Re: New Paper by Thomas Hertog and Stephen Hawking
On Wed, Dec 30, 2009 at 03:59:24PM +1100, Colin Hales wrote: > > > Jason Resch wrote: > > Described in this article: > > http://www.bioedonline.org/news/news.cfm?art=2617 > > > > "This summation of all paths, proposed in the 1960s by physicist > > Richard Feynman and others, is the only way to explain some of the > > bizarre properties of quantum particles, such as their apparent > > ability to be in two places at once. The key point is that not all > > paths contribute equally to the photon's behaviour: the straight-line > > trajectory dominates over the indirect ones. > > > > Hertog argues that the same must be true of the path through time that > > took the Universe into its current state. We must regard it as a sum > > over all possible histories." > > > > > > So we "must", must we? > > A mathematical construction by humans, happens to cohere to some extent > with reality. > A mere description. > > A million other descriptions, also constructed by humans, could be as > predictive of how the universe appears. > > What extra belief system must exist in order that someone conclude that > we 'must' chose a "sum of all histories" as "the" story? Why is the > universe compelled to be such a thing? > > Rhetorical question...don't answer. Just think. > > happy new year, everythingers. > > cheers > colin > "Must" in the sense that it is a "neat" idea, or even a "beautiful" idea. This will correlate to some notion of simplicity under Occam's razor. Then it is probably coupled in this case with giving the right answer. Of course it is always possible that it will be found wanting tomorrow :). -- Prof Russell Standish Phone 0425 253119 (mobile) Mathematics UNSW SYDNEY 2052 hpco...@hpcoders.com.au Australiahttp://www.hpcoders.com.au -- You received this message because you are subscribed to the Google Groups "Everything List" group. To post to this group, send email to everything-l...@googlegroups.com. To unsubscribe from this group, send email to everything-list+unsubscr...@googlegroups.com. For more options, visit this group at http://groups.google.com/group/everything-list?hl=en.
Re: New Paper by Thomas Hertog and Stephen Hawking
Jason Resch wrote: > Described in this article: > http://www.bioedonline.org/news/news.cfm?art=2617 > > "This summation of all paths, proposed in the 1960s by physicist > Richard Feynman and others, is the only way to explain some of the > bizarre properties of quantum particles, such as their apparent > ability to be in two places at once. The key point is that not all > paths contribute equally to the photon's behaviour: the straight-line > trajectory dominates over the indirect ones. > > Hertog argues that the same must be true of the path through time that > took the Universe into its current state. We must regard it as a sum > over all possible histories." > > So we "must", must we? A mathematical construction by humans, happens to cohere to some extent with reality. A mere description. A million other descriptions, also constructed by humans, could be as predictive of how the universe appears. What extra belief system must exist in order that someone conclude that we 'must' chose a "sum of all histories" as "the" story? Why is the universe compelled to be such a thing? Rhetorical question...don't answer. Just think. happy new year, everythingers. cheers colin -- You received this message because you are subscribed to the Google Groups "Everything List" group. To post to this group, send email to everything-l...@googlegroups.com. To unsubscribe from this group, send email to everything-list+unsubscr...@googlegroups.com. For more options, visit this group at http://groups.google.com/group/everything-list?hl=en.
