Re: Theory of Nothing available
HI Russell, Bravo for your publication. I hope you have take into account some remarks I made :) Like Stathis I will wait for an hard copy ... Regards, Bruno Le 14-juil.-06, à 04:16, [EMAIL PROTECTED] a écrit : Russell, Congratulations on the publication of your book! I look forward to getting the hard copy in my hands, as long PDF documents give me headaches. The Australian Booksurge website does not seem to be working, so I'll try again later and use one of the other sites to order the book if it's still a problem. Stathis Papaioannou Russell Standish wrote: I'm pleased to announce that my book Theory of Nothing is now for sale through Booksurge and Amazon.com. If you go to the Booksurge website (http://www.booksurge.com, http://www.booksurge.co.uk for Brits and http://www.booksurge.com.au for us Aussies) you should get the PDF softcopy bundled with the hardcopy book, so you can start reading straight away, or you can buy the softcopy only for a reduced price. The prices are USD 16 for the hardcopy, and USD 7.50 for the softcopy. In the book, I advance the thesis that many mysteries about reality can be solved by connecting ideas from physics, mathematics, computer science, biology and congitive science. The connections flow both ways - the form of fundamental physics is constrained by our psyche, just as our psyche must be constrained by the laws of physics. Many of the ideas presented in this book were developed over the years in discussions on the Everything list. I make extensive references into the Everything list archoives, as well as more traditional scientific and philosophical literature. This book may be used as one man's synthesis of the free flowing and erudite discussions of the Everything list. Take a look at the book. I should have Amazon's search inside feature wokring soon. In the meantime, I have posted a copy of the first chapter, which contains a precis of the main argument, at http://parallel.hpc.unsw.edu.au/rks/ToN-chapter1.pdf -- *PS: A number of people ask me about the attachment to my email, which is of type application/pgp-signature. Don't worry, it is not a virus. It is an electronic signature, that may be used to verify this email came from me if you have PGP or GPG installed. Otherwise, you may safely ignore this attachment. -- -- A/Prof Russell Standish Phone 8308 3119 (mobile) Mathematics 0425 253119 () UNSW SYDNEY 2052 [EMAIL PROTECTED] Australia http://parallel.hpc.unsw.edu.au/rks International prefix +612, Interstate prefix 02 -- -- 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 [EMAIL PROTECTED] For more options, visit this group at http://groups.google.com/group/everything-list -~--~~~~--~~--~--~---
Re: SV: Only Existence is necessary?
Le 12-juil.-06, à 18:06, 1Z a écrit : I mean that is what material exists regardless of any mathematical justification. So this is your main hypothesis: what is material exist. Now my problem is that a term like material is very vague in physics, and I would say experimentally vague since the birth of experimental quantum philosophy (EPR, Bell, Shimoni, Feynman, Deutsch, Bennett ...). The big problem with the notion of *primary* matter = how to relate 1-experiences with 3-experiments. The naïve idea of attaching consciousness to physical activity leads to fatal difficulties. Well, why not, if that is your definition. I understand better why you say you could introduce matter in Platonia. Plato would have disagree in the sense that matter is the shadow of the ideal intelligible reality. What is material exists. Whether Platonia exists is another matter. It is for Platonism to justify itslef in terms of the concrete reality we find oursleves in, not for concrete reality to be justify itself in terms of Platonia. It depends of the assumptions you start from. The intelligible is a quasi-empiricist mathematical epistemology. Mathematicians are supposed by Platonists to be able to perceive mathematical truth with some extra organ. That is naïve platonism. Already condemned by Plato himself and most of his followers. Read Plotinus for more on this (especially Ennead V). I don't understand what you mean by numbers don't exist at all. Well, I've never seen one. Again that would be a critics of naïve Platonism. As I have said: number n exists in Platonia means just that the proposition number n exists is true. For example I believe that the equation x^2 - 61y^2 = 1 admits integers solutions independently of any things related to me. Numbers exists in Platonia in the sense that the classical proposition 