The seven step-Mathematical preliminaries
Hi Kim, Hi Marty and others,
So it is perhaps time to do some math. Obviously, once we are open to
the idea that the fundamental reality could be mathematical, it is
normal to take some time to do some mathematics. Many people seems
also to agree here that the computationalist hypothesis could be
interesting, and this should motivate for some amount of theoretical
computer science, or recursion theory. This is a branch of mathematics
which study computability, and mainly uncomputability, as opposed to
computation theory which study all aspect of computation.
You can easily show that something is computable, by giving an
explicit procedure to compute it. But to show that something is NOT
computable, you need a very solid notion of computability. Now,
computationalism suggest that the interesting and fundamental things,
like life, consciousness, even matter, lives somehow on the border
of the computable and the non computable, so that it is perhaps time
to dig a little bit deeper in those direction.
I have already explain this on this list, but never from scratch,
having in mind those who are, for whatever reason, the mathematical
basis.
So I guess that many of view will find those preliminaries a bit too
much simple. yet, by experience, I know that difficulties will appear,
and it is frequent that I met people with very big baggages in
mathematics who have some difficulties to understand the final point.
So I would encourage everyone to be sure everything is clear. For
those who have already a thorough understanding of UDA1-7, and in
particular have grasped the difference between a computation and a
description of a computation, I ask them to just be patient with the
list. There will be nothing new here, nothing really original, and
nothing controversial. All what I will explain has been anticipate by
Emil Post in the 1920, and found or rediscovered by many
mathematicians independently in USA, and in the ex USSR.
The present post just give a preview, and the beginning . I intent to
send only short posts, or the shorter as possible. So here is the plan.
1) Set (probably a dozen of posts)
2) Function (I don't know how many posts, could depend on the replies.
It is the key notion of math)
3) Language, machine and computable function
4) Universal machine, universal language, universal function,
universal dovetailer, universal number ...
How will I proceed?
By exercise only. I will ask question, and I will wait for either the
answer. I expect that those who know wait for Kim's answer, or for
answer by those who are not supposed to know.
Kim seems to courageously accept the role of the candid, but if
someone else want to answer it is OK for me.
I will give only VERY SIMPLE exercise. The goal is to be short and
simple, and as informal as possible. I will explain by examples, and I
will avoid as much as possible tedious definitions. And when we will
meet a more difficult question, I will just solve it myself, and even
explain why it is difficult. So, those preliminaries will not be very
funny. I will of course adapt myself to the possible replies, and fell
free to comment or even metacomment.
Obviously this is a not a course in math, but it is an explanation
from scratch of the seven step of the universal dovetailer argument.
It is a shortcut, and most probably we will make some digression from
time to time, but let us try not to digress too much.
Kim, you are OK with this? I have to take into account the problem you
did have with math, and which makes this lesson a bit challenging for
me, and I guess for you too.
I begin with the very useful and elementary notion of set, as
explained in what is called naive set theory, and which is the base
of almost all part of math.
= begin
===
1) SET
Informal definition: a set is a collection of object, called elements,
with the idea that it, the collection or set, can be considered itself
as an object. It is a many seen as a one, if you want. If the set is
not to big, we can describe it exhaustively by listing the elements,
if the set is bigger, we can describe it by some other way. Usually we
use accolades {, followed by the elements, separated by commas, and
then }, in the exhaustive description of a set.
Example/exercise:
1) The set of odd natural numbers which are little than 10. This is a
well defined, and not to big set, so we can describe it exhaustively by
{1, 3, 5, 7, 9}. In this case we say that 7 belongs to {1, 3, 5, 7, 9}.
Exercise 1: does the number 24 belongs to the set {1, 3, 5, 7, 9}?
2) the set of even natural number which are little than 13. It is {0,
2, 4, 6, 8, 10, 12}. OK? Some people can have a difficulty which is
not related to the notion of set, for example they can ask themselves
if zero (0) is really an even number. We will come back to this.
