Good summary through the "Turing paradox" (quantum Zeno effect)
A good summary by Matt O’Dowd. https://youtu.be/SMPid7Sh0EE <https://youtu.be/SMPid7Sh0EE> The quantum many-world formulation of quantum mechanics (which is simply the wave equation without the collapse postulate) is similar to the "material" mode of self-reference of the arithmetically sound Turing (or Church, or ...) digital machines/programs/words/numbers, except that here the wave itself belongs to the phenomenology and is extracted from number relations. If they were a collapse here, the physical reality would become the arithmetical reality, but that is impossible: the physical (as part of the theological) is an inside view and as such is *far* greater than arithmetic. Same for the infinite: it does not exist ontologically (as only 0, s0, ss0, sss0, 0, ... exist), but the universal numbers can't avoid meeting the infinities in many shapes and colours, and this through their (universal) relations with other numbers due to the arithmetical laws... With mechanism, the wave or the matrix itself is brought from the internal view of the universal (Turing, Church, Kleene, Post, ...) numbers/words. This guy is good in comparing the Copenhagen theory and the Everett theory (which is just one axiom less), not just in this video. As a logician, I don’t consider Copenhagen as being a different interpretation, but as a different theory. Copenhagen postulates some reality (the wave?) and a collapse, Everett does not. Bruno -- You received this message because you are subscribed to the Google Groups "Everything List" group. To unsubscribe from this group and stop receiving emails from it, send an email to everything-list+unsubscr...@googlegroups.com. To view this discussion on the web visit https://groups.google.com/d/msgid/everything-list/4067B030-F8DC-4F5F-9429-2D5A3BC6DB71%40ulb.ac.be.
Re: Combinator 5 (Summary +solutions)
Hi, Before the Turing Universality, I give the solution of the exercises. I recall also all what we have seen so far. You can still get the wagon from this post only, but of course the preceding one can help. Still ask any question if there is any problem. What is a combinator? K and S are combinators, and all combinations of S and K are combinators. That means simply that if x and y are combinators (x y), written xy, is a combinator. So K, S, KK, KS, … KKK, K(KK), … are combinators. All combinators are build in that way. Note the parentheses in K(KK). They are obligatory: KKK is ((KK)K), that is KK married with K. K(KK) is (K(KK)), that is K married with KK … and those two combinators KKK and K(KK) are different, as we can see directly from the only two laws, or rules, or axioms, which constitute the theory: 1) Kxy = x 2) Sxyz = xz(yz). So for example KKK = K, but K(KK) remains itself, as it match Kx (with x = KK, but do not match Kxy. K(KK)K gives KK, because K(KK)K matches Kxy with x = KK, and y = K. ABCDEF means always (AB)C)D)E)F). I hope this is well understood. ABCDEF is the combinator ABCDE applied on F. It is not the combinator A applied on BCDEF. That one has to be written A(BCDEF). This convention makes much easier to read on the combinators. Then we have seen that the two axioms “1)” and “2)” make it possible to find many combinators, which each doing some combination. For example Ix = x. Indeed I = SKK does that job. SKKx = Kx(Kx) by rule “2)” Kx(Kx) = x by rule “1)”. OK? Similarly we found that Mx = xx is given by SII SIIx = Ix(Ix) by rule “2)” Ix(Ix) = xx, by Ix = x, with I abbreviating SKK. OK? We found Bxyz = x(yz). B introduces parentheses on the right of its “arguments” x y z. We have seen that B = S(KS)K. Then I have given some simple algorithms to synthesise a combinator from its specification. For example the algorithm ABCF: A) [x]N = KN B) [x]x = SKK = I C) [x]Nx = N F) [x](XY) = S ([x]X) ([x]Y) By specification, I mean the “Bxyz = x(yz)” description. The algorithm say that B = [x][y][z]x(yz). This means [x] ([y] ([z]x(yz)) ). We eliminate z, then y, then x. In that order. Let me do it, for the new one who take the wagon just now: [z]x(yz) = We use the rule “F)”, as x(yz) is the combinator XY with X = x, and Y = (yz), and the variable z appears somewhere (it is not a constant, with respect to z). Of course we use the algorithm with z playing the role of x. This needs some training, meaning a bit of home-work. That gives, [z]x(yz) = S [z]x [z]yz, which gives, by rule “A)” S (Kx) [z]yz, as x is a constant, when z is the variable to be eliminate, and that gives S (Kx) y, by using the rule “C)”. OK? z has been eliminated, and we need to eliminate [y] now: [y] S(Kx)y which is [y] ((S(Kx))y and that gives immediately, by rule “C)”: S(Kx). It remains to eliminate x: [x] S(Kx) The x is attached to K, so we CANNOT use the rule “C”, and need to use “F)” again: [x] S(Kx) = S [x]S [x]Kxby rule “F)” but [x]S = KS, by rule “A)”. And [x]Kx is K, so we get, for B S(KS)K OK? Then we have seen how to find a combinator defined in term of itself (recursion). Exemple: find A such that Ax = xA I gave two algorithm for this, using the “sage bird”, or "from scratch”. 1) from scratch: First I change the variable y for the variable x: we search A such that Ay = yA. By the elimination algorithm above we can find a combinator X such that Xxy = yxx, indeed it is [x][y]yxx We calledl it X. Then XX gives the solution. Indeed XXy = y(XX). 2) using the sage bird, aka the paradoxical combinator, to the fixed point combinator, often denoted by Y in the literature (Smullyan noted it “theta”). Y specification is Yx = x(Yx), from scratch we found a solution: Y = UU, with Uxy = y(xxy). U is the Turing combinator. Y gives the fixed point of its argument, that is a combinator remains invariant when we apply that argument on it. Having Y, we start again by changing the variable, we search A such that Ay = yA, Then, by the elimination algorithm, we can find a X such that Xxy = yx, indeed it is T, the permuting bird specified by Txy = yx. Then the solution is simply given by A = YT. Indeed YT gives a fixed point of T, that is A = TA, and then you see that Ay = TAy = yA. OK? This is crucially important. We can now define a combinators in term of itself, by a recursion equation where the unknown combinators appears in the right. The example above should convince you that all recursion equations admit solutions. This is what makes combinators quite powerful. All right? Then we have seen a bit of logic and arithmetic. I use, like Smullyan, the elegant idea of Barendrecht. “t” represents a boolean constant true, and “f” represents a constant boolean false. Barendrecht represent t by K, and f by KI. Note that Kxy = x, and KIxy = y, so that tyz = y, and fyz = yz and this means that we can represent “If x then y else
Re: Abridged summary of everything-list@googlegroups.com - 35 updates in 4 topics
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Re: Modal summary and new exercises + motivation
On 25 Mar 2014, at 03:02, LizR wrote:
Thank you for the above, for my diary!
On 24 March 2014 20:14, Bruno Marchal marc...@ulb.ac.be wrote:
New exercise:
show
(W,R) respects A - []A
iff
R is symmetrical.
OK, symmetrical means for all a and b, a R b implies b R a.
A - []A can (I hope) be read as the truth of A in one particular
world (which I will call this world) implies that for all worlds
accessible from this world, there exists at least one world in which
A is true.
I was about to write that you did that error again, but looking twice,
I saw you are entirely correct, OK.
Well, there is indeed one world accessible from those other worlds,
in which A is true - this one! Because all worlds accessible from
this one can access this world (due to symmetry) and in this world A
is true.
Excellent.
You proved that
If R is symmetrical then (W, R) respects A - []A.
What about finishing the work and prove the reciprocal? Hmm... Please
look at the iff in the quoted quote above.
You should still show that
if (W, R) respects A - []A then R is symmetrical.
The mere fact that A - []A is true in all worlds, whatever the
valuation is, imposes the symmetry for the binary accessibility.
To show that, you can reason by absurdum. You imagine that (W, R)
respects A - []A, and you consider that R is not symmetrical. Then
you have to find a valuation leading to a counterexample, a world in
which A is true and []A is false.
Let me do it, so that you can rest after the good work :)
*
If (W, R) is not symmetrical, there is two worlds a and b so that a R
b, and ~(b R a). OK?
Let us choose the valuation V which assign 0 to p in all the worlds
accessible from beta.
Well, but then if p is true in alpha, []p is true in alpha (as we
assume that (W, R) respects A - []A). But then p must be true in
beta, OK?
But beta accesses only to worlds with p false. Contradiction.
We say that the illuminated multiverse (W, R)
with W = {a, b, c},
together with the non symmetrical relation explicitly defined by aRb,
bRc. (so we have 'not bRa'), together with the valuation: p true in
a, and false in c, constitutes a counterexample, to the idea here that
a (W,R) with R non symmetrical can respect A - []A. Indeed, in that
illuminated multiverse the assymmetry break makes it possible to break
the law, and in the world a p is verified and []p is not
contradicting the law A - []A.
All right?
Bruno
PS I send this to FOAR as this is part of an answer to his question,
and a key (albeit tiny) part of the derivation of the physical laws,
notably giving clues on the reversibility on the bottom of the domain
of indeterminacy ( the true sigma_1 sentences).
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Modal summary and new exercises + motivation
Liz, Brent, others,
Just a revision, before you forget the definitions :)
A multiverse (W, R), or frame, is a set W, with a binary relation R.
The elements of the set are called world , and denoted often by
greek letter (alpha, beta, gamma, ...). The binary relation is called
accessibility relation, and constitutes the main ingredient in Kripke
semantic.
Now we are doing logic. So we suppose the usual classical
propositional language, with atomic sentence letters p, q, r, ..., and
the symbol - and f.
(The other logical symbols can be defined from - and f, for
example ~p can be defined as an abbreviation of (p - f), as it clear
when we give the usual classical semantic).
From them, we can build the usual formula like p, q, (p - q), ((p -
f) - (q - r)), etc.
A will denote such arbitrary formula. (It is a metavariable, and it
does not belong to the formal symbols).
A valuation or illumination is a function from {p, q, r, ...} to {1, 0}.
An multiverse becomes illuminated when each world get a valuation. So
in each world, the sentence letters can be said true, or false,
according to the valuation. Let us call V that valuation which
attributes a 1 or a 0 to each propositional letter, in each world of
the multiverse.
So we can denote an illuminated multiverse by (W, R, V).
We suppose that each world obeys classical logic. basically this
means, that:
- The propositional letter are true or false, according to the
valuation in the illuminated multiverse.
- f is false in alpha (any alpha)
- A - B is true in the world alpha iff A is false in alpha or B is
true in alpha.
Ah, but we do modal logic. So we have one unary connector symbol more:
[]p, and Kripke semantics for modal logic is that:
[]A is true in alpha iff A is true in all beta such that alpha
R beta. That is, such that beta is accessible from alpha.
p is defined as an abbreviation of ~[] ~, and you might enjoy
verifying that
A is true in alpha iff there exist a world beta, with A true
in beta, and beta accessible from
Last but not least definitions. Now that we know what it means for a
formula to be true in a world, we say that :
An illuminated multiverse (W, R, V) satisfies a formula if that
formula is true in all worlds of the multiverse.
A multiverse (W, R) respect a formula if that formula is true in all
worlds for any of its illumination (W, R, V).
That's all you have to know. Print this or recopy this by hand in the
diary, as this will remains with us, for sometime.
Then you have shown (Brent and Liz, at least):
(W, R) respects []A - A
iff
R is reflexive (that is for all alpha in W, alpha R alpha)
and
(W, R) respects []A - A
iff
R is ideal on W, that means that from any world you can access to some
world (another one or itself). It means that there is no cul-de-sac
worlds.
OK?
I will send one post with all the proofs.
Brent, in some post you tell me you were working on the proof of
(W, R) respects []A - [][]A
iff
R is transtive (aRb and bRc implies aRc) (writing quickly a b c for
alpha beta gamma, it is also clearer:
R is reflexive iff for all a aRa,
R is symmetrical iff for all a and B, aRb implies bRa.
I will prove the transtitive case soon, unless you ask some delay.
New exercise:
show
(W,R) respects A - []A
iff
R is symmetrical.
This one plays some role in the specific derivation of physics from
arithmetic. This is due to the fact that the logic B, with axioms:
[](A - B) -( []A - []B)
[]A - A
A - []A,
translates a minimal Quantum logic in modal terms. []A quantized
the truth of A in some sense. Quantum logic are usually handled by
the algebraical structure of the observable, and quantum proposition
are structured in terms of orthospace, that is a space with a
orthogonality notion (a scalar product). The complementary relation
(not-orthogonal) defines a proximity relation.
In arithmetic we will get a weakening/strengthening, of this as we
will get A-[]A, only for the atomic sentences (the arithmetical
interpretation of the letters p in the modal logic), and we will loss
the necessitation rules, losing some quantum tautologies, perhaps, but
not necessarily. It is a strengthening by the axioms corresponding to
the Löb formula, and the arithmetical reality (intensional and
extensional).
Bruno
http://iridia.ulb.ac.be/~marchal/
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Re: Modal summary and new exercises + motivation
Thank you for the above, for my diary! On 24 March 2014 20:14, Bruno Marchal marc...@ulb.ac.be wrote: New exercise: show (W,R) respects A - []A iff R is symmetrical. OK, symmetrical means for all a and b, a R b implies b R a. A - []A can (I hope) be read as the truth of A in one particular world (which I will call this world) implies that for all worlds accessible from this world, there exists at least one world in which A is true. Well, there is indeed one world accessible from those other worlds, in which A is true - this one! Because all worlds accessible from this one can access this world (due to symmetry) and in this world A is true. -- You received this message because you are subscribed to the Google Groups Everything List group. To unsubscribe from this group and stop receiving emails from it, send an email to everything-list+unsubscr...@googlegroups.com. To post to this group, send email to everything-list@googlegroups.com. Visit this group at http://groups.google.com/group/everything-list. For more options, visit https://groups.google.com/d/optout.
Re: Modal Logic (Part 3: summary + 1 exercise)
On 11 Feb 2014, at 04:27, LizR wrote: On 10 February 2014 01:49, Bruno Marchal marc...@ulb.ac.be wrote: So with - and f we can define all connectors. Is there a connector (like , V, -, ...) such that all connectors can be defined from it? You just said that ... oh do you mean without using 'f' ? This is a facultative exercise. Only for possible raining sundays. We will not use this in the sequel. OK. I don't know. It doesn't seem intuitively obvious, but if you can use ~ then we have already used and (I think) V that way... I think. Well, even though I did it, the result still looks rather strange to me! Cantor said I see but don't believe it. it is normal. von Neuman said nobody understand math, mathematicians get only used to it. I seem to be in good company. They laughed at Archimedes. The laughed at Einstein. The laughed at Bozo the clown... Understanding is good. Understanding and memorizing, even with the help of a well presented diary, is better, as it saves the future possible works. I agree. I'm sure I started one, too, but I can't find it now. (So sometimes I have to treat you as my diary...) Well, I hope you will not lost me too! Likewise. Gosh! Well, I lost myself myself, if I can say, and that as many times as they are numbers ... Memorizing is good, but only if you manage to keep the memory accessible. 'course. Yes OK. I hope you will not forget that. Some would disagree... I guess that they met the bad math teacher who kicks the student before math kick them, making it impossible for them to understand the real kicking back of math, and develop the appreciation. That's bad for the slow student, which sometimes are slow because they are more demanding in understanding, and it is good for the quick student, who can learn to solve problem by no more than pattern matching, without any understanding. Consumerist societies favour quick students, which aggravate the situation for slow students, and long term project. As a math teacher, I try to help the two kinds of student, but it is not always easy, and to be honest, I favor the slow one. For me, a valid reasoning with a false answer is better than a false reasoning with a correct answer. I know that in real life, the contrary is true. Wise words. Some others seems interested in the thread too, but might be less courageous for participating, as you need some courage to do a sort of persistent exam online. I can understand. But I know that if I explain everything ex-cathedra, everyone will be lost somewhere, and nobody will know where. I do hope some others will participate to make things lighter on your shoulders. It takes a bit of courage for me, certainly. Especially since it's all stuff that seems to melt away, even though I understood it at the time. But then I *can* switch back and understand it again. But I find popularised physics and biology easier to follow than popularised logic, even though we have most of Raymond Smullyan's books. Physics, biology, natural science are aristotelian, and use the natural intuitive (1p) conception of reality. Logicians and Classical logicians, well, first they insist not doing physics, nor philosophy, but mathematics, and this (most of the time unconsciously) makes them, with comp, exploring the highly counter-intuitive mindscape of the universal machine. Logicians like the counter-intuitive surprises, and they delight in invalidating prejudices. Smullyan's brother, if I remember well, told to the little Raymond --'tonight, I will surprise you, I promise!' Little Raymond waited all the night, but got nothing, so at morning he complained to his brother I thought you would surprise me!. The brother replied: If you thought that, are you not surprised by the fact that nothing happened. You got me!, Smullyan said, and he was very pleased, and surprised. Bruno Thank god he is still alive! I had to go and check - last time I checked on someone, he'd died. It was John Galbraith Graham, my hero of the crossword world. I never knew him yet I did in a way, just like Mr Smullyan and Ursula le Guin. -- You received this message because you are subscribed to the Google Groups Everything List group. To unsubscribe from this group and stop receiving emails from it, send an email to everything-list+unsubscr...@googlegroups.com. To post to this group, send email to everything-list@googlegroups.com. Visit this group at http://groups.google.com/group/everything-list. For more options, visit https://groups.google.com/groups/opt_out. http://iridia.ulb.ac.be/~marchal/ -- You received this message because you are subscribed to the Google Groups Everything List group. To unsubscribe from this group and stop receiving emails from it, send an email to everything-list+unsubscr...@googlegroups.com. To post to this group, send email to
Re: Modal Logic (Part 3: summary + 1 exercise)
On 12 February 2014 06:31, Bruno Marchal marc...@ulb.ac.be wrote: Smullyan's brother, if I remember well, told to the little Raymond --'tonight, I will surprise you, I promise!' Little Raymond waited all the night, but got nothing, so at morning he complained to his brother I thought you would surprise me!. The brother replied: If you thought that, are you not surprised by the fact that nothing happened. You got me!, Smullyan said, and he was very pleased, and surprised. Yes I remember that story. I think he said to Raymond the day before April Fool's day - Tomorrow I will fool you like you've never been fooled before. And the rest was as you said. -- You received this message because you are subscribed to the Google Groups Everything List group. To unsubscribe from this group and stop receiving emails from it, send an email to everything-list+unsubscr...@googlegroups.com. To post to this group, send email to everything-list@googlegroups.com. Visit this group at http://groups.google.com/group/everything-list. For more options, visit https://groups.google.com/groups/opt_out.
