The seven step series
Johnathan Corgan
I think you were off to a good start with your planned series of posts
about the seven step argument. I believe your first installment was a
discussion of set theory as one of the mathematical preliminaries to the
actual argument.
I am looking forward to your next installment.
Regards,
Johnathan Corgan
Bruno Marchal
Well, thanks. I am not sure Kim and Marty are there, but I can provide
a summary, and recall the motivation.
Marty, did you come back from holiday? Kim? still interested in
electronical summer's school on mathematics.
The goal of the seven step thread is to make clear the seventh step of
the UDA (Universal Dovetailer Argument). The purpose of the UDA is to
make clear that the mind-body problem (or the consciousness/reality
problem, or the first person/third person) problem is reduced, when we
do the computationalist assumption, to a pure body appearance or
discourse problem. UDA shows that if we assume the comp. hyp. then we
have to explain the appearance of matter from machine or number self-
reference only. The proof is constructive, it shows *how* the laws of
physics have to be extracted from self-reference.
Later, much later, I could explain, if everyone is OK with UDA, how we
can already extract from self-reference the general shape of physics,
so that we can already refute empirically, or confirm, the comp. hyp.
And it appears that the empirical quantum mechanics, currently,
confirms the comp. hyp. Quantum mechanics confirms the partial
indetermination of the outcomes of our possible experiences, and the
"high non booleanity" of the propositions describing those outcomes".
The object of the "seventh step thread' consists in making the seventh
step accessible to non mathematicians. So we have to start from zero.
I have decided to start from elementary "naive" set theory, without
which we cannot do anything in math. I will avoid all special
mathematical symbols, and use instead words with capital letters.
We have not yet done a lot. So I can sum up, with the new "notations".
Definition. A set is just a "many" considered, when clear enough, as a
"one". So a set is just a collection of objects, and those objects are
called the element, or the member, of the set. If some x is an element
of some set A, we write x BELONGS-TO A, or (x BELONGS-TO A).
A set can be described in extension or in intension. "in extension"
means that we give all elements of the set, enclosed in accolades.
When the set is not to complex (meaning big or infinite), we can use
the "...". We can give name to a set, to ease or talk about that set,
like we do all the times in mathematics. Most of the set we will
consider are set of mathematical object, mainly numbers in the
beginning, and then set of ... sets.
Example-exercise:
1°) Let A be the set {0, 1, 2, 3}. ("A" is said to be a local name for
the set {0, 1, 2, 3}. And local means that such a name is used in a
local context. One paragraph later "A" could designed another, so be
careful). If "A" names {0, 1, 2, 3}, we will write "A = {0, 1, 2, 3}".
OK, so with A = {0, 1, 2, 3}. Which of the following propositions are
true
1) the number 2 is a member of A
2) the number 12 is a member of A
3) the number 12 is not a member of A
4) (3 BELONGS-TO A)
5) all members of A are numbers
6) one element of A is not a number
7) A can be defined in intension in the following way A = {x SUCH-THAT
x is a positive integer little than 4}
2°) Same questions with the set A = {0, 1, 2, 3, ... , 61, 62, 63}
This makes 14 exercises, which should be easy. I intent to keep it
that way. I continue after I get either answers (correct or wrong), or
questions.
Everyone is welcome to participate. Yet, I ask those who are quick to
respect those who are slow. To be slow in the beginning usually help
for being deep in the sequel.
Best,
Bruno
John Mikes
m.a.
I'm still with you but I seriously wonder how far I can follow.
I have the sort of mind that groups of logical statement and propositions
cause to simply shut down. I am more than willing to accept that your proof
is consistent and I assume that others on the list will point out flaws if
there are any. What I would really appreciate would be a prose explanation
of the sequence of ideas that lead to the conclusion that physics can emerge
from and be (in some sense) actualized by math. If Kim and others wish to
continue the exposition of UDA, I'll try to keep up, but I won't ask you to
continue solely on my account. Best wishes,
marty a.
Bruno Marchal
Hi, Brunoyou know that I am in a different mindset, yet happy to read your train of thoughts. I consider a set a limited model of elements (and conclusions thereof are not applicable to wider domains) -
when I read your"A set can be described in extension or in intension. "in extension" means that we give all elements of the set, enclosed in accolades."I was really happy with the next sentence:"When the set is not to complex (meaning big or infinite), we can use the "...". - "(I missed here the exemption of the 'infinite' "set", really a contradiction, to which the 'set' considerations cannot apply - OR can they?
if you have something on that...)
"Many" cannot be infinite (by MY definition).
I loved your words on QM, the (linear) extension of the figment physical world as described in reductionist physical sciences.I also cannot wait for something more about your approach onthe "self reference" - the basis of physics? -
especially as to'self' of what (who)? I hope the answer will not be "machine" or comp, because then I have to continue "and what is that?"
(in more than a utilitarian explanation of what it does). ('it?')What boils down to my ignorance as to the originating and maintaining to ANY action we speak about. The 'theos' of a non-assumed and non-supernatural factor (system?) yet involved in conducting all we just find natural and proceeding.You may substitute 'numbers' for such, but so far did not reply (to my satisfaction at least) WHAT those 'numbers' may be.Sorry, I am not of the religious kind.
*Maybe my error is in 'believeing' that a REALITY may exist and 'we' have only access to part of it.
Inventing for our comfort (the D. Bohmian idea) 'numbers' at the human level of pre-Platonian thinking. If 'reality' exists only by 'comp' or 'consequences' then I may be in a reversed error, due to brainwashing by in college imprinted natural sciences - what I try to exceed yet it still sits there.Our 'perceived reality' (ColinH) may also provide the numbers.
Now that sounds heretical enough in this thread. Forgive me.*Waiting for the self-reference, (who's?)
- with thanks so far
Bruno Marchal
On 30 Jun 2009, at 18:57, m.a. wrote:
>
> Bruno,
> I'm still with you but I seriously wonder how far I can
> follow.
never know.
But I can imagine some shyness for doing so.
>
> I have the sort of mind that groups of logical statement and
> propositions
> cause to simply shut down.
propositions. Your problem is that you don't memorize what you
understand, so, especially after some break, you feel like you could
shut down.
This just means that you have to work a little more, and to stop
building negative self-prejudices, which are most of the time self-
fulfilling.
Now, I obvioulsy cannot ask you to do such a work, as life is rich and
full of consuming time opportunities, noir can I really provide the
motivation for doing so.
I appreciate very much your honesty, even if I suspect the reasons are
bad.
> I am more than willing to accept that your proof
> is consistent and I assume that others on the list will point out
> flaws if
> there are any.
remains to be explained is what is a mathematical computation. This is
not easy, and ask for some familiarity with basic mathematics.
> What I would really appreciate would be a prose explanation
> of the sequence of ideas that lead to the conclusion that physics
> can emerge
> from and be (in some sense) actualized by math.
seventies, before UDA. And I have developed UDA in the late seventies,
so as to provide some help for my friends. Since then I know that UDA
is not so simple, especially the seventh and eighth steps. I can think
about making a shorter attempt.
> If Kim and others wish to
> continue the exposition of UDA, I'll try to keep up, but I won't ask
> you to
> continue solely on my account. Best wishes,
Johnathan?
I know that to play the "candid" role in a public way, you need some
courage, and it is OK to remain silent. But people have to understand
than on those delicate matters, it is not really possible to proceed
without asking question/exercise. We can only continue "the seven step
series" thread if people provide answer to the exercises.
I will nevertheless try to explain things in a more "journalistic",
and shorter, way. The problem is that such a thing can easily be
misinterpreted. I will think about it. The "real thing" is more easy
to explain, but admittedly longer when we start from quasi zero.
Bruno
m.a.
Your faith in my ability to master the logic is encouraging. I
remain determined to wrestle with the steps in whichever form in which you
decide to present them. Tally ho,
m.a.
----- Original Message -----
From: "Bruno Marchal" <mar...@ulb.ac.be>
To: <everyth...@googlegroups.com>
Sent: Wednesday, July 01, 2009 12:00 PM
Subject: Re: The seven step series
I know that to play the "candid" role in a public way, you need some
courage, and it is OK to remain silent. But people have to understand
than on those delicate matters, it is not really possible to proceed
without asking question/exercise. We can only continue "the seven step
series" thread if people provide answer to the exercises.
I will nevertheless try to explain things in a more "journalistic",
and shorter, way. The problem is that such a thing can easily be
misinterpreted. I will think about it. The "real thing" is more easy
to explain, but admittedly longer when we start from quasi zero.
Bruno
m.a.
> From: "Bruno Marchal" <mar...@ulb.ac.be>
> To: <everyth...@googlegroups.com
> Sent: Tuesday, June 30, 2009 6:45 AM
> Subject: Re: The seven step series
>
>
>
> Hi Johnathan,
>
>
> 2) the number 12 is a member of A False
> 3) the number 12 is not a member of A True
> 4) (3 BELONGS-TO A) True: but you haven't told us whether the parenthesis cancels the locality of brackets.
> 5) all members of A are numbers True
> 6) one element of A is not a number False: we've established that zero is a number.
> 7) A can be defined in intension in the following way A = {x SUCH-THAT
>
> 2°) Same questions with the set A = {0, 1, 2, 3, ... , 61, 62, 63}
John Mikes
(JM)"Many" cannot be infinite (by MY definition).(BM):Well, I prefer to use the word in their most used and standard sense.-----which one is that? Can, or cannot?-----I did not identify "self-referenced" with computationalist. And that 'many' think so is no satisfying evidence in my view: scientific thinking is not a democratic voting formula. Even the "BIG" names... Al Gore and Jimmy Carter received Nobels.There is no "scientific" statement which could not be contrasted by two opposite ones of 'other' scientists.
--------------------------------------------------------------------------
Bruno Marchal
Hi Bruno,I'm responding to the quiz (see below). What does "high non booleanity" mean in the context of para.2?
>
> Example-exercise:
>
> 1°) Let A be the set {0, 1, 2, 3}. ("A" is said to be a local name for
> the set {0, 1, 2, 3}. And local means that such a name is used in a
> local context. One paragraph later "A" could designed another, so be
> careful). If "A" names {0, 1, 2, 3}, we will write "A = {0, 1, 2, 3}".
>
> OK, so with A = {0, 1, 2, 3}. Which of the following propositions are
> true
>
> 1) the number 2 is a member of A True
> 2) the number 12 is a member of A False
> 3) the number 12 is not a member of A True
> 4) (3 BELONGS-TO A) True: but you haven't told us whether the parenthesis cancels the locality of brackets.
> 5) all members of A are numbers True
> 6) one element of A is not a number False: we've established that zero is a number.
> 7) A can be defined in intension in the following way A = {x SUCH-THAT
> x is a positive integer little than 4} True...if zero is considered a positive integer.
>
> 2°) Same questions with the set A = {0, 1, 2, 3, ... , 61, 62, 63}1. True2. True3. False4. True: same question as 4 above.5. True6. False: zero is a number7. False
m.a.
Could you tell me if you understand and/or remember those definitions (where a and b denoting arbitrary sets):(a INTERSECTION b) = {x SUCH-THAT (x BELONGS-TO a) and (x BELONGS-TO b)}(a UNION b) = {x SUCH THAT (x BELONGS-TO a) or (x BELONGS-TO b)}Can you compute
{1, 2, 7, 789} UNION {1, 2, 7, 5678} = ? 1,2,7,789, 5678{1, 2, 7, 789} INTERSECTION {1, 2, 7, 5678} = ? 1, 2, 7, 789
Do you remember the empty set? Can you compute:
{1, 2} UNION { } = ? 1,2{1} UNION { } = ? { }{1, 2, 3} UNION {1, 2, 3} = ? 1,2,3{ } UNION { } = ? { }
{1, 2} INTERSECTION { } = ? { }{1} INTERSECTION { } = ? { }
{1, 2, 3} INTERSECTION {1, 2, 3} = ? 1, 2 3{ } INTERSECTION { } = ? { }
Now, an important distinction which will follow us through ... forever. I suggest you read attentively the next two paragraphs two times before breakfast, every day for one week. :), Really take all your time. It concerns the notion of operation, and relation.INTERSECTION and UNION, are operations on sets, like addition (+, or PLUS) and multiplication (*, or TIMES) are operation on numbers. This means, typically, that, if x and y denote numbers, then x + y, and x * y, will denote, or are equal to, numbers. For example 3 + 4 is equal to 7.Similarly, if x and y denotes, or are equal, to sets, then x INTERSECTION y denotes, or is equal to, some set. For example {1,2} INTERSECTION {2, 7} is equal to some set, actually the set {2}. OK?Operations are important, as you can guess, but relations are as well important. Operations lead to new elements, new objects. From the numbers 2 and 3, you get the element 5. Relations pertains or does not pertain, or equivalently, leads to true or false.Example. The relation LESS-THAN, among the numbers. (x LESS-THAN y) is true if x is less than y. So (3 LESS-THAN 56) is true, and (56 LESS-THAN 3) is false. An important relation pertaining on sets is the relation of inclusion, or of being a subset of a set.By definition a set x will be said included in y (or be said subset of y), when all the elements of x are among the elements of y. We will write (x INCLUDED-IN y) when the set x is included in the set y.For example, the set {1, 2} is included in the set {3, 2, 1}, but is not included in the set {3, 1}.Exercise: in the following, what is true or false?