3)
Re: The seven step-Mathematical preliminaries
Thank you for starting this discussion. I have only joined recently and have little knowledge of your research. To see it laid out in the sequence you describe should make it clear to me what it is all about. I'm particularly interested in the interaction between consciousness and computation. In Max Tegmark's Ensemble TOE paper he alludes to a self-aware structure. I take structure to be an object of study in logic (model theory, in particular) but am not at all sure how consciousness, which I envision self-awareness to be deeply tied to, connects to mathematics. It seems you're going to build up to a statement such as consciousness is computable OR consciousness is not computable, or something about consciousness, at least. In light of that it seems a prudent fundamental step would be to define what it means for one structure to be aware of another. This would apparently be some relation on the aggregate of all structures (which may be the entire level 4 multiverse in Tegmark's theory). Perhaps some basic fundamental step would be to provide some axioms on what this relation could be but I'm almost convinced this can't be done in a non-controversial way. I know I'm putting the cart before the horse here so I don't expect all to be revealed for some time when it occurs in your exposition. If there is some literature by yourself or others on the particular subjects and issues I mentioned, I'd appreciate links to them. -Brian Bruno Marchal wrote: Hi Kim, Hi Marty and others, So it is perhaps time to do some math. Obviously, once we are open to the idea that the fundamental reality could be mathematical, it is normal to take some time to do some mathematics. Many people seems also to agree here that the computationalist hypothesis could be interesting, and this should motivate for some amount of theoretical computer science, or recursion theory. This is a branch of mathematics which study computability, and mainly uncomputability, as opposed to computation theory which study all aspect of computation. You can easily show that something is computable, by giving an explicit procedure to compute it. But to show that something is NOT computable, you need a very solid notion of computability. Now, computationalism suggest that the interesting and fundamental things, like life, consciousness, even matter, lives somehow on the border of the computable and the non computable, so that it is perhaps time to dig a little bit deeper in those direction. I have already explain this on this list, but never from scratch, having in mind those who are, for whatever reason, the mathematical basis. So I guess that many of view will find those preliminaries a bit too much simple. yet, by experience, I know that difficulties will appear, and it is frequent that I met people with very big baggages in mathematics who have some difficulties to understand the final point. So I would encourage everyone to be sure everything is clear. For those who have already a thorough understanding of UDA1-7, and in particular have grasped the difference between a computation and a description of a computation, I ask them to just be patient with the list. There will be nothing new here, nothing really original, and nothing controversial. All what I will explain has been anticipate by Emil Post in the 1920, and found or rediscovered by many mathematicians independently in USA, and in the ex USSR. The present post just give a preview, and the beginning . I intent to send only short posts, or the shorter as possible. So here is the plan. 1) Set (probably a dozen of posts) 2) Function (I don't know how many posts, could depend on the replies. It is the key notion of math) 3) Language, machine and computable function 4) Universal machine, universal language, universal function, universal dovetailer, universal number ... How will I proceed? By exercise only. I will ask question, and I will wait for either the answer. I expect that those who know wait for Kim's answer, or for answer by those who are not supposed to know. Kim seems to courageously accept the role of the candid, but if someone else want to answer it is OK for me. I will give only VERY SIMPLE exercise. The goal is to be short and simple, and as informal as possible. I will explain by examples, and I will avoid as much as possible tedious definitions. And when we will meet a more difficult question, I will just solve it myself, and even explain why it is difficult. So, those preliminaries will not be very funny. I will of course adapt myself to the possible replies, and fell free to comment or even metacomment. Obviously this is a not a course in math, but it is an explanation from scratch of the seven step of the universal dovetailer argument. It is a shortcut, and most probably we will make some digression from
Re: The seven step-Mathematical preliminaries
Bruno Marchal skrev:
4) The set of all natural numbers. This set is hard to define, yet I
hope you agree we can describe it by the infinite quasi exhaustion by
{0, 1, 2, 3, ...}.
Let N be the biggest number in the set {0, 1, 2, 3, ...}.
Exercise: does the number N+1 belongs to the set of natural numbers,
that is does N+1 belongs to {0, 1, 2, 3, ...}?
--
Torgny Tholerus
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RE: The seven step-Mathematical preliminaries
Date: Tue, 2 Jun 2009 19:43:59 +0200
From: tor...@dsv.su.se
To: everything-list@googlegroups.com
Subject: Re: The seven step-Mathematical preliminaries
Bruno Marchal skrev:
4) The set of all natural numbers. This set is hard to define, yet I
hope you agree we can describe it by the infinite quasi exhaustion by
{0, 1, 2, 3, ...}.
Let N be the biggest number in the set {0, 1, 2, 3, ...}.
Exercise: does the number N+1 belongs to the set of natural numbers,
that is does N+1 belongs to {0, 1, 2, 3, ...}?