Re: Modal Logic (Part 3: summary + 1 exercise)
On 10 February 2014 01:49, Bruno Marchal marc...@ulb.ac.be wrote: So with - and f we can define all connectors. Is there a connector (like , V, -, ...) such that all connectors can be defined from it? You just said that ... oh do you mean without using 'f' ? This is a facultative exercise. Only for possible raining sundays. We will not use this in the sequel. OK. I don't know. It doesn't seem intuitively obvious, but if you can use ~ then we have already used and (I think) V that way... I think. Well, even though I did it, the result still looks rather strange to me! Cantor said I see but don't believe it. it is normal. von Neuman said nobody understand math, mathematicians get only used to it. I seem to be in good company. They laughed at Archimedes. The laughed at Einstein. The laughed at Bozo the clown... Understanding is good. Understanding and memorizing, even with the help of a well presented diary, is better, as it saves the future possible works. I agree. I'm sure I started one, too, but I can't find it now. (So sometimes I have to treat you as my diary...) Well, I hope you will not lost me too! Likewise. Memorizing is good, but only if you manage to keep the memory accessible. 'course. Yes Some would disagree... I guess that they met the bad math teacher who kicks the student before math kick them, making it impossible for them to understand the real kicking back of math, and develop the appreciation. That's bad for the slow student, which sometimes are slow because they are more demanding in understanding, and it is good for the quick student, who can learn to solve problem by no more than pattern matching, without any understanding. Consumerist societies favour quick students, which aggravate the situation for slow students, and long term project. As a math teacher, I try to help the two kinds of student, but it is not always easy, and to be honest, I favor the slow one. For me, a valid reasoning with a false answer is better than a false reasoning with a correct answer. I know that in real life, the contrary is true. Wise words. Some others seems interested in the thread too, but might be less courageous for participating, as you need some courage to do a sort of persistent exam online. I can understand. But I know that if I explain everything ex-cathedra, everyone will be lost somewhere, and nobody will know where. I do hope some others will participate to make things lighter on your shoulders. It takes a bit of courage for me, certainly. Especially since it's all stuff that seems to melt away, even though I understood it at the time. But then I *can* switch back and understand it again. But I find popularised physics and biology easier to follow than popularised logic, even though we have most of Raymond Smullyan's books. Thank god he is still alive! I had to go and check - last time I checked on someone, he'd died. It was John Galbraith Graham, my hero of the crossword world. I never knew him yet I did in a way, just like Mr Smullyan and Ursula le Guin. -- You received this message because you are subscribed to the Google Groups Everything List group. To unsubscribe from this group and stop receiving emails from it, send an email to everything-list+unsubscr...@googlegroups.com. To post to this group, send email to everything-list@googlegroups.com. Visit this group at http://groups.google.com/group/everything-list. For more options, visit https://groups.google.com/groups/opt_out.
Re: Modal Logic (Part 3: summary + 1 exercise)
On 08 Feb 2014, at 22:05, LizR wrote:
On 8 February 2014 08:43, Bruno Marchal marc...@ulb.ac.be wrote:
On 07 Feb 2014, at 02:29, LizR wrote:
On 7 February 2014 09:14, Bruno Marchal marc...@ulb.ac.be wrote:
On 06 Feb 2014, at 07:39, LizR wrote:
snip
OK, having had a look at what you say below, let's have another go.
Start from p - q being equivalent to (~p V q)
That gives us ~p - q equiv (p V q) and from the above ~p is (p -
f) so p V q is (p - f) - q which I seem to remember is what you
got. OK so far.
p q --- well, p - q is ~(p ~q), so ~(p - q) = (p ~q) and
~(p - ~q) = (p q)
so ~(p - (q - f)) which I guess is ((p - ( q - f)) - f) = (p
q)
Does it?!?! Looking below, I see that it does. Wow.
I knew you can do that.
With hints.
So with - and f we can define all connectors.
Is there a connector (like , V, -, ...) such that all
connectors can be defined from it?
This is a facultative exercise. Only for possible raining sundays. We
will not use this in the sequel.
But that doesn't make sense, because requires two arguments, so
it would have to be something like ... well, p - q is (~p V q)
and it's also ~(p ~q), which contain V and ... I'm not sure I
know what you mean.
Like for ~, to define and V to a machine which knows only -
and f. You can use the ~, as you have alredy see that you
can define it with - and f.
I reason aloud. Please tell me if you understand.
First we know that p - q is just ~p V q, OK?
So the V looks already close to -. Except that instead of ~p
V q (which is p - q) we want p V q.
May be we can substitute just p by ~p: and p V q might be then ~p -
q,
Well, you can do the truth table of ~p - q, and see that it is the
same as p V q.
To finish it of course, we can eliminate the ~, and we have that
p V q is entirely defined by (p - f) - q.
OK?
And the :
Well, we already know a relationship between the and the V,
OK? The De Morgan relations.
So, applying the de Morgan relation, p q is the same as ~(~p V
~q), (the same logically, not pragmatically, of course).
That solves the problem.
But we can verify, perhaps simplify. We can eliminate the V by
the definition above (A V B = ~A - B),
~(~p V ~q) becomes ~(~~p - ~q), that is ~(p - ~q). Or, to really
settle the things, and define from - and f:
p q = ((p - (q - f)) - f).
OK?
Apparently, yes.
OK. (Not sure what you mean by apparently, though).
Well, even though I did it, the result still looks rather strange to
me!
Cantor said I see but don't believe it. it is normal. von Neuman
said nobody understand math, mathematicians get only used to it.
Each world, once illuminated (that is once each proposition
letter has a value f or t) inherits of the semantics of classical
proposition logic.
This means that if p and q are true in some world alpha, then (p
q) is true in that world alpha, etc.
in particular all tautologies, or propositional laws, is true in
all illuminated multiverse, and this for all illuminations (that
for all possible assignment of truth value to the world).
OK?
Question: If the multiverse is the set {a, b}, how many
illuminated multiverses can we get?
I suppose 4, since we have a world with 2 propositions, and each
can be t or f?
Answer: there is three letters p, q, r, leading to eight
valuations possible in a, and the same in b, making a total of 64
valuations, if I am not too much distracted. I go quick. This is
just to test if you get the precise meanings.
Oh, OK. So a and b are worlds, not ... sorry. I see.
Good.
So that is 2^3 x 2^3 because a has p,q,r = 3 values, all t or f,
as does b. OK now I see what you meant.
OK.
Of course with the infinite alphabet {p, q, r, p1, q1, r1,
p2, ... } we already have a continuum of multiverses.
I can't quite see why it's a continuum. Each world has a countable
infinity of letters, and the number of worlds is therefore 2 ^
countable infinity! Is that a continuum?
Yes. We proved it, Liz.
Yes I had a sneaky suspicion we did. It seems familiar ... a bit.
Understanding is good.
Understanding and memorizing, even with the help of a well presented
diary, is better, as it saves the future possible works.
I agree. I'm sure I started one, too, but I can't find it now. (So
sometimes I have to treat you as my diary...)
Well, I hope you will not lost me too!
Memorizing is good, but only if you manage to keep the memory
accessible. 'course.
Take a the infinite propositional symbol letters {p, q, r, p1, q1,
r1, p2, ... } . They are well ordered. So a sequence of 1 and 0
(other common name for t and f) can be interpreted as being a
valuation. The valuation are the infinite sequences of 1 and 0. Or
the function from N to {0, 1}.
If such a set of function was in bijection with N, i - f_i, the
function g defined by g(n) = f_n(n) + 1 would be a function f_i,
let us sat f_k, and f_k, applied on k,
Re: Biology, Buddha and the irreflexive Multiverse (was Re: Modal Logic (Part 3: summary + 1 exercise)
On 07 Feb 2014, at 23:21, meekerdb wrote: On 2/7/2014 10:40 AM, Bruno Marchal wrote: On 06 Feb 2014, at 21:29, meekerdb wrote: On 2/6/2014 12:14 PM, Bruno Marchal wrote: In Kripke semantic all statements are relativized to the world you are in. []A can be true in some world and false in another. The meaning of [] is restricted, for each world, to the world they can access (through the accessibility relation available in the Kripke multiverse). []A still keep a meaning, but only in each world. So everything is said when we define the new meaning of [] by the rule []A is true in alpha, by definition, means that A is true in all world beta *accessible* from alpha. And A is true in alpha iff there is a world beta; where A is true, accessible from alpha. Suppose A is true in alpha, OK. Nice. but alpha is not accessible from alpha OK. and A is not true in any other world accessible from alpha. OK. Does it follow that A is not true in alpha? Yes. That does follow. How frustrating! A is true, but not possible. How could that makes sense? Well, this does not make sense ... in the Leibnizian multiverse. For sure. I don't see the point allowing that worlds may not be accesible from themselves? Does that have some application? Yes. First you prove to everybody that I can see in the future, as I announced yesterday the discovery of a Kripke multiverse violating the law []A - A. You just did. Well, in alpha, to be sure, []A - A is true (OK?), but []~A - ~A is falsified, as []~A is true (~A is true in all accessible world from alpha), and ~A is false in alpha, as A is true is true in alpha, and worlds obeys CPL). That amounts to the same, as the laws do not depend on the valuation. If []A - A is a law, []~A - ~A should follow. Note that []~A - ~A, is equivalent with (contraposition, double negation): ~~A - ~[]~A = A - A A - A is the dual formulation of []A - A. As law, they are equivalent. But as formula in one world, they can oppose to each other. So you did find a Kripke multiverse violating the *law* []A - A. And you did find the culprit: those bizarre world which does not access to themselves. Does that have some application? Yes. 1) An easy one, which plays some role in what I like to call the simplest buddhist theory of life ever! And that theory is a subtheory of G, and so will stay with us. That theory models life by worlds accessibility. To be alive at alpha means that t is true in alpha. It means that there is, at least, one world accessible from alpha. To die at alpha means that t is false in alpha. But t is true in alpha, as t is true in all worlds, so the only way to have t false, is that there are no accessible worlds from alpha, at all, including itself. That makes alpha into a cul-de-sac world. So in Kripke semantics, ~t, or equivalently []f, characterizes the cul-de-sac world. Then the simplest buddhist theory of life ever is just the statement, If you are alive, then you can die. It means that for all worlds alpha where you are alive (t is true), you can access to a cul-de- sac world. It means that everywhere, in all worlds we t - []f, or equivalently t - ~[]t. 2) If you interpret t by intelligent, and []f by stupid, you get with the same multiverse, my general theory of intelligence and stupidity. 3) if you interpret [] by provability (in PA, or in ZF), again, t - ~[]t is a law. Read: if I am consistent, then I can't prove that I am consistent. It is easy to see that the law t - ~[]t is a direct consequence of the formula of Löb []([]A - A) - []A. Just put t in place of A, and keep in mind that A - f is just ~A, and then contra-pose: []([]A - A) - []A []([]f - f) - []f [](~[]f) - []f ~[]f - ~[](~[]f) t - ~[]t The worlds in the Kripke mutiverse characterizing G are like that, they don't access to themselves. []A- A is not an arithmetical law from the 3p self-referential view of the machine, but that is why the Theaetetus idea is applicable and will give the non trivial S4Grz for the knower, or first person, fro which []A - A is indispensable. Some might be astonished that []f is true in a cul-de-sac world. But kripe semantics say that []f is true in alpha then f is true in all accessible worlds from alpha. This really means (for all beta): (alpha R beta) - (beta satisfy f). But (alpha R beta) is always false, and (beta satisfy f) is always false, so (alpha R beta) - (beta satisfy f). OK? Dunno. I'll have to think about it. Normally, we will discuss this a lot. One thing I find puzzling is that accessible seems ill defined. Of course, it means just binary relation, on some non empty set called multiverse (here). I have an intuitive grasp of what possible and necessary mean. But that is only the alethic modalities. In PA the modal box [] will represent provability, and the worlds will be non
Re: Modal Logic (Part 3: summary + 1 exercise)
On 8 February 2014 08:43, Bruno Marchal marc...@ulb.ac.be wrote:
On 07 Feb 2014, at 02:29, LizR wrote:
On 7 February 2014 09:14, Bruno Marchal marc...@ulb.ac.be wrote:
On 06 Feb 2014, at 07:39, LizR wrote:
On 6 February 2014 08:25, Bruno Marchal marc...@ulb.ac.be wrote:
Which among the next symbolic expressions is the one being a well formed
formula:
((p - q) - ((p (p V r)) - q))
))(p-)##à89- a - q)
OK?
I sure hope so.
Well, I will pray a little bit.
(to be sure the irst one might contain a typo, but I assure you there
are no typo in the second one (and there is no cat walking on the keyboard).
***
Then a set of worlds get alive when each proposition (p, q, r), in each
world get some truth value, t, or f. I will say that the mutiverse is
illuminated.
And we can decide to put f and t is the propositional symbol for the
boolean constant true and false.
(meaning that p - f is a proposition, or well formed formula).
In modal logic it is often simpler to use only the connector - and
that if possible if you have the constant f.
For example you can define ~p as an abbreviation for (p - f), as you
should see by doing a truth table. OK?
p - f is (~p V f), for which the truth table is indeed the same as ~p
OK.
(Can you define , V, with - and f in the same way? This is not
an exercise, just a question!).
I don't think I can define those *literally* with p, - and f if that's
what you mean.
That is what I mean, indeed.
OK, having had a look at what you say below, let's have another go. Start
from p - q being equivalent to (~p V q)
That gives us ~p - q equiv (p V q) and from the above ~p is (p - f) so p
V q is (p - f) - q which I seem to remember is what you got. OK so far.
p q --- well, p - q is ~(p ~q), so ~(p - q) = (p ~q) and ~(p -
~q) = (p q)
so ~(p - (q - f)) which I guess is ((p - ( q - f)) - f) = (p q)
Does it?!?! Looking below, I see that it does. Wow.
I knew you can do that.
With hints.
But that doesn't make sense, because requires two arguments, so it would
have to be something like ... well, p - q is (~p V q) and it's also ~(p
~q), which contain V and ... I'm not sure I know what you mean.
Like for ~, to define and V to a machine which knows only -
and f. You can use the ~, as you have alredy see that you can define
it with - and f.
I reason aloud. Please tell me if you understand.
First we know that p - q is just ~p V q, OK?
So the V looks already close to -. Except that instead of ~p V q
(which is p - q) we want p V q.
May be we can substitute just p by ~p: and p V q might be then ~p - q,
Well, you can do the truth table of ~p - q, and see that it is the same
as p V q.
To finish it of course, we can eliminate the ~, and we have that p V q
is entirely defined by (p - f) - q.
OK?
And the :
Well, we already know a relationship between the and the V, OK? The
De Morgan relations.
So, applying the de Morgan relation, p q is the same as ~(~p V ~q),
(the same logically, not pragmatically, of course).
That solves the problem.
But we can verify, perhaps simplify. We can eliminate the V by the
definition above (A V B = ~A - B),
~(~p V ~q) becomes ~(~~p - ~q), that is ~(p - ~q). Or, to really settle
the things, and define from - and f:
p q = ((p - (q - f)) - f).
OK?
Apparently, yes.
OK. (Not sure what you mean by apparently, though).
Well, even though I did it, the result still looks rather strange to me!
Each world, once illuminated (that is once each proposition letter has
a value f or t) inherits of the semantics of classical proposition logic.
This means that if p and q are true in some world alpha, then (p q) is
true in that world alpha, etc.
in particular all tautologies, or propositional laws, is true in all
illuminated multiverse, and this for all illuminations (that for all
possible assignment of truth value to the world).
OK?
Question: If the multiverse is the set {a, b}, how many illuminated
multiverses can we get?
I suppose 4, since we have a world with 2 propositions, and each can be t
or f?
Answer: there is three letters p, q, r, leading to eight valuations
possible in a, and the same in b, making a total of 64 valuations, if I am
not too much distracted. I go quick. This is just to test if you get the
precise meanings.
Oh, OK. So a and b are worlds, not ... sorry. I see.
Good.
So that is 2^3 x 2^3 because a has p,q,r = 3 values, all t or f, as does
b. OK now I see what you meant.
OK.
Of course with the infinite alphabet {p, q, r, p1, q1, r1, p2, ... } we
already have a continuum of multiverses.
I can't quite see why it's a continuum. Each world has a countable
infinity of letters, and the number of worlds is therefore 2 ^ countable
infinity! Is that a continuum?
Yes. We proved it, Liz.
Yes I had a sneaky suspicion we did. It seems familiar ... a bit.