Re: UDA query
Bruno: Is there a UD that is implemented in Fortran? Ronald On Dec 29, 4:55 am, Bruno Marchal wrote: > On 28 Dec 2009, at 21:24, Nick Prince wrote: > > > > >> Well, it is better to assume just the axiom of, say, Robinson > >> arithmetic. You assume 0, the successors, s(0), s(s(0)), etc. > >> You assume some laws, like s(x) = s(y) -> x = y, 0 ≠ s(x), the laws > >> of addition, and multiplication. Then the existence of the universal > >> machine and the UD follows as consequences. > > > Ok so the UD exists (platonically?) > > Yes. The UD exists, and its existence can be proved in or by very weak > (not yet Löbian) arithmetical theories, like Robinson Arithmetic. > The UD exists like the number 733 exists. The proof of its existence > is even constructive, so it exists even for an intuitionist (non > platonist). No need of the excluded middle principle. > > > > >> Better not to conceive them as living in some place. "where" and > >> "when" are not arithmetical predicate. The UD exists like PI or the > >> square root of 2. > >> (Assuming CT of course, to pretend the "U" in the UD is really > >> universal, with respect to computability). > > > Fine so the UD has an objective existence in spite of whatever else > > exists. > > It exists in the sense that we can prove it to exist once we accept > the statement that 0 is different from all successor (0 ≠ s(x) for > all x), etc. > If you accept high school elementary arithmetic, then the UD exists in > the same sense that prime numbers exists. > "exist" is used in sense of first order logic. This leads to the usual > philosophical problems in math, no new one, and the UDA reasoning does > not depend on the alternative way to solve those philsophical problem, > unless you propose a ultra-finitist solution (which I exclude in comp > by arithmetical realism). > > > > > > > > >> There is a "time order". The most basic one, after the successor law, > > >> is the computational steps of a Universal Dovetailer. > >> Then you have a (different) time order for each individual > >> computations generated by the UD, like > > >> phi_24 (7)^1, phi_24 (7)^2, phi_24 (7)^3, phi_24 (7)^4, ... > >> where "phi_i (j)^s" denotes the sth steps of the computation (by > >> the UD) of the ith programs on input j. > > > If the UD was a concrete one like you ran then it would start to > > generate all programs and execute them all by one step etc. But are > > you saying that because the UD exists platonically all these programs > > and each of their steps exist also and hence, by the existence of a > > successor law they have an implicit time order? > > Yes. The UD exist, and is even representable by a number. UD*, the > complete running of the UD does not exist in that sense, because it is > an infinite object, and such object does not exist in simple > arithmetical theories. But all finite parts of the UD* exist, and this > will be enough for "first person" being able to glue the computations. > For example, you could, for theoretical purpose, represent all the > running of the UD by a specific total computable function. For example > by the function F which on n gives the (number representing the) nth > first steps of the UD*. Then you can use the theorem which asserts > that all total computable functions are representable in Robinson > Arithmetic (a tiny fragment of Pean Arithmetic). That theorems is > proved in detail, for Robinson-ile arithmetic, in Boolos and Jeffrey, > or in Epstein and Carnielli. In Mendelson book it is done directly in > Peano Arithmetic. > > > > >> Then there will be the time generated by first person learning and > >> which relies eventually on a statistical view on infinities of > >> computations. > > > Is this because we are essentially constructs within these steps? > > It is because our "3-we", our bodies, or our bodies descriptions, are > constructed within these steps. But our first person are not, and no > finite pieces of the UD can give the "real experience". This is a > consequence of the first six steps: our next personal experience is > determined by the whole actual infinity of all the infinitely many > computations arrive at our current state. (+ step 8, where we abandon > explicitly the physical supervenience thesis for the computational one). > > > > >> Time is not difficult. It is right in the successor axioms of > >> arithmetic. > > > Here again you confirm the invocation of the successor axioms. > > Yes. It is fundamental. I cannot extract those from logic alone. No > more than I can define addition or multiplication without using the > successor terms s(-) : > > for all x x + 0 = x > for all x and y x + s(y) = s(x + y) > > You have to understand that all the talk on the phi_i and w_i, > including the existence of universal number > (EuAxAy phi_u() = phi_x(y)) can be translated in pure first order > arithmetic, using only s, + and *
Re: Definition of universe
> To me it would be that which is contained when you specify a number of > dimensions. 2d? The universe can be a piece of paper. But that implies that dimensionality is a fundamental property of reality. It is conceivable that dimensionality is not fundamental, but rather emergent. -- You received this message because you are subscribed to the Google Groups "Everything List" group. To post to this group, send email to everything-l...@googlegroups.com. To unsubscribe from this group, send email to everything-list+unsubscr...@googlegroups.com. For more options, visit this group at http://groups.google.com/group/everything-list?hl=en.