45 LESS-THAN 67 true0 LESS-THAN 1 true999 LESS-THAN 4 false{1, 2, 3} INCLUDED-IN {4, 1, 5, 2, 3, 8} true{1} INCLUDED-IN {1, 2} true
Bruno Marchal
Could you tell me if you understand and/or remember those definitions (where a and b denoting arbitrary sets):(a INTERSECTION b) = {x SUCH-THAT (x BELONGS-TO a) and (x BELONGS-TO b)}(a UNION b) = {x SUCH THAT (x BELONGS-TO a) or (x BELONGS-TO b)}Can you compute{1, 2, 7, 789} UNION {1, 2, 7, 5678} = ? 1,2,7,789, 5678
{1, 2, 7, 789} INTERSECTION {1, 2, 7, 5678} = ? 1, 2, 7, 789
Do you remember the empty set? Can you compute:{1, 2} UNION { } = ? 1,2
{1} UNION { } = { }
{1, 2, 3} UNION {1, 2, 3} = ? 1,2,3
{ } UNION { } = ? { }
{1, 2} INTERSECTION { } = ? { }
{1} INTERSECTION { } = ? { }
{1, 2, 3} INTERSECTION {1, 2, 3} = ? 1, 2 3
{ } INTERSECTION { } = ? { }
Now, an important distinction which will follow us through ... forever. I suggest you read attentively the next two paragraphs two times before breakfast, every day for one week. :), Really take all your time. It concerns the notion of operation, and relation.INTERSECTION and UNION, are operations on sets, like addition (+, or PLUS) and multiplication (*, or TIMES) are operation on numbers. This means, typically, that, if x and y denote numbers, then x + y, and x * y, will denote, or are equal to, numbers. For example 3 + 4 is equal to 7.Similarly, if x and y denotes, or are equal, to sets, then x INTERSECTION y denotes, or is equal to, some set. For example {1,2} INTERSECTION {2, 7} is equal to some set, actually the set {2}. OK?Operations are important, as you can guess, but relations are as well important. Operations lead to new elements, new objects. From the numbers 2 and 3, you get the element 5. Relations pertains or does not pertain, or equivalently, leads to true or false.Example. The relation LESS-THAN, among the numbers. (x LESS-THAN y) is true if x is less than y. So (3 LESS-THAN 56) is true, and (56 LESS-THAN 3) is false. An important relation pertaining on sets is the relation of inclusion, or of being a subset of a set.By definition a set x will be said included in y (or be said subset of y), when all the elements of x are among the elements of y. We will write (x INCLUDED-IN y) when the set x is included in the set y.For example, the set {1, 2} is included in the set {3, 2, 1}, but is not included in the set {3, 1}.Exercise: in the following, what is true or false?45 LESS-THAN 67 true
0 LESS-THAN 1 true
999 LESS-THAN 4 false
{1, 2, 3} INCLUDED-IN {4, 1, 5, 2, 3, 8} true
{1} INCLUDED-IN {1, 2} true
oops, I must go. You are lucky ;)
Bruno Marchal
On 02 Jul 2009, at 00:22, John Mikes wrote:
> I don't deny the practicality of applying 'numbers-based' science in
> sending a man to Mars, but it is NOT the numbers that does the job.
> It is the complexity of the state of the art we reached, which
> includes science, technology, skills, ideas AND of course numbers-
> application. Bohm's idea - as I understood it - was that searching
> nature, you do not bounce into numbers, you can observe 3-leaf or
> 4legged and manyshaped things, big and small, YOU (the human) can
> 'count them' if you invented the symbols 1 2 3 4 etc. but these
> refer to quantities and it required lots of abstracting in mental
> evolution to arrive in a numbers-based math - how humans think about
> nature.
I know well that theory. It is based on the idea that some primary
Nature exists. A common "superstition" among christians and atheists.
Which could be true, actually. I don't know.
But what I am almost completely sure, is that if comp is true, then it
is has to be supersitution. And that is what I try to explain.
> Thanks again and my mind works in crooked ways, if you can excuse me
> for that. It seems I need too much learning to catch up.
You are welcome. If you have the time and courage, I really encourage
you to follow the thread. You may be surprised ... soon!
Bruno
m.a.
----- Original Message -----From: Bruno MarchalSent: Thursday, July 02, 2009 1:44 PMSubject: Re: The seven step series
You are quick!
On 02 Jul 2009, at 18:42, m.a. wrote:
Could you tell me if you understand and/or remember those definitions (where a and b denoting arbitrary sets):(a INTERSECTION b) = {x SUCH-THAT (x BELONGS-TO a) and (x BELONGS-TO b)}(a UNION b) = {x SUCH THAT (x BELONGS-TO a) or (x BELONGS-TO b)}Can you compute{1, 2, 7, 789} UNION {1, 2, 7, 5678} = ? 1,2,7,789, 5678Almost OK.{1, 2, 7, 789} UNION {1, 2, 7, 5678} = {1,2,7,789, 5678}.Don't forget the accolades, which means that you have as result the SET {1,2,7,789, 5678}
{1, 2, 7, 789} INTERSECTION {1, 2, 7, 5678} = ? 1, 2, 7, 789Not correct. To belong to A INTERSECTION B, the element must belong to A, *and* must belong to B. 1, 2 and 7 does belong indeed to A and to B, in this case, with A = {1, 2, 7, 789}, and B = {1, 2, 7, 5678}), but neither 789, nor 5678 do belong to both A and B.So {1, 2, 7, 789} INTERSECTION {1, 2, 7, 5678} = {1, 2, 7}
Just tell me if you agree. I agree and can't understand how I could have been so careless.
Do you remember the empty set? Can you compute:{1, 2} UNION { } = ? 1,2
OK, but don't forget the accolades. Are accolades brackets?
{1, 2} UNION { } = ? {1,2}{1} UNION { } = { }You are too quick here, you forget to type the 1.
{1} UNION { } = {1 } Yes, I mistook the {1} for the number of the question...not part of the equation. I tend to overlook the fine points.
{1, 2, 3} UNION {1, 2, 3} = ? 1,2,3OK (my mind adds the accolades)
{ } UNION { } = ? { }Very good. You could eliminate the "?".
{1, 2} INTERSECTION { } = ? { }Excellent.
{1} INTERSECTION { } = ? { }Bravo.
{1, 2, 3} INTERSECTION {1, 2, 3} = ? 1, 2 3Exact. (well, I continue to add the accolades, and eliminate the "?")
{ } INTERSECTION { } = ? { }Exact. In this case you see how much it is important to not forget the accolades!
Now, an important distinction which will follow us through ... forever. I suggest you read attentively the next two paragraphs two times before breakfast, every day for one week. :), Really take all your time. It concerns the notion of operation, and relation.INTERSECTION and UNION, are operations on sets, like addition (+, or PLUS) and multiplication (*, or TIMES) are operation on numbers. This means, typically, that, if x and y denote numbers, then x + y, and x * y, will denote, or are equal to, numbers. For example 3 + 4 is equal to 7.
Similarly, if x and y denotes, or are equal, to sets, then x INTERSECTION y denotes, or is equal to, some set. For example {1,2} INTERSECTION {2, 7} is equal to some set, actually the set {2}. OK?......No!Why not the sets {1,2,7} if INTERSECTION means BOTH?
Operations are important, as you can guess, but relations are as well important. Operations lead to new elements, new objects. From the numbers 2 and 3, you get the element 5. Relations pertains or does not pertain, or equivalently, leads to true or false.Example. The relation LESS-THAN, among the numbers. (x LESS-THAN y) is true if x is less than y. So (3 LESS-THAN 56) is true, and (56 LESS-THAN 3) is false. An important relation pertaining on sets is the relation of inclusion, or of being a subset of a set.By definition a set x will be said included in y (or be said subset of y), when all the elements of x are among the elements of y. We will write (x INCLUDED-IN y) when the set x is included in the set y.For example, the set {1, 2} is included in the set {3, 2, 1}, but is not included in the set {3, 1}.Exercise: in the following, what is true or false?45 LESS-THAN 67 true
OK.
0 LESS-THAN 1 trueOK.
999 LESS-THAN 4 falseOK.
{1, 2, 3} INCLUDED-IN {4, 1, 5, 2, 3, 8} trueOK.
{1} INCLUDED-IN {1, 2} trueOK.
oops, I must go. You are lucky ;)
I'm back! I give you two last exercises to ponder about, just in case of insomnia. Again, take your time. I hope Kim follows, and does not look at the solution !1°) In the two relational formula below, one is true, the other is false. Which one are what?
a) { } INCLUDED-IN { }Trueb) { } BELONGS-TO { } True
2°) And I give you a slightly longer exercise. Can you give me all the subsets of the set {1, 2} ?. That is, can you give me all the sets which are included in the set {1, 2} ? In case of doubt, reread the definitions, reread the examples, and never panic! I give you a hint: the set {1, 2} has four subsets. Can you find them?
{1} {2} {1,2} {2,1} why not {3} ?
Bruno Marchal
Bruno,Comments and questions are interspersed below.marty
Just tell me if you agree. I agree and can't understand how I could have been so careless.
Do you remember the empty set? Can you compute:{1, 2} UNION { } = ? 1,2OK, but don't forget the accolades. Are accolades brackets?
You are too quick here, you forget to type the 1.{1} UNION { } = {1 } Yes, I mistook the {1} for the number of the question...not part of the equation. I tend to overlook the fine points.
Now, an important distinction which will follow us through ... forever. I suggest you read attentively the next two paragraphs two times before breakfast, every day for one week. :), Really take all your time. It concerns the notion of operation, and relation.INTERSECTION and UNION, are operations on sets, like addition (+, or PLUS) and multiplication (*, or TIMES) are operation on numbers. This means, typically, that, if x and y denote numbers, then x + y, and x * y, will denote, or are equal to, numbers. For example 3 + 4 is equal to 7.Similarly, if x and y denotes, or are equal, to sets, then x INTERSECTION y denotes, or is equal to, some set. For example {1,2} INTERSECTION {2, 7} is equal to some set, actually the set {2}. OK?......No!Why not the sets {1,2,7} if INTERSECTION means BOTH?
I'm back! I give you two last exercises to ponder about, just in case of insomnia. Again, take your time. I hope Kim follows, and does not look at the solution !1°) In the two relational formula below, one is true, the other is false. Which one are what?a) { } INCLUDED-IN { }True
b) { } BELONGS-TO { } True
2°) And I give you a slightly longer exercise. Can you give me all the subsets of the set {1, 2} ?. That is, can you give me all the sets which are included in the set {1, 2} ? In case of doubt, reread the definitions, reread the examples, and never panic! I give you a hint: the set {1, 2} has four subsets. Can you find them?{1} {2} {1,2} {2,1} why not {3} ?
Brent Meeker
> ...
> Due to Dirac, in Quantum Mechanics, I tend to believe that brackets
> are "<" and ">". parentheses are "(" and ")". I call "{" and "}"
> accolades, but perhaps they are called bracket. The terms are not
> important as far as we understand each other. How would you call "["
> and "]" ?
I've never seen "{" and "}" denoted "accolades" but I like it; they are
more commonly called "braces". I don't know a specific term for "["
and "]", I generally refer to them as "square brackets".
Brent
m.a.
For example {1,2} INTERSECTION {2, 7} is equal to some set, actually the set {2}. OK?......No!
Why not the sets {1,2,7} if INTERSECTION means BOTH?Ah, but the word "both" alone is ambiguous. You could say that the UNION of two sets is the merging of BOTH set, and the intersection is the given of the elements which are in both set. So the union of {1, 2} and {2, 7} is {1, 2, 7}, which indeed merges BOTH sets. But for computing the intersection, you must ask yourself, does this *element* belongs to BOTH set? So, for the intersection of {1, 2} and {2, 7}, you have to ask yourself the following question: does 1 belong to both set? well, the answer is NO. the 1 belongs to the first set but not to the second, and so 1 does not belong to the intersection. Does 2 belongs to both sets? The answer is yes. 2 belongs to {1, 2} and 2 belongs to {2, 7}. Does 7 belongs to both sets, the answer is no, 7 belongs to the second set, but does not belong to the first set, so 7 is not in the intersection.Tell me if you are OK with this.
Not OK. You previously defined UNION as one OR the other. Now you seem to be giving me the same definition for INTERSECTION.
Is the missing subset { } ?
Bruno Marchal
New comments in italics.For example {1,2} INTERSECTION {2, 7} is equal to some set, actually the set {2}. OK?......No!Why not the sets {1,2,7} if INTERSECTION means BOTH?Ah, but the word "both" alone is ambiguous. You could say that the UNION of two sets is the merging of BOTH set, and the intersection is the given of the elements which are in both set. So the union of {1, 2} and {2, 7} is {1, 2, 7}, which indeed merges BOTH sets. But for computing the intersection, you must ask yourself, does this *element* belongs to BOTH set? So, for the intersection of {1, 2} and {2, 7}, you have to ask yourself the following question: does 1 belong to both set? well, the answer is NO. the 1 belongs to the first set but not to the second, and so 1 does not belong to the intersection. Does 2 belongs to both sets? The answer is yes. 2 belongs to {1, 2} and 2 belongs to {2, 7}. Does 7 belongs to both sets, the answer is no, 7 belongs to the second set, but does not belong to the first set, so 7 is not in the intersection.Tell me if you are OK with this.Not OK. You previously defined UNION as one OR the other. Now you seem to be giving me the same definition for INTERSECTION.
2°) And I give you a slightly longer exercise. Can you give me all the subsets of the set {1, 2} ?. That is, can you give me all the sets which are included in the set {1, 2} ? In case of doubt, reread the definitions, reread the examples, and never panic! I give you a hint: the set {1, 2} has four subsets. Can you find them?{1} {2} {1,2} {2,1} why not {3} ?Not too bad. 3/4 correct:{1} is included in {1, 2}. Indeed.{2} is included in {1, 2}. Indeed.{1, 2} is included in {1, 2}. Indeed.{2, 1} is included in {1, 2}. Indeed, that is true, but you have to remember what you have already agree on: the set {1, 2} is equal to the set {2, 1}, so this is not a new solution. It is the preceding one in disguised!Why not {3}? {3} is not included in {1, 2} just because 3 does not belong to {1, 2}. Reread the definition of inclusion. A is included in B if all the elements of A belongs to B. OK?So you have found three subsets, among the four. Reading today's explanations I think you could find the missing subset. I let you search a little bit.So just one exercise: what is the missing subset?Is the missing subset { } ?
m.a.