4356667654090987890111 is prime or 4356667654090987890111 is not prime is true there. It's true here. why bring Platonia into it ? I don't understand what you mean by 4356667654090987890111 is prime or not is true here. Is it false or meaningless on the moon? is it false or meaningless beyond the solar system? is it false or meaningless beyond the Milky Way? they they cannot even produce the mere appearance of a physical world, as Bruno requires. Why? What doesn't exist at all cannot underpin the existence of anything -- even of an illusion. I do agree with you. But, once we assume comp, we can attach consciousness to sheaf of computational histories (abstract computations which can be defined precisely from the Fi and the Wi: more in the diagonalization posts). Those computations are entirely defined by infinite sets of true relations among numbers. You could perhaps wait I define the Kleene predicate in the diagonalization posts. or read the beautiful work of Matiazevitch on the diophantine equations. A set of numbers is RE, i.e. is a Wi set, if and only if it is given by the zero of a diophantine polynomial. In *all* situation, when I say a number exists, or when I say a sequence of numbers exists, I only mean that the proposition expressing that existence is true independently of me or you. With Church thesis all computations, as defined in computer science (not in physics), exists in Platonia, exactly in the same sense that for the prime numbers above. That is a most unhelpful remark. All you said above is that true mathematical sentences have truth-values independent of you. You have now started treating that as a claim about existence. It is as if your are using is true and exists as synonyms. You did not read carefully what I have said. I am just using exists as a quantifier (in first or second order logic). Exists n P(n) = truth of exists n P(n). I believe that there is an infinity of twin primes ... or not, independently of the fact that mathematicians on this planet or elsewhere will solve, or not, that (currently open) problem. And I do provide evidence that rational unitary transform could be the mathematical computations winning the measure-battle in Platonia. Huh How can you have a battle without time ? By using varieties of theoretical computer science notion of convergence. If you want, I am using the integers themselves for measuring complexity of computations. The UDA shows that if you are in the comp state S, then your consistent extensions are defined by a measure on all computations going through that state S. It is a static well defined mathematical set. A type of computation wins the measure-battle if it has a reasonable measure. This would explain not only the existence of computations with self-aware observers, but also they relative stability.@ But MUCH more can be said, from Solovay theorem (justifying the modal logics G and G* for the provable and non provable by a machine/entity self-referential truth) I get not only an arithmetical
Re: Diagonalization (solution-sequel)
Bruno Marchal wrote:
Le 10-juil.-06, à 21:55, Tom Caylor a écrit :
With Church thesis, Fortran is a Universal Language, and a fortran
interpreter is a Universal Machine, where Universal means it
computes
(at least) all computable functions. Fortran programs are recursively
(computably, mechanically, effectively) enumerable, so
G = Fn(n) + 1
is programmable, notably in fortran. So there is fortran code for G
and
it exists in the enumeration
of all fortran programs. So there is a number k such that G = Fk. So
G(k) = Fk(k) = Fk(k) + 1. So Fk(k) cannot be defined and it makes the
Universal Machine run for ever (crash). So, the notorious other
beasts are the *partial* recursive function. They are functions from
*subset* of N, called domain, in N.
OK. I noticed that you can get the Universal Machine (UM) to run for
ever even without the + 1. If I think of the program for G as a big
case statement with cases 1, 2, 3, to infinity, then the case for k
will contain the code for, or better yet a call to (hence the name
recursive?), Fk(k), but if we state by defining even G = Fn(n) (even
without the + 1) then this is equivalent to calling G(k)... But then
when we call G(k) we end up back in the k case again, calling G(k)
again,... forever.
I'm not sure. I'm afraid your argument could be machine or language
dependent.
But with Church's Thesis how could it be machine or language dependent?
Another way of arguing without the + 1 is this: Define G(n) = Fn(n)
for all n. If G is in the list of Fi's, then G=Fk for some fixed k.