Understanding is good.
Understanding and
Biology, Buddha and the irreflexive Multiverse (was Re: Modal Logic (Part 3: summary + 1 exercise)
On 06 Feb 2014, at 21:29, meekerdb wrote: On 2/6/2014 12:14 PM, Bruno Marchal wrote: In Kripke semantic all statements are relativized to the world you are in. []A can be true in some world and false in another. The meaning of [] is restricted, for each world, to the world they can access (through the accessibility relation available in the Kripke multiverse). []A still keep a meaning, but only in each world. So everything is said when we define the new meaning of [] by the rule []A is true in alpha, by definition, means that A is true in all world beta *accessible* from alpha. And A is true in alpha iff there is a world beta; where A is true, accessible from alpha. Suppose A is true in alpha, OK. Nice. but alpha is not accessible from alpha OK. and A is not true in any other world accessible from alpha. OK. Does it follow that A is not true in alpha? Yes. That does follow. How frustrating! A is true, but not possible. How could that makes sense? Well, this does not make sense ... in the Leibnizian multiverse. For sure. I don't see the point allowing that worlds may not be accesible from themselves? Does that have some application? Yes. First you prove to everybody that I can see in the future, as I announced yesterday the discovery of a Kripke multiverse violating the law []A - A. You just did. Well, in alpha, to be sure, []A - A is true (OK?), but []~A - ~A is falsified, as []~A is true (~A is true in all accessible world from alpha), and ~A is false in alpha, as A is true is true in alpha, and worlds obeys CPL). That amounts to the same, as the laws do not depend on the valuation. If []A - A is a law, []~A - ~A should follow. Note that []~A - ~A, is equivalent with (contraposition, double negation): ~~A - ~[]~A = A - A A - A is the dual formulation of []A - A. As law, they are equivalent. But as formula in one world, they can oppose to each other. So you did find a Kripke multiverse violating the *law* []A - A. And you did find the culprit: those bizarre world which does not access to themselves. Does that have some application? Yes. 1) An easy one, which plays some role in what I like to call the simplest buddhist theory of life ever! And that theory is a subtheory of G, and so will stay with us. That theory models life by worlds accessibility. To be alive at alpha means that t is true in alpha. It means that there is, at least, one world accessible from alpha. To die at alpha means that t is false in alpha. But t is true in alpha, as t is true in all worlds, so the only way to have t false, is that there are no accessible worlds from alpha, at all, including itself. That makes alpha into a cul-de-sac world. So in Kripke semantics, ~t, or equivalently []f, characterizes the cul-de-sac world. Then the simplest buddhist theory of life ever is just the statement, If you are alive, then you can die. It means that for all worlds alpha where you are alive (t is true), you can access to a cul-de-sac world. It means that everywhere, in all worlds we t - []f, or equivalently t - ~[]t. 2) If you interpret t by intelligent, and []f by stupid, you get with the same multiverse, my general theory of intelligence and stupidity. 3) if you interpret [] by provability (in PA, or in ZF), again, t - ~[]t is a law. Read: if I am consistent, then I can't prove that I am consistent. It is easy to see that the law t - ~[]t is a direct consequence of the formula of Löb []([]A - A) - []A. Just put t in place of A, and keep in mind that A - f is just ~A, and then contra-pose: []([]A - A) - []A []([]f - f) - []f [](~[]f) - []f ~[]f - ~[](~[]f) t - ~[]t The worlds in the Kripke mutiverse characterizing G are like that, they don't access to themselves. []A- A is not an arithmetical law from the 3p self-referential view of the machine, but that is why the Theaetetus idea is applicable and will give the non trivial S4Grz for the knower, or first person, fro which []A - A is indispensable. Some might be astonished that []f is true in a cul-de-sac world. But kripe semantics say that []f is true in alpha then f is true in all accessible worlds from alpha. This really means (for all beta): (alpha R beta) - (beta satisfy f). But (alpha R beta) is always false, and (beta satisfy f) is always false, so (alpha R beta) - (beta satisfy f). OK? Bruno Brent -- You received this message because you are subscribed to the Google Groups Everything List group. To unsubscribe from this group and stop receiving emails from it, send an email to everything-list+unsubscr...@googlegroups.com. To post to this group, send email to everything-list@googlegroups.com. Visit this group at http://groups.google.com/group/everything-list. For more options, visit https://groups.google.com/groups/opt_out. http://iridia.ulb.ac.be/~marchal/ -- You received this
Re: Biology, Buddha and the irreflexive Multiverse (was Re: Modal Logic (Part 3: summary + 1 exercise)
On 2/7/2014 10:40 AM, Bruno Marchal wrote: On 06 Feb 2014, at 21:29, meekerdb wrote: On 2/6/2014 12:14 PM, Bruno Marchal wrote: In Kripke semantic all statements are relativized to the world you are in. []A can be true in some world and false in another. The meaning of [] is restricted, for each world, to the world they can access (through the accessibility relation available in the Kripke multiverse). []A still keep a meaning, but only in each world. So everything is said when we define the new meaning of [] by the rule []A is true in alpha, by definition, means that A is true in all world beta *accessible* from alpha. And A is true in alpha iff there is a world beta; where A is true, accessible from alpha. Suppose A is true in alpha, OK. Nice. but alpha is not accessible from alpha OK. and A is not true in any other world accessible from alpha. OK. Does it follow that A is not true in alpha? Yes. That does follow. How frustrating! A is true, but not possible. How could that makes sense? Well, this does not make sense ... in the Leibnizian multiverse. For sure. I don't see the point allowing that worlds may not be accesible from themselves? Does that have some application? Yes. First you prove to everybody that I can see in the future, as I announced yesterday the discovery of a Kripke multiverse violating the law []A - A. You just did. Well, in alpha, to be sure, []A - A is true (OK?), but []~A - ~A is falsified, as []~A is true (~A is true in all accessible world from alpha), and ~A is false in alpha, as A is true is true in alpha, and worlds obeys CPL). That amounts to the same, as the laws do not depend on the valuation. If []A - A is a law, []~A - ~A should follow. Note that []~A - ~A, is equivalent with (contraposition, double negation): ~~A - ~[]~A = A - A A - A is the dual formulation of []A - A. As law, they are equivalent. But as formula in one world, they can oppose to each other. So you did find a Kripke multiverse violating the *law* []A - A. And you did find the culprit: those bizarre world which does not access to themselves. Does that have some application? Yes. 1) An easy one, which plays some role in what I like to call the /simplest buddhist theory of life ever/! And that theory is a subtheory of G, and so will stay with us. That theory models life by worlds accessibility. To be alive at alpha means that t is true in alpha. It means that there is, at least, one world accessible from alpha. To die at alpha means that t is false in alpha. But t is true in alpha, as t is true in all worlds, so the only way to have t false, is that there are no accessible worlds from alpha, at all, including itself. That makes alpha into a cul-de-sac world. So in Kripke semantics, ~t, or equivalently []f, characterizes the cul-de-sac world. Then the /simplest buddhist theory of life ever/ is just the statement, If you are alive, then you can die. It means that for all worlds alpha where you are alive (t is true), you can access to a cul-de-sac world. It means that everywhere, in all worlds we t - []f, or equivalently t - ~[]t. 2) If you interpret t by intelligent, and []f by stupid, you get with the same multiverse, my general theory of intelligence and stupidity. 3) if you interpret [] by provability (in PA, or in ZF), again, t - ~[]t is a law. Read: if I am consistent, then I can't prove that I am consistent. It is easy to see that the law t - ~[]t is a direct consequence of the formula of Löb []([]A - A) - []A. Just put t in place of A, and keep in mind that A - f is just ~A, and then contra-pose: []([]A - A) - []A []([]f - f) - []f [](~[]f) - []f ~[]f - ~[](~[]f) t - ~[]t The worlds in the Kripke mutiverse characterizing G are like that, they don't access to themselves. []A- A is not an arithmetical law from the 3p self-referential view of the machine, but that is why the Theaetetus idea is applicable and will give the non trivial S4Grz for the knower, or first person, fro which []A - A is indispensable. Some might be astonished that []f is true in a cul-de-sac world. But kripe semantics say that []f is true in alpha then f is true in all accessible worlds from alpha. This really means (for all beta): (alpha R beta) - (beta satisfy f). But (alpha R beta) is always false, and (beta satisfy f) is always false, so (alpha R beta) - (beta satisfy f). OK? Dunno. I'll have to think about it. One thing I find puzzling is that accessible seems ill defined. I have an intuitive grasp of what possible and necessary mean. And I know what provable means. But my intuitive idea of accessible say every world should be accessible from itself. Logic is about formal relations of sentences so I understand that accessible will have different applications, but what are some examples? Is Robinson arithmetic accessible from Peano? Is ZFC accessible from arithmetic? Brent -- You
Re: Modal Logic (Part 3: summary + 1 exercise)
On 06 Feb 2014, at 07:39, LizR wrote:
On 6 February 2014 08:25, Bruno Marchal marc...@ulb.ac.be wrote:
Which among the next symbolic expressions is the one being a well
formed formula:
((p - q) - ((p (p V r)) - q))
))(p-)##à89- a - q)
OK?
I sure hope so.
Well, I will pray a little bit.
(to be sure the irst one might contain a typo, but I assure you
there are no typo in the second one (and there is no cat walking on
the keyboard).
***
Then a set of worlds get alive when each proposition (p, q, r), in
each world get some truth value, t, or f. I will say that the
mutiverse is illuminated.
And we can decide to put f and t is the propositional symbol for the
boolean constant true and false.
(meaning that p - f is a proposition, or well formed formula).
In modal logic it is often simpler to use only the connector -
and that if possible if you have the constant f.
For example you can define ~p as an abbreviation for (p - f), as
you should see by doing a truth table. OK?
p - f is (~p V f), for which the truth table is indeed the same as ~p
OK.
(Can you define , V, with - and f in the same way? This is
not an exercise, just a question!).
I don't think I can define those *literally* with p, - and f if
that's what you mean.
That is what I mean, indeed.
But that doesn't make sense, because requires two arguments, so it
would have to be something like ... well, p - q is (~p V q) and
it's also ~(p ~q), which contain V and ... I'm not sure I know
what you mean.
Like for ~, to define and V to a machine which knows only -
and f. You can use the ~, as you have alredy see that you
can define it with - and f.
I reason aloud. Please tell me if you understand.
First we know that p - q is just ~p V q, OK?
So the V looks already close to -. Except that instead of ~p V
q (which is p - q) we want p V q.
May be we can substitute just p by ~p: and p V q might be then ~p - q,
Well, you can do the truth table of ~p - q, and see that it is the
same as p V q.
To finish it of course, we can eliminate the ~, and we have that p V
q is entirely defined by (p - f) - q.
OK?
And the :
Well, we already know a relationship between the and the V, OK?
The De Morgan relations.
So, applying the de Morgan relation, p q is the same as ~(~p V ~q),
(the same logically, not pragmatically, of course).
That solves the problem.
But we can verify, perhaps simplify. We can eliminate the V by the
definition above (A V B = ~A - B),
~(~p V ~q) becomes ~(~~p - ~q), that is ~(p - ~q). Or, to really
settle the things, and define from - and f:
p q = ((p - (q - f)) - f).
OK?
Each world, once illuminated (that is once each proposition letter
has a value f or t) inherits of the semantics of classical
proposition logic.
This means that if p and q are true in some world alpha, then (p
q) is true in that world alpha, etc.
in particular all tautologies, or propositional laws, is true in all
illuminated multiverse, and this for all illuminations (that for all
possible assignment of truth value to the world).
OK?
Question: If the multiverse is the set {a, b}, how many illuminated
multiverses can we get?
I suppose 4, since we have a world with 2 propositions, and each can
be t or f?
Answer: there is three letters p, q, r, leading to eight valuations
possible in a, and the same in b, making a total of 64 valuations,
if I am not too much distracted. I go quick. This is just to test if
you get the precise meanings.
Oh, OK. So a and b are worlds, not ... sorry. I see.
Good.
So that is 2^3 x 2^3 because a has p,q,r = 3 values, all t or f, as
does b. OK now I see what you meant.
OK.
Of course with the infinite alphabet {p, q, r, p1, q1, r1, p2, ... }
we already have a continuum of multiverses.
I can't quite see why it's a continuum. Each world has a countable
infinity of letters, and the number of worlds is therefore 2 ^
countable infinity! Is that a continuum?
Yes. We proved it, Liz.
Take a the infinite propositional symbol letters {p, q, r, p1, q1, r1,
p2, ... } . They are well ordered. So a sequence of 1 and 0 (other
common name for t and f) can be interpreted as being a valuation. The
valuation are the infinite sequences of 1 and 0. Or the function from
N to {0, 1}.
If such a set of function was in bijection with N, i - f_i, the
function g defined by g(n) = f_n(n) + 1 would be a function f_i, let
us sat f_k, and f_k, applied on k, would gives both f_k(k) + 1 and
f_k(k), and be well defined, making 0 = 1.
My transfinite maths may not be quite up to that one.
The infinite sequence of 0, and 1, if you put 0. at the front, you get
0.1101111011010000...
for all sequences of 0 and 1, that is you get the real numbers,
written in binary, belonging to the interval (0, 1].
That is the continuum. 2^aleph_0.
Well, that was Leibniz sort of multiverse, with
Re: Modal Logic (Part 3: summary + 1 exercise)
On 2/6/2014 12:14 PM, Bruno Marchal wrote: In Kripke semantic all statements are relativized to the world you are in. []A can be true in some world and false in another. The meaning of [] is restricted, for each world, to the world they can access (through the accessibility relation available in the Kripke multiverse). []A still keep a meaning, but only in each world. So everything is said when we define the new meaning of [] by the rule []A is true in alpha, by definition, means that A is true in all world beta *accessible* from alpha. And A is true in alpha iff there is a world beta; where A is true, accessible from alpha. Suppose A is true in alpha, but alpha is not accessible from alpha and A is not true in any other world accessible from alpha. Does it follow that A is not true in alpha? I don't see the point allowing that worlds may not be accesible from themselves? Does that have some application? Brent -- You received this message because you are subscribed to the Google Groups Everything List group. To unsubscribe from this group and stop receiving emails from it, send an email to everything-list+unsubscr...@googlegroups.com. To post to this group, send email to everything-list@googlegroups.com. Visit this group at http://groups.google.com/group/everything-list. For more options, visit https://groups.google.com/groups/opt_out.
Re: Modal Logic (Part 3: summary + 1 exercise)
On 7 February 2014 09:14, Bruno Marchal marc...@ulb.ac.be wrote:
On 06 Feb 2014, at 07:39, LizR wrote:
On 6 February 2014 08:25, Bruno Marchal marc...@ulb.ac.be wrote:
Which among the next symbolic expressions is the one being a well formed
formula:
((p - q) - ((p (p V r)) - q))
))(p-)##à89- a - q)
OK?
I sure hope so.
Well, I will pray a little bit.
(to be sure the irst one might contain a typo, but I assure you there are
no typo in the second one (and there is no cat walking on the keyboard).
***
Then a set of worlds get alive when each proposition (p, q, r), in each
world get some truth value, t, or f. I will say that the mutiverse is
illuminated.
And we can decide to put f and t is the propositional symbol for the
boolean constant true and false.
(meaning that p - f is a proposition, or well formed formula).
In modal logic it is often simpler to use only the connector - and
that if possible if you have the constant f.
For example you can define ~p as an abbreviation for (p - f), as you
should see by doing a truth table. OK?
p - f is (~p V f), for which the truth table is indeed the same as ~p
OK.
(Can you define , V, with - and f in the same way? This is not
an exercise, just a question!).
I don't think I can define those *literally* with p, - and f if that's
what you mean.
That is what I mean, indeed.
OK, having had a look at what you say below, let's have another go. Start
from p - q being equivalent to (~p V q)
That gives us ~p - q equiv (p V q) and from the above ~p is (p - f) so p
V q is (p - f) - q which I seem to remember is what you got. OK so far.
p q --- well, p - q is ~(p ~q), so ~(p - q) = (p ~q) and ~(p - ~q)
= (p q)
so ~(p - (q - f)) which I guess is ((p - ( q - f)) - f) = (p q)
Does it?!?! Looking below, I see that it does. Wow.
But that doesn't make sense, because requires two arguments, so it would
have to be something like ... well, p - q is (~p V q) and it's also ~(p
~q), which contain V and ... I'm not sure I know what you mean.
Like for ~, to define and V to a machine which knows only -
and f. You can use the ~, as you have alredy see that you can define
it with - and f.
I reason aloud. Please tell me if you understand.
First we know that p - q is just ~p V q, OK?
So the V looks already close to -. Except that instead of ~p V q
(which is p - q) we want p V q.
May be we can substitute just p by ~p: and p V q might be then ~p - q,
Well, you can do the truth table of ~p - q, and see that it is the same
as p V q.
To finish it of course, we can eliminate the ~, and we have that p V q
is entirely defined by (p - f) - q.
OK?
And the :
Well, we already know a relationship between the and the V, OK? The
De Morgan relations.
So, applying the de Morgan relation, p q is the same as ~(~p V ~q),
(the same logically, not pragmatically, of course).
That solves the problem.
But we can verify, perhaps simplify. We can eliminate the V by the
definition above (A V B = ~A - B),
~(~p V ~q) becomes ~(~~p - ~q), that is ~(p - ~q). Or, to really settle
the things, and define from - and f:
p q = ((p - (q - f)) - f).
OK?
Apparently, yes.
Each world, once illuminated (that is once each proposition letter has
a value f or t) inherits of the semantics of classical proposition logic.