Here we met a set of sets.The set of subsets of a set, can only be, of course, a set of sets. The set {2, 21, 14} is a set of numbers. The set { { }, {4, 78, 56} } is a set of sets. It has two elements: the empty set {}, and the set of numbers {4, 78, 56}. Do not confuse a number, like 24, and a set, like {24}, which is a set having a number has elements. In particular it is the case that {4, 78, 56} belongs to { { }, {4, 78, 56} }. Take it easy, and meditate on the following exercise:Which of the following are true
{3, 5} included-in {3, 5} True
Bruno Marchal
Bruno,Can you provide definitions of "belongs-to" and "included-in" that distinguish them from "union" and "intersection"?
Here we met a set of sets.The set of subsets of a set, can only be, of course, a set of sets. The set {2, 21, 14} is a set of numbers. The set { { }, {4, 78, 56} } is a set of sets. It has two elements: the empty set {}, and the set of numbers {4, 78, 56}. Do not confuse a number, like 24, and a set, like {24}, which is a set having a number has elements. In particular it is the case that {4, 78, 56} belongs to { { }, {4, 78, 56} }. Take it easy, and meditate on the following exercise:Which of the following are true{3, 5} included-in {3, 5} True
{3, 5} belongs-to {3, 5}{3, 5} included-in { {3, 5} }{3, 5} belongs-to { {3, 5} }Take your time,Bruno
m.a.
Bruno,Can you provide definitions of "belongs-to" and "included-in" that distinguish them from "union" and "intersection"?
Here we met a set of sets.The set of subsets of a set, can only be, of course, a set of sets. The set {2, 21, 14} is a set of numbers. The set { { }, {4, 78, 56} } is a set of sets. It has two elements: the empty set {}, and the set of numbers {4, 78, 56}. Do not confuse a number, like 24, and a set, like {24}, which is a set having a number has elements. In particular it is the case that {4, 78, 56} belongs to { { }, {4, 78, 56} }. Take it easy, and meditate on the following exercise:Which of the following are true{3, 5} included-in {3, 5} True
{3, 5} belongs-to {3, 5} True{3, 5} included-in { {3, 5} } False{3, 5} belongs-to { {3, 5} } True
Bruno Marchal
My answers. m.a.
Here we met a set of sets.The set of subsets of a set, can only be, of course, a set of sets. The set {2, 21, 14} is a set of numbers. The set { { }, {4, 78, 56} } is a set of sets. It has two elements: the empty set {}, and the set of numbers {4, 78, 56}. Do not confuse a number, like 24, and a set, like {24}, which is a set having a number has elements. In particular it is the case that {4, 78, 56} belongs to { { }, {4, 78, 56} }. Take it easy, and meditate on the following exercise:Which of the following are true{3, 5} included-in {3, 5} True
{3, 5} belongs-to {3, 5} True
{3, 5} included-in { {3, 5} } False
{3, 5} belongs-to { {3, 5} } True
m.a.
----- Original Message -----From: Bruno MarchalSent: Monday, July 06, 2009 12:14 PMSubject: Re: The seven step series
On 06 Jul 2009, at 16:12, m.a. wrote (in bold):
My answers. m.a.Here we met a set of sets.The set of subsets of a set, can only be, of course, a set of sets. The set {2, 21, 14} is a set of numbers. The set { { }, {4, 78, 56} } is a set of sets. It has two elements: the empty set {}, and the set of numbers {4, 78, 56}. Do not confuse a number, like 24, and a set, like {24}, which is a set having a number has elements. In particular it is the case that {4, 78, 56} belongs to { { }, {4, 78, 56} }. Take it easy, and meditate on the following exercise:Which of the following are true{3, 5} included-in {3, 5} True
OK.
{3, 5} belongs-to {3, 5} True
Not OK. The elements of {3, 5} are 3 and 5. {3, 5} is not an *element* of {3, 5}.Why not? They look like elements to me. Please define "elements" as applies to this example..
Ask in case you are not OK with this, of course.
{3, 5} included-in { {3, 5} } FalseOK. Very good.
{3, 5} belongs-to { {3, 5} } TrueOK. {3, 5} is even the *only* element of { {3, 5} }No exercise today. Just a question, a suggestion, and a plan.The question is: have you the feeling to learn something?The suggestion: I think the best way to answer the preceding question consists in trying to explain what you learn to someone else. It is the best way to see if you remember and understand the definition. You could try to explain what you learn to some gentle "victim" in your neighborhood (wife, friend, child, parent, ...).I give you a plan, and some more motivation. To get the seventh step in some proper way, there is a need to understand the mathematical notion of "universal machine".
I've read about Turing machines if that's what you're referring to.
For this I need to explain what is a computable function. For this I need to explain what is a function, and for this I need to explain what is a set, given that functions can more easily be explained through sets relating sets. Once you will have a good grip of what is a universal machine, or what is a universal number, and what really means "universal", we will be able to tackle the notion of universal dovetailing, and especially the "mathematical universal dovetailing" (which is really important for the whole approach, and for the step eight). I am hesitating to work quickly on the notion of function, or to do some pieces of number theory and geometry to provide examples before.As I said recently to John, the discovery of the notion universal machine is one of the most astonishing and gigantic discovery made by the humans, and what I do is just an exploitation of that discovery. Universes, cells, brains and computers are example of universal machine, and the notion of universal machine are a key to understand why eventually, once we say "yes to the doctor", and believe we can survive "qua computatio", we have to redefine physics as an invariant for the permutation of all possible observers, and how physics can be recovered from an invariant among all universal machines point-of-views ...Feel free to slow me down, or to accelerate me, and to ask any question at whichever level of details you want. Feel free to ask any question that you have already asked.Have a good day, and thanks for your effort and seriousness,BrunoPS. It should be obvious for everyone that if there are still questions, critics, objections, problems, feeling of dizziness, whatever, with the first six steps of the UDA, please, feel free to ask. And people should not hesitate to discuss other everything-like subject, I don't want to monopolize the list of course. But the UDA reasoning really changes the perspective on all possible TOEs, so I will feel free myself to point on UDA on each discussion where I find it relevant (of course also).
Bruno Marchal
Questions and comments interspersed below (in bold)
{3, 5} belongs-to {3, 5} True
Not OK. The elements of {3, 5} are 3 and 5. {3, 5} is not an *element* of {3, 5}.Why not? They look like elements to me. Please define "elements" as applies to this example..
I give you a plan, and some more motivation. To get the seventh step in some proper way, there is a need to understand the mathematical notion of "universal machine".I've read about Turing machines if that's what you're referring to.
m.a.
----- Original Message -----From: Bruno MarchalSent: Tuesday, July 07, 2009 3:30 AMSubject: Re: The seven step series
Bruno Marchal
Thanks, Bruno. I think I've got it now. Sorry to be such a slow learner.marty
m.a.
----- Original Message -----From: Bruno MarchalSent: Tuesday, July 07, 2009 2:07 PMSubject: Re: The seven step series
On 07 Jul 2009, at 16:18, m.a. wrote:
Bruno Marchal
----- Original Message -----From: Bruno MarchalSent: Tuesday, July 07, 2009 2:07 PMSubject: Re: The seven step seriesOn 07 Jul 2009, at 16:18, m.a. wrote:Bruno,I'm not entirely sure of these answers, but I think I learn more from your corrections than from pondering the rules to the point where I confuse myself. m.a.
Do you remember, I asked you to give me all the subsets of {1, 2}. That is, all the sets which are included in {1, 2}. You gave me the correct answer: those subsets are { }, {1}, {2}, {1, 2}. You see that the set {1, 2} has 2 elements, and 4 subsets. But then I asked to give me the set of all subsets of {1, 2}.{1, 2} has four subsets, and it is natural to make that many a one, by considering *the* set of all subsets of {1, 2}. The answer is:{{ }, {1}, {2}, {1, 2}}Considering all subsets of a set is a rather important operation, which we will meet more than one times in the sequel. Given its importance mathematicians gave it a name. It is the power operation. Later I will be able to explain why it is called power.It is an UNARY operation, which means it applies on ONE set. (Intersection, and union are BINARY operations, they need two sets to work on).So (power x) = {y such-that y is included in x}, by definition.For example:(power {1, 2}) = {{ }, {1}, {2}, {1, 2}}Here are the three promised exercises. Compute(power {1}) = ? {{ }, {1}}
(power { }) = ? {{ }}
(power {1, 2, 3}) = ? {{ }, {1}, {2}, {3}}
m.a.
(power {1, 2, 3}) = ? {{ }, {1}, {2}, {3}, {1,2}, {2,3}, {1,2,3}}
.And I give you a little subject research: if a set x has n elements, how many elements are in (power x)?
How's this for a wild guess? I have a feeling that it's missing the accolades, but I have no idea where to put them.(power {x}) = n(n-1) (n-2)...(n-x+1)x!Bruno
Bruno Marchal
Second try:(power {1, 2, 3}) = ? {{ }, {1}, {2}, {3}, {1,2}, {2,3}, {1,2,3}}
And I give you a little subject research: if a set x has n elements, how many elements are in (power x)?How's this for a wild guess? I have a feeling that it's missing the accolades, but I have no idea where to put them
(power {x}) = n(n-1) (n-2)...(n-x+1)x!
m.a.
----- Original Message -----From: Bruno MarchalSent: Wednesday, July 08, 2009 1:31 PMSubject: Re: The seven step series
On 08 Jul 2009, at 15:43, m.a. wrote:
Second try:(power {1, 2, 3}) = ? {{ }, {1}, {2}, {3}, {1,2}, {2,3}, {1,2,3}}
Third try:= {{ }, {1}, {2}, {3}, {1,2}, {2,3}, {1,2,3}, {{ },1,2,3}}
Bruno Marchal
Here's my third try. I'll continue working on the (power x) problem. m.a.----- Original Message -----From: Bruno MarchalSent: Wednesday, July 08, 2009 1:31 PMSubject: Re: The seven step seriesOn 08 Jul 2009, at 15:43, m.a. wrote:Second try:(power {1, 2, 3}) = ? {{ }, {1}, {2}, {3}, {1,2}, {2,3}, {1,2,3}}Third try:= {{ }, {1}, {2}, {3}, {1,2}, {2,3}, {1,2,3}, {{ },1,2,3}}
I gave you the hint that there are 8 elements. Let us count:The empty set { } ..................................1Three singletons {1}, {2}, {3}................3Two doubletons {1,2 }, {2,3 }................2The biggest subset {1,2,3}..................11 + 3 + 2 + 1 = 7A subset is missing! Can you see which one?
Bruno Marchal
I suddenly feel sorry putting too much burden on just one
correspondent in the list, and I would appreciate if someone else
could propose answers or any remarks to the exercises.
I am also a bit anxious about Kim, who is the one who suggested me the
initial explanations, but who seems to have disappear right now.
There is also some sort of burden onto me, because it is hard to
explain "the real thing" concerning the seventh step, without
explaining or just illustrating at least some relevant portion of the
mathematical reality: mainly the unexpected mathematical discovery of
the universal functions, sets, numbers, systems, language, machine ...
I don't mention the absence of drawing ability which does not help.
Given that the list raised from a critical approach toward Tegmark and
Schmidhuber, I was usually assuming some knowledge in math and
physics. What was harder for me in the beginning was to motivate the
use of "philosophy of mind" notions, notably the key distinction
between the first and third person point of view. Then UDA should make
you realize how non obvious the relation between the first person and
the third person can be once we assume comp (= work in the theory
comp). My original goal was to illustrate that once we assume digital
mechanism, we can build a "scientific formulation" of the mind-body
problem or the consciousness-reality problem. We probably depart from
Tegmark and Schmidhuber, or Wolfram, by taking into account that
making comp explicit entails a delocalisation of the 1-person
relatively to the third person computations, and makes the identity
thesis, a most complex equivalence relations.
The knowledge of most people participating to the discussion is very
varied, due to the extreme transdiciplinarity of the subject, and the
interest it can evidently have for the layman (and indeed, for any
universal machine).
Marty asked me to make an attempt toward a "journalistic" description
of "how physics has to become part of number theory". This is very
difficult, and risky due to inevitable misunderstanding.
And I feel like I have to explain in what deep sense the mathematical
discovery of the "universal machine", made by Post, Turing, ... is
already a quite utterly astonishing, yet subtle, discovery. Gödel
himself took time to swallow it and he described Church thesis as an
epistemological miracle.
My intention was to derive properly Cantor theorem, and then Kleene
theorem, which was the object of my old "diagonalization posts".
I feel important that people understand how unbelievable Church thesis
is, and why most startling propositions, including incompleteness, are
easy consequences of it.
Typically I am happy to share my enthusiasm about all theorems in
computer science which leads to the reversal, but knowing myself I
know that I could accelerate too much and makes too much burden for
the correspondent especially if he is alone.
So before becoming an harasser myself I invite Marty to let other
people trying to answer the exercises.
Marty has fully agreed to this proposal and is happy the pressure is
off him to represent all those who are following anonymously.
Eventually I can show the solution and proceed in addressing the post
to everyone.
Note that this is what I have done with the combinators, feedback were
made out-of-line, then. But this lead to difficulties too. I cannot
solve all the exercise out-of-line. By experience this ends up with
finding myself writing too many posts with almost the same info to
different people.
Yet I can imagine how much it is to be the only public target of what
could look like an perpetual exam, and I really want to proceed in a
cooler way.
Some people have encouraged me, out-of-line, to proceed, but now I
think they should participate a little bit, if only to witness they
are following the thread. I will probably stop to propose "easy" (a
quite relative notion) exercise, but then it is important to stop me
once anything is unclear. This is a problem with math, if you miss a
piece, everything becomes senseless.
Understanding implies some self-implication in the reasoning. So,
either someone else try to participate, or I continue impersonally and
eventually I will try some "non technical summary". I recall that one
of the goal consists in explaining the difference between a
computation and a description of a computation (beyond just doing the
step 7).
Any remark to improve the communication or to design a better
methodology is welcome,
Best regards to all of you, and thanks for letting me know your
interests,
Bruno
Johnathan Corgan
> I suddenly feel sorry putting too much burden on just one
> correspondent in the list, and I would appreciate if someone else
> could propose answers or any remarks to the exercises.