So Fk(n) = Fn(n) for all n. Now if all you're thinking of is a matrix
of numbers Fx(y) (a lookup table if you will) with rows numbered by x,
and columns numbered by y, then this doesn't seem problematic (unless
you introduce the + 1). But such a lookup table is infinite and
therefore is not allowed as the code of a computable function. You
need code for the functions Fi. Specifically, you need code for the
function Fk (=G). What does this code look like, even in a universal
sense? Well Fk(n) = G(n) = Fn(n) for all n, so Fk would have to have
some code to compute Fk(1)=F1(1), Fk(2)=F2(2), Fk(3)=F3(3), ...Fk(k)=?,
...
How does Fk *ever* know what to compute for Fk(k)? This is actually
rather funny to me. It's like me being my own grandpa. It seems that
there already is a case of G(n) not being defined for n=k, even without
the + 1.
The key point now, is that the recursively enumerable sequence Fi
give us a sort of coordinate system for reasoning about programs. To
fix some Universal machine or language is the equivalent of fixing
some
reference frame in geometry. And then we can reason in a way which
does
not depend on which Universal Machine has been chosen.
Now Fi denotes just the partial function programmed by the ith fortran
program. So Fi has a domain. It is written Wi. That is: Wi = domain of
Fi.
Exercises:
1) show that
A) all the Wi are recursively enumerable (mechanically generable = in
the range of a total computable function, or empty).
B) all the recursively enumerable sets are among the Wi
I don't have time to word together arguments for all of these, but I
drew pictures. Let's see. Each Wi is a subset of N, so it is easy to
see how each Wi could be in (a subset of) the range (output) of a
function from N to N, so A follows.
You are too much quick here. The set R of codes of total computable
functions is also a subset of N, but this does not entail R is the
range of a total computable function. The diag2 and diag3 already
showed that R cannot be such a range. Oops, I realize I should not have
said in the range of some Fi but the range of some Fi (my fault).
OK. For 1A I'm not sure whether you mean {the whole set of Wi's} is
RE, or each and every {Wi, for a given i} is RE. I think you mean
the first one. I think that we have to use the fact that the set of
Fi's is RE (=the range of a total computable function). However, I
can't see how that would make the set of domains Wi for all i, RE. I
was thinking along the lines of a composition of two total computable
functions would be a total computable function, but it seems that the
Fi's and Wi's apples and oranges, since the Fi's are the (code of?)
functions, and the Wi's are the domains of the Fi's.
Each RE set is a subset of N. But it is not just any subset of N, is
it? Likewise the set of all Wi's cannot be the set of *all* subsets of
N, can it? This would be not enumerable.
Right.
This was trying to address 1B. I have a feeling that all the RE sets
being among the Wi has something to do with the Church Thesis
statement (below) about all partial computable functions being among
the Fi.
2) Conclude that the following version of Church thesis are
equivalent:
A set is intuitively (effectively, mechanically) generable iff it is
recursively enumerable (RE)
This one seems
RE: Re: A calculus of personal identity
I am not so sure that the standard model of personal identity with which we are familiar would be a universal standard. Imagine intelligent beings evolved from hive insects whichgo through several radically different life stages, frequently share genetic informationwith each other like bacteria, identify self and othersvia pheromones which can change or be transferred to other individuals... the possibilities are endless. These beings would have an utterly alien psychology, ethics, aesthetics, and probably also an utterly alien sense of what it means to be a person, including what it means to be the same person from one life stage to another.However, if they were intelligent, they would come up with the same scientific