-dim) --> 2-dim system, and only then you can show a
"2-d Math picture" of mandelbrot set (which has a fractal-dim between 1-dim and 2-dim).
Shortly speaking, in a 1-dim universe any y-dim reduced to x-dim.
Therefore d reduced to c and b0 reduced to a0, and you have no 2-dim Math picture of some Julia set.
Even if one system is a special case of another system, the first system can still be just as powerful as the second system.
You did not show it yet.
AND and XOR connectives are independed and can be changed according to the "logical environment" that using them.
Incorrect. The representations may be the same (e.g. they're both called AND and XOR), but your AND and XOR are certainly very different from the AND and XOR from boolean logic.
In Complementary Logic, probability is a first-order property that changing the results of AND and XOR connectives.
For example:
f=dead cat
t=alive cat
r=redundancy
u=uncertainty
When probability is a first-order property then AND connective is used whenever a no-unique result can be found:
<--r--> ^
t t |
# # u
f f |
| | v
|&__|_
|
When probability is a first-order property then XOR connective is used whenever a unique result can be found:
f t
| |
|#__|
|
Simple as that.
For example:
Let XOR be #
Let AND be &
Let a,b,c,d stands for uniqueness, therefore logical forms of 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}
{a, b, c, d}
. . . .
| | | |
|#_| | |
| | | <--(Last 4-valued logical form)
|#____| |
| |
|#_______|
|
={{{{x},x},x},x}
============>>>
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}
A 2-valued logic is:
b b
# #
a a
. .
| |
|&__|_
|
a b
. .
| | <--- (Standard Math logical system fundamental building-block)
|#__|
|
matt grime
Apr23-04, 08:47 AM
I know it, but in 1-dim all you can get is the shadow of what you can find between 1-dim and 2-dim, isn't it?
Please explain Why do you think they are not defined?
in order the answers are:
that's at best wrong, at worst completely meaningless.
where in this thread have you offered a definition of uncertainty or redundancy? and i don't mean their plain english meanings.
it is a courtesy whenever you introduce non-standard terms to explain them.
you have i believe offered a vague idea of one of them in some other thread, but it didn't explain it fully.
matt grime
Apr23-04, 08:51 AM
No you did not, because in 1-dim(=x-dim) universe no point can be found as a result of (x-dim,y-dim) system.
Your (c,d)(a0,b0) example is a (x-dim,y-dim) --> 2-dim system, and only then you can show a
"2-d Math picture" of mandelbrot set (which has a fractal-dim between 1-dim and 2-dim).
Shortly speaking, in a 1-dim universe any y-dim reduced to x-dim.
Therefore d reduced to c and b0 reduced to a0, and you have no 2-dim Math picture of some Julia set.
You did not show it yet.
sentence one ahs no content as far as i can tell.
the mandelbrot set is a subset of R^2, or C, so the second bit is silly too.
third part? nope, nothing there that makes sense either (reduces?)
and finally, show what?
It is very simple matt,
Any b0 or d in Hurkyl's example is always 0 in a 1-dim universe, and if not then we are no longer in a 1-dim but in a 2-dim universe.
Therefore he gets (a0,0) or (c,0) 1-dim representation, which is definitely not a 2-dim representation of some Julia set.
Matt,
You ommited parts of it, so here is all of it:
mandelbrot's set doesn't have integer dimension....
I know it, but in 1-dim all you can get is the shadow of what you can find between 1-dim and 2-dim, isn't it?
still not defined uncertainty and redundancy, non-standard terms.
Please explain Why do you think they are not defined?
Your answer to the first part is:
that's at best wrong, at worst completely meaningless.
Please give more details why do you think so?
Your answer to the second part is:
where in this thread have you offered a definition of uncertainty or redundancy? and i don't mean their plain english meanings.
Please show us an example of how a definition of r and u when:
f=dead cat
t=alive cat
r=redundancy
u=uncertainty
When probability is a first-order property then AND connective is used whenever a no-unique result can be found:
<--r--> ^
t t |
# # u
f f |
| | v
|&__|_
|
will look like?
matt grime
Apr23-04, 10:52 AM
you haven't said what you mean by "shadow between 1-d and 2-d". sorry but it makes no sense as a sentence, elaborate, explain and clarify. nor have you explained what you mean be representation of a julia set, and what that has to do with the ambient space. seeing as RxR and R are in bijective correspondence I can encode the points of a 2-dim space in a 1-dim one, and by induction, any n-dim space. And the same goes for fractals as it's just some subset of some space.
how can i tell you what a definition of u and r will look like. they're your objects to define.