Bruno--you're doing great. I think it is the case where silence means
"I understand, continue", rather than disinterest.
> There is also some sort of burden onto me, because it is hard to
> explain "the real thing" concerning the seventh step, without
> explaining or just illustrating at least some relevant portion of the
> mathematical reality: mainly the unexpected mathematical discovery of
> the universal functions, sets, numbers, systems, language, machine ...
> I don't mention the absence of drawing ability which does not help.
The derivation of your thesis from first principles is a very compelling
idea. The somewhat startling and unorthodox conclusions you espouse are
bound to cause confusion unless all their underpinnings are well
understood. The arguments from others then can have a much more
specific target than the top-level conclusions; instead they will come
out earlier in the derivation process and at the time of introduction of
the controversial subject.
> The knowledge of most people participating to the discussion is very
> varied, due to the extreme transdiciplinarity of the subject, and the
> interest it can evidently have for the layman (and indeed, for any
> universal machine).
While I do have training in math and physics, I still benefit from your
targeting the motivated layman. Personally, I'm not interested in doing
the exercises on the list, but they are still useful to check my
understanding.
> Best regards to all of you, and thanks for letting me know your
> interests,
By all means, proceed. Personally, if I don't understand something or
have an objection, you'll hear about it on the list, but I think you
should take silence as assent.
Johnathan Corgan
John Mikes
Bruno Marchal
On 10 Jul 2009, at 22:52, Johnathan Corgan wrote:
>
> On Fri, 2009-07-10 at 22:24 +0200, Bruno Marchal wrote:
>
>> I suddenly feel sorry putting too much burden on just one
>> correspondent in the list, and I would appreciate if someone else
>> could propose answers or any remarks to the exercises.
>
> Bruno--you're doing great. I think it is the case where silence means
> "I understand, continue", rather than disinterest.
Well, thanks, OK, perhaps.
>
>
>> There is also some sort of burden onto me, because it is hard to
>> explain "the real thing" concerning the seventh step, without
>> explaining or just illustrating at least some relevant portion of the
>> mathematical reality: mainly the unexpected mathematical discovery of
>> the universal functions, sets, numbers, systems, language,
>> machine ...
>> I don't mention the absence of drawing ability which does not help.
>
> The derivation of your thesis from first principles is a very
> compelling
> idea. The somewhat startling and unorthodox conclusions you espouse
> are
> bound to cause confusion unless all their underpinnings are well
> understood.
There are two things. Understanding the conclusions, and understanding
how we get to them.
Many variations are possible in between are possible for varied
audience.
> The arguments from others then can have a much more
> specific target than the top-level conclusions; instead they will come
> out earlier in the derivation process and at the time of
> introduction of
> the controversial subject.
But what is controversial? I have never heard about something
controversial seen in the reasoning. The conclusion are astonishing,
and certainly annoying for someone who believes "religiously" in both
physicalism and digital mechanism.
The subject matter was controversial a long time ago, but today, it is
no more, I think. Well, it depends on which circle. That something
appears in the academy (like studies on consciousness, does not mean
that all academicians understand the questioning there, alas).
I have heard that the first person indeterminacy, which is my mean
early contribution, is controversial, but I have never seen any
controversy on it, just sometimes, some discussion on the vocabulary
or definition, which does not change any conclusion.
The subject matter is difficult, so it easier for the "religious"
people (like convinced atheists, to be clear) to speculate about some
difficulties they don't even try to single out.
I proceed in the scientific way, which means that I just ask
questions, and anyone can verify what follows from what, or interrupt
and present an objection. Up to now, none of the "real" objections
presented were fatal, and eventually those reduce also to a problem of
vocabulary.
>
>
>> The knowledge of most people participating to the discussion is very
>> varied, due to the extreme transdiciplinarity of the subject, and the
>> interest it can evidently have for the layman (and indeed, for any
>> universal machine).
>
> While I do have training in math and physics, I still benefit from
> your
> targeting the motivated layman. Personally, I'm not interested in
> doing
> the exercises on the list, but they are still useful to check my
> understanding.
OK.
>
>
>> Best regards to all of you, and thanks for letting me know your
>> interests,
>
> By all means, proceed. Personally, if I don't understand something or
> have an objection, you'll hear about it on the list, but I think you
> should take silence as assent.
If only silence could be assent!
But I am willing to take yours as such and I will proceed.
Best,
Bruno
Bruno Marchal
Dear Bruno,when I looked at the set-analysis it immediately popped up that {1,3} was missing, -
YET - this fantastic<G> discovery of mine did not bring me closer to the idea "what are numbers".
It seems I can win the battle and still lose the war.JohnOn Wed, Jul 8, 2009 at 9:05 PM, m.a. <mart...@bellsouth.net> wrote:
Here's my third try. I'll continue working on the (power x) problem. m.a.Sent: Wednesday, July 08, 2009 1:31 PMSubject: Re: The seven step seriesOn 08 Jul 2009, at 15:43, m.a. wrote:
Second try:(power {1, 2, 3}) = ? {{ }, {1}, {2}, {3}, {1,2}, {2,3}, {1,2,3}}Third try:= {{ }, {1}, {2}, {3}, {1,2}, {2,3}, {1,2,3}, {{ },1,2,3}}This is far better! Not yet correct though.I gave you the hint that there are 8 elements. Let us count:The empty set { } ..................................1Three singletons {1}, {2}, {3}................3Two doubletons {1,2 }, {2,3 }................2The biggest subset {1,2,3}..................11 + 3 + 2 + 1 = 7A subset is missing! Can you see which one?
Mirek Dobsicek
I'd like to let you know that I'm following the serie of your letters.
While I have the background you are covering right now, I still enjoy
your insights.
I joined the list like two years ago and from that time I've read most
of your key papers. Honestly, it is not the easiest stuff to read
style-wise. You try to precise, define well, etc. yet it cannot really
be compared to the quality of, let us say, Physical Review Letters and
alike articles. In my opinion, that is why it is hard to either agree or
disagree with your thesis.
I can imagine that right now you are tempted to write something along
the lines
a\ I just propose to take Church thesis seriously
b\ All I ask you is 'Do you say yes to the doctor?
While valid proposal and question, there is really not much to agree
with/disagree with/critize unless one is willing to undertake long
discussions, clarifications and position adjustments.
Anyway, your papers and letters are really a great source of ideas to
think about and that is exactly what I do. From the day one on the list
I keep myself busy with the question of "Why should I believe in the
Church thesis" (you see, I don't write "Why do I ..."). I've got into
the writing of Bernard Bolzano (I consider his work cruicial in order to
keep an open mind about the Cantor diagonal argument) ..
- and now back to the beginning of my letter -
Bolzano (Cantor), your insights and thinking about alternatives at any
moment make me pretty happy. Thanks!
Mirek
PS: I'd love to read a book by Bruno Marchal.
Kim Jones
On 11/07/2009, at 6:24 AM, Bruno Marchal wrote:
>
> I am also a bit anxious about Kim, who is the one who suggested me the
> initial explanations, but who seems to have disappear right now.
OK - I'm back. Since May 27 to two days ago I have been without
Internet access.
I made the mistake of upgrading my Broadband plan to add Internet
phone. It took two telcos a month to complete this ridiculously basic
operation with mistakes made and attendant extra waiting times.
Then, just as the connection was restored at the beginning of July,
the plumbing in this block of apartments fell apart and a major
excavation work went ahead and this time the plumbers cut the phone
cable and didn't realise it which meant I wasted another week trying
to get the problem diagnosed.
So now finally everything is back to normal. I have just started
reading this thread and can see that the class is a very exclusive
one! I will try my best to follow through on the exercises and the
comments, corrections. I feel I have access to the correct
mathematical symbols on my Mac now but *time* is the thing that I
don't have much of anymore, so I feel a bit depressed about the level
of effort I can devote to it. If only we didn't have to work for a
living things would be vastly easier.
The notion of sets is indeed a tricky one. I am just now going over
the initial exercises again. Do not wait for me. I am also trying to
catch up on about 4,000 emails.
Bruno - my sincerest apologies for this hiatus. You seem eager to get
to the seventh and eighth steps. Why wouldn't you be.
regards,
Kim
Bruno Marchal
On 13 Jul 2009, at 00:46, Mirek Dobsicek wrote:
>
> I'd like to let you know that I'm following the serie of your letters.
> While I have the background you are covering right now, I still enjoy
> your insights.
Thanks for letting me know.
>
>
> I joined the list like two years ago and from that time I've read most
> of your key papers. Honestly, it is not the easiest stuff to read
> style-wise. You try to precise, define well, etc. yet it cannot really
> be compared to the quality of, let us say, Physical Review Letters and
> alike articles.
I work on a subject which is not usually approach in any reviews of
physics.
And then my english is sometimes a bit hazardous. But I have never get
referee or any feedback about the rigors, which in my field (machine
theology) is far more developed than usual. It is part of the problem
for some people I think. It is just unusual.
> In my opinion, that is why it is hard to either agree or
> disagree with your thesis.
I disagree. It is simple. Just say a number between 1 and 8, with a
justification of what you don't understand.
Perhaps between 0 and 8, if you have a problem with the definition of
comp.
>
>
> I can imagine that right now you are tempted to write something along
> the lines
> a\ I just propose to take Church thesis seriously
> b\ All I ask you is 'Do you say yes to the doctor?
yes, for the sake of the argument. A non computationalist can just
consider someone else saying yes to the doctor. A bit more is needed,
and it is necessary to recall the definition of comp: it exists a
level of description of my (generalized) such that I survive through a
digital functional substitution made at that level.
>
> While valid proposal and question, there is really not much to agree
> with/disagree with/critize unless one is willing to undertake long
> discussions, clarifications and position adjustments.
Indeed, there is nothing to disagree. Only to understand, and who
knows? to clarify. But then it is up to those who try to understand to
say what they don't understand, besides the intrinsic difficulty with
the subject. In the seventies some people argued that any sentence
containing the word "consciousness" was automatically crackpot. Of
course this is an illustration of the complete absence of
understanding of the axiomatic method. We never know what we are
talking about, we can only agree on starting propositions and method
of reasoning, and then see if the conclusion follows from what we have
admitted.
>
>
>
> Anyway, your papers and letters are really a great source of ideas to
> think about and that is exactly what I do.
I am happy with that.
> From the day one on the list
> I keep myself busy with the question of "Why should I believe in the
> Church thesis" (you see, I don't write "Why do I ...").
Good question. A lot of my work consists in showing that CT is a very
strong principle. It is far stronger than most computer scientist
imagine.
> I've got into
> the writing of Bernard Bolzano (I consider his work cruicial in
> order to
> keep an open mind about the Cantor diagonal argument) ..
> - and now back to the beginning of my letter -
> Bolzano (Cantor), your insights and thinking about alternatives at any
> moment make me pretty happy. Thanks!
You are welcome.
> PS: I'd love to read a book by Bruno Marchal.
I have already written three books, and one was ordered by a publisher
after getting a price. The two other one were disputed by different
publishers, and then suddenly, without explanation, all those projects
were abandoned. I have lost my trust in that kind of world I'm afraid.
I don't think the reason of that abandon has any relationship with my
work which is really of the type: "find the error". I will surely
write one paper and one book. Recently I have submitted a paper, and
the referees were quite enthusiast, but the paper has been refused for
being out of the topic, which it was not (unless you don't believe
that observers are person).
We will see. My work is simple in two senses: UDA is simple because
you need nothing more than a very tiny amount of understanding on
numbers, set, computable functions and consciousness/kowledge. AUDA is
relatively "simple" because you need only to understand Solovay's
theorem and the Theaetetical definitions of knowledge. It makes comp
hard to believe, no doubt, but here the work of Everett in quantum
mechanics can provide a big help. Ah, ok, all this at once needs works
and time, but nothing more.
Bruno Marchal
On 13 Jul 2009, at 09:38, Kim Jones wrote:
>
>
> On 11/07/2009, at 6:24 AM, Bruno Marchal wrote:
>
>>
>> I am also a bit anxious about Kim, who is the one who suggested me
>> the
>> initial explanations, but who seems to have disappear right now.
>
>
>
> OK - I'm back.
I am quite glad you are back, and alive!
> Since May 27 to two days ago I have been without
> Internet access.
Well, that is form of death nowadays. I wish you an happy resurrection.
>
>
> I made the mistake of upgrading my Broadband plan to add Internet
> phone. It took two telcos a month to complete this ridiculously basic
> operation with mistakes made and attendant extra waiting times.
>
> Then, just as the connection was restored at the beginning of July,
> the plumbing in this block of apartments fell apart and a major
> excavation work went ahead and this time the plumbers cut the phone
> cable and didn't realise it which meant I wasted another week trying
> to get the problem diagnosed.
Matter kicks back, even with comp!
>
>
> So now finally everything is back to normal. I have just started
> reading this thread and can see that the class is a very exclusive
> one! I will try my best to follow through on the exercises and the
> comments, corrections. I feel I have access to the correct
> mathematical symbols on my Mac now but *time* is the thing that I
> don't have much of anymore, so I feel a bit depressed about the level
> of effort I can devote to it. If only we didn't have to work for a
> living things would be vastly easier.
You are right. Take your time and ask any question. I will try to sum
up through the next posts.
>
>
> The notion of sets is indeed a tricky one. I am just now going over
> the initial exercises again. Do not wait for me. I am also trying to
> catch up on about 4,000 emails.
Gosh!
>
>
> Bruno - my sincerest apologies for this hiatus. You seem eager to get
> to the seventh and eighth steps. Why wouldn't you be.
No, we have all the time. I just answered to posts, and made
elementary recalls of math. But I realize that some people have not
follow any course in set theory. In Belgium it was taught as "modern
math" during 10 years in high school, and then it has disappeared
(alas 1000X). The discrepancy between the math level of different
people is very big, but the shortcut I am pursuing now, should not be
insurmountable. I will make the next posts as self-contained as
possible, but please interrupt when you feel I am esoteric.