This means that if p and q are true in some world alpha, then (p q) is
true in that world alpha, etc.
in particular all tautologies, or propositional laws, is true in all
illuminated multiverse, and this for all illuminations (that for all
possible assignment of truth value to the world).
OK?
Question: If the multiverse is the set {a, b}, how many illuminated
multiverses can we get?
I suppose 4, since we have a world with 2 propositions, and each can be t
or f?
Answer: there is three letters p, q, r, leading to eight valuations
possible in a, and the same in b, making a total of 64 valuations, if I am
not too much distracted. I go quick. This is just to test if you get the
precise meanings.
Oh, OK. So a and b are worlds, not ... sorry. I see.
Good.
So that is 2^3 x 2^3 because a has p,q,r = 3 values, all t or f, as does
b. OK now I see what you meant.
OK.
Of course with the infinite alphabet {p, q, r, p1, q1, r1, p2, ... } we
already have a continuum of multiverses.
I can't quite see why it's a continuum. Each world has a countable
infinity of letters, and the number of worlds is therefore 2 ^ countable
infinity! Is that a continuum?
Yes. We proved it, Liz.
Yes I had a sneaky suspicion we did. It seems familiar ... a bit.
Take a the infinite propositional symbol letters {p, q, r, p1, q1, r1, p2,
... } . They are well ordered. So a sequence of 1 and 0 (other common name
for t and f) can be interpreted as being a valuation. The valuation are the
infinite sequences of 1 and 0. Or the function from N to {0, 1}.
If such a set of function was in
Re: Modal Logic (Part 3: summary + 1 exercise)
Hi Liz,
Logician have a large notion of world. A world is a element of a
set, called the set of worlds, or multiverse.
Statisticians do the same, with the notion of population, which is
also just a set. In fact classical logic and classical statistics have
a sufficiently large common base so that George Boole deemed them both
under the label the laws of thought.
Propositional logician have variable propositional letter, supposed to
assign to true or false in each worlds. I will assume the letter p, q,
r.
They have some grammar for the formula. I hope you can solve the
following exercise:
Which among the next symbolic expressions is the one being a well
formed formula:
((p - q) - ((p (p V r)) - q))
))(p-)##à89- a - q)
OK?
(to be sure the irst one might contain a typo, but I assure you there
are no typo in the second one (and there is no cat walking on the
keyboard).
***
Then a set of worlds get alive when each proposition (p, q, r), in
each world get some truth value, t, or f. I will say that the
mutiverse is illuminated.
And we can decide to put f and t is the propositional symbol for the
boolean constant true and false.
(meaning that p - f is a proposition, or well formed formula).
In modal logic it is often simpler to use only the connector - and
that if possible if you have the constant f.
For example you can define ~p as an abbreviation for (p - f), as you
should see by doing a truth table. OK?
(Can you define , V, with - and f in the same way? This is
not an exercise, just a question!).
Each world, once illuminated (that is once each proposition letter
has a value f or t) inherits of the semantics of classical proposition
logic.
This means that if p and q are true in some world alpha, then (p q)
is true in that world alpha, etc.
in particular all tautologies, or propositional laws, is true in all
illuminated multiverse, and this for all illuminations (that for all
possible assignment of truth value to the world).
OK?
Question: If the multiverse is the set {a, b}, how many illuminated
multiverses can we get?
Answer: there is three letters p, q, r, leading to eight valuations
possible in a, and the same in b, making a total of 64 valuations, if
I am not too much distracted. I go quick. This is just to test if you
get the precise meanings.
Of course with the infinite alphabet {p, q, r, p1, q1, r1, p2, ... }
we already have a continuum of multiverses.
Well, that was Leibniz sort of multiverse, with all worlds quite
independent of each other.
With Kripke, we introduce a binary relation R on the set of world.
That's all. We read alpha R beta, as beta is accessible from alpha.
OK. Time for the main recall:
We add then new unary connector [], and define by ~[]~
In Leibniz semantics, []A is true (absolutely) means that A is true in
all worlds.
In Kripke semantics []A is true in a world alpha means that A is true
in all worlds accessible from alpha.
And the only one exercise:
prove that []A - A is true in all worlds of a multiverse, for all
illumination possible (choice of valuation for the letter)
iff the relation is reflexive (that is: all world can access
themselves).
Hint: this should be easy. Any difficulty here is due to my probable
unclarity, or my excess of verbosity, or a lack of familiarity with
math of your part. I suggest you might search for counterexample.
And yes, this is truly two exercises, because to prove an iff, you
have to prove two if. You must prove:
1) if a multiverse is reflexive, then, whatever the illumination is,
each world satisfy []A - A (for all formula A).
2) If, whatever the illumination is, each world satisfy []A - A (for
all formula A), then the multiverse is reflexive.
whatever the illumination is important: for example in the simple
multiverse with one world: {alpha}, and the empty accessibility
relation (so that alpha does not access to itself, ~ (alpha R alpha),
and with p valuated to 1 in alpha, you have that []p is true, p is
true, so []p - p is true in alpha, yet the mutiverse is not reflexive.
OK?
Please, ask any question to clarify. Note in passing the beauty: a
modal formula, made into a law, impose some structure on a Kripke
multiverse, and inversely, an accessibility structure on a multiverse
impose a modal law.
And now a free subject of reflexion :) (to prepare the sequel)
If reflexivity in Kripke multiverse characterizes []A - A
Which relations can characterize the following formula?
The Leibnizian one:
[]A - [][]A
[]A - A
p - []A
A - []A
[](A-B) - ([]A - []B)
And what about (more hard) the non Leibnizian one, which will play
some role (as scheme of some machines discourses)
A - ~[]A (related to Gödel)
[]([]A - A) - []A (related to Löb)
[]([](p - []p) - p) - p (related to Grzegorczyk, the Grz of S4Grz).
Bruno
On 29 Jan 2014, at 11:23, Bruno Marchal wrote:
On 29 Jan 2014, at 01:05,
Re: Modal Logic (Part 3: summary + 1 exercise)
On 6 February 2014 08:25, Bruno Marchal marc...@ulb.ac.be wrote:
Hi Liz,
Logician have a large notion of world. A world is a element of a set,
called the set of worlds, or multiverse.
Statisticians do the same, with the notion of population, which is also
just a set. In fact classical logic and classical statistics have a
sufficiently large common base so that George Boole deemed them both under
the label the laws of thought.
Propositional logician have variable propositional letter, supposed to
assign to true or false in each worlds. I will assume the letter p, q, r.
They have some grammar for the formula. I hope you can solve the following
exercise:
Which among the next symbolic expressions is the one being a well formed
formula:
((p - q) - ((p (p V r)) - q))
))(p-)##à89- a - q)
OK?
I sure hope so.
(to be sure the irst one might contain a typo, but I assure you there are
no typo in the second one (and there is no cat walking on the keyboard).
***
Then a set of worlds get alive when each proposition (p, q, r), in each
world get some truth value, t, or f. I will say that the mutiverse is
illuminated.
And we can decide to put f and t is the propositional symbol for the
boolean constant true and false.
(meaning that p - f is a proposition, or well formed formula).
In modal logic it is often simpler to use only the connector - and that
if possible if you have the constant f.
For example you can define ~p as an abbreviation for (p - f), as you
should see by doing a truth table. OK?
p - f is (~p V f), for which the truth table is indeed the same as ~p
(Can you define , V, with - and f in the same way? This is not
an exercise, just a question!).
I don't think I can define those *literally* with p, - and f if that's
what you mean. But that doesn't make sense, because requires two
arguments, so it would have to be something like ... well, p - q is (~p V
q) and it's also ~(p ~q), which contain V and ... I'm not sure I know
what you mean.
Each world, once illuminated (that is once each proposition letter has a
value f or t) inherits of the semantics of classical proposition logic.
This means that if p and q are true in some world alpha, then (p q) is
true in that world alpha, etc.
in particular all tautologies, or propositional laws, is true in all
illuminated multiverse, and this for all illuminations (that for all
possible assignment of truth value to the world).
OK?
Question: If the multiverse is the set {a, b}, how many illuminated
multiverses can we get?
I suppose 4, since we have a world with 2 propositions, and each can be t
or f?
Answer: there is three letters p, q, r, leading to eight valuations
possible in a, and the same in b, making a total of 64 valuations, if I am
not too much distracted. I go quick. This is just to test if you get the
precise meanings.
Oh, OK. So a and b are worlds, not ... sorry. I see. So that is 2^3 x 2^3
because a has p,q,r = 3 values, all t or f, as does b. OK now I see what
you meant.
Of course with the infinite alphabet {p, q, r, p1, q1, r1, p2, ... } we
already have a continuum of multiverses.
I can't quite see why it's a continuum. Each world has a countable infinity
of letters, and the number of worlds is therefore 2 ^ countable infinity!
Is that a continuum? My transfinite maths may not be quite up to that one.
Well, that was Leibniz sort of multiverse, with all worlds quite
independent of each other.
With Kripke, we introduce a binary relation R on the set of world. That's
all. We read alpha R beta, as beta is accessible from alpha.
This seems like a way of getting subsets from the multiverse, but I'm not
completely sure what accessible means here.
OK. Time for the main recall:
We add then new unary connector [], and define by ~[]~
In Leibniz semantics, []A is true (*absolutely*) means that A is true in
all worlds.
In Kripke semantics []A is true *in a world alpha* means that A is true
in all worlds *accessible from alpha*.
And the only one exercise:
prove that []A - A is true in all worlds of a multiverse, for all
illumination possible (choice of valuation for the letter)
So []A means the proposition A is true in all worlds accessible from ...
somewhere.
Oh dear. I don't seem to be able to get my head around this. Maybe because
I'm not sure what accessible means here...
iff the relation is reflexive (that is: all world can access themselves).
Hint: this should be easy. Any difficulty here is due to my probable
unclarity, or my excess of verbosity, or a lack of familiarity with math of
your part. I suggest you might search for counterexample.
And yes, this is truly two exercises, because to prove an iff, you have to
prove two if. You must prove:
1) if a multiverse is reflexive, then, *whatever the illumination* is,
each world satisfy []A - A (for all formula A).
2) If, *whatever the illumination* is, each world satisfy []A - A (for
all
A Summary of Peirce, Leibniz and Sheldrake on habits
A Summary of Peirce, Leibniz and Sheldrake on habits Habits are the results of of the taming of random ensembles -- Leibniz states that there are two kinds of logic, a) necessary logic, which is always true (the timeless logic of Heaven or Platonia), and b) contingent logic, which is the time-based logic of earth , also called modal logic. One puzzle is if there are a) things always true and b) other things only sometimes (or somewhere) true, couldn't there be a conflict ? Your concept of morphic resonance, or Peirce's Thirdness, might be the solution, namely that habits are link between these two fields above: Habits are the tendencies (but not quite the necessity) of things to be true down here. From a Christian perspective, the presence of the Kingdom of God in the Kingdom of Earth. -- Also, you can think of Peirce's categories using a classical black box model. Firstness = input signal (contingent world, chance) Secondness = the black box or filter, which does the signal processing or convolution of input signal of contingent phenomena against law or mechanism. Thirdness = output signal (habit or tendencyt for lawful behavior) This also works for evolution, one verswion of which might be: Firstness = randomly chosen gene Secondness = that gene tested in a real situation Thirdness= surviving gene or new habit IMHO morphic resonance could be understood as modification of lawless behavior subjected to a lawful universe They've put hidden optical speed detectors on my neighborhood streets to slow down traffic. If you don't see the detectors and speed through, the detectors will flash photo your license plate and electronically issue you a speed ticket. Gradually everybody tends to slow down to meet the legal speed limit. A wild speculation is perhaps quantum mechanics behavior gradually adapts to enstein behavior in such a way. [Roger Clough], [rclo...@verizon.net] 1/3/2013 Forever is a long time, especially near the end. - Woody Allen - Receiving the following content - From: chris kramer Receiver: mindbr...@yahoogroups.com Time: 2013-01-02, 17:43:50 Subject: Re: Re: [Mind and Brain] The Triune World: Various RepresentationsofPeirce's Triads Thanks Roger. What are your thoughts on his notions of belief and habit? Chris From: Roger Clough To: - mindbr...@yahoogroups.com Sent: Wednesday, January 2, 2013 6:14 AM Subject: Re: Re: [Mind and Brain] The Triune World: Various Representations ofPeirce's Triads Hi chris kramer Keep in mind that pragmatism doesn't have a metaphysics or ontology (an overall picture of reality, in which the particular can be obtained analytuically from the general). Instead, Peirce's praqgmatism is an epistempology (a method of synthetically obtaining a general from a particular). Only the method is defined (the categories), not what they obtain. The method is essentially that of experimental science, not scientific theory. Closer to Aristotle. So I would class Peirce's statements on ontology or God as typical of any scientist today, namely pretty much of a personal belief. I prefer Leibniz for a more ontological picture of God as that platonic entity (the One) which views and works on reality through the Supreme Monad (which could possibly be Jesus, Leibniz doesn't say). [Roger Clough], mailto:rclo...@verizon.net] 1/2/2013 Forever is a long time, especially near the end. - Woody Allen - Receiving the following content - From: chris kramer Receiver: mailto:mindbr...@yahoogroups.com,everything-list@googlegroups.com Time: 2013-01-01, 14:00:11 Subject: Re: [Mind and Brain] The Triune World: Various Representations ofPeirce's Triads The attachments of the original message is as following: (1). peirce Roger, Does Peirce provide an understandable account of that which can exist wholly independent of anything (everything) else? I am assuming he refers to God; that which is ontologically independent. But what exactly can this mean? I am also interested in Peirce's notion of the fixation of belief and its contrast with the irritation of doubt. When habits form, it seems we are less inclined to experiecne that irritation of doubt; this could be good, the they are good habits, or beliefs that somehow track the truth, or more pragmatically, provide a cash value for the believer; but could it not also lead to dogmatism? Thanks, Chris From: Roger Clough To: everything-list ; - mindbr...@yahoogroups.com Sent: Monday, December 31, 2012 6:39 AM Subject: [Mind and Brain] The Triune World: Various Representations of Peirce's Triads
Re: A summary I just wrote for my blog
Kim,
I presume you have clear ideas about what 'life' may be (to live?) and the
a-temporal distinction of 'ever'. (It is definitely not = 'a long long
time').
I paraphrase you wisdom as:
time in our opinion goes as long as we live(?) so 'after that' is not
identified.
My reasons for not including afterlife or reincarnation (or even the other
sci-fi concepts on this list (replicas, teleportation etc.) is my view of
the 'existence' (also a hard word): the 'world' (use whatever is you beef:
nature, totality, even existence) is a complexity of everything in
relational unity. We observe parts of it (according to our capabilities,
we can't encompass all) and select 'models' for our views. In our
understanding (limited as it is) we identify our relation to such models
('our' is similarly a figment to be explained - person, self, you name it)
and realize (partial) complexities constituting our world, our life,
ourselves. When relations change by interference from (maybe even out of
model) participants, we talk about a process. Maybe in form of a 'zipp'.
When a complexity reorganizes in a major(?) process it vanishes (=death) and
there is no further continuation of the complexity that was reorganized.
The complexity us is more than physically describable (Aris-total) and by
major reorganization all is gone as identifiable as pertinent to the
vanished - reorganized - complexity (us). (3rd person memories ABOUT are not
to be mistaken for the complexity's 1st person assignable processes.)
Forever means it just stopped dead. It stepped out from the time concept.
Time is a coordinating factor how our universe 'orders' its happenings and
space is the other one.
All this pertains to my NARRATIVE (not theory!) to make our world a bit
easier to handle logically (commonsensically) in our mind(?).
If you like it, use it, if not, delete.
So a less verbose reflection to your pretty laconic maxim:
there is one instance for the entire complexity 'us' to function (processes)
- whith the major instrumental components in unchanged relationships. Once
such relationships are changed the process-complexity is over. Time
space) are our coordinating figments to make relational changes palatable
for our limited understanding.