truths as us, even if they thought about them very differently, because such truths are in a fundamental sense observer-independent. Perhaps we have reached a consensus of sorts (Brent and Lee, let me know if you disagree): evolution has given us brains hardwired witha sense of continuity of personal identity over time for very good reasons, but it could have been otherwise, and it would not have been inconsistent with any logical or empirical fact about the world had it been otherwise. On the other hand, evolution has also given us brains which tend to believe that the Earth is flat and that there is an absolute up and down in the universe, also for fairly good reasons. However, in the latter case, the received belief *is* inconsistent with empirical facts about the world. This is a basic, and I think not immediately obvious, difference between beliefs about personal identity and logical or empirical facts. Stathis Papaioannou Date: Wed, 12 Jul 2006 17:25:07 -0700 From: [EMAIL PROTECTED] To: everything-list@googlegroups.com Subject: Re: A calculus of personal identity StathisPapaioannouwrote: BrentMeekerwrites: Iwouldsaythatwhatmakesastatementlike"we'rethesamepersonfrommomenttomoment"true isthatit'saninferencefrom,orapartof,amodeloftheworldthatis"true"inthe provisionalsenseofscientifictheories,i.e.itsubsumesandpredictsmanyemprically verifiedobservations(e.g.ifIwakeyouupinthemiddleofthenightandaskyouyourname you'llreply'Stathis')andithasnotmadeanyfalsifiedpredictions.Sointhissensewe couldsaythatourmodelofpersonhoodisbetterthanthatoftheday-people-notinthesense thatwecanshowtheirsisfalse,butinthesensethatourshasgreaterpredictivepowerand scope. IfIwereaday-personandyouwokemeinthemiddleofthenight,Iwouldsaythattheperson whowenttobedlastnightwasStathis-1andthepersonnowawakeisStathis-2.Iwouldagree thatStathis-1andStathis-2arecomprisedofmostlythesamematterandhavesimilarmental attributes,butthefactremains,thebrainsofmyspecieshaveevolvedsothatwakingupfrom sleepmakesthembelievetheyareanewperson.Thisisn'tamodeloratheory;it'smorelike reportingthatI'mhungry,orfrightened.Philosophicalproblemsarisewhenthisfeelingof continuityofidentity(orlackofit)isequatedwithsomeempiricalfact.Ithappensthatin ourownevolutionphysicalandmentalcontinuityhasbeenstronglycorrelatedwiththesubjective feelingofcontinuityofidentity,anditistemptingtosaythatthereforephysicalandmental continuityisequivalenttoor(slightlyweaker)necessitatescontinuityofidentity.However, thisdefaultmodelthatweallusedaytodayisflawedontwocounts.Firstly,thecorrelation isnotnecessary,butcontingentonevolutionarycircumstances.Itiseasyenoughtoimagine rationalbeingsliketheday-peoplewhohaveacompletelydifferentapproachtopersonal identity.Secondly,thedefaultmodelisnoteveninternallyconsistent,asshowninduplication thoughtexperiments.IfIamtobeduplicatedtomorrowandoneofthecopiestortured,Iam worried;butwhentomorrowcomes,andIamnottortured,Iamrelieved.HowisitthatI"become" oneorothercopywhenmymentalcontinuitywithbothisthesame?Thereisnoambiguityinthe empiricalfacts,butthereisambiguityinhowIexperiencecontinuityofidentity-because thesearetwodifferentthingsandthereisnosimple,consistentrelationshipbetweenthem. Well,thedefaultmodel,personalcontinuity,isconsistentabsentduplications...andthereain't anyyet. Myexampleofwakingyouupandaskingyournamewasaweakone.IagreewithLeethatthetestof amodelisinthebehavoiritpredicts(andnotjustthevocalbehavoir).AndonthatbasisIthink themodelofpersonalcontinuitywouldbeabetterone,andyoumightevenconvinceaday-personof it;Justthereverseoftryingconvincepeopleherethatthereisn't*really*continuity.Ofcourse iftheydidn'tactasiftherewerepersonalcontinuity,theirphysicalcontinuitywouldlikelyend. BrentMeeker --~--~-~--~~~---~--~~ 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 [EMAIL PROTECTED] For more options, visit this group at http://groups.google.com/group/everything-list -~--~~~~--~~--~--~---
Re: Infinities, cardinality, diagonalisation
Hi Quentin, Tom and List,
Of course, N is the set of finite positive integers:
N = {0, 1, 2, 3, ...}.