Re: Definition of universe
On Wed, Dec 30, 2009 at 1:07 AM, Mindey wrote: > Hello, > > I was just wondering, we are talking so much about universes, but how > do we define "universe"? Sorry if that question was answered > somewhere, but after a quick search I didn't find it. To me it would be that which is contained when you specify a number of dimensions. 2d? The universe can be a piece of paper. > Inyuki > http://www.universians.org > > -- -- silky http://www.mirios.com.au/ http://island.mirios.com.au/t/rigby+random+20 drape experimentation COLDBLOODED, verisimilitude: fragment-mum gloriously? CONTRACTOR prickl... -- You received this message because you are subscribed to the Google Groups "Everything List" group. To post to this group, send email to everything-l...@googlegroups.com. To unsubscribe from this group, send email to everything-list+unsubscr...@googlegroups.com. For more options, visit this group at http://groups.google.com/group/everything-list?hl=en.
Re: Robotic Scientist
Russell, I made my WEB-acquaintance with Hod - his interview-picture with his students reminded me of my then Cornelian son and friends, (before he was for 17 yrs in IBM's development) and saw that the 'inductive' you mention is still based on the already known(?) elements. The "creative" is relative, just as the "new" I referred to in my today's post to Bruno. Your: *"But it certainly feasible that humans are really just more so of what this machine does."* is a fair assumption, it may even be reversed. We cannot know as long as we 'think' with our human mind - restricted into the (perceived? -cf, Colins) reality of ours. Hod Lipson's good quotes include: " The central act of the coming era is to connect everything to everything." "Complexity must be grown from simple systems that already work - none pointing to previously unknowables". >From his interview: about "Machine Creativity": "I have been interested in machine creativity for many years. A lot of work has been done to make computers smart. They can play chess or drive a vehicle across the dessert. These are hallmarks of intelligence but not the ultimate artificial intelligence we are looking for. What uniquely defines human intelligence? I think the ultimate challenge is creativity and curiosity. Trying to understand what it means to be creative or curious in a way that we can imitate has been a long fascination. Can computers augment creativity or curiosity? Can computers ask intelligent questions? Generate new ideas? This is the epitome of AI. How can we make computers with these characteristics?" (I think the 'Koza' circuits are unusual combinations of elements so far applied in different patterns. So I figure also the 'Virtual Reality' games. A 'shuffling' in the memory banks). - I was undecided about your above quoted statement (Italics) because I was perplexed years ago by a question: "How do we learn 'brand-new' ideas? I could not come up with anything better than 'try the opposites of the known' or 'try the unrelated' - still ALL within our knowledge-base. A clever machine can do just that. What a clever AI machine cannot do though, while I can: Imagine a machine "trained" in solving math problems, to receive a math problem to solve it and sais unexpectedly: "OK, I will do it, but first I want to play a fugue by J.S.Bach, then I will address the problem". (To satisfy an emotional joy before going to work as constructed). This may be added to Hod's 'curiosity' (and who know how many more?) Happy 2010 John M On Sun, Dec 27, 2009 at 5:16 PM, russell standish wrote: > On Sun, Dec 27, 2009 at 10:54:53AM -0500, John Mikes wrote: > > I wonder if a 'robot' can produce a "noch nie dagewesen" (Ger. for brand > > new) unrelated idea? > > I do know Hod Lipson from the ALife community, but am not familiar > with this particular piece of research. From the WIRED article, I > understand this to be a particular implementation of inductive > reasoning by machine. It is impressive enough that this is possible, > but I don't for one minute think that they have approached the > creative power of a human being. But it certainly feasible that humans > are really just more so of what this machine does. > > Still, the whole area of machine learning, and minimum length > description has some very interesting surprises in store, which is why > I've never bought Colin's argument. For instance John Koza's genetic > programs have created several electronic circuits, some of which were > patentable, so fit the requirement of noch nicht dagewesen. > > Cheers > > -- > > > > Prof Russell Standish Phone 0425 253119 (mobile) > Mathematics > UNSW SYDNEY 2052 hpco...@hpcoders.com.au > Australiahttp://www.hpcoders.com.au > > > > -- > > You received this message because you are subscribed to the Google Groups > "Everything List" group. > To post to this group, send email to everything-l...@googlegroups.com. > To unsubscribe from this group, send email to > everything-list+unsubscr...@googlegroups.com > . > For more options, visit this group at > http://groups.google.com/group/everything-list?hl=en. > > > -- You received this message because you are subscribed to the Google Groups "Everything List" group. To post to this group, send email to everything-l...@googlegroups.com. To unsubscribe from this group, send email to everything-list+unsubscr...@googlegroups.com. For more options, visit this group at http://groups.google.com/group/everything-list?hl=en.