Not every well-ordered set has a largest member. Every well-ordered set has a
size represented by an ordinal (see
http://en.wikipedia.org/wiki/Ordinal_number ) and there is a particular type of
ordinal called a limit ordinal which has no largest member, as discussed in
the section of that article at
http://en.wikipedia.org/wiki/Ordinal_number#Successor_and_limit_ordinals
Of course this is just how it works in set theory, I think you have said you
are some type of finitist so unlike a set theorist you may not want to allow
sets with no largest member, but in this case you shouldn't even use notation
like {0, 1, 2, 3, ...} that does not specify the largest member. I suppose
instead you could write something like {0, 1, 2, 3, ..., N} but in this case
you should specify what N is supposed to represent...the largest finite number
that any human has conceived of up to the present date? The number of distinct
physical entities in the universe (or the observable universe)? For a finitist
what defines largest, and can it change over time?
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Re: The seven step-Mathematical preliminaries
On 02 Jun 2009, at 18:54, Brian Tenneson wrote: Thank you for starting this discussion. I have only joined recently and have little knowledge of your research. To see it laid out in the sequence you describe should make it clear to me what it is all about. I'm particularly interested in the interaction between consciousness and computation. In Max Tegmark's Ensemble TOE paper he alludes to a self-aware structure. I take structure to be an object of study in logic (model theory, in particular) but am not at all sure how consciousness, which I envision self-awareness to be deeply tied to, connects to mathematics. It seems you're going to build up to a statement such as consciousness is computable OR consciousness is not computable, or something about consciousness, at least. In UDA, I avoid the use of consciousness. I just use the hypothesis that consciousness, or first person experience remains unchanged for a functional substitution made at the correct comp substitution level (this is the comp hypothesis). Then the UD Argument is supposed to show, that physicalism cannot be maintained and that physics is a branch of computer science, or even just number theory. In AUDA, I refine the constructive feature of UDA to begin the extraction of physics. You can read my paper here, and print the UD slides, because I currently refer often to the steps of that reasoning: http://iridia.ulb.ac.be/~marchal/publications/SANE2004MARCHALAbstract.html I have written a better one, but I must still put it in my webapge. It seems to me that Tegmark is a bit fuzzy about the way he attaches the first person experience with the universes/bodies. Like many physicists, he is a bit naive about the mind-body problem. The computationalist hypothesis is not a solution per se, just a tool making it possible to reformulate the problem. Indeed it forces a reduction of the mind-body problem to a highly non trivial body problem. It is my whole point. UDA shows that if I am a machine then the universe, whatever it may be, cannot be a machine. An apparent physical universe can, and actually must emerge, from inside, but this one too cannot be entirely described by a machine. In light of that it seems a prudent fundamental step would be to define what it means for one structure to be aware of another. In AUDA, the arithmetical and more constructive version of UDA, consciousness, like truth, will appear to be undefinable, except by some fixed point of the doubting procedure. It is then show equivalent to an instinctive bet on a reality. It has a relative self-speeding role. This would apparently be some relation on the aggregate of all structures (which may be the entire level 4 multiverse in Tegmark's theory). Perhaps some basic fundamental step would be to provide some axioms on what this relation could be but I'm almost convinced this can't be done in a non-controversial way. Computationalism is not controversal, nor is my deduction, but few people get both the quantum difficulties and the mathematical logic. I am more ignored than misunderstood, and then I don't publish so much. But I love to explain to people with a genuine interest in those issues. I know I'm putting the cart before the horse here so I don't expect all to be revealed for some time when it occurs in your exposition. If there is some literature by yourself or others on the particular subjects and issues I mentioned, I'd appreciate links to them. Almost all my papers dig on that issue. See my url http://iridia.ulb.ac.be/~marchal/ or search in this list for explanations. What is new, and counterintuitive is that computationalism entails a reversal between physics and machine's biology/psychology/theology See my paper on Plotinus for a presentation of AUDA in term of Plotinus (neo)platonist theology. We cannot define consciousness (nor the notion of natural numbers), but we don't have to define those things to reason about, once we agree on some principles (like the yes doctor and Church thesis). Welcome aboard on the train Brian, Bruno --~--~-~--~~~---~--~~ You received this message because you are subscribed to the Google Groups Everything List group. To post to this group, send email to everything-list@googlegroups.com To unsubscribe from this group, send email to everything-list+unsubscr...@googlegroups.com For more options, visit this group at http://groups.google.com/group/everything-list?hl=en -~--~~~~--~~--~--~---
Re: The seven step-Mathematical preliminaries
On 02 Jun 2009, at 19:43, Torgny Tholerus wrote:
Bruno Marchal skrev:
4) The set of all natural numbers. This set is hard to define, yet I
hope you agree we can describe it by the infinite quasi exhaustion by
{0, 1, 2, 3, ...}.