Have a good day,
Bruno
Bruno Marchal
Mirek Dobsicek
I am in a good mood and a bit picky :-) Do you know how many entries
google gave me upon entering
Theaetetical -marchal -bruno
Mirek
> for some people I think. It is just unusual.
Bruno Marchal
Hi Mirek,
>
> Hi Bruno,
>
> I am in a good mood and a bit picky :-) Do you know how many entries
> google gave me upon entering
> Theaetetical -marchal -bruno
Well 144?
Good way to find my papers on that. The pages refer quickly to this
list or the FOR list.
Tomorrow I will not be there.
I let those interested to meditate on two questions (N is {0, 1, 2, 3,
4, ...}):
1) What is common between the set of all subsets of a set with n
elements, and the set of all finite sequences of "0" and "1" of length
n.
2) What is common between the set of all subsets of N, and the set of
all infinite sequences of "0" and "1".
Just some (finite and infinite) bread for surviving the day :)
Bruno
John Mikes
m.a.
----- Original Message -----From: Bruno MarchalSent: Tuesday, July 14, 2009 4:40 AMSubject: Re: The seven step series
Hi Kim, Marty, Johnathan, John, Mirek, and all...
Bruno: May I advise you about an instance of English usage? The word "supposed" in the next sentence is often used as sarcasm to imply serious doubt about the statement. In this context it can be interpreted as a slight. I think you meant to say "assumed" which implies an evident fact. Please don't apologize, we are most grateful for your efforts in using English and are happy to make allowances for minor slips.
B = {Kim, Marty, Russell, Bruno, George, Jurgen} is a set with 5 elements which are supposed to be humans.
I also have a question: see below:
We have seen INTERSECTION, and UNION.The intersection of the two sets S1 = {1, 2, 3} and S2 = {2, 3, 7, 8} will be written (S1 \inter S2), and is equal to the set of elements which belongs to both S1 and S2. We have(S1 \inter S2) = {2, 3}We can define (S1 \inter S2) = {x such-that ((x belongs-to S1) and (x belongs-to S2))}2 belongs to (S1 \inter S2) because ((2 belongs-to S1) and (2 belongs-to S2))8 does not belongs to (S1 \inter S2) because it is false that ((2 belongs-to S1) and (2 belongs-to S2)). Indeed 8 does not belong to S1.
Doesn't the statement in bold (above) contradict the statement immediately preceding (also in bold)?
Kim Jones
On 14/07/2009, at 6:40 PM, Bruno Marchal wrote:
> The intersection of the two sets S1 = {1, 2, 3} and S2 = {2, 3, 7,
> 8} will be written (S1 \inter S2), and is equal to the set of
> elements which belongs to both S1 and S2. We have
>
> (S1 \inter S2) = {2, 3}
>
> We can define (S1 \inter S2) = {x such-that ((x belongs-to S1) and
> (x belongs-to S2))}
>
> 2 belongs to (S1 \inter S2) because ((2 belongs-to S1) and (2
> belongs-to S2))
> 8 does not belongs to (S1 \inter S2) because it is false that ((2
> belongs-to S1) and (2 belongs-to S2)). Indeed 8 does not belong to S1.
>
Quick (silly) questions:
1.
why do you have to write "\inter" ? Why not just write "inter" ?
Typing "\" causes me to make use of a key on my keyboard I have never
used before which is scary ;-)
2.
"such-that" is surely "such that" but the hyphen might just mean
something
(this is mathematics; there are dots and dashes and slashes all over
the place so you have to know what they all mean)
likewise
"belongs-to" would still mean the same thing if we wrote "belongs to"
would it not?
best
K
Bruno Marchal
Bruno,I appreciate your grade-school teaching. We (I for one) can use it.I still find that whatever you explain is an 'extract' of what can be thought of a 'set' (a one representing a many).Your 'powerset' is my example.All those elements you put into { }s are the same as were the physical objects to Aristotle in his 'total' - the SUM of which was always MORE than the additives of those objects.Relations!The set is not an inordinate heap (correct me please, if I am off) of the elements, the elements are in SOME relation to each other and the "set"-idea of their ensemble, to form a SET.You stop short at the naked elements together, as I see.
They wear cloths and hold hands. Mortar is among them.Maybe your math-idea can tolerate any sequence and hiatus concerning to the 'set', and it still stays the same, as far as the "math-idea you need" goes,
but if I go further (and you indicated that ANYTHING can form a set)
the relations of the set-partners comes into play. Not only those which WE choose for 'interesting' to such set, but ALL OF THEM influencing the character of that "ONE".Just musing.
Bruno Marchal
----- Original Message -----From: Bruno MarchalSent: Tuesday, July 14, 2009 4:40 AMSubject: Re: The seven step seriesHi Kim, Marty, Johnathan, John, Mirek, and all...Bruno: May I advise you about an instance of English usage? The word "supposed" in the next sentence is often used as sarcasm to imply serious doubt about the statement. In this context it can be interpreted as a slight. I think you meant to say "assumed" which implies an evident fact. Please don't apologize, we are most grateful for your efforts in using English and are happy to make allowances for minor slips.B = {Kim, Marty, Russell, Bruno, George, Jurgen} is a set with 5 elements which are supposed to be humans.
I also have a question: see below:We have seen INTERSECTION, and UNION.The intersection of the two sets S1 = {1, 2, 3} and S2 = {2, 3, 7, 8} will be written (S1 \inter S2), and is equal to the set of elements which belongs to both S1 and S2. We have(S1 \inter S2) = {2, 3}We can define (S1 \inter S2) = {x such-that ((x belongs-to S1) and (x belongs-to S2))}2 belongs to (S1 \inter S2) because ((2 belongs-to S1) and (2 belongs-to S2))8 does not belongs to (S1 \inter S2) because it is false that ((2 belongs-to S1) and (2 belongs-to S2)). Indeed 8 does not belong to S1.Doesn't the statement in bold (above) contradict the statement immediately preceding (also in bold)?
Bruno Marchal
On 14/07/2009, at 6:40 PM, Bruno Marchal wrote:The intersection of the two sets S1 = {1, 2, 3} and S2 = {2, 3, 7,8} will be written (S1 \inter S2), and is equal to the set ofelements which belongs to both S1 and S2. We have(S1 \inter S2) = {2, 3}We can define (S1 \inter S2) = {x such-that ((x belongs-to S1) and(x belongs-to S2))}2 belongs to (S1 \inter S2) because ((2 belongs-to S1) and (2belongs-to S2))8 does not belongs to (S1 \inter S2) because it is false that ((2belongs-to S1) and (2 belongs-to S2)). Indeed 8 does not belong to S1.
Quick (silly) questions:
1.
why do you have to write "\inter" ? Why not just write "inter" ?
Typing "\" causes me to make use of a key on my keyboard I have never
used before which is scary ;-)
2.
"such-that" is surely "such that" but the hyphen might just mean
something
(this is mathematics; there are dots and dashes and slashes all over
the place so you have to know what they all mean)
likewise
"belongs-to" would still mean the same thing if we wrote "belongs to"
would it not?
John Mikes
On 15 Jul 2009, at 00:50, John Mikes wrote:
Bruno,I appreciate your grade-school teaching. We (I for one) can use it.I still find that whatever you explain is an 'extract' of what can be thought of a 'set' (a one representing a many).Your 'powerset' is my example.All those elements you put into { }s are the same as were the physical objects to Aristotle in his 'total' - the SUM of which was always MORE than the additives of those objects.Relations!The set is not an inordinate heap (correct me please, if I am off) of the elements, the elements are in SOME relation to each other and the "set"-idea of their ensemble, to form a SET.You stop short at the naked elements together, as I see.You get the idea.We can add structure to sets, by explicitly endowing them with operations and relations.
Furhter below you also expose the contrary (to simplify) - I am afraid your "operations and relations" are restricted to the numbers-based (math?) domain, which is not what I mean by 'totality'.
They wear cloths and hold hands. Mortar is among them.Maybe your math-idea can tolerate any sequence and hiatus concerning to the 'set', and it still stays the same, as far as the "math-idea you need" goes,Yes, it is the methodology.
but if I go further (and you indicated that ANYTHING can form a set)More precisily, we can form a set of multiple thing we can conceive or defined.
I would not restrict 'a set' to what WE can conceive, or define now. (Not even within the 'math'-related domain).
the relations of the set-partners comes into play. Not only those which WE choose for 'interesting' to such set, but ALL OF THEM influencing the character of that "ONE".Just musing.
It is OK. The idea consists in simplifying the things as much as possible, and then to realize that despite such simplification we are quickly driven to the unprovable, unnameable, un-reductible, far sooner than we could have imagine.
BrunoJohn
Bruno Marchal
On 14 Jul 2009, at 10:40, Bruno Marchal wrote:
<snip>
> So the subsets of {a, b} are { }, {a}, {b}, {a, b}.
>
> But set have been invented to make a ONE from a MANY, and it is
> natural to consider THE set of all subsets of a set. It is called
> the powerset of that set.
>
> So the powerset of {a, b} is THE set {{ }, {a}, {b}, {a, b}}. OK?
>
> Train yourself on the following exercises:
>
> What is the powerset of { }
> What is the powerset of {a}
> What is the powerset of {a, b, c}
I give the answer, and I continue slowly.
1) What is the powerset of {a, b, c}?
By definition, the powerset of {a, b, c} is the set of all subsets of
{a, b, c}.
I go slowly.
Is the set {d, e, f} a subset of {a, b, d}? No. None of the elements
of {d, e, f} are elements of {a, b, c}. The question was ridiculous.
Is the set {a, b, d} a subset of {a, b, c}? No. One element of {a, b,
d}, indeed, d, does not belong to {a, b, c}, so {a, b, d} cannot be a
subset of {a, b, c}. The question was ridiculous again, but less
obviously so.
Is the set {a, b, c} a subset of {a, b, c}. Yes. All elements of {a,
b, c} are elements of {a, b, c}. {a, b, c} is included in {a, b, c}.
Can we conclude from this that the powerset of {a, b, c} is {{a, b,
c}}. No. We can conclude only that {{a, b, c}} is included in the
powerset. It is very plausible that there are other subsets!
Indeed,
Is {a, b} included in {a, b, c}? Yes, all elements of {a, b} are
elements of {a, b, c}. This take two verifications: we have to verify
that a belongs to {a, b, c}. And that b belongs to {a, b, c}.
Can we conclude from this that the powerset of {a, b, c} is {{a, b, c}
{a, b}}. No, we could still miss other subsets.
I accelerate a little bit.
Is {a, c} a subset of {a, b, c}? Yes, by again two easy verification.
Is there another doubleton (set with two elements) having elements in
{a, b, c}? Yes. {c, b}. It is easy to miss them, so you have to be
careful. All two elements of {c, b} are elements of {a, b, c}, as can
be verified by two easy verification.
Is {b, c} a subset of {a, b, c}. Yes, but we have already consider
it. Indeed the set {b, c} is the same set as {c, b}.
Is there another doubleton? No. Why? I search and don't find it.
is there yet some subset to find?
Yes, the set with one element, notably. They are called singleton.
Here it is easy to guess that there will be as many singletons
included in (a, b, c} that there is elements in {a, b, c}. So the
singletons are {a}, {b}, and {c}. This can be verified by one
verification for each.
Are there still subset? Yes. We have seen that the empty set { } is
included in any set. This can be (re)verify by 0 verifications, given
that there is 0 element in { }..
Conclusion:
There are 8 subsets in {a, b, c}, which are { }, {a}, {b}, {c}.{a, b},
{a, c}, {c, b} and {a, b, c}. And thus,
The powerset of {a, b, c} is the set { { }, {a}, {b}, {c}.{a, b}, {a,
c}, {c, b} {a, b, c}}.
2) What is the powerset of {a}?
Answer {{ } {a}}. It has two elements.
3) What is the powerset of { }
We could think at first sight that there are no subsets, given that
{ } is empty. But we have seen that { } is included in any set. So { }
is included in { }. Again you can verify this by zero verification!
But then the powerset of { }, which is the set of sets included in { }
is not empty: It has one element, the empty set. It is {{ }}. Think
that {{ }} is a box containing that empty box.
Attempt toward a more general conclusion.
The powerset of a set with 0 element has been shown having 1 elements,
and no more.
The powerset of a set with 1 element has been shown having 2 elements,
and no more.
The powerset of a set with 2 elements has been shown having 4
elements, and no more. (preceding post)
The powerset of a set with 3 elements has been shown having 8
elements, and no more.
The powerset of a set with 4 elements has been shown having 16
elements, and no more. (older post)
In math we like to abstract things. Let us look at the preceding line
with all the words dropped! This gives
--- 0 ---- 1 ---
--- 1 ---- 2 ---
--- 2 ---- 4 ---
--- 3 ---- 8 ---
--- 4 ---- 16 ---
On the left, we see, vertically disposed, the natural numbers,
appearing with their usual order.
On the right, we see, vertically disposed, some natural numbers, which
seems to depend in some way from what numbers appears on the left. It
looks like there is a functional relation, that is a function.
The notion of function is the most important and pervading notion in
math, physics, science in general, and we will have to come back on
that very notion soon enough.
The idea that there is a function lurking there, is the idea that we
can guess a general law, capable of providing the answer to the
general line:
"The powerset of a set with n element has been shown having ?
elements, and no more."
Can we determine ? from n. Surely it depends on n.
In this case, a simple guess can be made, by meditating on the
sequence of numbers which appear on the right. Those are, when written
horzontally:
1, 2, 4, 8, 16, ...
I guess you see what happens. Each number in that sequence is the
double of the preceding one. So the next one can be obtained, by just
continuing the multiplication by two.
1, 2, 4, 8, 16, 32, 64, 128, 256, 512, 1024, 2048, 4096, 8192, 16384,
32768, 65536, 131072, ... that is,
1, 2x2, 2x2x2, 2x2x2x2, 2x2x2x2x2, 2x2x2x2x2x2, ...
We will write a possibly lengthy expression like 2x2x2x2x2x ... x2, as
2^n, where n is the number of occurence of "2" in the expression.