Bruno - puzzle
Dear Bruno, for those on the list who like to solve puzzles and got stuck with the question of my grandson: it is the word 'incorrectly' spelled as such. John -- You received this message because you are subscribed to the Google Groups "Everything List" group. To post to this group, send email to everything-l...@googlegroups.com. To unsubscribe from this group, send email to everything-list+unsubscr...@googlegroups.com. For more options, visit this group at http://groups.google.com/group/everything-list?hl=en.
Re: Robotic Scientist
Bruno,
excuse me if I suggest some circularity in you reply. A "learning machine"
is by def. learning SOMETHING and that SOMETHING comes from its inside, if
we do not specify an 'outside' it may explore (which would not be *learning*,
rather *exploring* - a quite different ballgame - maybe followed by 'and
learning *IT'*).
*The applied (ball)game of 'machine' (substituted for 'learning machine',
excluded per se from the 'exploring' function) reminds me of the puzzle of
my midle-school grandkid: which word is the ONE spelled always incorrectly
in every good dictionary? (My wife found it out, immediately, not me). For
the lucky guessers I allow a Coke on NewYear's Eve at his own expense, of
course.*
It depends on 'machine'. Independent? that. too, has to be explained. Maybe
B&B did.
Your question: "Can a machine find a new thing(?)". I refer to Russell's
"patentable" which I wanted to address: a 'new' ('patentably new'?) thing is
not necessarily a (sorry for the Ger.) "noch nie dagewesen" - it can be not
yet described (but knowable - a new combination of elements usually applied
for different patterns etc.). A good example is in this thread about
"electricity" as NOT describable to a medieval scientist: it might have been
"brand new" and unknown, but it still fits into the 'knowables', so I think
about more 'real' novelty.
E.g. cousins of the Milky Way in outer space before the telescope. That did
not fit into the Flat Earth views. - A 'better mousetrap' IS 'patentable
and new'.
I agree with your ending: " How to define "new", [for example]. It is a
relative concept."
Happy 2010
John M
On Tue, Dec 29, 2009 at 6:32 AM, Bruno Marchal wrote:
>
> On 27 Dec 2009, at 23:16, russell standish wrote:
>
> > On Sun, Dec 27, 2009 at 10:54:53AM -0500, John Mikes wrote:
> >> I wonder if a 'robot' can produce a "noch nie dagewesen" (Ger. for
> >> brand
> >> new) unrelated idea?
> >
> > I do know Hod Lipson from the ALife community, but am not familiar
> > with this particular piece of research. From the WIRED article, I
> > understand this to be a particular implementation of inductive
> > reasoning by machine. It is impressive enough that this is possible,
> > but I don't for one minute think that they have approached the
> > creative power of a human being. But it certainly feasible that humans
> > are really just more so of what this machine does.
> >
> > Still, the whole area of machine learning, and minimum length
> > description has some very interesting surprises in store, which is why
> > I've never bought Colin's argument. For instance John Koza's genetic
> > programs have created several electronic circuits, some of which were
> > patentable, so fit the requirement of noch nicht dagewesen.
>
>
> And there is the whole computational learning theory which shows that
> machine learners exists.
> Even universal learners exists, but the proofs are necessarily non
> constructive. We cannot recognize such machine even if we are in front
> of them.
> There are a lot of amazing theorems in that field. For example the
> theorem of Blum and Blum, which says that there is something
> infinitely (even non computably) more clever (in learning) than any
> machine: a couple of (independent) machines!
> Learning machines exist, and the theory explains why we cannot build
> them from scratch. Some form of learning-competence can need
> intrinsically long computations/histories, but once there, they can
> multiplied.
>
> Can a machine find a new thing. Of course, from I said above. Can we
> judge if a machine has find something new? This is hard to say. It is
> even hard to judge this in a definitive way with the discoveries made
> by humans. It would need many formal criteria in a place where
> formalization is difficult. How to define "new", for example. It is a
> relative concept.