Let N be the biggest number in the set {0, 1, 2, 3, ...}.
Exercise: does the number N+1 belongs to the set of natural numbers,
that is does N+1 belongs to {0, 1, 2, 3, ...}?
Yes. N+1 belongs to {0, 1, 2, 3, ...}.
This follows from classical logic and the fact that the proposition N
be the biggest number in the set {0, 1, 2, 3, ...} is always false.
And false implies all propositions.
But this is a bit advanced matter, Torgny. The math I am explaining to
Kim and some others are typical classical mathematics.
Bruno
http://iridia.ulb.ac.be/~marchal/
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Re: The seven step-Mathematical preliminaries
Thanks for the links. I'll look over them and hopefully I'll understand what I see. At least if I have questions I can ask though maybe not in this thread. I don't yet know precisely what you mean by a machine but I do have superficial knowledge of Turing machines; I'm assuming there is a resemblance between the two concepts. I surmise that a machine can have an input like a question and if it halts then the question has a decidable answer, else it has no decidable answer. What about posing the following question am I a machine or the statement I am a machine and maybe some machines halt on an answer and some don't. Ie, if X is a machine, then have it attempt to compute the statement X is a machine. (I know I'm a bit fuzzy on the details.) For machines X that return X is a machine I would be inclined to think such machines possess at least some form of self-awareness, a kind of abstract self-awareness devoid of sensation (or so it would appear). -Brian Bruno Marchal wrote: On 02 Jun 2009, at 18:54, Brian Tenneson wrote: Thank you for starting this discussion. I have only joined recently and have little knowledge of your research. To see it laid out in the sequence you describe should make it clear to me what it is all about. I'm particularly interested in the interaction between consciousness and computation. In Max Tegmark's Ensemble TOE paper he alludes to a self-aware structure. I take structure to be an object of study in logic (model theory, in particular) but am not at all sure how consciousness, which I envision self-awareness to be deeply tied to, connects to mathematics. It seems you're going to build up to a statement such as consciousness is computable OR consciousness is not computable, or something about consciousness, at least. In UDA, I avoid the use of consciousness. I just use the hypothesis that consciousness, or first person experience remains unchanged for a functional substitution made at the correct comp substitution level (this is the comp hypothesis). Then the UD Argument is supposed to show, that physicalism cannot be maintained and that physics is a branch of computer science, or even just number theory. In AUDA, I refine the constructive feature of UDA to begin the extraction of physics. You can read my paper here, and print the UD slides, because I currently refer often to the steps of that reasoning: http://iridia.ulb.ac.be/~marchal/publications/SANE2004MARCHALAbstract.html I have written a better one, but I must still put it in my webapge. It seems to me that Tegmark is a bit fuzzy about the way he attaches the first person experience with the universes/bodies. Like many physicists, he is a bit naive about the mind-body problem. The computationalist hypothesis is not a solution per se, just a tool making it possible to reformulate the problem. Indeed it forces a reduction of the mind-body problem to a highly non trivial body problem. It is my whole point. UDA shows that if I am a machine then the universe, whatever it may be, cannot be a machine. An apparent physical universe can, and actually must emerge, from inside, but this one too cannot be entirely described by a machine. In light of that it seems a prudent fundamental step would be to define what it means for one structure to be aware of another. In AUDA, the arithmetical and more constructive version of UDA, consciousness, like truth, will appear to be undefinable, except by some fixed point of the doubting procedure. It is then show equivalent to an instinctive bet on a reality. It has a relative self-speeding role. This would apparently be some relation on the aggregate of all structures (which may be the entire level 4 multiverse in Tegmark's theory). Perhaps some basic fundamental step would be to provide some axioms on what this relation could be but I'm almost convinced this can't be done in a non-controversial way. Computationalism is not controversal, nor is my deduction, but few people get both the quantum difficulties and the mathematical logic. I am more ignored than misunderstood, and then I don't publish so much. But I love to explain to people with a genuine interest in those issues. I know I'm putting the cart before the horse here so I don't expect all to be revealed for some time when it occurs in your exposition. If there is some literature by yourself or others on the particular subjects and issues I mentioned, I'd appreciate links to them. Almost all my papers dig on that issue. See my url http://iridia.ulb.ac.be/~marchal/ or search in this list for explanations. What is new, and counterintuitive is that computationalism entails a reversal between physics and machine's biology/psychology/theology See my paper on Plotinus for a presentation of AUDA in term of