Matt,
seeing as RxR and R are in bijective correspondence I can encode the points of a 2-dim space in a 1-dim one,
You can ecode any {x} with any {x,y} so what.
By {x}_only 1-dim data you cannot represent {x,y} 2-dim data.
how can i tell you what a definition of u and r will look like. they're your objects to define.
Now it is clearly understood that you have nothing but a 'NO'_reflex in this r u case.
For example, your response to:
<--r--> ^
t t |
# # u
f f |
| | v
|&__|_
|
was "what is v"? and from this question we can learn that your abstraction's ability totally depends on the standard way.
My abstraction's ability totally depends on my non-standard way.
So, the problem of translation is twice difficult in our case.
But I think that there is a deeper problem here, with is:
You simply do not understand my ideas, and therefore they are "non sense" for you.
So I think it is the time to say good bye to each other because I cannot help you and you cannot help me.
matt grime
Apr23-04, 06:45 PM
given you inability to explain clearly any of your objects i can't see how asking what the v is in the diagram is a bad thing in any sense.
My logic is an included-middle yours is excluded-middle.
You say that included-middle can be defined by excluded-middle, I say it cannot, simply because probabilty is a first-order property in included-middle system, and in excluded-middle system it is not a first-order property.
If you can show how probabilty is a first-order property in excluded-middle logical system, then it will be the gate between our different worlds.
In Complementary Logic, probability is a first-order property that changing the results of AND and XOR connectives.
For example:
f=dead cat
t=alive cat
r=redundancy (more then one copy of the same value can be found)
u=uncertainty (more than one unique value can be found)
When probability is a first-order property then AND connective is used whenever a no-unique result can be found:
<--r--> ^
t t |
# # u
f f |
| | v
|&__|_
|
When probability is a first-order property then XOR connective is used whenever a unique result can be found:
f t
| |
|#__|
|
Simple as that.
For example:
Let XOR be #
Let AND be &
Let a,b,c,d stands for uniqueness, therefore logical forms of 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}
{a, b, c, d}
. . . .
| | | |
|#_| | |
| | | <--(Last 4-valued logical form)
|#____| |
| |
|#_______|
|
={{{{x},x},x},x}
============>>>
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}
A 2-valued logic is:
b b
# #
a a
. .
| |
|&__|_
|
a b
. .
| | <--- (Standard Math logical system fundamental building-block)
|#__|
|
If this time your response is "non-sense" then good-bye.
Russell E. Rierson
Apr24-04, 02:24 AM
What is 'distance' from your point of view?
For me 'distance' is the preventing side of some perevent/complement system for example: http://www.geocities.com/complementarytheory/4BPM.pdf
By Complementary Logic any existing element that can be changed, is the result of at least two opposites that simultaneously pereventing/defining each other.
Distance is a property between objects in space. Space is a structure, which is constructed of discrete units. The structure of space is a distributive lattice. A set of properties, being a "complementary logic?", expressing difference in wholeness.
Distance is a property between objects in space...
So to define distance we need at least two states local(= a unique object) and global(=a space).
Therefore any existing thing is at least a product of the interactions between the local and the global.
When we research a QM product then this is exactly what we find: a product which is both particle(=strong locality) and wave(=strong non-locality).
Shortly speaking, Complementary logic is the logic of interaction between opposite properties, which means: any distance (logical or physical) is the preventing property, where any non-distance is the complementing property.
Form this point of view, the evolution of concessions is the story of the increasing ability of communication between the global and the local in a cybernetic way, for example:
http://www.geocities.com/complementarytheory/CK.pdf
matt grime
Apr24-04, 06:02 AM
My logic is an included-middle yours is excluded-middle.
You say that included-middle can be defined by excluded-middle, I say it cannot, simply because probabilty is a first-order property in included-middle system, and in excluded-middle system it is not a first-order property.
If you can show how probabilty is a first-order property in excluded-middle logical system, then it will be the gate between our different worlds..[/B]
i'd like to see you explain where i said any of that.
the rest is starting to be readable. see what happens when you actually explain the meanings of the terms you use?
now, the main thing you need to demonstrate is that there is any point to all this.
For instance, are the axioms of ZF(C) consistent in this "logical world"
f=dead cat
t=alive cat
r=redundancy (more then one copy of the same value can be found)
u=uncertainty (more than one unique value can be found)
When probability is a first-order property then AND connective is used whenever a no-unique result can be found:
<--r--> ^
t t |
# # u
f f |
| | v
|&__|_
|
When probability is a first-order property then XOR connective is used whenever a unique result can be found:
f t
| |
|#__|
|
Simple as that.