An infinite set A is countable or enumerable if there is a computable bijection between A and N.
Forgetting temporarily the number zero, all finite number can be put in the shapes:
|
||
|||
|
||
|||
...
This raises already an infinitely difficult problem: how to define those finite numbers to someone who does not already have some intuition about them. The answer is: IMPOSSIBLE. This is part of the failure of logicism, the doctrine that math can be reduced to logic. technically this can be explained through mathematical logic either invoking the incompleteness phenomenon, or some result in model theory (for example Lowenheim-Skolem results).
But it is possible to experience somehow that impossibility by oneself without technics by trying to define those finite sequence of strokes without invoking the notion of finiteness.
Imagine that you have to explain the notion of positive integer, or natural number greater than zero to some extraterrestrials A and B. A is very stubborn, and B is already too clever (as you will see).
So, when you present the sequence:
| || ||| ...
A replies that he has understood. Numbers are the object |,and the object ||, and the object |||, and the object ...”. So A conclude there are four numbers. You try to correct A by saying that ... does not represent a number, but does represent some possibility of getting other numbers by adding a stroke | at their end. From this A concludes again that there is four numbers: |, ||, |||, and . You try to explain A that ... really mean to can continue to add the |; so A concludes that there are five numbers now. So you will try to explain to A that ... means you can continue to add the I as many times as you want. From this A will understand that the set of numbers is indefinite: it is
{|, ||, |||, , |} or {|, ||, |||, , |, ïï} or some huge one but similarly ... finite.
Apparently A just doesn't grasp the idea of the infinite.
B is more clever. After some time he seems to grasp the idea of ..., and apparently he does understand the set {|, ||, |||, , |, ...}. But then, like in Tom's post, having accepted the very idea of
infinity through the use of ..., B, in some exercise, can accept the infinite object
|...
itself as a number. How will you explain him that he has not the right to take this as a finite number. He should add that the rule, consisting in adding a stroke | at the end of a number (like |||), can only be applied a finite number of times. OK, but the problem was just that: how to define a finite number
The modern answer is that this is just impossible. The set of positive integers N cannot be defined univocally in any finite way.
This can take the form of some theorem in mathematical logic. For example: it is not possible to define the term finite in first order classical logic. There is not first order logic theory having finite model for each n, but no infinite models.
You can define finite in second order logic, but second order logic are defined through the intuition of finiteness/non-finiteness, so this does not solve the problem.
This can be used to show that comp will make the number some absolute mystery.
Now, note that B, somehow, can consider the generalized number:
|...
as a number. Obviously, this corresponds to our friend the *ordinal omega*. From the axiom that you get a number by adding a stroke at its end: you will get
omega+1, as
|...|
omega+2, as
|...||
omega+3
|...|||
...
omega+omega
||...||...
...
omega+omega+omega
||...||...||...
...
omega*omega
|||...|||...|||...|||...|||...|||...|||...|||...|||...|||...|||...|||...|||...|||...|||...|||...|||...|||... ...
... and this generate a part of the constructive countable ordinals.
And we stay in the domain of the countable structure, unless you decide to go up to the least non countable ordinals and beyond. For doing this properly you need some amount of (formal) set theory. In all case, what ... expressed is unavoidably ambiguous.
Bruno
Le 13-juil.-06, à 15:29, Quentin Anciaux a écrit :
Hi, thank you for your answer.
But then I have another question, N is usually said to contains positive integer number from 0 to +infinity... but then it seems it should contains infinite length integer number... but
Re: Infinities, cardinality, diagonalisation (errata)
Le 14-juil.-06, à 14:34, Bruno Marchal a écrit :
Hi Quentin, Tom and List,
Of course, N is the set of finite positive integers:
N = {0, 1, 2, 3, ...}.
An infinite set A is countable or enumerable if there is a computable
bijection between A and N.
Please suppress the computable in that last sentence.