Re: The seven step-Mathematical preliminaries
The beauty of all this, Brian, is that the correct (arithmetically) universal machine will never been able to answer the question are you a machine?, but she (it) will be able to bet she is a (unknown) machine. She will never know which one, and she will refute all theories saying which machine she could be, unless she decided to identify herself with the virgin, never programmed, universal one. There is a way to attribute a first person view to a machine, but then, from that first person view, the machine will be correct in saying I am not a machine. The consequence of computationalism are so much counterintuitive that even machines cannot really believe in comp. Yet, those machine which believe in the numbers and induction will be able to explain exactly all this. Machines can prove that if they are a correct machine, then they cannot believe that they are a correct machine. It is related to the incompleteness phenomenon and the logic of self- reference which is exploited in the AUDA. Actually it works also for Turing hypermachines, and a vast collection of machine extensions, and even self-aware structures completly unrelated to machine, which unfortunately needs a lot of model theory to be described (like truth in all transitive model of ZF, if this rings a bell). But here we anticipate a lot. Hope this can open your appetite. Bruno On 02 Jun 2009, at 21:08, Brian Tenneson wrote: Thanks for the links. I'll look over them and hopefully I'll understand what I see. At least if I have questions I can ask though maybe not in this thread. I don't yet know precisely what you mean by a machine but I do have superficial knowledge of Turing machines; I'm assuming there is a resemblance between the two concepts. I surmise that a machine can have an input like a question and if it halts then the question has a decidable answer, else it has no decidable answer. What about posing the following question am I a machine or the statement I am a machine and maybe some machines halt on an answer and some don't. Ie, if X is a machine, then have it attempt to compute the statement X is a machine. (I know I'm a bit fuzzy on the details.) For machines X that return X is a machine I would be inclined to think such machines possess at least some form of self-awareness, a kind of abstract self-awareness devoid of sensation (or so it would appear). -Brian Bruno Marchal wrote: On 02 Jun 2009, at 18:54, Brian Tenneson wrote: Thank you for starting this discussion. I have only joined recently and have little knowledge of your research. To see it laid out in the sequence you describe should make it clear to me what it is all about. I'm particularly interested in the interaction between consciousness and computation. In Max Tegmark's Ensemble TOE paper he alludes to a self-aware structure. I take structure to be an object of study in logic (model theory, in particular) but am not at all sure how consciousness, which I envision self-awareness to be deeply tied to, connects to mathematics. It seems you're going to build up to a statement such as consciousness is computable OR consciousness is not computable, or something about consciousness, at least. In UDA, I avoid the use of consciousness. I just use the hypothesis that consciousness, or first person experience remains unchanged for a functional substitution made at the correct comp substitution level (this is the comp hypothesis). Then the UD Argument is supposed to show, that physicalism cannot be maintained and that physics is a branch of computer science, or even just number theory. In AUDA, I refine the constructive feature of UDA to begin the extraction of physics. You can read my paper here, and print the UD slides, because I currently refer often to the steps of that reasoning: http://iridia.ulb.ac.be/~marchal/publications/SANE2004MARCHALAbstract.html I have written a better one, but I must still put it in my webapge. It seems to me that Tegmark is a bit fuzzy about the way he attaches the first person experience with the universes/bodies. Like many physicists, he is a bit naive about the mind-body problem. The computationalist hypothesis is not a solution per se, just a tool making it possible to reformulate the problem. Indeed it forces a reduction of the mind-body problem to a highly non trivial body problem. It is my whole point. UDA shows that if I am a machine then the universe, whatever it may be, cannot be a machine. An apparent physical universe can, and actually must emerge, from inside, but this one too cannot be entirely described by a machine. In light of that it seems a prudent fundamental step would be to define what it means for one structure to be aware of another. In AUDA, the arithmetical and more constructive version of UDA, consciousness, like truth, will appear to be undefinable, except by
Re: Consciousness is information?