JohnM
On Wed, Feb 11, 2009 at 6:04 AM, Kim Jones kimjo...@ozemail.com.au wrote:
We only live once, but we live forever
There is no afterlife - only life eternal
Kim Jones
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Re: A summary I just wrote for my blog
We only live once, but we live forever There is no afterlife - only life eternal Kim Jones On 11/02/2009, at 4:27 AM, Michael Rosefield wrote: I wrote it for my friends, but feel free to criticise! http://rosyatrandom.livejournal.com/35445.html _ Perhaps it's time I had another go at explaining all that weird stuff I believe in and why. Well, for those few that don't know, I reckon that all possible universes exist and that everyone's immortal. I admit, this does sound rather odd. It would have sounded odd to me about 10 years ago, too. Since about the age of 8 I was a pretty hardcore rational scientific naturalist: everything is simply matter and energy, and we but its dreams. What was real? Well, a chair. An atom. Something you can touch. After all, when you think of reality, you think of something... there. Something that sits there, quietly existing to itself. But what does that mean, really? Everyone knows that matter is almost entirely empty space, anyway - the solidity is just the feather-touch of far-extended electromagnetic fields. Electrons popping in and out of existence as the energy fields knot so charge can be transferred in quantised lumps. Particles do not behave as billiard balls - they are ghosts, obeying strange equations, lacking hard and fast surfaces or reliable locations. Matter, energy, space, time... they all begin to seem a bit ethereal when you look at them. Time. There's another one. I don't really believe in that, either. Spacetime is just a barely distinguishable fabric woven by the universe. Events do not occur at a time or a place - most of the observables we see arise kaleidoscope like out of an intricate web of possibilities, their form imposed by our own consciousness. And by that, I mean that our minds are embedded within the universe, constructed in such a way that the metaphysical structure of the cosmos is implied by our design - the word without reflects the world within. This has an aspect of the anthropic principle to it - that we observe a world capable of supporting our existence because if it didn't, we wouldn't. But this still has no bearing on how I started thinking things like this, so I shall get that out of the way. The short story is that I read some stories by a science-fiction author called Greg Egan. Before you laugh too much, a lot of sci-fi is essentially just window-dressing to convey an idea - the implications of some item of technology, turn of events or scientific/philosophical argument. And Greg Egan is a 'hard' science- fiction author, an ideas merchant. Well, you get the drift. The first story I read was called Wang's Carpets (later included as a chapter of the book Diaspora), in which some spacefarers (themselves software) find a planet whose major life-form are floating mats that take the form of Wang Tiles - tesselating objects whose patterns can implement a universal turing machine. But that's just the set-up for the idea: when someone analyses the Carpets, by taking various abstract variables (appearance of certain tiles and features, etc) and putting them through frequency transforms, it turns out that the computations the Carpets encode as part of their reproductive habits give rise to a fully realised n-dimensional space containing self-aware creatures. The thought-provoking part here was not that consciousness could be digitalised and run as software - I had already pretty much accepted that - but that the mathematical transformations necessary to do this could be pretty strange, and come from processes that were essentially plucked arbitrarily from the environment. That, largely, consciousness could be a matter of perspective. The second story was the book, Permutation City. A great deal of this book concerns one of the protagonists who wakes up one day and finds he is simply a downloaded copy - and that the 'real' him is running experiments. After being run at different speeds, and distributed in space and time, backwards, in chunks of different sizes, etc., the argument becomes that it doesn't matter what or how the program is run - it is all the same from the perspective of the consciousness being implemented, and that this is so abstract that one can find the relevant computational processes within any physical substrate. That all consciousnesses can be found within a grain of sand. That there is not even any physical bedrock to fall back upon - there is no way ever to verify, even in principle, that one is on the 'fundamental' metapysical level. At the end of the book, the characters have escaped into their own computational world, completely divorced from any physical hardware. Their universe contains a simulation of another world, whose very alien inhabitants find their own
A summary I just wrote for my blog
I wrote it for my friends, but feel free to criticise! http://rosyatrandom.livejournal.com/35445.html _ Perhaps it's time I had another go at explaining all that weird stuff I believe in and why. Well, for those few that don't know, I reckon that all possible universes exist and that everyone's immortal. I admit, this does sound rather odd. It would have sounded odd to me about 10 years ago, too. Since about the age of 8 I was a pretty hardcore rational scientific naturalist: everything is simply matter and energy, and we but its dreams. What was *real*? Well, a chair. An atom. Something you can * touch*. After all, when you think of reality, you think of something... there. Something that sits there, quietly existing to itself. But what does that mean, really? Everyone knows that matter is almost entirely empty space, anyway - the solidity is just the feather-touch of far-extended electromagnetic fields. Electrons popping in and out of existence as the energy fields knot so charge can be transferred in quantised lumps. Particles do not behave as billiard balls - they are ghosts, obeying strange equations, lacking hard and fast surfaces or reliable locations. Matter, energy, space, time... they all begin to seem a bit ethereal when you look at them. Time. There's another one. I don't really believe in that, either. Spacetime is just a barely distinguishable fabric woven by the universe. Events do not occur at a time or a place - most of the observables we see arise kaleidoscope like out of an intricate web of possibilities, their form imposed by our own consciousness. And by that, I mean that our minds are embedded within the universe, constructed in such a way that the metaphysical structure of the cosmos is implied by our design - the word without reflects the world within. This has an aspect of the anthropic principle to it - that we observe a world capable of supporting our existence because if it didn't, we wouldn't. But this still has no bearing on how I started thinking things like this, so I shall get that out of the way. The short story is that I read some stories by a science-fiction author called Greg Egan. Before you laugh too much, a lot of sci-fi is essentially just window-dressing to convey an idea - the implications of some item of technology, turn of events or scientific/philosophical argument. And Greg Egan is a 'hard' science-fiction author, an ideas merchant. Well, you get the drift. The first story I read was called Wang's Carpets (later included as a chapter of the book Diaspora), in which some spacefarers (themselves software) find a planet whose major life-form are floating mats that take the form of Wang Tiles - tesselating objects whose patterns can implement a universal turing machine. But that's just the set-up for the *idea*: when someone analyses the Carpets, by taking various abstract variables (appearance of certain tiles and features, etc) and putting them through frequency transforms, it turns out that the computations the Carpets encode as part of their reproductive habits give rise to a fully realised n-dimensional space containing self-aware creatures. The thought-provoking part here was not that consciousness could be digitalised and run as software - I had already pretty much accepted that - but that the mathematical transformations necessary to do this could be pretty strange, and come from processes that were essentially plucked arbitrarily from the environment. That, largely, consciousness could be a matter of perspective. The second story was the book, Permutation City. A great deal of this book concerns one of the protagonists who wakes up one day and finds he is simply a downloaded copy - and that the 'real' him is running experiments. After being run at different speeds, and distributed in space and time, backwards, in chunks of different sizes, etc., the argument becomes that it doesn't matter what or how the program is run - it is all the same from the perspective of the consciousness being implemented, and that this is so abstract that one can find the relevant computational processes within *any*physical substrate. That *all* consciousnesses can be found within a grain of sand. That there is not even any physical bedrock to fall back upon - there is no way ever to verify, even in principle, that one is on the 'fundamental' metapysical level. At the end of the book, the characters have escaped into their own computational world, completely divorced from any physical hardware. Their universe contains a simulation of another world, whose very alien inhabitants find their own physical principles for the cosmos they observe - principles radically different from the computational ones 'running' it, and so compelling they start to take over the character's world, too. So when you get down to it, I no longer believe in the physical world - or rather, I believe in all of them. While I used to require reasons to
Re: A summary I just wrote for my blog
Not too much here that would raise hackles on the everything-list, but (IMHO) for the first sentence-- Perhaps it's time I had another go at explaining all that weird stuff I believe in and why. The word believe can mean many things but in my parlance it means to attach a very high confidence to a proposition. I believe that eating satiates my hunger, that the Pacific Ocean lies a few miles to my west, that if I sit in a chair I will not fall through it to the ground, etc. I also from time to time *entertain* notions similar to the ones you've written about, and admit the possibility of some, but I don't believe any of it with anything like the confidence with which I believe that water will freeze at -10°C. I suspect the same is true for you too. Or is it really the case that in the few years since you've read those stories you have really thought things through to the point where you believe it? -Pete --~--~-~--~~~---~--~~ You received this message because you are subscribed to the Google Groups Everything List group. To post to this group, send email to everything-l...@googlegroups.com To unsubscribe from this group, send email to everything-list+unsubscr...@googlegroups.com For more options, visit this group at http://groups.google.com/group/everything-list?hl=en -~--~~~~--~~--~--~---
Re: A summary I just wrote for my blog
I did think about what word to use there - and while I don't _believe_ believe it, I would be _very_ surprised to be proved wrong :D . And besides, any other word seems like a bit of a fudge. -- - Did you ever hear of The Seattle Seven? - Mmm. - That was me... and six other guys. 2009/2/10 Pete Carlton pmcarl...@mac.com Not too much here that would raise hackles on the everything-list, but (IMHO) for the first sentence-- Perhaps it's time I had another go at explaining all that weird stuff I believe in and why. The word believe can mean many things but in my parlance it means to attach a very high confidence to a proposition. I believe that eating satiates my hunger, that the Pacific Ocean lies a few miles to my west, that if I sit in a chair I will not fall through it to the ground, etc. I also from time to time *entertain* notions similar to the ones you've written about, and admit the possibility of some, but I don't believe any of it with anything like the confidence with which I believe that water will freeze at -10°C. I suspect the same is true for you too. Or is it really the case that in the few years since you've read those stories you have really thought things through to the point where you believe it? -Pete --~--~-~--~~~---~--~~ You received this message because you are subscribed to the Google Groups Everything List group. To post to this group, send email to everything-l...@googlegroups.com To unsubscribe from this group, send email to everything-list+unsubscr...@googlegroups.com For more options, visit this group at http://groups.google.com/group/everything-list?hl=en -~--~~~~--~~--~--~---
Re: 'White Rabbit' solution summary (+ simplicity explanation)
One of my references did not 'HTMLize' properly for some reason. This one should: www.physica.freeserve.co.uk/pa01.htm - Original Message - From: Alastair Malcolm To: [EMAIL PROTECTED] Sent: Saturday, April 19, 2008 9:48 AM Subject: 'White Rabbit' solution summary (+ simplicity explanation) Since there is a distinct possibility that readers of Russell's 'Theory of Nothing' book will be left with the wrong impression that my approach to the White Rabbit problem is essentially the same as that of the author, I feel I should at least record here a brief summary of the relevant part of my own ideas, which are in essence very simple and straightforward. My starting point is a consideration of the potentially fatal 'failure of induction' (WR) challenge to the 'all logically possible universes' (alpu) solution to the question of our existence (a solution that general arbitrariness and abstract symmetry arguments appear more-or-less to ultimately require): even if the world happened to be ordered up to now, why should we happen to be in that world that continues in an ordered way, if all logically possible futures do in fact occur, as alpu requires. The solution to this challenge that is outlined here also explains why we live in a relatively simple world, and is arrived at by a general consideration of the most compressed fully accurate representation of our (past/present/future) world (which in that most compressed form may well need to include other worlds, for example those of (what would be the rest of) an Everett multiverse), conceptually in the form of Tegmark's 'bird view'; whether the form of this representation is some standard interpretation of a bit string, or an axiom list (under some common theorem-generating inference rules), the two key points are the same: first, there is nothing logically to prevent some worlds themselves (including ours) being more 'compressed' than as we would perceive them to be, and second, any difference from the world to be represented (which must also exist under alpu) has to be reflected in a difference in that representation - it then follows that in any comparison of all possible combinations of bit/axiom strings up to any equal finite (long) length (many representing not only a world but also (using 'spare' string segments inside the total length) extraneous features such as other worlds, nothing in particular, or perhaps 'invisible' intra-world entities), it is reasonable to suppose that the simplest worlds (ie those with the shortest representing string segments) will occur most often across all strings, since they will have more 'spare' irrelevant bit/axiom combinations up to that equal comparison length, than those of more complex worlds (and so similarly for all long finite comparison lengths). Thus out of all worlds inhabitable by SAS's, we are most likely to be in one of the simplest (other things being equal) - any physics-violating events like flying rabbits or dragons would require more bits/axioms to (minimally) specify their worlds, and so we should not expect to find ourselves in such a world, at any time in its history. (It also seems to me that for at least some of the scenarios where the above analysis could conceivably be considered inaccurate/incorrect (eg in comparing uncountably infinite quantities), the necessary assumptions for these scenarios render the White Rabbit problem void anyway.) These ideas are fleshed out in: www.physica.freeserve.co.uk/pa01.htm (which enlarges on the 'compressed' objective reality that corresponds to the more compressed representations), and www.physica.freeserve.co.uk/pb01.htm (a more general and informal read). (Comments welcome - particularly if any problems are spotted in the above.) Alastair Malcolm --~--~-~--~~~---~--~~ You received this message because you are subscribed to the Google Groups Everything List group. To post to this group, send email to [EMAIL PROTECTED] To unsubscribe from this group, send email to [EMAIL PROTECTED] For more options, visit this group at http://groups.google.com/group/everything-list?hl=en -~--~~~~--~~--~--~---
Re: 'White Rabbit' solution summary (+ simplicity explanation)
Apologies - still some technical problem (it worked when I tested it out). If anyone's interested in the ref it's best to edit the URL line or retype. - Original Message - From: Alastair Malcolm To: [EMAIL PROTECTED] Sent: Saturday, April 19, 2008 6:06 PM Subject: Re: 'White Rabbit' solution summary (+ simplicity explanation) One of my references did not 'HTMLize' properly for some reason. This one should: www.physica.freeserve.co.uk/pa01.htm --~--~-~--~~~---~--~~ You received this message because you are subscribed to the Google Groups Everything List group. To post to this group, send email to [EMAIL PROTECTED] To unsubscribe from this group, send email to [EMAIL PROTECTED] For more options, visit this group at http://groups.google.com/group/everything-list?hl=en -~--~~~~--~~--~--~---
Re: 'White Rabbit' solution summary (+ simplicity explanation)
Perhaps if you added the protocol part of the URL (http://)? Without it, client programs cannot know that it is even a URL, unless it is something like the address bar of a web browser, where it is assumed to be a URL, and http:// is the default protocol if not specified. Being a bit old-fashioned, I always put the http:// on URLs, even though convention has come to leave it assumed. When I mention a bare web address, it usually is _not_ a URL. On Sat, Apr 19, 2008 at 06:24:31PM +0100, Alastair Malcolm wrote: Apologies - still some technical problem (it worked when I tested it out). If anyone's interested in the ref it's best to edit the URL line or retype. - Original Message - From: Alastair Malcolm To: [EMAIL PROTECTED] Sent: Saturday, April 19, 2008 6:06 PM Subject: Re: 'White Rabbit' solution summary (+ simplicity explanation) One of my references did not 'HTMLize' properly for some reason. This one should: www.physica.freeserve.co.uk/pa01.htm -- A/Prof Russell Standish Phone 0425 253119 (mobile) Mathematics UNSW SYDNEY 2052 [EMAIL PROTECTED] Australiahttp://www.hpcoders.com.au --~--~-~--~~~---~--~~ You received this message because you are subscribed to the Google Groups Everything List group. To post to this group, send email to [EMAIL PROTECTED] To unsubscribe from this group, send email to [EMAIL PROTECTED] For more options, visit this group at http://groups.google.com/group/everything-list?hl=en -~--~~~~--~~--~--~---
Re: SUMMARY
Time for the Kleene diagonal argument. Opps, a language L that I dreamt of does not exist. I have to relax from the condition that M on E_i always return a number in a finite time. Well, what to return if not a number ... nothing - M experiences an infinite loop. What a world, ok, my language has to describe total functions from N to N and as well as strict partial functions from N to N. And it is clear that I cannot know whether E_i corresponds to a total function or a strict partial function. f' stands for any function descriable by L. 0 --- E_0 ~ f'_0 1 --- E_1 ~ f'_1 2 --- E_2 ~ f'_2 3 --- E_3 ~ f'_3 N, E and C are enumerable, moreover obviously effectively enumerable. Any subset of C is at least enumerable. A subset of C corresponding to total functions is no effectively enumerable. It cannot be. Correction: N and E are enumerable, moreover obviously effectively enumerable. C is enumerable and thus any subset of C is at least enumerable. A subset of C corresponding to total functions is not effectively enumerable. It cannot be. Neither C as such is effectively enumerable. It cannot be. Mirek --~--~-~--~~~---~--~~ You received this message because you are subscribed to the Google Groups Everything List group. To post to this group, send email to [EMAIL PROTECTED] To unsubscribe from this group, send email to [EMAIL PROTECTED] For more options, visit this group at http://groups.google.com/group/everything-list?hl=en -~--~~~~--~~--~--~---
Re: SUMMARY