So you can guess that in the "general line":
"The powerset of a set with n elements has been shown having ?
elements, and no more."
? does indeed depend on n, and is actually equal to 2^n.
Here is the general law, that we have guessed by experience (counting
in the simple case) and generalized by intution:
The powerset of a set with n elements has been shown having 2^n
elements, and no more.
When we found a law in such a way, we could ask if we couldn't prove
it from facts we are already believing (or guessing).
Here, the theory is the intuitive basic knowledge of logic, numbers
and sets. And the question is really: can you prove, or justify, or
explain WHY the powerset of a set of n elements has 2^n elements?
What happens which could explain why, when we add an element to a set,
its powerset becomes two time bigger. is there a reason for that
special happening. Can I convince myself that it has to be like that?
I let you think.
Hints. We have already see a "doubling scenario". Indeed, I often
mention at the step 3 of the UDA, the iterated self-duplication. Some
is cut and paste in Brussel, and copy at place W and M. Then both
individuals come back, by train, in Brussels and both do the
duplication experience again, and again, and again. The number of
individuals grows like 2, 4, 8, ... that is 2^n, with n the number of
iteration of the experience. After n iteration, each has written in
its "first person diary" the sequence of places they have visit, which
look like a string of W and M of length n.
No question? If everything is clear, I continue asap.
Oh! I have a question myself. If the law above is correct, it looks
like 2^0 = 1. It is the laws applied to the first line---the case of
the powerset of the empty set. Is it true that 2x2x2x...x2 is equal to
1 in case the number of occurence of 2 is 0? How come?
Bruno
m.a.
> I let those interested to meditate on two questions (N is {0, 1, 2, 3,
> 4, ...}):
>
> 1) What is common between the set of all subsets of a set with n
> elements, and the set of all finite sequences of "0" and "1" of length
> n.
> 2) What is common between the set of all subsets of N, and the set of
> all infinite sequences of "0" and "1".
>
>
Bruno Marchal
I would not restrict 'a set' to what WE can conceive, or define now. (Not even within the 'math'-related domain).
the relations of the set-partners comes into play. Not only those which WE choose for 'interesting' to such set, but ALL OF THEM influencing the character of that "ONE".Just musing.It is OK. The idea consists in simplifying the things as much as possible, and then to realize that despite such simplification we are quickly driven to the unprovable, unnameable, un-reductible, far sooner than we could have imagine.I may suggest (or: assume?) that instead of "despite" it would make more sense to write: "AS A CONSEQUENCE"- think about it.
Bruno Marchal
Bruno Marchal
m.a.
----- Original Message -----From: Bruno Marchal
Sent: Thursday, July 16, 2009 3:56 PMSubject: Re: The seven step series
Brian Tenneson
http://arxiv.org/abs/0904.0867
Abstract
I discuss some problems related to extreme mathematical realism, focusing on a recently proposed "shut-up-and-calculate" approach to physics (arXiv:0704.0646, arXiv:0709.4024). I offer arguments for a moderate alternative, the essence of which lies in the acceptance that mathematics is (at least in part) a human construction, and discuss concrete consequences of this--at first sight purely philosophical--difference in point of view.
-Brian
Bruno Marchal
Bruno,I don't know about Kim, but I'm ready to push on. I'm waiting for the answer to problem 2) see below. And could you please retstate that problem as I'm not sure which one it is? Thanks, marty a.
>
> 1) What is common between the set of all subsets of a set with n
> elements, and the set of all finite sequences of "0" and "1" of length
> n.
> 2) What is common between the set of all subsets of N, and the set of
> all infinite sequences of "0" and "1".
>
Bruno Marchal
Brian Tenneson
Bruno Marchal wrote:
Exercise: criticize the following papers mentioned below in the light of the discovery of the universal machine and its main consequences from incompleteness to first person indeterminacy. Think of the identity thesis. To be sure Tegmark is less "wrong" than Jannes.
Solution: search in the archive of this list where I have already explained this, or use directly UDA, or wait for what will (perhaps) follow.
I should send some of my papers on arXiv, but up to now, only logicians understand the whole "trick", so I have to better appreciated what physicians don't understand in logic, before making a version free of references to mathematical logical baggage. Logicians are not interested in mind, nor really matter, and physicians are still naïve on the link consciousness/reality, I would say.
To be sure Tegmark is closer than most physicists except perhaps Wheeler.
Also, Tegmarks' argument for mathematicalism is invalid (even with strong non-comp axioms). But I prefer to help you to understand this by yourself through the understanding of what a universal machine is, than trying a direct argument.
According of the part of UDA (or perhaps AUDA) you understand, you can already see the weakness of such direct mathematical approach. Note that comp makes physics much more fundamental, and separate it much clearly from possible geograpies. Above all comp does not eliminate the person, which Tegmark is still doing: the frog view is not yet a first person view, in the comp sense.
Interesting stuff, still. Thanks for the references.
Bruno Marchal
Comments below.
Bruno Marchal wrote:Exercise: criticize the following papers mentioned below in the light of the discovery of the universal machine and its main consequences from incompleteness to first person indeterminacy. Think of the identity thesis. To be sure Tegmark is less "wrong" than Jannes.I need to get a better grasp on what a universal machine is, yes. I am interested in finding out how Tegmark's argument for mathematicalism is invalid, especially since I'm using it to motivate my research.
Solution: search in the archive of this list where I have already explained this, or use directly UDA, or wait for what will (perhaps) follow.
I should send some of my papers on arXiv, but up to now, only logicians understand the whole "trick", so I have to better appreciated what physicians don't understand in logic, before making a version free of references to mathematical logical baggage. Logicians are not interested in mind, nor really matter, and physicians are still naïve on the link consciousness/reality, I would say.
To be sure Tegmark is closer than most physicists except perhaps Wheeler.
Also, Tegmarks' argument for mathematicalism is invalid (even with strong non-comp axioms). But I prefer to help you to understand this by yourself through the understanding of what a universal machine is, than trying a direct argument.
I'll have to think more on Jannes' paper. As I basically resting the motivation of my research on the correctness of "ERH implies MUH," I'm trying to formulate a good refutation to his paper.
According of the part of UDA (or perhaps AUDA) you understand, you can already see the weakness of such direct mathematical approach. Note that comp makes physics much more fundamental, and separate it much clearly from possible geograpies. Above all comp does not eliminate the person, which Tegmark is still doing: the frog view is not yet a first person view, in the comp sense.
Interesting stuff, still. Thanks for the references.
m.a.
----- Original Message -----From: Bruno MarchalSent: Monday, July 20, 2009 3:17 PMSubject: Re: The seven step series
On 20 Jul 2009, at 15:34, m.a. wrote:
And then we have seen that such cardinal was given by 2^n.
Brent Meeker
n+1, one of them by appending a 0 and one of them by appending a 1. So
to get all binary strings of length n+1 take each string of length n,
make two copies, to one copy append a 0 and to the other copy append a 1.
Brent
m.a. wrote:
> Hi Bruno,
> I'm not clear on the sentence in bold below,
> especially the word "correspondingly". The example of Mister X only
> confuses me more. Could you please give some simple examples? Thanks,
>
>
>
>
> marty a.
>
>
>
> ----- Original Message -----
> *To:* everyth...@googlegroups.com
> <mailto:everyth...@googlegroups.com>
> *Sent:* Monday, July 20, 2009 3:17 PM
> *Subject:* Re: The seven step series
>
> On 20 Jul 2009, at 15:34, m.a. wrote:
>
>> And then we have seen that such cardinal was given by 2^n.
> You can see this directly by seeing that adding an element in a
> set, double the number of subset, due to the dichotomic choice in
> creating a subset "placing or not placing" the new element in the
> subset.
>
> This is also illustrated by the iterated self-duplication W, M.
> Mister X is cut and paste in two rooms containing each a box, in
> which there is a paper with zero on it, in room W, and 1 on it in
> room M. After the experience, the 'Mister X' coming out from room
> W wrote 0 in his diary, and the 'Mister X' coming out from room M
> wrote 1 in his diary. And then they redo each, the experiment. The
> Mister-X with-0-in-his-diary redoes it, and gives a Mister-X
> with-0-in-his-diary coming out from room W, and adding 0 in its
> diary and a Mister-X with-0-in-his-diary coming out from room M,
> adding 1 in its diary: they have the stories
>
>
> Bruno
>
>
>
> http://iridia.ulb.ac.be/~marchal/
>
>
>
>
>
> >
m.a.
Brent Meeker
00 01 10 11
Make two copies of each
00 00 01 01 10 10 11 11
Add a 0 to the first and a 1 to the second
000 001 010 011 100 101 110 111
and you have all strings of length 3.
Brent
m.a. wrote:
> *Thanks Brent,*
> * Could you supply some illustrative examples? *
> *
> marty a.*
> **
m.a.
> 00 01 10 11
> Make two copies of each
> 00 00 01 01 10 10 11 11
> 000 001 010 011 100 101 110 111
> and you have all strings of length 3.
Bruno Marchal
Take all strings of length 2
00 01 10 11
Make two copies of each
00 00 01 01 10 10 11 11
Add a 0 to the first and a 1 to the second
000 001 010 011 100 101 110 111
and you have all strings of length 3.
I can see where adding 0 to the first and 1 to the second gives 000 and 001 and I think I see how you get 010 but the rest of the permutations don't seem obvious to me. P-l-e-a-s-e explain, Best,m. (mathematically hopeless) a.
00
Make two copies of each
first copy:
The example of Mister X only confuses me more.
Brent Meeker
> Hi Brent,
> I really appreciate the help and I hate to impose on
> your patience but...(see below)
>
> ----- Original Message -----
> From: "Brent Meeker" <meek...@dslextreme.com
> <mailto:meek...@dslextreme.com>>
> To: <everyth...@googlegroups.com
> <mailto:everyth...@googlegroups.com>>
> Sent: Tuesday, July 21, 2009 5:24 PM
> Subject: Re: The seven step series
>
> >
> > Take all strings of length 2
> > 00 01 10 11
> > Make two copies of each
> > 00 00 01 01 10 10 11 11
>
> > Add a 0 to the first and a 1 to the second
> > 000 001 010 011 100 101 110 111
> > and you have all strings of length 3.
> **
>
>
>
> * m. (mathematically hopeless) a.*
They aren't permutations. They're just sticking a 0 or 1 on the end. One copy
of 01 becomes 010 and the other become 011.
Brent
m.a.
> of 01 becomes 010 and the other become 011.
>
> Brent
>
>
Brent Meeker
> *Going a step further... (see below)*
> **
> **
> *000 **000 001 001 010 010 011 011
> **
> *...and sticking a 0 or 1 at the end:*
> **
> *0000 0001 0010 0011 0100 0101 0110 0111
> **
> *and this is the binary sequence of length 4.*
Right, it's all the binary strings of length 4
> **
> *How do these translate into ordinary numerals? 1,2,3,4...*
Bruno's using them to represent sets and subsets. So if we have a set {a b c}
we can represent the subset {a c} by 101 and {a b} by 110, etc. That's quite
different from using a binary string to represent a number in positional
notation. I'll leave it to Bruno whether he wants to go into that.
Brent
m.a.
m.a.
From: Bruno Marchal
Sent: Wednesday, July 22, 2009 12:20 PMSubject: Re: The seven step series
Bruno Marchal
Bruno Marchal
Bruno,Yes, yours and Brent's explanations seem very clear. I hate to ask you to spell things out step by step all the way, but I can tell you that when I'm confronted by a dense hedge or clump of math symbols, my mind refuses to even try to disentangle them and reels back in terror. So I beg you to always advance in baby steps with lots of space between statements. I want to assure you that I'm printing out all of your 7-step lessons and using them for study and reference. Thanks for your patience, m.a.
ronaldheld
I am following, but have not commented, because there is nothing
controversal.
When you are done, can your posts be consolidated into a paper or a
document that can be read staright through?
Ronald
>
> - Show quoted text -
Bruno Marchal
On 27 Jul 2009, at 16:07, ronaldheld wrote:
>
> I am following, but have not commented, because there is nothing
> controversal.
>
> When you are done, can your posts be consolidated into a paper or a
> document that can be read staright through?
Bruno
Bruno Marchal
Hi,
OK, I will come back on the square root of 2 later.
We have talked on sets.
Sets have elements, and elements of a set define completely the set,
and a set is completely defined by its elements.
Example: here is a set of numbers {1, 2, 3}
and a set of sets of numbers {{1, 2}, {3}, { }}.
We can do some operations, like their union, or their intersection.
Examples:
{1,2,3} union {3,4,5} = {1,2,3,4,5},
{1,2,3} intersection {3,4,5} = { }.
We can verify if some relation hold for them, like equality, or
inclusion.
{1, 2} = {2, 1, 1} (yes!)
{1, 2} included-in {3, 2, 1}
{1, 2} not-included-in {1, 3}
We can compute their powerset.
Powerset {1, 2} = {{ }, {1}, {2}, {1, 2}}
We have discovered SBIJECTION between powersets of a set with cardinal
n, and the set of binary strings of length n.
And we have presented reasons for the existence of a bijection between
the powerset of N = {0, 1, 2, ...} and the set of infinite binary
strings.
OK?
Today, I suggest we look at two new operations on sets. The product of
sets, and the exponentiation of sets. Well, I will probably do only
the product today.
First I have to introduce a new, well actually *very* well known, and
absolutely important, notion: the couple.
A couple is when there is two things, but with some order. It looks
like a pair, but the order counts.
Usually a couple of things a , b is designated, in math, like this:
(a, b).
It looks like a pair {a, b}, but it is not. Indeed, {a, b} = {b, a},
but the couple (a, b) is NOT equal to the couple (b, a).
When are two couples (a, b) and (c, d) equal? Only when a = c and b = d.
Examples. the couple of number (2, 3) is not equal to the couple (3,
2), but the couple (0, 666) is equal to the couple (0, 666).
OK?