On 02 Jun 2009, at 18:46, Kelly Harmon wrote: First, in the multiplication experience, the question of your choice is not addressed, nor needed. The question is really: what will happen to you. You give the right answer above. You're saying that there are no low probability worlds? Or only that they're outnumbered by the high probability worlds? The last. Low probability world exists but not only it is rare to access them, but it super-rare to remain in them, well, if comp succeeds! I guess I'm not clear on what you're getting at with this pixel thought-experiment. The UD is the many-world, or many-histories. The 2^big movies multiplication is a tiny trivial part of the UD, and being immaterialist you should understand that we are doing all the time this thought experiment. If we don't succeed in justifying why things look normal, comp has to be abandoned. We have to explain why the computational histories win when the UD plays the trick of generating a continuum of non computational histories. The computational histories which will win are those who entangled with the non computational histories so as to make normality inherited by the computational one. Somehow. Have you understand UDA1-6?, because I think most get those steps. I will soon explain in all details UDA-7, which is not entirely obvious. If you take your own philosophy seriously, you don't need UDA8. But it can be useful to convince others, of the necessity of that philosophy, once we bet on the comp hyp. I think I have a good grasp of 1 through 6. Cool, I am just explaining UDA-7, in all details, from scratch. Bruno http://iridia.ulb.ac.be/~marchal/ --~--~-~--~~~---~--~~ You received this message because you are subscribed to the Google Groups Everything List group. To post to this group, send email to everything-list@googlegroups.com To unsubscribe from this group, send email to everything-list+unsubscr...@googlegroups.com For more options, visit this group at http://groups.google.com/group/everything-list?hl=en -~--~~~~--~~--~--~---
Re: The seven step-Mathematical preliminaries
What is the definition of a machine? I have a sense that there is an intuitive one but not an explicit one, appropriate to the discussions here. James - Original Message From: Bruno Marchal marc...@ulb.ac.be To: everything-list@googlegroups.com Sent: Tuesday, June 2, 2009 12:29:47 PM Subject: Re: The seven step-Mathematical preliminaries The beauty of all this, Brian, is that the correct (arithmetically) universal machine will never been able to answer the question are you a machine?, but she (it) will be able to bet she is a (unknown) machine. She will never know which one, and she will refute all theories saying which machine she could be, unless she decided to identify herself with the virgin, never programmed, universal one. There is a way to attribute a first person view to a machine, but then, from that first person view, the machine will be correct in saying I am not a machine. The consequence of computationalism are so much counterintuitive that even machines cannot really believe in comp. Yet, those machine which believe in the numbers and induction will be able to explain exactly all this. Machines can prove that if they are a correct machine, then they cannot believe that they are a correct machine. It is related to the incompleteness phenomenon and the logic of self- reference which is exploited in the AUDA. Actually it works also for Turing hypermachines, and a vast collection of machine extensions, and even self-aware structures completly unrelated to machine, which unfortunately needs a lot of model theory to be described (like truth in all transitive model of ZF, if this rings a bell). But here we anticipate a lot. Hope this can open your appetite. Bruno On 02 Jun 2009, at 21:08, Brian Tenneson wrote: Thanks for the links. I'll look over them and hopefully I'll understand what I see. At least if I have questions I can ask though maybe not in this thread. I don't yet know precisely what you mean by a machine but I do have superficial knowledge of Turing machines; I'm assuming there is a resemblance between the two concepts. I surmise that a machine can have an input like a question and if it halts then the question has a decidable answer, else it has no decidable answer. What about posing the following question am I a machine or the statement I am a machine and maybe some machines halt on an answer and some don't. Ie, if X is a machine, then have it attempt to compute the statement X is a machine. (I know I'm a bit fuzzy on the details.) For machines X that return X is a machine I would be inclined to think such machines possess at least some form of self-awareness, a kind of abstract self-awareness devoid of sensation (or so it would appear). -Brian Bruno Marchal wrote: On 02 Jun 2009, at 18:54, Brian Tenneson wrote: Thank you for starting this discussion. I have only joined recently and have little knowledge of your research. To see it laid out in the sequence you describe should make it clear to me what it is all about. I'm particularly