Hi Mirek, Le 30-janv.-08, à 13:42, Mirek Dobsicek a écrit : Hi Bruno and everybody, I hope to send my comments and/or 'OK' sign :-) on Monday. Take it easy. There is no deadline on the list. Making a declaration helps me to get things done. Yet I'm late. Whenever you see such sentences in my posts, you can skip it, they are mostly for me :-) Am I suppose to skip this one too? :) --- I'll try to write a summary in my own words. Let's see how much I did understand. Prepositions: A finite alphabet A* finite words over A (it is enumerable, moreover effectively enumerable) L . a language over A. E a subset of A*, a set of valid expressions in L (obviously, it is at least enumerable) M . a machine which understands to L f . a total function from N to N. Goal: I want to develop a universal language L which describes all and only all functions f. Given an expression from E, M computes the result in finite time. Given the restrictions on L, the result is a number and nothing else. The set of all functions f from N to N is not enumerable (by Cantor's diagonal). Thus there is no bijection to E. Thus, I have to find a smaller set of functions. I will call this set a set of computable functions, C. Inevitably, this is computability by definition, by definition of L. So L should be 'really good' in order to encompass as much functions f from N to N as possible. Now, I think of a bijection between E and C. 0 --- E_0 ~ f_0 1 --- E_1 ~ f_1 2 --- E_2 ~ f_2 3 --- E_3 ~ f_3 Since E are efficiently enumerable, C are efficiently enumerable ... I guess you want to say effectively enumerable. Typo error. ... as well. Yes, it might happen that f_i = f_j for i != j, but is does not matter as long as all unique f_i are in the enumeration. Time for the Kleene diagonal argument. Opps, a language L that I dreamt of does not exist. I have to relax from the condition that M on E_i always return a number in a finite time. Well, what to return if not a number ... nothing - M experiences an infinite loop. What a world, ok, my language has to describe total functions from N to N and as well as strict partial functions from N to N. OK. And it is clear that I cannot know whether E_i corresponds to a total function or a strict partial function. Clear for you, apparently. Is it clear for everybody? This follows from the Diagonal argument applied on the programs in L. f' stands for any function descriable by L. 0 --- E_0 ~ f'_0 1 --- E_1 ~ f'_1 2 --- E_2 ~ f'_2 3 --- E_3 ~ f'_3 N, E and C are enumerable, moreover obviously effectively enumerable. Any subset of C is at least enumerable. A subset of C corresponding to total functions is no effectively enumerable. It cannot be. Well, a language L which describes both total and strict partial functions and hereby defines a set of computable functions is not refuted by Kleene's diagonal. OK. I guess that you see now that is by allowing a universal machine to do infinite task which makes CT consistent (possible). OK? It is not obvious to me how strong sort of evidence that universal L exists, it is to survive Kleene's diagnonal. Droping an apple on the floor is in favor of Newton's law but not very convincing :-) Because the diagonal argument is *the* tool for demolishing the idea that such or such set is universal or complete, but then it does not work on language like Lambda, Turing, etc. This entails a strong form of incompleteness. Then, Judson Webb, argued that Godel's incompleteness confirmed such incompleteness, and thus CT, and thus Mechanism, Comp ... Oh, now I realize that my question is actually weird. Since the set of computable functions is defined by L, and L is said to be universal if it describes exactly these functions - it is simple to develop a trivial L - it defines a set of computable functions ... and of course universal L exists. OK, but this is due to your definition above of the set of computable functions. Recall that we do have some starting intuition of what is an intuitively computable functions. Church thesis is the assertion that LAMBDA (or FORTRAN, or whatever programming system you love) does capture that starting intuition. CT is both philosophical (linking an epistemic concept with a mathematical class of functions) and scientific (refutable in Popper sense: CT would be refuted in case we find a clearly humanly computable function which would be uncomputable in Lambda (and thus in Turing, Java, or any modern all-purpose computer, etc.). In this sence a universal language L always exists. So I write it rather like this: If we develop many sufficiently different programming languages and they turn out to be all equivalent, it might convince me that the set of 'computable functions' is fixed. Yes, and that is the case. I guess (from your
Re: SUMMARY
Hi Bruno and everybody, I hope to send my comments and/or 'OK' sign :-) on Monday. Take it easy. There is no deadline on the list. Making a declaration helps me to get things done. Yet I'm late. Whenever you see such sentences in my posts, you can skip it, they are mostly for me :-) --- I'll try to write a summary in my own words. Let's see how much I did understand. Prepositions: A finite alphabet A* finite words over A (it is enumerable, moreover effectively enumerable) L . a language over A. E a subset of A*, a set of valid expressions in L (obviously, it is at least enumerable) M . a machine which understands to L f . a total function from N to N. Goal: I want to develop a universal language L which describes all and only all functions f. Given an expression from E, M computes the result in finite time. Given the restrictions on L, the result is a number and nothing else. The set of all functions f from N to N is not enumerable (by Cantor's diagonal). Thus there is no bijection to E. Thus, I have to find a smaller set of functions. I will call this set a set of computable functions, C. Inevitably, this is computability by definition, by definition of L. So L should be 'really good' in order to encompass as much functions f from N to N as possible. Now, I think of a bijection between E and C. 0 --- E_0 ~ f_0 1 --- E_1 ~ f_1 2 --- E_2 ~ f_2 3 --- E_3 ~ f_3 Since E are efficiently enumerable, C are efficiently enumerable as well. Yes, it might happen that f_i = f_j for i != j, but is does not matter as long as all unique f_i are in the enumeration. Time for the Kleene diagonal argument. Opps, a language L that I dreamt of does not exist. I have to relax from the condition that M on E_i always return a number in a finite time. Well, what to return if not a number ... nothing - M experiences an infinite loop. What a world, ok, my language has to describe total functions from N to N and as well as strict partial functions from N to N. And it is clear that I cannot know whether E_i corresponds to a total function or a strict partial function. f' stands for any function descriable by L. 0 --- E_0 ~ f'_0 1 --- E_1 ~ f'_1 2 --- E_2 ~ f'_2 3 --- E_3 ~ f'_3 N, E and C are enumerable, moreover obviously effectively enumerable. Any subset of C is at least enumerable. A subset of C corresponding to total functions is no effectively enumerable. It cannot be. Well, a language L which describes both total and strict partial functions and hereby defines a set of computable functions is not refuted by Kleene's diagonal. It is not obvious to me how strong sort of evidence that universal L exists, it is to survive Kleene's diagnonal. Droping an apple on the floor is in favor of Newton's law but not very convincing :-) Oh, now I realize that my question is actually weird. Since the set of computable functions is defined by L, and L is said to be universal if it describes exactly these functions - it is simple to develop a trivial L - it defines a set of computable functions ... and of course universal L exists. In this sence a universal language L always exists. So I write it rather like this: If we develop many sufficiently different programming languages and they turn out to be all equivalent, it might convince me that the set of 'computable functions' is fixed. Although, written like this I can think of educated (math) people who will tell me: This is all you have So, what are like 2-3 most direct consequences of CT which make CT to seem rock-solid? Here I assume that CT basically says that the set of functions descriable by the lambda calculus is all what I can ever compute. - Regarding points 3)-5) of your summary, I am lost on terms such as Absolute ontic TOE, Observer Moments, Aristotelian principles, Machine theology, ... - I wrote down a list of short-term goals on what I would like to have some background/knowledge with a help from this list: 1\ I saw somewhere a sentence saying approximately this: so universe is performing a computation. Is then universe a big computer? No. I would like to know in a broad sense what it tries to say a why one shoud rather accept it or reject. 2\ Bruno, you recently wrote that you do not agree with Wolfram's Principle of Computational Equivalence. As I understand to that principle, Wolfram says that universe is a big cellular automata. What is the evidence that it is unlikely this way? Sincerely, Mirek --~--~-~--~~~---~--~~ You received this message because you are subscribed to the Google Groups Everything List group. To post to this group, send email to [EMAIL PROTECTED] To unsubscribe from this group, send email to [EMAIL PROTECTED] For more options, visit this group at http://groups.google.com/group/everything-list?hl=en -~--~~~~--~~--~--~---
Re: SUMMARY
David, Please take your time. As I said to Mirek we have all the time on this list. I know that your motivation consists in clarifying the notion and role of the first person in the unravelling of everything, and I feel a bit uneasy that I have to go back on Cantor and most importantly on Kleene's diagonal argument for explaining that. You can also ask me to go back on the motivations in case I am too technical or in case you loose the main line, which is something easy to do in a long multi-conversation. What I have to do, before getting the math of the 1-person, is to explain the nuance between computability and provability, and then the nuance between provability which, like computability, is a third person notion, and many 1-person (singular and/or plural) notions like knowability, observability and sensibility or sensitivity (I am still searching some words). And I have to make clear that all those notions are quite different from the notion of truth. But thanks telling me that you are still thinking on my posts, despite your short-of-time-ness (hmmm... that's not english). I will perhaps give soon the solution of how to write a combinator which makes the system looping, though. It is not necessary to understand this to get the main point, but it is helpful for some people in providing example of universal system/language. Have a good day, Bruno Le 28-janv.-08, à 18:16, David Nyman a écrit : Bruno, I'm sorry that I'm very short of time just now, but just to let you know I'm still trying to read and think about, if not comment, on these posts. David On 28/01/2008, Bruno Marchal [EMAIL PROTECTED] wrote: Le 25-janv.-08, à 21:10, Mirek Dobsicek a écrit : Bruno Marchal wrote: Title: SUMMARY (was: OM = SIGMA_1) I send to David Nyman (the 06 Nov 2007) a little planning: 1) Cantor's diagonal 2) Does the universal digital machine exist? 3) Lobian machines, who and what are they? 4) The 1-person and the 3- machine. 5) Lobian machines' theology 6) Lobian machines' physics 7) Lobian machines' ethics Let me summarize what has been done and what remains to be done. Hi Bruno, just want to let you know that I am still following your CT posts. Thanks for saying. Don't hesitate to ask anything in case some points are unclear. I hope to send my comments and/or 'OK' sign :-) on Monday. Take it easy. There is no deadline on the list. Nice weekend to everyone, Best, Bruno http://iridia.ulb.ac.be/~marchal/ http://iridia.ulb.ac.be/~marchal/ --~--~-~--~~~---~--~~ You received this message because you are subscribed to the Google Groups Everything List group. To post to this group, send email to [EMAIL PROTECTED] To unsubscribe from this group, send email to [EMAIL PROTECTED] For more options, visit this group at http://groups.google.com/group/everything-list?hl=en -~--~~~~--~~--~--~---
Re: SUMMARY
Le 25-janv.-08, à 21:10, Mirek Dobsicek a écrit : Bruno Marchal wrote: Title: SUMMARY (was: OM = SIGMA_1) I send to David Nyman (the 06 Nov 2007) a little planning: 1) Cantor's diagonal 2) Does the universal digital machine exist? 3) Lobian machines, who and what are they? 4) The 1-person and the 3- machine. 5) Lobian machines' theology 6) Lobian machines' physics 7) Lobian machines' ethics Let me summarize what has been done and what remains to be done. Hi Bruno, just want to let you know that I am still following your CT posts. Thanks for saying. Don't hesitate to ask anything in case some points are unclear. I hope to send my comments and/or 'OK' sign :-) on Monday. Take it easy. There is no deadline on the list. Nice weekend to everyone, Best, Bruno http://iridia.ulb.ac.be/~marchal/ --~--~-~--~~~---~--~~ You received this message because you are subscribed to the Google Groups Everything List group. To post to this group, send email to [EMAIL PROTECTED] To unsubscribe from this group, send email to [EMAIL PROTECTED] For more options, visit this group at http://groups.google.com/group/everything-list?hl=en -~--~~~~--~~--~--~---
Re: SUMMARY
Bruno, I'm sorry that I'm very short of time just now, but just to let you know I'm still trying to read and think about, if not comment, on these posts. David On 28/01/2008, Bruno Marchal [EMAIL PROTECTED] wrote: Le 25-janv.-08, à 21:10, Mirek Dobsicek a écrit : Bruno Marchal wrote: Title: SUMMARY (was: OM = SIGMA_1) I send to David Nyman (the 06 Nov 2007) a little planning: 1) Cantor's diagonal 2) Does the universal digital machine exist? 3) Lobian machines, who and what are they? 4) The 1-person and the 3- machine. 5) Lobian machines' theology 6) Lobian machines' physics 7) Lobian machines' ethics Let me summarize what has been done and what remains to be done. Hi Bruno, just want to let you know that I am still following your CT posts. Thanks for saying. Don't hesitate to ask anything in case some points are unclear. I hope to send my comments and/or 'OK' sign :-) on Monday. Take it easy. There is no deadline on the list. Nice weekend to everyone, Best, Bruno http://iridia.ulb.ac.be/~marchal/ --~--~-~--~~~---~--~~ You received this message because you are subscribed to the Google Groups Everything List group. To post to this group, send email to [EMAIL PROTECTED] To unsubscribe from this group, send email to [EMAIL PROTECTED] For more options, visit this group at http://groups.google.com/group/everything-list?hl=en -~--~~~~--~~--~--~---
Re: SUMMARY
Bruno Marchal wrote: Title: SUMMARY (was: OM = SIGMA_1) I send to David Nyman (the 06 Nov 2007) a little planning: 1) Cantor's diagonal 2) Does the universal digital machine exist? 3) Lobian machines, who and what are they? 4) The 1-person and the 3- machine. 5) Lobian machines' theology 6) Lobian machines' physics 7) Lobian machines' ethics Let me summarize what has been done and what remains to be done. Hi Bruno, just want to let you know that I am still following your CT posts. I hope to send my comments and/or 'OK' sign :-) on Monday. Nice weekend to everyone, Mirek --~--~-~--~~~---~--~~ You received this message because you are subscribed to the Google Groups Everything List group. To post to this group, send email to [EMAIL PROTECTED] To unsubscribe from this group, send email to [EMAIL PROTECTED] For more options, visit this group at http://groups.google.com/group/everything-list?hl=en -~--~~~~--~~--~--~---
SUMMARY (was: OM = SIGMA_1)
Title: SUMMARY (was: OM = SIGMA_1) I send to David Nyman (the 06 Nov 2007) a little planning: 1) Cantor's diagonal 2) Does the universal digital machine exist? 3) Lobian machines, who and what are they? 4) The 1-person and the 3- machine. 5) Lobian machines' theology 6) Lobian machines' physics 7) Lobian machines' ethics Let me summarize what has been done and what remains to be done. 1) Cantor's diagonal I tend to consider that the point 1) is finished. Cantor's argument is that if there is a bijection between natural numbers, that is: 0, 1, 2, 3, 4, ..., and sequences of numbers, that is a bijection like 0 --- 45, 7, 8976, 4, 32, ... 1 --- 0, 0, 67, 78, 0, ... 2 --- 27, 1, 24, 24, 23, ... 3 --- 1, 1, 1, 345, 7, ... ... then the antidiagonal sequence 46, 1, 25, 346, ... cannot be in the list, because by construction it differs from each sequence in the list. See below how to make explicit the contradiction. The reasoning does not depend on the particular sequences exhibited, and it shows that no enumerable set of sequences can be put in 1-1 correspondence with the natural numbers. The conclusion is that the set of all sequences of natural numbers is innumerable (not enumerable, not countable, uncountable, etc. Important concept have many synonym in math). Let me recall the same proof, but with usual mathematical notation. A sequence of numbers, like f_0 = 56, 7897876, 67, 89, 1, 1, 45, ... is really just a function from N to N: f_0(0), f_0(1) f_0(2), f_0(3), f_0(4), ... with here: f_0(0) = 56, f_0(1) = 7897876, f_0(2) = 67, f_0(3) = 89, f_0(4) = 1, etc. So the bijection above becomes: 0 --- f_0 = f_0(0), f_0(1) f_0(2), f_0(3), f_0(4), ... 1 --- f_1 = f_1(0), f_1(1) f_1(2), f_1(3), f_1(4), ... 2 --- f_2 = f_2(0), f_2(1) f_2(2), f_2(3), f_2(4), ... 3 --- f_3 = f_3(0), f_3(1) f_3(2), f_3(3), f_4(4), ... ... You can see that the diagonal sequence can be described by: f_0(0), f_1(1), f_2(2), f_n(n), ... Then the antidiagonal sequence (function) g is given by f_0(0)+1, f_1(1)+1, f_2(2)+1, f_n(n)+1, ... That is: g(n) = f_n(n)+1 (definition of g) Now we can make the contradiction explicit. Suppose that g is in the list f_i. Then it exists a number k such that g = f_k. This means of course that for all numbers n we have g(n) = f_k(n). In particular g(k) = f_k(k). But by the definition of g: g applied on k = g(k) = f_k(k)+1. Thus (by Leibniz identity rule): f_k(k) = f_k(k)+1 Now, all f_i are functions from N to N, so they are defined on all natural numbers, so f_k(k) is a number. We have seen in high school that identical numbers can be subtract on both sides of an equation leading to 0 = 1. (contradiction). Thus the f_i cannot enumerate all functions from N to N. We say: N^N is innumerable. This was point 1). Hope it is ok for every one. Please be sure you get the point before proceeding. 2) Does the universal digital machine exist? I recall the informal notion of what is an (intuitively) computable function (from N to N). Def: A function f from N to N is computable if we can describe in some formal language L, in a finite way, how to compute, in a finite time, its value f(n) on each natural number n. Def. I will call code of f such a description of how to compute f. Def. A language L is said universal if all computable functions can be described in the language. Def. A machine is universal if she understands a universal language, (and thus can indeed compute all computable functions from N to N, at least in Platonia, where Platonia is defined by a place where you can always ask and get more time and more space/memory: we don't put deadline to the (universal) machine. Church thesis is the statement that a universal language (and machine) exists, and indeed that in particular lambda-calculus provides such a universal language. Church's thesis is not obvious. Indeed, when Church defined the computable functions by those capable of being computed by a lambda-expression (a symbolic expression or code written in the lambda-calculus), Stephen Cole Kleene thought at first that a reasoning similar to Cantor's proof of the non enumerability of N^N (see above) could be made against Church's pretension. Kleene's reasoning is the following, and works for any pretension that there is a universal language (so we have not to even define what lambda-calculus). Indeed, suppose that there is a universal machine and thus a universal language in which all computable functions from N to N can be given a code. Now the set of codes in the language L is enumerable, being a subset of all possible expression written in the language (which we have seen to be enumerable). Thus there is an enumeration of all computable functions from N to N f_1, f_2, f_3, f_4, f_5, f_6, f_7, f_8, f_9, ... but then the antidiagonal function g defined by g(n) = f_n(n) + 1 is computable
The Mathematico-Cognition Reality Theory (MCRT) -Updated abstract summary
An abstract summary of my Mathematico-Cognition Reality Theory (MCRT) has now been published. The final summary comes in at over 6 500 words. I hope to publish a fuller paper with some actual mathematics in it at a later date. This summary is intended to be a statement of the *general* conceptual principles behind my theory of everything. My advice to every-one here: I'd read this *very* careful if I were you! Here's the link: http://www.toequest.com/forum/showthread.php?t=1360 -- Till shade is gone, till water is gone, into the shadow with teeth bared, screaming defiance with the last breath, to spit in Sightblinder's eye on the last day
Re: Summary of seed ideas for my developing TOE - 'The Sentient Centered Theo...