APARTE: Are couples sets? No. Nor are numbers. But yes, you can easily
represent them by sets, so we could work only with sets, but we will
not do that. Much later we will work only with numbers, in fact. The
very notion of representation will be important, though.
Now we are ready to define the so called "cartesian" product of sets.
It is indeed a cousin of Descartes' discovery that you can represent a
point of the plane by a couple of (real) numbers. I read somewhere
that Descartes discovered this by trying to describe a spider walking
on a window with squared little piece of glass. But such a
localization works also for cities like Los Angeles where you address
is something like 15th avenue 61th street. The whole field of
analytical geometry is founded on this idea.
That cartesian idea generalises on sets A and B. It is written A X B,
and it is defined by
the set of couples (x, y) such that x belongs to A, and y belongs to B.
AXB = {(x, y) such-that x belongs-to A, and y belongs to B}.
(compare with the preceding definitions).
Example: what is the product {0, 1} X {a, b}? Well it is the set of
all the couples made from elements of A in company of elements of B,
and in that order, with A = {0, 1}, and B = {a, b}.
So (0, a) is in there, and there are others. The product of {0, 1)
with {a, b} is equal to
{(0,a), (0, b), (1, a), (1, b)}
The convenient usual cartesian drawing is, for AXB, with A = {0, 1},
and B = {a, b} :
a (0, a) (1, a)
b (0, b) (1, b)
0 1
A product of numbers a and b, ab, can be conceived as the area of a
rectangle of sides a and b. Here you can see that the product of sets
AXB can fit in a rectangle when you dispose horizontally the elements
of A, and vertically the elements of B. By convention, usually A is
put horizontally, and B vertically.
But note that if the number ab is equal to the number ba, it is not
the case that the set AXB is equal to the set BXA. (0, a) does not
belong to BXA, for example.
Exercise: to the cartesian drawing for BXA.
1)
Compute
{a, b, c} X {d, e} =
{d, e} X {a, b, c} =
{a, b} X {a, b} =
{a, b} X { } =
2)
Convince yourself that the cardinal of AXB is the product of the
cardinal of A and the cardinal of B.
A and B are finite sets here. Hint: meditate on their cartesian drawing.
3) Draw a piece of NXN.
Solution and sequel tomorrow.
Any question?
Bruno
ronaldheld
I meant the mathematical formalism you are teaching us. When we
eventually get to the UDA steps, I wil be better able to do that
assessment.
Ronald
m.a.
>
>
>
Bruno Marchal
Bruno,I have searched my notes for an exposition of BIJECTION and found only one mention in an early email which promises to define it in a later lesson. Do you have a reference to that lesson or perhaps an instant explanation of it? Thanks,Chief Ignoramus
Bruno Marchal
On 28 Jul 2009, at 12:51, ronaldheld wrote:
>
> Bruno:
> I meant the mathematical formalism you are teaching us. When we
> eventually get to the UDA steps, I wil be better able to do that
> assessment.
>
OK.
Note that the first 6 steps have already be done recently, with Kim,
and even before. But there is no problem to come back on this, later.
The key point there consists in explaining the first person
indeterminacy, and its invariance for set of transformations (adding
delays in the computation, going from "real" to "virtual", etc.).
You may prepare yourself by reading the relevant portion of the sane04
paper. Eventually the seventh step itself somehow recapitulates the 6
preceding steps, so it is OK.
Best,
Bruno
Bruno Marchal
In this case, A X B = {(0,a), (0, b), (1, a), (1, b)}
The cartesian drawing is, for AXB :
a (0, a) (1, a)
b (0, b) (1, b)
0 1
0 (a, 0) (b, 0)
a b
1)
Compute
{a, b, c} X {d, e} =
{d, e} X {a, b, c} = {(d, a), (d, b), (d, c), (e, a), (e, b), (e, c)}
{a, b} X {a, b} = {(a, a), (a, b), (b, a), (b, b)}
{a, b} X { } = { }.
2)
Convince yourself that the cardinal of AXB is the product of the
cardinal of A and the cardinal of B.
A and B are finite sets here. Hint: meditate on their cartesian drawing.
3) Draw a piece of NXN. (with, as usual, N = {0, 1, 2, 3, ...}):
John Mikes
Bruno Marchal
Hi, Bruno,let me skip the technical part
and jump on the following text.F u n c t i o n as I believe is - for you - the y = f(x) form. For me: the activity - shown when plotting on a coordinate system the f(x) values of the Y-s to the values on the x-axle resulting in a relation (curve). And here is my problem: who does the plotting? (Do not say: YOU are, or Iam, that would add to the function concept the homunculus to make it from a written format into a F U N C T I O N ).
F u n c t i o n as I believe is - for you - the y = f(x) form.
Mirek Dobsicek
>> I am in a good mood and a bit picky :-) Do you know how many entries
>> google gave me upon entering
>> Theaetetical -marchal -bruno
>
>
> Well 144?
>
> Good way to find my papers on that. The pages refer quickly to this
> list or the FOR list.
I am sorry for the delay, I've just got back from my vacation.
Hmm. The above written search should not return any references to your
papers/letters as the minus sign in front of your name asks for an
exclusion.
Given that it works as supposed google then gives only 1 hit in my
location (Sweden). That hit is a translation of the word "Theaetetical"
into some eastern characters. Thus, I end up with zero meaningful hits
and a feeling that you might be the only one using this word.
That makes me insists a little bit more (in a very polite way) that,
occasionally, your work is
"difficult to read unless one is willing to undertake long
discussions, clarifications and position adjustments."
I am writing this in a reference to your complains that sometimes you
have troubles to get enough relevant feedback to your work.
> I let those interested to meditate on two questions (N is {0, 1, 2, 3,
> 4, ...}):
>
> 1) What is common between the set of all subsets of a set with n
> elements, and the set of all finite sequences of "0" and "1" of length
> n.
> 2) What is common between the set of all subsets of N, and the set of
> all infinite sequences of "0" and "1".
>
> Just some (finite and infinite) bread for surviving the day :)
I am going to catch up with the thread ...
Cheers,
mirek
Bruno Marchal
I am in a good mood and a bit picky :-) Do you know how many entriesgoogle gave me upon enteringTheaetetical -marchal -brunoWell 144?Good way to find my papers on that. The pages refer quickly to thislist or the FOR list.
I am sorry for the delay, I've just got back from my vacation.
Hmm. The above written search should not return any references to your
papers/letters as the minus sign in front of your name asks for an
exclusion.
Given that it works as supposed google then gives only 1 hit in my
location (Sweden). That hit is a translation of the word "Theaetetical"
into some eastern characters. Thus, I end up with zero meaningful hits
and a feeling that you might be the only one using this word.
That makes me insists a little bit more (in a very polite way) that,
occasionally, your work is
"difficult to read unless one is willing to undertake long
discussions, clarifications and position adjustments."
I am writing this in a reference to your complains that sometimes you
have troubles to get enough relevant feedback to your work.
I let those interested to meditate on two questions (N is {0, 1, 2, 3,4, ...}):1) What is common between the set of all subsets of a set with nelements, and the set of all finite sequences of "0" and "1" of lengthn.2) What is common between the set of all subsets of N, and the set ofall infinite sequences of "0" and "1".Just some (finite and infinite) bread for surviving the day :)
I am going to catch up with the thread ...
John Mikes
Mirek Dobsicek
> Come on Mirek: "Theaetetical" is an adjective I have forged from
> "Theatetus".
> "Theatetus" gives 195.000 results on Google.
> "Theatetus" wiki 4310.
Of course, after all you reference the dialogue Theaetetus in your
papers thus one can easily match the word Theaetetical agains it.
Let me quickly summarize the experience I had with "theatetical notion
of knowledge" while reading one of your papers for the first time.
Maybe I am an ignorant, then shame on me, but I have not read the
Theaetetus. So I took a look at the Wikipedia and read
"In this dialogue, Socrates and Theaetetus discuss three definitions of
knowledge: knowledge as nothing but perception, knowledge as true
judgment, and, finally, knowledge as a true judgment with an account.
Each of these definitions are shown to be unsatisfactory."
Hmm that really helps .., I told to myself and continued with reading.
With an uneasy feeling of stepping into the water I eventually settled
down to conclusion that you likely mean something as "true justified
belief".
I really wished you wrote it more straightforwardly without turning your
readers quite unnecessarily down to the Theaetetus and inventing new
words such as "Theaetetical".
Anyway, I'd like to stop discussing this issue :-) since my only point
was to give you a hint why I said that it is not easy to read your
papers/letters.
> Feel free to ask for any clarification, position
> adjustments, question, at any level ...Do you understand what is the
> comp hypothesis?
Let us see if I get it right. Your comp hypothesis is
1) I'm a machine,
2) Each possible computation is Turing-computable,
3) Natural numbers and their relations do exist.
This should not be confused with other quite common comp hypothesis that
the universe is a big computer. This hypothesis entails the existence of
a physical computer.
Ad 1) I take the position that "I" is only a convenient temporary
pointer to a part of universe. The pointer "Socrates' thoughts" is of
the same quality.
Ad 2) Breath taking. While 1) and 3) are assumptions of the kind "OK,
let's think for a while that ...", 2) has the status of a thesis. I
don't have any firm position on what could an objective reality be (and
without a justification I tend to think it is inaccessible to us), but
if there is any objective reality, 2) could be a statement about it.
Ad 3) If natural numbers and their relations are the only entities which
do exist then me, you, everything is a recipe of a Turing-computable number.
OK, that is it. This is how I understand to your starting assumptions.
Mirek
Bruno Marchal
Bruno and Mirek,concerning Theateticus vs. Theaeteticus:in my strange linguistic background I make a difference betwee ai and ae - the spelling in Greek and Latin of the name. As far as I know, nobody knows for sure how did the 'ancient' Greeks pronounce their ai - maybe as the flat 'e' like in German "lehr" while the 'e' pronounciation might have been clsoer to (between) 'make' and 'peck' - the reason why the Romans transcribed it by their ONE letter "ae", (lehr) and not as English would read: 'a'+'ee'. The spelling you gave points to this latter. The Latin 'ae' is not TWO separate letters (a+e), it is a twin, as marked in the Wiki article..."Theætetus"... and not Theaetetuswhich looked strange to me from the beginning .(I wonder if the e-mail reproduces the (ae) one sign? look up in Wiki's Theaetetus Dialogue (in the title with the wrong spelling) the 1st line brings the merged-together double 'æ'.)*English spelling always does a job on classical words, the Greek 'oi' has been transcribed into Latin sometimes as 'oe' and pronounced as in "girl" (oeuvre) while many think it was a sound like what the pigs say: as "oy". then comes America, with it's Phoenix (pron: feenix)....I don't think the Romans were much better off, centuries after and a world apart from the ancient (classical for them) Greeks.And who knows today if the great orator was Tzitzero or Kikero to turn later into Tchitchero?*"The Old Man" did quite a job on us at the tower of Babel.*[[ - I am enjoying your 'other' post where you spelled out my own vocabulary as indeed thinking functions as relations, lately not as a static description, but also the interchanging factor - ]]John
Bruno Marchal
Come on Mirek: "Theaetetical" is an adjective I have forged from"Theatetus"."Theatetus" gives 195.000 results on Google."Theatetus" wiki 4310.
Of course, after all you reference the dialogue Theaetetus in your
papers thus one can easily match the word Theaetetical agains it.
Let me quickly summarize the experience I had with "theatetical notion
of knowledge" while reading one of your papers for the first time.
Maybe I am an ignorant, then shame on me, but I have not read the
Theaetetus.
So I took a look at the Wikipedia and read
"In this dialogue, Socrates and Theaetetus discuss three definitions of
knowledge: knowledge as nothing but perception, knowledge as true
judgment, and, finally, knowledge as a true judgment with an account.
Each of these definitions are shown to be unsatisfactory."
Hmm that really helps .., I told to myself and continued with reading.
With an uneasy feeling of stepping into the water I eventually settled
down to conclusion that you likely mean something as "true justified
belief".
I really wished you wrote it more straightforwardly without turning your
readers quite unnecessarily down to the Theaetetus and inventing new
words such as "Theaetetical".
Anyway, I'd like to stop discussing this issue :-) since my only point
was to give you a hint why I said that it is not easy to read your
papers/letters.
Feel free to ask for any clarification, positionadjustments, question, at any level ...Do you understand what is thecomp hypothesis?
Let us see if I get it right. Your comp hypothesis is
1) I'm a machine,
2) Each possible computation is Turing-computable,
3) Natural numbers and their relations do exist.
This should not be confused with other quite common comp hypothesis that
the universe is a big computer.
This hypothesis entails the existence of
a physical computer.
Ad 1) I take the position that "I" is only a convenient temporary
pointer to a part of universe. The pointer "Socrates' thoughts" is of
the same quality.
Ad 2) Breath taking. While 1) and 3) are assumptions of the kind "OK,
let's think for a while that ...",
2) has the status of a thesis. I
don't have any firm position on what could an objective reality be (and
without a justification I tend to think it is inaccessible to us), but
if there is any objective reality, 2) could be a statement about it.
Ad 3) If natural numbers and their relations are the only entities which
do exist then me, you, everything is a recipe of a Turing-computable number.
OK, that is it. This is how I understand to your starting assumptions.
Mirek Dobsicek
Bruno Marchal wrote:
> Hi Mirek,
>
> Long and perhaps key post.
Thank you a lot for a prompt and long reply. I am digesting it :-)
Just some quick comments.
> There is no shame in being ignorant. Only in staying ignorant :)
I've ordered the dialogue from a second-hand book shop :-) The Stanford
encyclopedia says
"Arguably, it is his (Plato) greatest work on anything."
So I'll give it a try :-)
>> judgment, and, finally, knowledge as a true judgment with an account.
> The remarkable thing is that if you accept to modelize "account" by
> "sound machine provability",
This is probably the key problem for me. I know next to nothing about
provability, the logic of provability, PA/ZF provers.