interested in the interaction between consciousness and computation. In Max Tegmark's Ensemble TOE paper he alludes to a self-aware structure. I take structure to be an object of study in logic (model theory, in particular) but am not at all sure how consciousness, which I envision self-awareness to be deeply tied to, connects to mathematics. It seems you're going to build up to a statement such as consciousness is computable OR consciousness is not computable, or something about consciousness, at least. In UDA, I avoid the use of consciousness. I just use the hypothesis that consciousness, or first person experience remains unchanged for a functional substitution made at the correct comp substitution level (this is the comp hypothesis). Then the UD Argument is supposed to show, that physicalism cannot be maintained and that physics is a branch of computer science, or even just number theory. In AUDA, I refine the constructive feature of UDA to begin the extraction of physics. You can read my paper here, and print the UD slides, because I currently refer often to the steps of that reasoning: http://iridia.ulb.ac.be/~marchal/publications/SANE2004MARCHALAbstract.html I have written a better one, but I must still put it in my webapge. It seems to me that Tegmark is a bit fuzzy about the way he attaches the first person experience with the universes/bodies. Like many physicists, he is a bit naive about the mind-body problem. The computationalist hypothesis is not a solution per se, just a tool making it possible to reformulate the problem. Indeed it forces a reduction of the mind-body problem to a highly non trivial body problem. It is my whole point. UDA shows that if I am a machine then the universe, whatever it may be, cannot be a machine. An apparent physical universe can, and actually must
Re: The seven step-Mathematical preliminaries
Bruno,
I appreciate the simplicity of the examples. My answers follow the
questions.marty a.
- Original Message -
From: Bruno Marchal marc...@ulb.ac.be
= begin
===
1) SET
Informal definition: a set is a collection of object, called elements,
with the idea that it, the collection or set, can be considered itself
as an object. It is a many seen as a one, if you want. If the set is
not to big, we can describe it exhaustively by listing the elements,
if the set is bigger, we can describe it by some other way. Usually we
use accolades {, followed by the elements, separated by commas, and
then }, in the exhaustive description of a set.
Example/exercise:
1) The set of odd natural numbers which are little than 10. This is a
well defined, and not to big set, so we can describe it exhaustively by
{1, 3, 5, 7, 9}. In this case we say that 7 belongs to {1, 3, 5, 7, 9}.
Exercise 1: does the number 24 belongs to the set {1, 3, 5, 7, 9}?NO
2) the set of even natural number which are little than 13. It is {0,
2, 4, 6, 8, 10, 12}. OK? Some people can have a difficulty which is
not related to the notion of set, for example they can ask themselves
if zero (0) is really an even number. We will come back to this.
3) The set of odd natural numbers which are little than 100. This set
is already too big to describe exhaustively. We will freely describe
such a set by a quasi exhaustion like {1, 3, 5, 7, 9, 11, ... 95, 97,
99}.
Exercise 2: does the number 93 belongs to the set of odd natural
numbers which are little than 100, that is: does 93 belongs to {1, 3,
5, 7, 9, 11, ... 95, 97, 99}?
YES
4) The set of all natural numbers. This set is hard to define, yet I
hope you agree we can describe it by the infinite quasi exhaustion by
{0, 1, 2, 3, ...}.
Exercise 3: does the number 666 belongs to the set of natural numbers,
that is does 666 belongs to {0, 1, 2, 3, ...}.
YES
Exercice 4: does the real number square-root(2) belongs to {0, 1, 2,
3, ...}?
NO (a guess)
5) When a set is too big or cumbersome, mathematician like to give
them a name. They will usually say: let S be the set {14, 345, 78}.
Then we can say that 14 belongs to S, for example.
Exercise 5: does 345 belongs to S?
YES
A set is entirely defined by its elements. Put in another way, we will
say that two sets are equal if they have the same elements.
Exercise 6. Let S be the set {0, 1, 45} and let M be the set described
by {45, 0, 1}. Is it true or false that S is equal to M?
YES
Exercise 7. Let S be the set {666} and M be the set {6, 6, 6}. Is is
true or false that S is equal to M?
NO
Seven exercises are enough. Are you ready to answer them. I hope you
don't find them too much easy, because I intend to proceed in a way
such that all exercise will be as easy, despite we will climb toward
very much deeper notion. Feel free to ask question, comments, etc. I
will try to adapt myself.
SO FAR SO GOOD
Next: we will see some operation on sets (union, intersection), and
the notion of subset. If all this work, I will build a latex document,
and make it the standard reference for the seventh step for the non
mathematician, or for the beginners in mathematics.
Bruno
http://iridia.ulb.ac.be/~marchal/
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