--- Hal Ruhl [EMAIL PROTECTED] wrote: Hi John: I do not know if one should use the word Theory but what strikes me is the convergence I see in numerous lines of thought. I see my model as having many features in common with Russell's even though some of the differences may not be subject to complete reconciliation. I also see a place for Bruno's consistent histories, consistent extensions computational hypothesis approach as a sub component of mine. I have been made aware of others that fit the same pattern of convergence towards what appears to me to be a single simple model. The apparent convergence from such different starting places and ensuing seemingly incompatible lines of thought I find remarkable. It makes me believe that the model at the apex of this convergence is the correct one as far as we can ever know it. Hal Ruhl That is all fine, but all those 'convergent' thinking comes from the limited minds of present day humans. If nature is not restricted to our understanding (watch for the word: UNDER) then we have no right to speak about 'everything' (without due identification). Similar to possible (which includes our deemed impossibilities as well, restricted to our feeble imagination). Even imaginable is a restriction. Nature is not limited to that - as I like to think about her. Not even in her 'logic'. With friendly greetings from our perceived universe(?) John M
Rép : Summary of seed ideas for my developing T OE - 'The Sentient Centered Theo...
Le 24-sept.-05, à 18:14, John M a écrit : Dear Logician G: I copy from your post below: But if there are features of reality not explained by the TOE, we still can expect that the TOE will be able to justify---or meta-justify--- why it cannot explain those features. * 'Everything' includes segments galore we have no idea about (so far undisputed statement). This is already debatable. If the 3-everything are just numbers, there is a sense in saying we know them all, even if we cannot know personally each number, and totally ignore some interesting subsets of the natural numbers. The first person plenitude is bigger. So big that no first person can even just name the whole of it. We really cannot go into any analysis of unknown features, I think. Actually I was talking on knowable feature that we cannot explain. There are a lot of things we can know without being able to explain. Even the sky or the sun could belong to that. Consciousness is more typical. Do you simply mean a 'generalitiy' that SUCH (unkown) features cannot be explained? (saying: what we don't know we don't know, or something similar?) I may expect better from you (ha ha). From me? or from the lobian machine. (Unless, of course, we think that it is not OUR TOE, but the infinite comp's = the entire (and thinking) WORLD, or whatever you call it, and it is ITS TOE, not ours - in which case I am out.) What do you mean by our TOE ? And what do you mean by our ? Could you perhaps elaborate on your last two paragraphs? Not so clear for me. Have a nice weekend Have a nice weekstart :-) Bruno http://iridia.ulb.ac.be/~marchal/
Re: My All Nothing model compacted, was: Re: Summary of seed ideas ...
Le 23-sept.-05, à 16:29, Hal Ruhl a écrit : Hi Bruno: At 06:04 AM 9/23/2005, you wrote: Le 22-sept.-05, à 18:12, John M a écrit : John Bruno: according to your (and Marc's?) definition, is Hal's work a TOEandTON? Bruno The problem, for me, is with the T (both in TOE and TON). I cannot judge. Hal's talk is still too much vague for me. I appreciate and perhaps share soime intuitions, though.I certainly appreciate the role of logical incompleteness. Below I have tried to compact my model to see if it helps: It does not really help. We have already discuss this. If you want to communicate to other you need to find a language others can understand. Or you should be able to represent what you want to say in some theory by others. I have not yet the slightest clue of what you ask us to take as primitive and what you derive from it, nor if what you do can help to solve the measure problem, and so one. There are three levels of existence in the model: 1) The list of all possible aspects of objects and ideas [and its representative one to one correspondence with the natural numbers]. In our context all those terms have no obvious interpretation. What is an object, what is an idea? What are the assumption making it possible to build a correspondence with the natural numbers. What do you mean by possible aspect. Is aspect a first or a third person notion. The next levels contain parsings of this list as objects or the descriptions of such parsings. 2) The [Nothing:All] parsing of the list. [As a pair of resulting objects and at this level due to the unavoidability of there being such objects.] Please refer to places where you define all those terms. Your posts does not help. 3) The descriptions of all the possible parsings of the list [kernels] [including level 2] all of which are placed in the All [along with the power set based representation as a one to one correspondence with the real numbers] and none of which are placed in the Nothing. ? The incompleteness of the Nothing produces a dynamic at levels 2 and 3. Proving machines or theories can be incomplete. What do you mean by Nothing. What is a dynamic? Do you assume some time? The inconsistency and content of the All and the imperative for level 2 makes this dynamic random and perpetual. ? The resulting dynamic is a repeated extinction/establishment of Nothings and evolving Somethings in the All. The part of the this dynamic that is within the All [evolving Somethings] provides repeated instantations of physical reality to all the kernels in the All. [In a random fashion in keeping with the inconsistent nature of the All.] Some kernels are descriptions of states of universes so states of universes are given perpetual repetitions of instantations of physical reality in random sequences. The result is that all sequences of all states of all universes experience a flow of instantations of physical reality [over and over] some of which bridge states thus giving any Self Aware Structures described in these states a flow of awareness [consciousness]. You lost me, sorry. I really think you should try to recast what you want to explain in some known theory (with sets or numbers, or programs, or automata, or waves, whatever ... I can understand the basic). Although formalization could be useless, it could help, in this difficult subject, to realize the many implicit assumptions that you are using or not. It could help if you were able to put light on questions already discussed. For example in your 1, it looks you are using comp, but then in 2 you introduce randomness, but then you should know that comp implies strong form of 1-randomness. Are those randomness related? Also, how do you relate third person description and first person description (this have to be explained once comp is assumed). Bruno http://iridia.ulb.ac.be/~marchal/
Re: My All Nothing model compacted, was: Re: Summary of seed ideas ...
Hi, Hal and Bruno, the T is my problem as well (I swalloed it) because ALL (pardon the pun) we may know is within the feeble capabilities of our little minds and I have no right to assume that 'nature' does not include much more than this little segment. This is why I call whatever I find out in my speculations a narrative. Hal's language is the music of the spheres for me: it is way above my head, and although I may 'feel' the harmony in it, the simplistic (vocabulary) meaning of the terms he is using - if I look them up - does not give HIS topical 'substance' to my understanding; besides, even such 'substance' would be weak, considering my background. One has to be comfortable with the meaning of the words to follow the strange meanings of someone else'e ideas. Ten years ago Hal, you were much simpler in your descriptions but then you started to chisle out problems ie. include explanatory or modifying clauses into the 'new' versions. Now I don't know if I agree with it or not. (Don't misunderstand me: the 'not' is not disagreement, only a lack of an enthusiastic agreement.) Regards John M --- Hal Ruhl [EMAIL PROTECTED] wrote: Hi Bruno: At 06:04 AM 9/23/2005, you wrote: Le 22-sept.-05, à 18:12, John M a écrit : John Bruno: according to your (and Marc's?) definition, is Hal's work a TOEandTON? Bruno The problem, for me, is with the T (both in TOE and TON). I cannot judge. Hal's talk is still too much vague for me. I appreciate and perhaps share some intuitions, though.I certainly appreciate the role of logical incompleteness. Below I have tried to compact my model to see if it helps: There are three levels of existence in the model: 1) The list of all possible aspects of objects and ideas [and its representative one to one correspondence with the natural numbers]. The next levels contain parsings of this list as objects or the descriptions of such parsings. 2) The [Nothing:All] parsing of the list. [As a pair of resulting objects and at this level due to the unavoidability of there being such objects.] 3) The descriptions of all the possible parsings of the list [kernels] [including level 2] all of which are placed in the All [along with the power set based representation as a one to one correspondence with the real numbers] and none of which are placed in the Nothing. The incompleteness of the Nothing produces a dynamic at levels 2 and 3. The inconsistency and content of the All and the imperative for level 2 makes this dynamic random and perpetual. The resulting dynamic is a repeated extinction/establishment of Nothings and evolving Somethings in the All. The part of the this dynamic that is within the All [evolving Somethings] provides repeated instantations of physical reality to all the kernels in the All. [In a random fashion in keeping with the inconsistent nature of the All.] Some kernels are descriptions of states of universes so states of universes are given perpetual repetitions of instantations of physical reality in random sequences. The result is that all sequences of all states of all universes experience a flow of instantations of physical reality [over and over] some of which bridge states thus giving any Self Aware Structures described in these states a flow of awareness [consciousness]. Hal Ruhl
Re: Summary of seed ideas for my developing TOE - 'The Sentient Centered Theo...
Le 22-sept.-05, à 06:27, Marc Geddes a écrit : What I'd like is a *logical scaffolding* - a *finite* system which is *universal* in scope - or at least applying everywhere in reality where sentient minds can exist and which explains the relationship between Mind and Reality. That for me is a TOE. I don't require that the theory literally explains everything. I agree and I agree with your other statement according to which a TOE must explain the relation between mind and reality (what most physicalist put under the rug). But if there are features of reality not explained by the TOE, we still can expect that the TOE will be able to justify---or meta-justify--- why it cannot explain those features. Bruno http://iridia.ulb.ac.be/~marchal/
Re: Summary of seed ideas for my developing TOE - 'The Sentient Centered Theo...
Bruno: according to your (and Marc's?) definition, is Hal's work a TOEandTON? Or would you include Nothing into the relations of Mind (again: wat is it really?) and reality (same question really!)? (I mean: defined in less than 1000 words G) John M --- Bruno Marchal [EMAIL PROTECTED] wrote: Le 22-sept.-05, à 06:27, Marc Geddes a écrit : What I'd like is a *logical scaffolding* - a *finite* system which is *universal* in scope - or at least applying everywhere in reality where sentient minds can exist and which explains the relationship between Mind and Reality.That for me is a TOE. I don't require that the theory literally explains everything. I agree and I agree with your other statement according to which a TOE must explain the relation between mind and reality (what most physicalist put under the rug). But if there are features of reality not explained by the TOE, we still can expect that the TOE will be able to justify---or meta-justify--- why it cannot explain those features. Bruno http://iridia.ulb.ac.be/~marchal/
Re: Summary of seed ideas for my developing TOE - 'The Sentient Centered Theory Of Metaphysics' (SCTOM)
On 9/20/05, [EMAIL PROTECTED] [EMAIL PROTECTED] wrote: OK, you said All comments welcome.You asked for it.First, there's a lot to read here, so I assumed you were presenting the basic gist of your ideas in the first few paragraphs, and so I have afew comments about those paragraphs.I commend you for trying to explain values as part of the framework.I've whinced before when I've read some thought experiments on this list that depended on accepting the existence of such ideas as good andbad.I believe in the existence of good and bad, but one needs tosupport his/her belief in good and bad and not take them as a given. It seems that your limitation of reality to meaningful existence isactually rejecting Mathematical Platonism.Why is consciousnessrequired to make a mathematical truth real?I thought that you aretrying to deal with all of existence, not just meaningful existence, since your theory tries to explain how the most fundamental propertiesof existence facts fit together into a unified metaphysical framework.And yet here you limit existence to what we can perceive. The core assumption is that existence without perception ismeaningless. Reality requires not only raw data but something to*interpret* that data, to supply meaning to it. This can only be done by consciousness of *some* kind. If something was hypothesized to existthat could in no way directly or indirectly affect the consciousperceptions of *any* possible observer, then in what sense could it besaid to exist at all? Even if it could be successfully argued that it did have some kind of abstract philosophical existence, it could neverhave any possible value to sentient minds. For the purposes ofunderstanding general intelligence, it suffices to define that whichexists as that which could directly or indirectly ( i.e. in principle)affect the perceptions of *some* possible conscious observer.So you've eliminated the whole realm of unperceived reality in thesuperset of existence.You've eliminated the motivation to bring unperceived reality into the realm of perceived reality, since theformer does not exist.Reading these metaphysical theories doesn't really impress me when Irealize that these theories really don't have anything new in them that the ancient Greeks (for instance) didn't have.Of course the big gap in all of these theories, which I believe willnever be filled, is the integration of consciousness (in general) intophysics.Even if we integrate human consciousness into it (which I don't think is going to happen), that doesn't cover the whole gammit ofwhat consciousness is in the whole universe.Who knows, there's somuch we don't know about stars (and they are so big) that perhaps some stars have consciousness of some kind that is outside of the definitionof how we would define it, but may be even more enlightened about theuniverse, and yet we may never know.Tom What I wrote there may be misleading. By 'perceivable' I don't necessarily mean 'perceived by humans', what I mean is 'perceivable *in principle* ( i.e. by some mind, somewhere in the universe). Reality can only ever be understood from the perspective of a mind. Therefore only things capable of (in principle) making a difference to perceived reality need to be taken into account when devising ultimate theories of metaphysics. If you read what I wrote I made it pretty clear that I believe in a kind of mathematical Platonism. My proposed noumenon (raw fabric) of reality was something I called 'Mathematico-Cognition' (a hybrid of mathematics and information processing). I don't think the 'perceivable in principle' requirement contradicts mathematical Platonism. What makes you think that mathematical objects aren't perceivable? True, most *humans* can't perceive mathematical things, but that's probably just a limitation of the human mind. I think that a mind sufficiently talented at math *could* in principle directly perceive mathematical objects. Kurt Godel claimed that it was possible to directly perceive mathematical objects. He even thought the mind was capable of directly perceiving infinite sets.-- Please vist my website: http://www.riemannai.orgScience, Sci-Fi and Philosophy---THE BRAIN is wider than the sky,For, put them side by side,The one the other will includeWith ease, and you beside. -Emily Dickinson'The brain is wider than the sky'http://www.bartleby.com/113/1126.html
Re: Summary of seed ideas for my developing TOE - 'The Sentient Centered Theory Of Metaphysics' (SCTOM)
On 9/22/05, [EMAIL PROTECTED] [EMAIL PROTECTED] wrote: By 'perceivable' I don't necessarily mean 'perceived by humans', whatI mean is 'perceivable *in principle* ( i.e. by some mind, somewhere inthe universe).I admit my misunderstanding, and that you are talking about theunperceivable rather than the unperceived, so the argument abouteliminating the motivation to discover does not apply, although it does apply to those that reject the existence of an objective reality. Reality can only ever be understood from the perspective of a mind.Are you willing to admit that you have to be agnostic (by definition!) about the fact that there could be reality that can't be understood bya mind? Yes. But only minds can perceive and comprehend reality. Only minds can value. The parts of reality that are beyond the comprehension of all possible minds cannot by definition be directly dealt with by any metaphysical theory. And what value could they possibly beto us? That's why I called my theory the 'Sentient Centered' theory.A mind is the most important thing inthe universe because without mind there can be no value (values come from minds). What I'm asking is: Why do you limit metaphysics, at the outset, tobeing for the purposes of understanding general intelligence? On the other hand, how do we know what general intelligence is if all wehave is our human understanding?Thus my example of conscious starswhich are enlightened about the universe in ways that don't even fit into our mind's capability of understanding what enlightened can mean. Only a general intelligence (a mind capable of fully reflective reasoning) can value things, perceive things and comprehend things. Therefore any metaphysical theory needs to deal with those aspects of reality that can in principle impinge on the mind of a general intelligence. You make a good point about kinds of consciousness that may be beyond human understanding. But my theory does not attempt to provide a full explanation of what general intelligence is. It is simply meant to serve as a logical scaffolding to which new scientific and philosophical information would continue to be added. In order for the words 'intelligence' and 'consciousness' to have an unitary meaning, there would have to be *some* general properties that all possible minds had in common. A metaphysical theory intended to serve as a 'logical scaffolding' simply has to deal with these general properties. Therefore only things capable of (in principle) making a differenceto perceived reality need to be taken into account when devising ultimate theories of metaphysics.Is not there a difference between things that (in principle) cannever make a difference to perceived reality (i.e. unperceivable bysome logical contradiction to perceivability, but yet existing somehow), and things that never will make a difference to perceivedreality because of the limitations of minds (in general)?I admit thatwe can't include the former, but what about the latter? The latter possibilitywould mean that there'san unbridgeable seperation betweenthething in itself and a mind's conception of a thing aka Kant. It's a logical possibility of course but I note that many modern philosophers reject Kant's idea. I don't think the 'perceivable in principle' requirement contradictsmathematical Platonism. What makes you think that mathematical objects aren't perceivable?True, most *humans* can't perceivemathematical things, but that's probably just a limitation of the humanmind. I think that a mind sufficiently talented at math *could* inprinciple directly perceive mathematical objects.Kurt Godel claimed that it was possible to directly perceive mathematical objects. Heeven thought the mind was capable of directly perceiving infinite sets.What if the proof of Goldbach's Conjecture was such that it could not be perceived by a mind?Doesn't our incomplete picture of the mindallow for such a possibility? I suppose so. But it seems unlikely to me. What does the word 'proof' *mean* if not that there are a series of logical connections each of which is capable of being comprehended (in principle) by *some* mind? Of course, there are likely proofs beyond human understanding but such proofs should not be beyond the understanding of *some* (in principle) sufficiently powerful mind. THE BRAIN is wider than the sky, For, put them side by side, The one the other will include With ease, and you beside.-Emily DickinsonIn all of the history of humans' exploration of the universe, theperpetual message that keeps coming back to us from the universe isthat the brain is not as wide as the sky.I think that trying to make an end run around everything and starting with the doctrine that itis, is not a new thing (even to the ancient Greeks), but it contradictsthe evidence.Tom *Given* that we want a metaphysical 'Theory Of Everything' (the name of this mailing list after all!) we must *assume* as a starting point that mind can comprehend reality.
Re: Summary of seed ideas for my developing TOE - 'The Sentient Centered Theo...