For example:
Let XOR be #
Let AND be &
Let a,b,c,d stands for uniqueness, therefore logical forms of 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}
{a, b, c, d}
. . . .
| | | |
|#_| | |
| | | <--(Last 4-valued logical form)
|#____| |
| |
|#_______|
|
={{{{x},x},x},x}
============>>>
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}
A 2-valued logic is:
b b
# #
a a
. .
| |
|&__|_
|
a b
. .
| | <--- (Standard Math logical system fundamental building-block)
|#__|
|
Can you show us a ZF(C) axiom whare probability included?
now, the main thing you need to demonstrate is that there is any point to all this.
Please give more details.
matt grime
Apr24-04, 07:34 AM
es, you keep reposting that but odn't actually demonstrate that it is useful at any point.
As you don't define what you mean by probability we cannot answer your last request. Seriously, mathematics is a formal construction; you cannot just informally use words and expect it to be meaningful. if we think of a proper quantum system (ie not the stupid cat experiment) then all of the things in it are modelled using properly defined mathematical objects. so why don't you demonstrate a way of producing pure states, say, within your system. hint, you'll have to construct the real numbers, the complex numbers, in fact everything if you want to do mathematics. if youy merely want to argue about philosophy then do so, but don't get angry and change your user name so that we might think you were pretending to do maths seriously.
I did not choose to change my name. I actually had no choice because PF mentors shut me down twice in the last 2 years, and as you now, if you want to register again you have no choice but to do it under a new name.
QM element is naturally included-middle element, because it is based on two opposite properties that preventing from us to know exactly both of them simultaneously, as we can do in macro systems.
If we want to develop some formal language that deal with QM world, we have to do it by changing our logical reasoning from excluded-middle to an included-middle.
And here Complementary Logic entering to the picture, and using probability as first-order property.
Through CL (Complementary Logic) any n>1 has several variations of internal structures based on interactions between its integral side (root-like side) and its differential side (leaf-like side).
These internal structures can be ordered by their vagueness degrees, which vagueness is a combination between redundancy_AND_uncertainty properties that give us the "cloud of probability" of each ordered information form.
Redundancy exists if more then one copy of the same value can be found.
Uncertainty exists if more than one unique value can be found.
For example: http://www.geocities.com/complementarytheory/ComplexTree.pdf
Pay attention that I used the words "information forms" because these information forms, which are ordered by their vagueness degrees, can be used as general building-blocks that can help us to develop much more fine models that have to deal with included-middle problems.
From the pdf example we can learn that the standard base value expansion method is actually based on 0_redundancy_AND_0_uncertainty building blocks, which are a very small part of infinitely many different building blocks that can be used by us to construct and explode a very complex information models with variety of combinations of vagueness.
Another thing is that some experimental result is actually some single section which is cut out of 0_redundancy_AND_0_uncertainty building blocks that are ordered in several scales.
My system suggesting a much more complex information form as a result, as can be found in the last page of the pdf example.
Shortly speaking, because any information form in my system is at least structural/quantitative, it can be used straightly as it is, and we don't have to translate it to quantity before we can use it in our system.
matt grime
Apr24-04, 11:04 AM
why are all your information forms only numbers?
Please see post #79 and also please read again my last post, thank you.
If I am more understood to you then please read:
http://www.geocities.com/complementarytheory/Complex.pdf
Why are all your information forms only numbers?
I need help to develop it, it is only in its first stages, and it is definitely cannot be done by a one person.
Actually, you should be able to post here as Organic.
QM element is naturally included-middle element, because it is based on two opposite properties that preventing from us to know exactly both of them simultaneously, as we can do in macro systems.
This is a common misunderstanding about QM. Things aren't simultaneously (classical) particles and (classical) waves... they're neither; instead they're some new quantum mechanical thing that given the right circumstances, can approximate a (classical) particle or a (classical) wave.
Since quantum mechanical things aren't classical particles, it should be unsurprising that they cannot be represented exactly as classical particles.
If we want to develop some formal language that deal with QM world, we have to do it by changing our logical reasoning from excluded-middle to an included-middle.
As I've tried to point out with your Mandelbrot example, one can use weaker systems (e.g. a "1-dim system") to build new systems (e.g. a "2-dim system"). Even if you are right, it is not necessarily the case that the old logical reasoning is incapable of building the new logical reasoning.