>
>
> 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-l...@googlegroups.com.
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> everything-list+unsubscr...@googlegroups.com
> .
> For more options, visit this group at
> http://groups.google.com/group/everything-list?hl=en.
>
>
>
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Re: Definition of universe
Mindey, I hurry to reply before some smarter guys do so on this list, so here is MY opinion: I consider this OUR universe a part of the Multiverse (unknown, unknowable, but assumed) with its 'physical' (so far discovered!) built (similarly assumed) and described as (our) so called 'physical world' in (our) conventional sciences. I wrote a 'narrative' in 2000 (partly obsolete in my today's views) which is best findable in my Karl Jaspers Forum publication ( www.kjf.ca look up TA-62 - Networks-2003 under [A4] - ) which contains "my" assumptions, not agreeable to the topics on most of this list. It outlines a view about (our and other) universes in a not-so-scientific manner. Good luck to it and to other views John Mikes On Tue, Dec 29, 2009 at 9:07 AM, Mindey wrote: > Hello, > > I was just wondering, we are talking so much about universes, but how > do we define "universe"? Sorry if that question was answered > somewhere, but after a quick search I didn't find it. > > Inyuki > http://www.universians.org > > -- > > You received this message because you are subscribed to the Google Groups > "Everything List" group. > To post to this group, send email to everything-l...@googlegroups.com. > To unsubscribe from this group, send email to > everything-list+unsubscr...@googlegroups.com > . > For more options, visit this group at > http://groups.google.com/group/everything-list?hl=en. > > > -- You received this message because you are subscribed to the Google Groups "Everything List" group. To post to this group, send email to everything-l...@googlegroups.com. To unsubscribe from this group, send email to everything-list+unsubscr...@googlegroups.com. For more options, visit this group at http://groups.google.com/group/everything-list?hl=en.
Definition of universe
Hello, I was just wondering, we are talking so much about universes, but how do we define "universe"? Sorry if that question was answered somewhere, but after a quick search I didn't find it. Inyuki http://www.universians.org -- You received this message because you are subscribed to the Google Groups "Everything List" group. To post to this group, send email to everything-l...@googlegroups.com. To unsubscribe from this group, send email to everything-list+unsubscr...@googlegroups.com. For more options, visit this group at http://groups.google.com/group/everything-list?hl=en.
Re: Robotic Scientist
On 27 Dec 2009, at 23:16, russell standish wrote: > On Sun, Dec 27, 2009 at 10:54:53AM -0500, John Mikes wrote: >> I wonder if a 'robot' can produce a "noch nie dagewesen" (Ger. for >> brand >> new) unrelated idea? > > I do know Hod Lipson from the ALife community, but am not familiar > with this particular piece of research. From the WIRED article, I > understand this to be a particular implementation of inductive > reasoning by machine. It is impressive enough that this is possible, > but I don't for one minute think that they have approached the > creative power of a human being. But it certainly feasible that humans > are really just more so of what this machine does. > > Still, the whole area of machine learning, and minimum length > description has some very interesting surprises in store, which is why > I've never bought Colin's argument. For instance John Koza's genetic > programs have created several electronic circuits, some of which were > patentable, so fit the requirement of noch nicht dagewesen. And there is the whole computational learning theory which shows that machine learners exists. Even universal learners exists, but the proofs are necessarily non constructive. We cannot recognize such machine even if we are in front of them. There are a lot of amazing theorems in that field. For example the theorem of Blum and Blum, which says that there is something infinitely (even non computably) more clever (in learning) than any machine: a couple of (independent) machines! Learning machines exist, and the theory explains why we cannot build them from scratch. Some form of learning-competence can need intrinsically long computations/histories, but once there, they can multiplied. Can a machine find a new thing. Of course, from I said above. Can we judge if a machine has find something new? This is hard to say. It is even hard to judge this in a definitive way with the discoveries made by humans. It would need many formal criteria in a place where formalization is difficult. How to define "new", for example. It is a relative concept. 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-l...@googlegroups.com. To unsubscribe from this group, send email to everything-list+unsubscr...@googlegroups.com. For more options, visit this group at http://groups.google.com/group/everything-list?hl=en.