Bruno
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RE: Re: A calculus of personal identity
Stathis writes I am not so sure that the standard model of personal identity with which we are familiar would be a universal standard. Imagine intelligent beings evolved from hive insects which go through several radically different life stages, frequently share genetic information with each other like bacteria, identify self and others via pheromones which can change or be transferred to other individuals... the possibilities are endless. These beings would have an utterly alien psychology, ethics, aesthetics, and probably also an utterly alien sense of what it means to be a person, including what it means to be the same person from one life stage to another. Yes, I think that that is right. However, if they were intelligent, they would come up with the same scientific truths as us, even if they thought about them very differently, because such truths are in a fundamental sense observer-independent. Right. Perhaps we have reached a consensus of sorts (Brent and Lee, let me know if you disagree): evolution has given us brains hardwired with a sense of continuity of personal identity over time for very good reasons, but it could have been otherwise, Otherwise in the sense that if we were like insects (instead of mammals, or maybe just large primate-like creatures), yes, we might not have this lingering notion that we are the same people from day to day. And the sense that (I claim) young people have that they will not be the same people when they are old. and it would not have been inconsistent with any logical or empirical fact about the world had it been otherwise. Yes, that seems so too: though no tribe of humans (or even lions, for that matter) would ever develop the notion of day-persons (see Mike Perry's book, Forever For All for his independent discussion of day-persons), that is indeed a contingent fact of evolution. On the other hand, evolution has also given us brains which tend to believe that the Earth is flat and that there is an absolute up and down in the universe, also for fairly good reasons. However, in the latter case, the received belief *is* inconsistent with empirical facts about the world. Only inconsistent, of course, when data became available that was not available in the EEA (Environment of Early Adapteness). This is a basic, and I think not immediately obvious, difference between beliefs about personal identity and logical or empirical facts. I would agree. Lee --~--~-~--~~~---~--~~ 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 [EMAIL PROTECTED] For more options, visit this group at http://groups.google.com/group/everything-list -~--~~~~--~~--~--~---
Re: A calculus of personal identity
Stathis Papaioannou wrote: I am not so sure that the standard model of personal identity with which we are familiar would be a universal standard. Imagine intelligent beings evolved from hive insects which go through several radically different life stages, frequently share genetic information with each other like bacteria, identify self and others via pheromones which can change or be transferred to other individuals... the possibilities are endless. These beings would have an utterly alien psychology, ethics, aesthetics, and probably also an utterly alien sense of what it means to be a person, including what it means to be the same person from one life stage to another. However, if they were intelligent, they would come up with the same scientific truths as us, even if they thought about them very differently, because such truths are in a fundamental sense observer-independent. Perhaps we have reached a consensus of sorts (Brent and Lee, let me know if you disagree): evolution has given us brains hardwired with a sense of continuity of personal identity over time for very good reasons, but it could have been otherwise, and it would not have been inconsistent with any logical or empirical fact about the world had it been otherwise. I agree, except I don't see how evolution could have worked it out otherwise for our kind of animal. Your thought-experimental day-people took supernatural intervention to evolve. Assuming that their outward behavoir comported with personal continuity; I'm not sure how much their inner narrative could differ from our own. To what degree could they really worry (an emotion) about their future circumstance without feeling that they would *be* that future person. Is there anything to continuity of personal identity besides a) the third-person continuity of body, memory, personality and b) the emotions related to anticipation of pain, pleasure, etc. You make a good point though that a species like the social insects must have a different kind of feeling of identity - if any. Richard Hofstader imagined an intelligent ant colony in which the mind supervened on the spatial and chemical interactions among individuals of the colony. This has also been addressed in fiction. Greg Egan wrote a short story about a person who woke up in a different body every morning. Stanislaw Lem, in one of his Star Diaries stories, has the hero land on a planet where everyone changes societal roles each day. Brent Meeker --~--~-~--~~~---~--~~ 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 [EMAIL PROTECTED] For more options, visit this group at http://groups.google.com/group/everything-list -~--~~~~--~~--~--~---