I know that quite often you reference Boolos 1993 - The Logic of
Provability. I took a look at it at Google Books preview but ... there
is something missing in my education. From the beginning I am puzzled
with "Why?, what?". What a headache :-)
> In french students are burned alive if they dare to create new
> adjective, and I thought that in English we have more freedom, but I may
> be wrong. Sorry.
I'd grant this freedom to rational native speakers only :-)
> x divides y if and only if it exists a number z such that y = x*z.
I don't dare to correct your english but "there is/exists a number ..."
is what I would write.
>> Ad 3) If natural numbers and their relations are the only entities which
>> do exist then me, you, everything is a recipe of a Turing-computable
>> number.
>
> No. Not at all. Sorry. Gosh, you will be very surprised if you follow
> the UDA-7. On the contrary. Arithmetical truth VASTLY extends the
> computable domain. Most relations between numbers are not Turing emulable.
Aha! Then I really have a wrong mental picture of your work. I
understood to arithmetical realism along the lines of this quotation
from the Stanford article on realism:
"According to a platonist about arithmetic, the truth of the sentence '7
is prime' entails the existence of an abstract object, the number 7.
This object is abstract because it has no spatial or temporal location,
and is causally inert. A platonic realist about arithmetic will say that
the number 7 exists and instantiates the property of being prime
independently of anyone's beliefs, linguistic practices, conceptual
schemes, and so on."
So I thought that you essentially take
a) Numbers and their properties and relations exists.
b) Now, since you don't assume existence of anything else => your body,
your bike and coffee must emerge as patterns in the world of numbers.
c) Taking the Church-Turing thesis, these patterns are Turing-computable.
d) Definitely, the vast majority of all patterns is not Turing-computable.
This is how I have thought about your working framework. Notice, that I
don't talk about what you try to show, argue for, want to end up with etc.
Cheers,
Mirek
Bruno Marchal
On 05 Aug 2009, at 00:52, Mirek Dobsicek wrote:
> I've ordered the dialogue from a second-hand book shop :-) The
> Stanford
> encyclopedia says
> "Arguably, it is his (Plato) greatest work on anything."
> So I'll give it a try :-)
I love that book, and it is also my favorite piece of Plato.
To be sure, I don't think it is needed to understand neither UDA nor
AUDA, but it can help.
> This is probably the key problem for me. I know next to nothing about
> provability, the logic of provability, PA/ZF provers.
>
> I know that quite often you reference Boolos 1993 - The Logic of
> Provability. I took a look at it at Google Books preview but ... there
> is something missing in my education. From the beginning I am puzzled
> with "Why?, what?". What a headache :-)
You miss an introductory course on mathematical logic.
Have you herad about Gödel's incompletness theorem. Boolos book
explains the sequel.
I thought that, after Hofstadter best selling book on Gödel's theorem
(Gödel, Escher, Bach), it would be possible to talk on mathematical
logic to the layman, like we can talk on physics to the layman. But I
was wrong. Gödel's theorem is not yet part of the common knowledge,
and when it is used by non mathematician, in general it is abused.
>> x divides y if and only if it exists a number z such that y = x*z.
>
> I don't dare to correct your english but "there is/exists a
> number ..."
> is what I would write.
Thanks.
>
>
>>> Ad 3) If natural numbers and their relations are the only entities
>>> which
>>> do exist then me, you, everything is a recipe of a Turing-computable
>>> number.
>>
>> No. Not at all. Sorry. Gosh, you will be very surprised if you follow
>> the UDA-7. On the contrary. Arithmetical truth VASTLY extends the
>> computable domain. Most relations between numbers are not Turing
>> emulable.
>
> Aha! Then I really have a wrong mental picture of your work. I
> understood to arithmetical realism along the lines of this quotation
> from the Stanford article on realism:
>
> "According to a platonist about arithmetic, the truth of the
> sentence '7
> is prime' entails the existence of an abstract object, the number 7.
> This object is abstract because it has no spatial or temporal
> location,
> and is causally inert. A platonic realist about arithmetic will say
> that
> the number 7 exists and instantiates the property of being prime
> independently of anyone's beliefs, linguistic practices, conceptual
> schemes, and so on."
That is quite correct. All mathematicians are realist about
arithmetic, and most are realist about sets. But set realism is a much
more stronger belief than arithmetical realism.
Comp necessitates arithmetical realism if only to be able to state
Church thesis. Theoretical computer scientist are realist, because
they belief that all machine either stop or not stop.
>
>
> So I thought that you essentially take
> a) Numbers and their properties and relations exists.
Yes, but some people put to much sense in "exists". It is the
mathematical usual sense, like when you derive "there exists a prime
number" from the statement "17 is a prime number". No need to invoke
Plato Heaven, in the assumption.
>
> b) Now, since you don't assume existence of anything else => your
> body,
> your bike and coffee must emerge as patterns in the world of numbers.
I am agnostic. I assume neither that something else exists nor that it
does not exist, and then I prove from the assumption that we are
turing emulable, that physics is no more the fundamental science. I
prove that if we are machine then matter has to be an emerging
epistemological concept, and physics is a branch of machine biology/
psychology/theology, or mathematical computer science.
"b)" is obviously non valid. The fact that bike an coffee must emerge
from numbers is really the conclusion of the whole UD reasoning. It is
not because I don't assume them, it is because their independent
existence is shown contradictory.
I show that mechanism makes physicalism epistemologically
inconsistent. Even if matter really exists, it cannot be used to
justify our belief in matter. A slight application of Occam razor
eliminates matter, at that stage.
>
> c) Taking the Church-Turing thesis, these patterns are Turing-
> computable.
Not at all. The world of number is provably not Turing-computable.
Only a very tiny part of the world of number is computable. There is a
whole branch of mathematical logic devoted to the study of the degree
of non computability of the relations existing among the numbers.
Church thesis asserts only that the *computable* patterns are Turing
computable. It is just the assertion that Turing computability can be
used to define computability.
>
> d) Definitely, the vast majority of all patterns is not Turing-
> computable.
I don't understand.
>
>
> This is how I have thought about your working framework. Notice,
> that I
> don't talk about what you try to show, argue for, want to end up
> with etc.
My framework, comp, is just the hypothesis that I can survive with an
artificial digital brain (even material, if you want). That's all.
The negation of comp is "my soul/person/consciousness" is not Turing
emulable. Or I say "no" to all doctors. Or "I don't survive classical
teleportation done at any level".
I use the term "computationalism" in its standard usual traditional
sense. If you assume explicitly that computationalism needs the brain
to be a material object, then the UDA can be seen as a reductio ad
absurdo.
The conclusion of UDA is that Materialism is incompatible with
Computationalism. It is not obvious, but given that the numerous
attempts by materialist to solve the mind body problem have failed, it
is not so astonishing that a solution of the mind body problem needs
some "scientific revolution". The "revolution" is the reversal between
physics and the "theology of numbers" (the study of what numbers can
believe in, can know, can bet on, etc.).
Bruno
John Mikes
John
Bruno Marchal
Exercise:1) how many functions and what are they, from the set {0, 1} to himself. What are the functions from {0, 1) to {0, 1}?Solution:{(0,0), (1,0)} the constant function which associates zero to any value of its argument.{(0,1), (1,1)} the constant function which associates one to any value of its argument.{(0,0), (1,1)} the identity function, which output its argument as value.{(0,1), (1,0)}, the NOT function, which associate 0 to 1, and 1 to 0.There is four functions from {0, 1} to {0, 1}.
2) how many functions, and what are they, from the set cartesian product {0, 1} X {0, 1} to {0, 1}Among them many are celebrities, you know. The AND, the OR, and many (how many?) others.For a beginner in math, this is not at all an easy exercise. The real useful exercise is to try to understand the enunciation of the question. We will take the time needed.
3) A bit tricky perhaps: how many functions exist from { } to { } ?
Mirek Dobsicek
> 3) compute { } ^ { } and card({ } ^ { })
> If card(A) = n, and card(B) = m. What is
> card(A^B)?
I find it neat to write | {} ^ {} | = | { {} } | = 1 :-)
It's almost like ASCII art. Just wanted to signal that I'm following.
mirek
Bruno Marchal
On 11 Aug 2009, at 15:32, Mirek Dobsicek wrote:
>
>
>> 3) compute { } ^ { } and card({ } ^ { })
>
>> If card(A) = n, and card(B) = m. What is
>> card(A^B)?
>
> I find it neat to write | {} ^ {} | = | { {} } | = 1 :-)
You will make panic those who are not familiar with symbols!
>
> It's almost like ASCII art. Just wanted to signal that I'm following.
Thank for telling me.
OK, people, good time to solve the problems. Please don't read this
post, unless you find it is the good time for you to do some math. If
not, postpone until a good time. Technical posts have to be studied,
not read. Take the time needed. Tell me if I am too quick.
The solution of "3)" has been given.
Let us look at:
>> If card(A) = n, and card(B) = m. What is
>> card(A^B)?
card(A) = n
This means A is a finite set with n elements.
card(B) = m
This means B is a finite set with m elements.
Let us simplify by supposing that m = 3, and n = 2. Hoping that the
reasoning done for finding the solution on the particular case will
inspire the reasoning for finding the solution in the general case.
Let us imagine that A is the set {a, b, c}, with its three elements,
and that B is the set {1, 2}, with its two elements.
And let us try now to remember what is the question.
The question is: what is card(A^B)?
Well, card(A^B) is the number of elements of A^B. By definition of
the cardinal.
What is A^B?
Well, A^B is the set of functions from B to A. By definition of set
exponentiation.
Well, if the question was just "what is card(A^B)?", this would
provide the best solution, or the best note if you want. But the
teacher provided the information that A has n elements, and that B has
m elements, and intuitively we can bet that the number of functions
from a set to another can depend on the number of elements of each
sets involved, so that "what is card( A^B)?" meant probably how to
compute card(A^B) in function of card(A) and card(B).
Ok, we decided to look on the particular case with A = {a, b, c}, and
B = {1, 2}.
A^B = the set of all functions from B to A.
That is the set of functions from {1, 2} to {a, b, c}.
Well, let us try to find, or to build, one function from B to A.
But , here a moment of panic can occur, (empirical observation). For
the unnameable sake, what *is* a function? What is a function from B
to A.
Well, if it is an open manual home work, such panic can be eased by
looking in the math notes. You may remember the motivation or the
informal sense of what a function represents, which is a relation of
dependency, and this is in the most general sense, so that all
possible dependency are tolerated. For a function from B to A, it
means the element of A depends in function of the elements of B. Such
a dependency is well described by a couple (x, y) with x in B and y in
A.
we have (x,y) belongs-to F representing the meaning that y depends "in
the function F" of x.
Think about x as time and y as temperature.
So, a function from B to A is just a set of couples (x, y) with x in B
and y in A, with the functional restriction that x is not send to two
different values y. At each time x, you can have only one temperature y.
That is, here: a set of couples (x, y) with x in {1, 2} and y in {a,
b, c}, and such that if (1, x) belongs-to F, no other (1, y) belongs
to F.
Let us build one function from {1, 2} to {a, b, c}.
OK, 1, from B, can determine what in A ? Well, we have three
possibilities a, b and c. OK, i will use my free will to decide that
for this function I want now, 1 will determine a. So I put the couple
(1, a) in the function.
At this stage, the "function" looks like {(1, a)}.
Finished?
No, a function from a set to another one gives a values, outcomes,
outputs for all elements of its domain. I have to say what is
determine by 2, in B. OK, I will use my free will again, and decide to
add the couples (2, a).
At this stage, the function looks like {(1, a) (2, a)}.
Finished?
Yes.
We do have a function from B to A. The set {(1, a) (2, a)} describes
completely a function from B to A, a so-called "constant function".
think of 1 and 2 as moment of times, and think of a, b, c, as possible
temperature. The function {(1, a) (2, a)} describe a case here the
temperature is constant and equal to a.
Finished? No, we have to find all functions from B to A. All functions
from {1, 2} to {a, b, c}.
Well actually, we need to find only the number of such functions. For
1 I have three choices, then for 2, I have still three choices, and
the choices are independent, so that for each choice the remaining
three choice will lead to distinct functions, this make 3 X 3
functions = 9 functions:
{(1, a) (2, a)}
{(1, a) (2, b)}
{(1, a) (2, c)}
{(1, b) (2, a)}
{(1, b) (2, b)}
{(1, b) (2, c)}
{(1, c) (2, a)}
{(1, c) (2, b)}
{(1, c) (2, c)}
so A^B = {{(1, a) (2, a)}, {(1, a) (2, b)}, {(1, a) (2, c)}, ... ,
{(1, c) (2, c)}}, and card(A^B) = 9. In this case. This give all the
way the a, b, c can depend on 1 and 2.
I stop here. I let you train on the following question:
How many functions from {a, b, c} to {1, 2}?
How many functions from {1, 2, 3, 4, 5} to {a, b, c, d, e}?
What is the general solution, in term of cardinal n and m of the sets
involved ? (the original question).
Take your time, and ask any question. This is the type of stuff rather
easy for exact scientists, and rather new for those who buried math in
their unconscious in high school, so take each your own time.
I hope I am not too long. We will see many many examples of functions.
Bruno
Mirek Dobsicek
> Well, A^B is the set of functions from B to A. By definition of set
> exponentiation.
I'd just like to point out that Bruno in his previous post in the seven
step serii made a small typo
"A^B - the set of all functions from A to B."
It should have been from B to A. The latest post is correct in this respect.
mirek
Bruno Marchal
On 11 Aug 2009, at 22:24, Mirek Dobsicek wrote:
>
>
>> Well, A^B is the set of functions from B to A. By definition of set
>> exponentiation.
>
> I'd just like to point out that Bruno in his previous post in the
> seven
> step serii made a small typo
>
> "A^B - the set of all functions from A to B."
I wrote that? I was wrong. Thanks for saying.
>
>
> It should have been from B to A.
Yes!
> The latest post is correct in this respect.
Thank God!
Apologies for typos, mispelling, and believe me, I can do even bigger
mistakes. I will. Be vigilant.
Bruno
Bruno Marchal
If card(A) = n, and card(B) = m. What is
card(A^B)?