Re: Why I am I?
On 28 Dec 2009, at 21:22, benjayk wrote: > > > Bruno Marchal wrote: >> >> I have never claim it explains something fundamental, it explains a >> "new" problem, the problem of justifying how machine dreams "glue" >> enough to stabilize first person plural sharable observation. > "The theory > explains what exists, and how the rest emerges from it."... Sounds > pretty > fundamental to me ;). I think your wording was just a bit absolute > for me > here, maybe you should be more careful there, maybe I just took you > too > serious. After all you're talking in the context of a theory, so I > should > take "The theory > explains what exists, and how the rest emerges from it." as ""The > theory > explains what exists as formalizable in the theory, and explains > from it how > there must be more than this, which trascends the formalities of this > theory.". OK. > > > > > > Bruno Marchal wrote: >> >>> >>> >>> Bruno Marchal wrote: The theory explains what exists, and how the rest emerges from it. >>> But then doesn't the "rest" exist, too? I just see a problem with >>> claiming >>> to explain what exists, when it is really not clear what existance >>> could >>> mean apart from the relatively meaningful, but vague, every day use. >> >> In that context existence is the same as in the expression "it exists >> a number having this or that property". Among the property there will >> be property like "relatively to that number this number observe this >> phenomenon". the rest belongs to the dream of numbers, and they do >> those dream because they describe computations. We assume >> mechanism, I >> recall. > Okay, though I still think it's advisable to not use simply > "existence" as a > word a here, because it sounds too exclusive. "What exists" sounds > like > "Everything that exists". > And I find "dreams of numbers" sounds as if the dreams where less > fundamental than the numbers. They are. Numbers are primitive. The variable x and y represents excusively those numbers. Finite pieces of computation are speical numbers, like prime numbers. To be a (finite piece of a) computation is a property of number, a relation which has to be defined in term of addition and multiplication of numbers. To be a computation are emergent property (emerging from addition and multiplication). > But since you don't only assume mechanism, but > also conciousness (like all theories) Digitam mechanism (comp) assumes consciousness explicitly (cf the sense of the "yes doctor"). Most theories does not assume "consciousness". The word does not appear in the description of the theories. > and consensual reality (the dreams in > which the representations of numbers appear), I don't see how it > makes sense > to put numbers "before" conciousness and (perceived) reality. Well, it is a bit like "addition" comes before "being prime". You need addition in Robinson arithmetic to define what a prime number is. Then you need addition, and prime, before defining when a number represent a finite piece of computation. And you need that to eventually attach consciousness to computations. The "before" is logical, not temporal. > > > > Bruno Marchal wrote: >> >>> Really we only discuss semantics here... I just find "theory of >>> everything" >>> sounds authorative, because it seems to claim there is nothing >>> else to >>> explain. Basically that is my only problem with a "theory of >>> everything" - >>> it is either a confusing name or disingenious, >> >> >> And what do you think about "theology". The idea is to unify >> knowledge >> in a coherent realm, which does not eliminate the person nor the >> appearances, but help to figure them out. > Not so good. Theology sounds too big. After all, there is no science > or any > other practice that does not study spirituality or god in some > sense. By > calling it theology it sounds like "your" theory is especially close > to > grasping god. But I don't think it's any good to ever invoke > closeness to > god in any theory. > I would like "theory of relationship of numbers and that which > trascends > them" or something more precise and modest, without using > "everything" or > some appeal to god. That is a vocabulary problem. I like "theology" for three reasons: 1) comp is a belief in a form of possible technological reincarnation, leading to notions of afterlife, or after-annihilation. 2) the gap between G and G* provides a gap between science and theology-proper. 3) It necessitates an unprovable belief in the universal machine (the little god, Plotinus' man). This is Church thesis. This is made clear by the arithmetical interpretation of Plotinus. God (the ONE) = arithmetical truth, the NOUS = arithmetical provability, the third god (universal soul) = provability in company of truth, matter = ... etc. > > > > > > Bruno Marchal wrote: >> >>> >>> >>> Bruno Marchal wrote: > > > Bruno Marc
Re: UDA query