And here Complementary Logic entering to the picture, and using probability as first-order property.
In particular, you seem to indicate now that probability is the key ingredient in your vision of the new way to do things. Well, the old way has known how to do probability for a long time, why do you think it is inadequate now?
matt grime
Apr24-04, 02:47 PM
Good luck with convincing him that quantum objects such as photons are neither waves nor particles. I seem to remember posting a long sequence emphasizing the difference between "displaying wave like properties" and "being a a wave". I don't think it got through.
Hurkyl,
Actually, you should be able to post here as Organic.
First, I really hope that it is not you who shut me down as Organic, because it is a west of time to speck with mentors which closing members because they have different point of view than them.
...instead they're some new quantum mechanical thing that given the right circumstances,...
And this is exactly my point of view which is: wave/particle properties are under a probability state, and they are not physical realm until we change this probability according to our measurements tools, to some accurate particle-like XOR wave-like results.
Shortly speaking , I have Max Born's probability point of view ( http://www.chembio.uoguelph.ca/educmat/chm386/rudiment/tourquan/born.htm ).
In particular, you seem to indicate now that probability is the key ingredient in your vision of the new way to do things. Well, the old way has known how to do probability for a long time, why do you think it is inadequate now?
The probability of convetional Math is not a first-order property, therefore a natural first-order system is much better in this case, even if the old way works.
As I've tried to point out with your Mandelbrot example, one can use weaker systems (e.g. a "1-dim system") to build new systems (e.g. a "2-dim system"). Even if you are right, it is not necessarily the case that the old logical reasoning is incapable of building the new logical reasoning.
The old logical reasoning is capable of building the new logical reasoning if probabilty is a first-order property of it.
If you don't think so then please show us how we can represnt Complementary Logic by an excluded-middle logical system.
To help you, please read this first:
http://www.geocities.com/complementarytheory/BFC.pdf
matt grime
Apr24-04, 05:19 PM
present here and now a rigorous explanation/definition of probability uaing only "first order" objects, whatever theyu may be. Or are you confusing the warm fuzzy idea of probability with its rigorous axiomatic abstraction?
can i suggest that the reasons you aren't allowed to post in the maths forum are your refusals to deal in mathematics and hijacking of threads to espouse your unmathematical views?
wave/particle properties are under a probability state
The QM point of view is that wave / particle properties (when they appear) are approximate, not under a "probability state".
can i suggest that the reasons you aren't allowed to post in the maths forum are your refusals to deal in mathematics and hijacking of threads to espouse your unmathematical views?
I was shut down after I put http://www.physicsforums.com/showthread.php?t=18972 in General Math.
Math , in my opinion, is not an unchagable monolitic objective state, but a living form of language.
present here and now a rigorous explanation/definition of probability uaing only "first order" objects, whatever theyu may be. Or are you confusing the warm fuzzy idea of probability with its rigorous axiomatic abstraction?
please give more details, because I do not really understand what do you looking for.
The QM point of view is that wave / particle properties (when they appear) are approximate, not under a "probability state".
By saying "under a probability state" (sorry about my poor English) I speak about wave / particle properties before they appear through our experiment tools.
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
Apr24-04, 06:18 PM
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
Apr25-04, 01:48 AM
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
Apr25-04, 04:29 AM
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?)
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 any one 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}
In the first case each accord can be one of 4^4 different possibilities.
{a, b, c, d}
. . . .
| | | |
|#_| | |
| | | <--(Last 4-valued logical form)
|#____| |
| |
|#_______|
|
={{{{x},x},x},x}
In the last case each accord is a one and only one possibility.
The ordered possibilities between 4^4 and 1 is:
============>>>
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}
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:
<--r--> ^
t t |
# # u
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
Apr25-04, 05:54 AM
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.
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" statment.
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
Apr25-04, 03:44 PM
If all the statements must be true you're even omre off beam than you first appear.
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?
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
Apr26-04, 09:51 AM
"Take each note as a "true" statment"
and now you contradict that...
Take each note as a "true" statment
I wrote "true" and not true, which means that there is no true XOR false in my system.
matt grime
Apr27-04, 03:45 AM
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.
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
Apr27-04, 06:09 AM
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.
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
Apr27-04, 07:03 AM
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.
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
Apr27-04, 08:40 AM
"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 sytem 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.
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
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
Apr27-04, 02:07 PM
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.
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.
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...
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:
http://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.
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