THE BRAIN is wider than the sky, For, put them side by side, The one the other will include With ease, and you beside.-Emily DickinsonIn all of the history of humans' exploration of the universe, theperpetual message that keeps coming back to us from the universe isthat the brain is not as wide as the sky.I think that trying to make an "end run" around "everything" and starting with the doctrine that itis, is not a new thing (even to the ancient Greeks), but it contradictsthe evidence.Tom *Given* that we want a metaphysical 'Theory Of Everything' (the name of this mailing list after all!) we must *assume* as a starting point that mind can comprehend reality. Our assumption could be wrong.That's why it's called a *theory* ofeverything ;) Why couldn't the theory be that the mind can comprehend reality, but not all of reality. Wouldn't that be a theory of everything? What if that's the actual truth? We would be doing ourselves a disservice by theorizing otherwise.
Re: Summary of seed ideas for my developing TOE - 'The Sentient Centered Theo...
THE BRAIN is wider than the sky, For, put them side by side, The one the other will include With ease, and you beside.-Emily DickinsonIn all of the history of humans' exploration of the universe, theperpetual message that keeps coming back to us from the universe isthat the brain is not as wide as the sky.I think that trying to make an "end run" around "everything" and starting with the doctrine that itis, is not a new thing (even to the ancient Greeks), but it contradictsthe evidence.Tom *Given* that we want a metaphysical 'Theory Of Everything' (the name of this mailing list after all!) we must *assume* as a starting point that mind can comprehend reality. Our assumption could be wrong.That's why it's called a *theory* ofeverything ;) Why couldn't the theory be that the mind can comprehend reality, but not all of reality. Wouldn't that be a theory of everything? What if that's the actual truth? We would be doing ourselves a disservice by theorizing otherwise.And I'm saying (see above) that the evidence is against the assumption that the mind can comprehend everything. The message we get from the universe is that its paradigm is always beyond our minds.
Re: Summary of seed ideas for my developing TOE - 'The Sentient Centered Theo...
On 9/22/05, [EMAIL PROTECTED] [EMAIL PROTECTED] wrote: *Given* that we want a metaphysical 'Theory Of Everything' (the name of this mailing list after all!) we must *assume* as a starting point that mind can comprehend reality. Our assumption could be wrong.That's why it's called a *theory* ofeverything ;) Why couldn't the theory be that the mind can comprehend reality, but not all of reality. Wouldn't that be a theory of everything? What if that's the actual truth? We would be doing ourselves a disservice by theorizing otherwise. Well, of course, the question that arises is: what actually *is* a 'theory of everything'? By TOE I don't require that the mind can literally comprehend *all* of reality. Ijust think thatthere's some way to integratemental and physical concepts into afinite unified explanatory framework which *is* comprehensible. So for me, a TOE is a theory which explains the relationshipbetween Mind on the one hand, and Reality on the other. M (Mind) relationship - R (Reality).My theory is attempting to explain that relationship. What I'd likeis a *logical scaffolding* - a *finite* system whichis *universal* in scope -or at least applying everywhere in reality where sentient minds can exist and which explains the relationship between Mind and Reality. That for me isaTOE. I don't require that the theory literally explains everything. -- Please vist my website:http://www.riemannai.orgScience, Sci-Fi and Philosophy---THE BRAIN is wider than the sky,For, put them side by side, The one the other will includeWith ease, and you beside. -Emily Dickinson'The brain is wider than the sky'http://www.bartleby.com/113/1126.html
Re: Summary of seed ideas for my developing TOE - 'The Sentient Centered Theory Of Metaphysics' (SCTOM)
OK, you said All comments welcome. You asked for it. First, there's a lot to read here, so I assumed you were presenting the basic gist of your ideas in the first few paragraphs, and so I have a few comments about those paragraphs. I commend you for trying to explain values as part of the framework. I've whinced before when I've read some thought experiments on this list that depended on accepting the existence of such ideas as good and bad. I believe in the existence of good and bad, but one needs to support his/her belief in good and bad and not take them as a given. It seems that your limitation of reality to meaningful existence is actually rejecting Mathematical Platonism. Why is consciousness required to make a mathematical truth real? I thought that you are trying to deal with all of existence, not just meaningful existence, since your theory tries to explain how the most fundamental properties of existence facts fit together into a unified metaphysical framework. And yet here you limit existence to what we can perceive. The core assumption is that existence without perception is meaningless. Reality requires not only raw data but something to *interpret* that data, to supply meaning to it. This can only be done by consciousness of *some* kind. If something was hypothesized to exist that could in no way directly or indirectly affect the conscious perceptions of *any* possible observer, then in what sense could it be said to exist at all? Even if it could be successfully argued that it did have some kind of abstract philosophical existence, it could never have any possible value to sentient minds. For the purposes of understanding general intelligence, it suffices to define that which exists as that which could directly or indirectly ( i.e. in principle) affect the perceptions of *some* possible conscious observer. So you've eliminated the whole realm of unperceived reality in the superset of existence. You've eliminated the motivation to bring unperceived reality into the realm of perceived reality, since the former does not exist. Reading these metaphysical theories doesn't really impress me when I realize that these theories really don't have anything new in them that the ancient Greeks (for instance) didn't have. Of course the big gap in all of these theories, which I believe will never be filled, is the integration of consciousness (in general) into physics. Even if we integrate human consciousness into it (which I don't think is going to happen), that doesn't cover the whole gammit of what consciousness is in the whole universe. Who knows, there's so much we don't know about stars (and they are so big) that perhaps some stars have consciousness of some kind that is outside of the definition of how we would define it, but may be even more enlightened about the universe, and yet we may never know. Tom
Re: Summary of seed ideas for my developing TOE - 'The Sentient Centered Theory Of Metaphysics' (SCTOM)
Marc seems unclear between unperceivable and unperceived, maybe clearing that up would help. If everything real needs some sort of perceivability, then everything real would need not only to be interpretable and decodable, but also to be verifiable, confirmable, corroborable, etc., by interpreted signs' (not symbols per se, just anything significant) recipients on the basis of earlier/current/later experiences. Evolution confirms/disconfirms in a way; but percipient intelligent organisms prefer to check our interpretations before evolution gets a chance to find them wrong and to discard them by discarding us from the gene pool. If reality needs perceivability, not merely decodability by something plantlike and unlearning, then it needs not only interpretability (meaning, value, etc.), but also observability-in-light-of-interpretations and verifiability (validity, cogency, soundness, etc.) as to meaning. This seems more or less the view of typical working scientists (of whom I'm not one) -- if it's beyond all observability by anything whatsoever, even in principle,! then is it even real? One can argue about it. But if we're talking about a requirement for actual perception, then we're talking about a need by reality for actual observation, verification, etc. (and ultimately more science than seems possible for us finite creatures to produce). Bishop Berkeley might like it, though. Regards, Ben Udell - Original Message - From: [EMAIL PROTECTED] To: [EMAIL PROTECTED]; everything-list@eskimo.com Sent: Monday, September 19, 2005 6:07 PM Subject: Re: Summary of seed ideas for my developing TOE - 'The Sentient Centered Theory Of Metaphysics' (SCTOM) OK, you said All comments welcome. You asked for it. First, there's a lot to read here, so I assumed you were presenting the basic gist of your ideas in the first few paragraphs, and so I have a few comments about those paragraphs. I commend you for trying to explain values as part of the framework. I've whinced before when I've read some thought experiments on this list that depended on accepting the existence of such ideas as good and bad. I believe in the existence of good and bad, but one needs to support his/her belief in good and bad and not take them as a given. It seems that your limitation of reality to meaningful existence is actually rejecting Mathematical Platonism. Why is consciousness required to make a mathematical truth real? I thought that you are trying to deal with all of existence, not just meaningful existence, since your theory tries to explain how the most fundamental properties of existence facts fit together into a unified metaphysical framework. And yet here you limit existence to what we can perceive. The core assumption is that existence without perception is meaningless. Reality requires not only raw data but something to *interpret* that data, to supply meaning to it. This can only be done by consciousness of *some* kind. If something was hypothesized to exist that could in no way directly or indirectly affect the conscious perceptions of *any* possible observer, then in what sense could it be said to exist at all? Even if it could be successfully argued that it did have some kind of abstract philosophical existence, it could never have any possible value to sentient minds. For the purposes of understanding general intelligence, it suffices to define that which exists as that which could directly or indirectly ( i.e. in principle) affect the perceptions of *some* possible conscious observer. So you've eliminated the whole realm of unperceived reality in the superset of existence. You've eliminated the motivation to bring unperceived reality into the realm of perceived reality, since the former does not exist. Reading these metaphysical theories doesn't really impress me when I realize that these theories really don't have anything new in them that the ancient Greeks (for instance) didn't have. Of course the big gap in all of these theories, which I believe will never be filled, is the integration of consciousness (in general) into physics. Even if we integrate human consciousness into it (which I don't think is going to happen), that doesn't cover the whole gammit of what consciousness is in the whole universe. Who knows, there's so much we don't know about stars (and they are so big) that perhaps some stars have consciousness of some kind that is outside of the definition of how we would define it, but may be even more enlightened about the universe, and yet we may never know. Tom
Re: Summary of seed ideas for my developing TOE - 'The Sentient Centered Theory Of Metaphysics' (SCTOM)
Whether it's ignoring the unperceived or unperceivable, what I'm asking is: Why do you limit metaphysics, at the outset, to being for the purposes of understanding general intelligence? On the other hand, how do we know what general intelligence is if all we have is our human understanding? Tom
Good summary of Bogdanov controversy
A good summary of the Bogdanov controversy is in the New York Times today. URL is http://www.nytimes.com/2002/11/09/arts/09PHYS.html Some of the folks we like to quote here are quoted in the article, including Lee Smolin, John Baez, Carlo Rovelli, etc. Also, the latest Wired print issue has a fairly good survey article by Kevin Kelly about theories of the universe as a cellular automaton. Konrad Zuse gets prominent mention, along with Ed Fredkin. I didn't read the article closely, so I didn't notice if either Tegmark or Schmidhuber were mentioned. The usual stuff about CA rules, Wolfram's book, etc. Things have been quiet here on the Everything list. I haven't been commenting on my own reading, which is from books such Physics Meets Philosophy at the Planck Scale and Entanglement. Isham's collection of essays on QM should arrive momentarily at my house. My interest continues to be in topos theory, modal logic, and quantum logic. --Tim May (.sig for Everything list background) Corralitos, CA. Born in 1951. Retired from Intel in 1986. Current main interest: category and topos theory, math, quantum reality, cosmology. Background: physics, Intel, crypto, Cypherpunks
Re: Good summary of Bogdanov controversy
From: Osher Doctorow [EMAIL PROTECTED], Sunday Nov. 10, 2002 5PM Thanks to Tim May for the site reference. I read the story, and it's quite interesting. It's the first time I've looked at this in detail, although I heard a rumor about it. I have a few comments that I'd like to make now. 1. The acceptance of nonsense for publishing or Ph.D.s or M.A.s or M.S.s is obviously wrong. 2. The cause of the acceptance needs to be investigated by scientists and philosophers and others. 3. History tells us a few things about nonsense if we study it carefully, especially the history of Creative Geniuses like Beethoven, Shakespeare, Paul Dirac, Einstein, Schrodinger, Socrates, Plato, Mozart, etc. I will itemize these below beginning with 4, but I'll just mention that they fall under Mediocrity, Ingenious Imitation, and Creative Genius. 4. Mediocre scientific people in my definition don't even have the ability to imitate (see below). 5. Ingenious Imitators in science (and similarly for music, literature, etc.) imitate other scientists but only go 0 or 1 step ahead of whomever they are imitating. 6. Creative Geniuses go more than 1 step ahead of anybody else working on the same or similar problem or anybody else in the field or subfield. 7. Having spent most of my 63 years of life in Academia, both as a student and as a teacher/researcher in mathematics including statistics and mathematical physics, it is my opinion that more than 99% of mathematicians and physicists are Ingenious Imitators, and I have a stong suspicion that this is the case in most other academic fields. 8. Peer review is the usual way of determining which papers are published in scientific journals, and it follows from 7 if I am correct that most peer reviewers are Ingenious Imitators, and therefore that what gets published in most journals is at most one step ahead of the previous person (and possibly 0 steps ahead). 9. The solution to the problem of 8 and similar difficulties with Ph.D. and Masters Degrees is in my opinion a positive one rather than a negative one, namely, to foster more Creative Geniuses in Mathematics and Physics (and other fields). 10. In my opinion, Ingenious Imitators can become Creative Geniuses with sufficient education, tolerance, practice in accepting and thinking up new ideas, learning tranquility rather than anger or fear, and guidance from other Creative Geniuses or Creative Problem Solvers (a sort of borderline type between Creative Genius and Igenious Imitators, which I'll explain another time hopefully). Giving up Materialism, including Money-Related Materialism, Power Materialism, and Sensation Materialism, which includes giving up bureaucracy or the interest in becoming part of it, is key in this. Osher Doctorow, Ph.D. One or more of California State Universities and Community Colleges (Mathematics, Statistics) - Original Message - From: Tim May [EMAIL PROTECTED] To: [EMAIL PROTECTED] Sent: Sunday, November 10, 2002 12:44 PM Subject: Good summary of Bogdanov controversy A good summary of the Bogdanov controversy is in the New York Times today. URL is http://www.nytimes.com/2002/11/09/arts/09PHYS.html Some of the folks we like to quote here are quoted in the article, including Lee Smolin, John Baez, Carlo Rovelli, etc. Also, the latest Wired print issue has a fairly good survey article by Kevin Kelly about theories of the universe as a cellular automaton. Konrad Zuse gets prominent mention, along with Ed Fredkin. I didn't read the article closely, so I didn't notice if either Tegmark or Schmidhuber were mentioned. The usual stuff about CA rules, Wolfram's book, etc. Things have been quiet here on the Everything list. I haven't been commenting on my own reading, which is from books such Physics Meets Philosophy at the Planck Scale and Entanglement. Isham's collection of essays on QM should arrive momentarily at my house. My interest continues to be in topos theory, modal logic, and quantum logic. --Tim May (.sig for Everything list background) Corralitos, CA. Born in 1951. Retired from Intel in 1986. Current main interest: category and topos theory, math, quantum reality, cosmology. Background: physics, Intel, crypto, Cypherpunks
Re: Good summary of Bogdanov controversy
From: Osher Doctorow [EMAIL PROTECTED], Sunday Nov. 10, 2002 5:45PM Duraid, Well said! I am very happy that some Australians have a sense of humor, which I hadn't realized until now. I know that British and Irish humor are excellent. USA humor varies between the mediocre and the sublime. This reminds me of the last time that I wrote similarly about Creative Genius on the internet to a forum of rather incompetent (mostly) teachers, after which one teacher replied with a hysterical email accusing me of implying that I am a Creative Genius and everybody else is ___ (expletive deleted). Her argument was that teachers are so dedicated and loving and kind and generous and...etc., that to criticize them was tantamount to blasphemy. I hesitated to tell her (and I did not) that expletives deleted as a way of life are more common among the Mediocre than other categories in my opinion. My wife, Marleen J. Doctorow, Ph.D., a licensed clinical psychologist for over 30 years, would be very proud of me if she had any time left after her patients. Oops! Did I imply anything about her? If so, I withdraw my last sentence. : ) Osher - Original Message - From: Duraid Madina [EMAIL PROTECTED] To: [EMAIL PROTECTED] Cc: [EMAIL PROTECTED] Sent: Sunday, November 10, 2002 5:39 PM Subject: Re: Good summary of Bogdanov controversy 1. The acceptance of nonsense for publishing or Ph.D.s or M.A.s or M.S.s is obviously wrong. 4. Mediocre scientific people in my definition don't even have the ability to imitate (see below). Why are you being so hard on yourself?? Tongue firmly pressed against cheek, Duraid
Re: Summary
I guess I'm not too interested in worlds that are not self-observed, but admit their existence for the sake of various arguments. Clearly formal systems without concious entities exist in a mathematical sense - for most purposes, I don't believe it actually matters whether they physically exist. Cheers In a message dated 99-10-05 01:45:10 EDT, Russell Standish writes: Hmm... I would for the most part follow the many perspective interpretation, however I consider that perspectives without conscious observers may also be considered to exist, (in as much as they are self-consistent) in that they may be able to be imagined by conscious observers elsewhere in the plenitude. A perspective world without a conscious observer, seems to be a contradiction in terms. Yet you make the point that such a world can exist in the imagination of an observer elsewhere in the Plenitude. This world then exists or is simulated or dreamed in the observer's mind and is in fact observed by the observer's mind's eye. Is there an identical world out of his mind and in the Plenitude? If there is, we must go back to Leibniz Identity Principle (LIP). Are the world in the mind and the world in the Plenitude one and the same or are they different? If they are the same, then in a sense these worlds are observed by the observer's mind's eye. If they are different then what is the nature of this difference? The difference is not inherent in the worlds themselves. It lies in the presence or absence of a simulating observer, property which is outside these worlds! This is a contradiction. Thus, it appears that the only way out is to accept LIP for this particular case. The other case of a perspective world without a conscious observer, and which does not exist in any observer's mind is definitely a contradiction in terms. Such a world is just portion of the Plenitude which is out of reach of consciousness possibly because its inherent self contradictions prevents consciousness from arising within it or from imagining it. Do such worlds exist? In other words are there portions of the Plenitude which are inaccessible? I think that in this case, the verb to be loses its meaning and I rather not discuss it further. George Levy Dr. Russell StandishDirector High Performance Computing Support Unit, University of NSW Phone 9385 6967 Sydney 2052 Fax 9385 6965 Australia [EMAIL PROTECTED] Room 2075, Red Centre http://parallel.hpc.unsw.edu.au/rks
Re: Summary
Sorry Russell I am not good with acronyms. In a message dated 99-09-30 22:37:01 EDT, you write: the QTI issue. I assume it stands for Quantum Theory Interpretation??? George