On 28 Dec 2009, at 21:24, Nick Prince wrote: > > >> Well, it is better to assume just the axiom of, say, Robinson >> arithmetic. You assume 0, the successors, s(0), s(s(0)), etc. >> You assume some laws, like s(x) = s(y) -> x = y, 0 ≠ s(x), the laws >> of addition, and multiplication. Then the existence of the universal >> machine and the UD follows as consequences. > > Ok so the UD exists (platonically?) Yes. The UD exists, and its existence can be proved in or by very weak (not yet Löbian) arithmetical theories, like Robinson Arithmetic. The UD exists like the number 733 exists. The proof of its existence is even constructive, so it exists even for an intuitionist (non platonist). No need of the excluded middle principle. > >> Better not to conceive them as living in some place. "where" and >> "when" are not arithmetical predicate. The UD exists like PI or the >> square root of 2. >> (Assuming CT of course, to pretend the "U" in the UD is really >> universal, with respect to computability). > > Fine so the UD has an objective existence in spite of whatever else > exists. It exists in the sense that we can prove it to exist once we accept the statement that 0 is different from all successor (0 ≠ s(x) for all x), etc. If you accept high school elementary arithmetic, then the UD exists in the same sense that prime numbers exists. "exist" is used in sense of first order logic. This leads to the usual philosophical problems in math, no new one, and the UDA reasoning does not depend on the alternative way to solve those philsophical problem, unless you propose a ultra-finitist solution (which I exclude in comp by arithmetical realism). > > >> There is a "time order". The most basic one, after the successor law, > >> is the computational steps of a Universal Dovetailer. >> Then you have a (different) time order for each individual >> computations generated by the UD, like > >> phi_24 (7)^1, phi_24 (7)^2, phi_24 (7)^3, phi_24 (7)^4, ... >> where"phi_i (j)^s" denotes the sth steps of the computation (by >> the UD) of the ith programs on input j. > > If the UD was a concrete one like you ran then it would start to > generate all programs and execute them all by one step etc. But are > you saying that because the UD exists platonically all these programs > and each of their steps exist also and hence, by the existence of a > successor law they have an implicit time order? Yes. The UD exist, and is even representable by a number. UD*, the complete running of the UD does not exist in that sense, because it is an infinite object, and such object does not exist in simple arithmetical theories. But all finite parts of the UD* exist, and this will be enough for "first person" being able to glue the computations. For example, you could, for theoretical purpose, represent all the running of the UD by a specific total computable function. For example by the function F which on n gives the (number representing the) nth first steps of the UD*. Then you can use the theorem which asserts that all total computable functions are representable in Robinson Arithmetic (a tiny fragment of Pean Arithmetic). That theorems is proved in detail, for Robinson-ile arithmetic, in Boolos and Jeffrey, or in Epstein and Carnielli. In Mendelson book it is done directly in Peano Arithmetic. > > > >> Then there will be the time generated by first person learning and >> which relies eventually on a statistical view on infinities of >> computations. > > Is this because we are essentially constructs within these steps? It is because our "3-we", our bodies, or our bodies descriptions, are constructed within these steps. But our first person are not, and no finite pieces of the UD can give the "real experience". This is a consequence of the first six steps: our next personal experience is determined by the whole actual infinity of all the infinitely many computations arrive at our current state. (+ step 8, where we abandon explicitly the physical supervenience thesis for the computational one). > >> Time is not difficult. It is right in the successor axioms of >> arithmetic. > > Here again you confirm the invocation of the successor axioms. Yes. It is fundamental. I cannot extract those from logic alone. No more than I can define addition or multiplication without using the successor terms s(-) : for all x x + 0 = x for all x and yx + s(y) = s(x + y) You have to understand that all the talk on the phi_i and w_i, including the existence of universal number (EuAxAy phi_u() = phi_x(y)) can be translated in pure first order arithmetic, using only s, + and *. I could add some nuances. "To be prime" is an intrinsic property of a number. To be a universal number is not intrinsic. To define a universal number I have to "arithmetize" the theory. The theory uses variables x, y, z, ..., so I will have to represent "to be a variable" in the theory. The
